ABSTRACT. LI, LAN. Two-person Investment Game. (Under the direction of Dr. Shu-Cherng Fang.)

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1 ABSTRACT LI, LAN. Two-person Investment Game. (Under the direction of Dr. Shu-Cherng Fang.) In this thesis, we design and study a new game for investment consideration. Two investors, each with an individual budget, bid on a common pool of potential projects. Due to the economic development consideration, these projects are packed into multiple sets for investors to select. Associated with each project, there is a potential market profit that can be taken by the only investor or shared proportionally between both of them. The objective function for each investor is assumed to be a linear combination of two investors profits. In the game, both investors act in a selfish manner with the best-response to each other to optimize their own objective functions by choosing portfolios under the budget constraints. We show that a pure Nash equilibrium exists under certain conditions. In this case, no investor can improve the objective by changing individual strategy unilaterally. A dynamic programming algorithm is presented to generate a pure Nash equilibrium in special cases. For general situations, we design a genetic-based algorithm to find pure Nash equilibrium solutions. Also, we investigate the price of anarchy associated with a simplified two-person investment game.

2 Two-person Investment Game by Lan Li A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Industrial and Systems Engineering Raleigh, North Carolina 2010 APPROVED BY: Dr. Yahya Fathi Dr. Russell E. King Dr. Simon Hsiang Dr. Shu-Cherng Fang Chair of Advisory Committee

3 DEDICATION Dedicated to My parents Pengyue Li and Jinhui Wang And my boyfriend Kun Huang ii

4 BIOGRAPHY Lan Li, the only daughter of Pengyue Li and Jinhui Wang, was born on June 28th, 1983 in Tianjin city, China. She was taught at an early age the values of hard work and focus. She attended Nankai University in the fall of 2001, where she joined the school of Mathematics and built strong mathematical background. The training and study in the college helps her greatly in the later graduate study and research. After receiving her Bachelor of Science degree, she came to the United State and became a Ph.D. student in the department of Industrial and Systems Engineering at North Carolina State University. In the winter of 2006, she received her master degree in Industrial Engineering. From the summer of 2008, Lan started working for SAS Institute, Inc. as a technical student. She will continue her full-time employment with SAS after her graduation. In addition, Lan is a board member and dancer of Sunny Performing Arts Group, which consists of people who love and devote themselves to the performing arts of Chinese classic and folk dances in RTP area, North Carolina. iii

5 ACKNOWLEDGEMENTS I would like to express my deep and sincere appreciation to my advisor Dr. Shu- Cherng Fang for his great guidance and support throughout my Ph.D. study. Also I would like to thank my committee members - Dr. Yahya Fathi, Dr. Russell E. King, and Dr. Simon Hisang for their valuable help and comments. My appreciations also go to Dr. Zhenbo Wang for his helpful suggestions to this work. I am lucky to receive the support and encouragement from all my friends in Fuzzy and Neural Group (FANGroup) and the ISE department. We have had some great moments together and I would like to thank them for sharing the happy school life with me. Finally I thank my parents and my boyfriend for all their love and support. They always have confidence in me and always encourage me to overcome any difficulties I faced during my Ph.D. study. Also, I would like to thank my first cat - Nash. He made me forget being unhappy and filled my life with laughter during his short but sweet life. iv

6 TABLE OF CONTENTS List of Tables vii List of Figures viii Chapter 1 Introduction The problem Game theory Price of anarchy Model and notations Outline of dissertation Chapter 2 Literature Review Spare allocation problem Existence of Nash equilibrium Congestion games and potential games Complexity of determining the existence of Nash equilibrium Algorithms for finding Nash equilibrium Complexity for finding Nash equilibrium Polynomial-time algorithms for finding Nash equilibrium Heuristics for finding Nash equilibrium Price of anarchy Chapter 3 Existence of Nash Equilibrium Basic model Extensions Case 1 - same proportional profits between two players Case 2 - different proportional profits between two players Summary Chapter 4 Finding Nash Equilibrium - Dynamic Programming Approach N P-hardness A dynamic programming algorithm for a special case Algorithm A numerical example Extension of the dynamic programming algorithm Algorithm Proof and complexity Difficulty of extension for general games Summary v

7 Chapter 5 Finding Nash Equilibrium - Genetic-based Algorithm Introduction to genetic algorithms A genetic algorithm for the two-person investment game Components of our genetic algorithm Proposed genetic algorithm Experimental results - numerical analysis Experimental results - tuning parameters Comparison of the genetic algorithm and random search method Summary Chapter 6 Price of Anarchy Preliminary Main results Assumption and definition Conservative investment game Aggressive investment game Mixed investment game Summary and discussion Chapter 7 Conclusion and Future Research Conclusion Future research References vi

8 LIST OF TABLES Table 3.1 Two players objective values at all the states - Example Table 3.2 Two players objective values at all the states - Example Table 4.1 The profits of all the projects - Example Table 4.2 Dynamic programming algorithm - Example Table 4.3 The profits of all the projects - Example Table 5.1 Number of runs to achieve optimality for each instance - Example Table 5.2 Number of runs to achieve optimality for each instance - Example Table 5.3 Number of runs to achieve optimality for each instance - Example Table 5.4 Number of runs to achieve optimality for each instance - Example Table 5.5 Number of runs to achieve optimality Table 5.6 Tuning experiment: levels of the parameters Table 5.7 GA vs. RS, Problem Table 5.8 GA vs. RS, Problem Table 5.9 GA vs. RS, Problem Table 5.10 GA vs. RS, Problem Table 5.11 GA vs. RS, Problem vii

9 LIST OF FIGURES Figure 1.1 A 7 9 array with 2 spare rows and 3 spare columns Figure 1.2 A chain-service investment problem Figure 2.1 Bipartite description of the example Figure 2.2 Flow chart of the repair-analysis algorithm Figure 5.1 Mutation Figure 5.2 Before crossover Figure 5.3 After crossover Figure 5.4 Performance analysis of Example Figure 5.5 Performance analysis of Example Figure 5.6 Performance analysis of Example Figure 5.7 Performance analysis of Example Figure 5.8 Tuning parameters viii

10 Chapter 1 Introduction In many business management or optimization problems, the market is shared among multiple agents who are selfish and noncooperative. The decision of one agent may affect the profits of other competitors, which results in a strategic interaction among the decision making of all agents. Noncooperative game theory provides a normative framework for analyzing strategic interactions. In this dissertation, we propose a two-person investment game model for investment decision making considerations. Specifically, we consider that there exist two noncooperative investors who bid on a common pool of resources which are packed into groups. We apply game theory to analyze the model and study the best known solution concept - Nash equilibrium, in order to provide a business view not only for the investors but also for the social benefits. 1.1 The problem At the beginning of this thesis, we introduce the background and motivation of our research work for readers to understand the problem better. The idea of the two-person investment game was initially inspired by the work of Kuo and Fuchs who considered an efficient spare allocation problem for reconfigurable arrays in [43]. As stated in their work, with the current trend of technology advancement, a rapidly increasing 1

11 number of devices have been placed on some integrated circuits and wafers. It is now possible to build an integrated system on a chip or wafer by interconnecting a large number of identical elements, such as memory cells or processors. The issue of yield degradation due to physical failures in large memory and processor array is of significant importance to semiconductor manufacturers. One important method to increase the yield is through the restructurable VLSI (Very Large Scale Integration) technology, which employs redundancy that can be used to replace faulty modules. Their work considers the employment of spare rows and columns for tolerating failures in rectangular arrays of identical computational elements. When a faulty computational element is detected, either the row or column containing this element is disconnected and a spare is programmed to replace it. However, there are a limited number of spares. Therefore, the goal is to effectively utilize the spares to perfect the system whenever it is repairable. The problem can be modeled as follows. A rectangular array has M N cells, SR spare rows and SC spare columns. For example, if the array has 7 9 cells with 9 being faulty, 2 spare rows and 3 spare columns, then the configuration is shown below SC=3 Figure 1.1: SR=2 A 7 9 array with 2 spare rows and 3 spare columns. 2

12 In Figure 1.1, the coordinates of faulty cells are (1, 1), (1, 4), (1, 9), (3, 1), (4, 3), (4, 4), (4, 7), (7, 4) and (7, 9). The spare allocation problem is to choose the minimum number of spare rows and/or columns to repair all the faulty cells. The spare allocation problem is a combinatorial optimization problem and of great importance to the semiconductor industry. However, this model not only can be used for semiconductor manufacturing, but also may enhance our understanding of some business practices. For example, suppose a service company considers providing some chain service to a number of locations. We assume that some of the locations are lined up by individual chains as shown in Figure 1.2 on the next page. The company will provide service to all the locations in each selected chain. With each location, there is an estimated profit associated with the customer market. If the company invests to run a particular chain, it will get all the market profits of the locations in the chain. And the profit of a location only counts once towards the total profits even it is covered by multiple chains. However, the company has its budget constraint, it can only invest on a limited number of chains to earn the maximal total market profits. The problem is selecting a combination of the service chains under the budget constraint then becomes an interesting issue for the company s investment consideration. In the business world, the market is almost always shared among multiple firms that compete each other for profits and market shares. Specifically, we consider that there exist two noncooperative service companies who are interested in investing in the service chains as described above. Assume that both companies have limited budgets. Should a company cover the service of a location solely, an expected market profit of this location will be rewarded to the firm. If both companies share the same chain or different ones but with some common locations overlapped, then the expected market profits of these overlapped locations will be shared in proportion to each company s market power. It is not hard to observe that one company s strategy may affect the decision of the other, which results in strategic interactions between the 3

13 Figure 1.2: A chain-service investment problem. decision making of both companies. Assume that each company has an objective, in relation to both firms profits, to be optimized and the company acts in a selfish manner with best response to the opponent s bidding. We are interested in knowing how these two companies would play their strategies to optimize their objectives under each other s influence. We are also interested in learning if there exists a state in which no company can derive any additional benefit by deviating from its strategy unilaterally. There are many interesting business problems that fall in this framework. Here is another example. Suppose there exist two noncooperative investors who bid on a common pool of potential business projects. For each project, there is a potential market profit associate with it. Some projects are popular ones which have high profit of return, while others may have very low or even zero profit such as welfare projects. To seek a balance of economic development, the government may pack these projects into multiple sets for investors to select. The profit of a project set is the sum of the profits of all the projects in it. Each investor has an individual budget. Should an investor be the sole bidder of a particular project set, an expected market profit will be rewarded to this investor. If both investors bid on the same project set or differ- 4

14 ent ones but share some common projects, then the expected market profits of these specific projects will be shared in proportion to each investor s market power. Moreover, we assume this proportion will not be changed no matter how many times these projects are selected by one investor. Suppose that each investor has an objective function which is a function of his own profit as well as his opponent s. For example, the objective may be to maximize the investor s own profit, or to minimize his opponent s profit, or to maximize their profit gap. Every investor acts in a selfish manner with the best response to his opponent s choice. We are interested in knowing how should an investment decision be made by each investor. We are also interested in learning whether this decision making process would terminate with a stable state in which no investor can improve individual objective value by changing bids unilaterally. In general, for the above mentioned optimization problems involving two investors, we aim to investigate if there exists a state in which no investor has any incentive to unilaterally change his/her actions. Moreover, we are interested in finding such a state when it exists. This naturally leads us to the realm of Nash equilibrium [52] in Game Theory. In this thesis, we adopt a game theoretical approach to study this class of optimization problems called two-person investment game. 1.2 Game theory In this section, we introduce some basic concepts and knowledge of game theory related to our research. Game theory is developed to model situations in which multiple agents interact or affect each other s outcomes. The different agents in a game are called players in game theory, while the profit function of a player is called a payoff function. A player s best response is his/her best strategy given the strategies of all other players. The concept of Nash equilibrium is 5

15 used to represent a solution to the game in which all players make decisions simultaneously. In this solution, no player can change his/her chosen strategy from one state to another to improve the payoff, assuming that all other players stick to the strategies they have chosen. Observe that such a solution is a stable state in the sense that once the players reach such a solution, it is in every player s best interest to stick to his or her own strategy. There are two types of Nash equilibrium. One is called pure strategy Nash equilibrium, implying that each player deterministically plays his/her chosen strategy. The other is mixed strategy Nash equilibrium, in which the players choices are not deterministic but regulated by a probability distribution. Nash [53] proved that, every game with a finite number of players, each having a finite set of strategies, always has a mixed Nash equilibrium. However, in the setting of pure strategies, Nash equilibrium may not exist. There is a class of games called potential games [51], in which pure Nash equilibrium is guaranteed to exist. The well-known congestion games form a special subclass of potential games. These games are of particular interests to theoretical computer scientists in recent years due to their applications to the research on Internet. Internet is viewed as a very complicated system for research. A fusion of algorithmic ideas with concepts and techniques from game theory and computer science may advance our understanding [59]. This also leads to a new interdisciplinary area named algorithmic game theory, involving the research of existence, complexity and algorithms for pure Nash equilibrium in games. In our thesis, we adopt some ideas of algorithmic game theory to investigate the two-person investment problem. If it is not specifically mentioned, in this thesis, a Nash equilibrium solution simply means a pure Nash equilibrium. We are interested in investigating pure Nash equilibrium because in many practical situations, players are likely to adopt pure strategies. And in game theory, pure Nash equilibria are fundamental to the understanding of more complicated 6

16 strategies. 1.3 Price of anarchy Nash equilibria are known for not always optimizing social optimum, with the Prisoner s Dilemma being the best-known example. Conditions under which Nash equilibria can achieve or approximate the overall optimum have been studied extensively especially in the early 1990s. Examples include the optimal flow control problem by Korilis and Lazar (1995) and optimal routing control problem by La and Anantharam (1997). This type of research compares the overall optimum with the best equilibrium and aims at achieving or approximating the social optimum by employing implicit acts of coordination. For noncooperative games, we are interested in evaluating the loss to the system due to its deliberate lack of coordination. The price of anarchy is a popular measure of the inefficiency of equilibria. It resolves the issue of multiple equilibria by adopting a worst-case approach. To be more specific, the price of anarchy [39, 59] is defined as the ratio of the worst objective function value of an equilibrium and that of an optimal solution. We need to note that the price of anarchy of a game is defined with respect to the choice of objective function and equilibrium concept. It is easy to understand that if the price of anarchy of a game is 1, then its equilibria are fully efficient. The price of anarchy also tells us the worst-possible inefficiency of equilibria in the games. The concept has been adopted in studying the selfish routing game [68], load balancing game [39] and network game [20]. For a two-person investment game, the noncooperative nature of the game and the selfish manner of each player may prevail at the sacrifice of social welfare. In order to study the impact of investment decisions to the whole society led by different objectives, we shall quantify Nash equilibrium solutions by using the concept of the price of anarchy. 7

17 1.4 Model and notations In the following, we present a game theoretical model corresponding to the project investment problem we described in Section 1.1. We define players, strategies, states, payoff functions and Nash equilibrium for the two-person investment game accordingly. In a two-person investment game model, there exist two players (investors), say player i, i = 1, 2. They are both interested in investing in n potential projects, say project P k for k = 1, 2,..., n. Let p k represent the potential profit of return associated with the project k. Without loss of generality, we assume p k are nonnegative integers for k = 1, 2,..., n. Now these k projects are packed into m sets for players investment consideration, say sets C 1, C 2,..., and C m. We should note that the ways to pack the projects into sets may be very flexible. For each project, it can be included in one or multiple sets. For any multiple sets, they may have none or one or multiple projects in common. We view each set as a strategy for a player to choose, the profit of which is the sum of profits associated with all the projects in it. Assume each investor has a budget constraint, denoted b i, i = 1, 2, which is simply a limit on the total number of the strategies he/she can choose. Assume that b 1 and b 2 are nonnegative integers smaller than m. For each project set added to a player s portfolio, the budget in the player s hand is reduced by 1. Let S i denote the set of strategies chosen by player i, i = 1, 2, and (S 1, S 2 ) a state of the two-person investment game. If player i is the sole investor of a project set, the profit of this set will be rewarded to the player, no matter i = 1 or 2. If both players select the same strategy or different ones but share some common projects, then the expected market profits of these specific projects will be divided between two players. We use α i to represent the profit reduction of player i, i = 1, 2. Then player 1 will receive (1 α 1 ) p k and player 2 P k S 1 S2 will receive (1 α 2 ) p k, 0 α i 1, for the commonly selected projects. Notice that P k S 1 S2 α 1 + α 2 is not required to be 1 in our model, because the presence of two similar services in a location may either stimulate new demands for a bigger market or reduce the total needs for 8

18 the market. For example, two fast food companies consider open their new stores in the same shopping mall. They may attract more demands by presenting multiple food choices to the potential customers. But it may also possible for them to reduce the demands on the market because of the competition. Also we should note that, the market power of a player is reflected by the parameter α i : a smaller α i goes with an investor i with higher market power. We say a state (S 1, S 2 ) is feasible to the two-person investment game if S 1 and S 2 satisfy the budget constraints of both players, that is S 1 b 1 and S 2 b 2. For a given feasible state (S 1, S 2 ), we use P i (S 1, S 2 ) to represent the total obtainable profit for player i, i = 1, 2, at this state. Let p(s i ) represent the sum of profits in the strategy set S i for player i. Then it is not difficult to verify that P 1 (S 1, S 2 ) = p(s 1 ) α 1 p(s 1 S2 ), P 2 (S 1, S 2 ) = p(s 2 ) α 2 p(s 1 S2 ). We assume the payoff (or objective) function of a player is a linear combination of his profit as well as his opponent s. It may differ a lot as we mentioned in section one may consider maximizing his own profit while the other may consider suppressing opponent s profit or a combination of both. It depends on the market position and mind set of a particular investor (player). A new comer or a conservative player tends to survive by maximizing his own profit, but an aggressive player cares more about the gap of profits between two players, and a wellestablished dominating player may care about minimizing the opponent s profit. As long as each player acts in a selfish manner in response to the opponent s decision to optimize a given objective function, we are interested in knowing how should an investment decision be made by each investor. To capture this idea, we define our objective functions as follows, O 1 (S 1, S 2 ) = β 11 P 1 (S 1, S 2 ) β 12 P 2 (S 1, S 2 ), O 2 (S 1, S 2 ) = β 22 P 2 (S 1, S 2 ) β 21 P 1 (S 1, S 2 ), (1.1) 9

19 where β ii 0 and β ij can be an arbitrary real number. The nonnegativity of β ii indicates that both players value their own profits. When β ij > 0, player i is concerned about the weighted gap between his and opponent s obtainable profits. When β ij < 0, that means player i partly hopes to improve the total profits. But we assume β ii β ij, i.e. an investor s own profit is a dominant factor of his objective. Substituting P 1 (S 1, S 2 ) and P 2 (S 1, S 2 ) into (1.1), we get O 1 (S 1, S 2 ) = β 11 p(s 1 ) β 12 p(s 2 ) 1 p(s 1 S2 ), O 2 (S 1, S 2 ) = β 22 p(s 2 ) β 21 p(s 1 ) 2 p(s 1 S2 ), (1.2) where i = α i β ii α j β ij for i = 1, 2 and j i. This format of the players objective functions will be constantly used in our thesis. In this thesis, we are interested in studying the properties of Nash equilibrium of the twoperson investment game model. Recall that a state of the two-person investment game is called a Nash equilibrium if it is a feasible state at which no player can improve his objective value by changing his strategy unilaterally. We give a formal definition of Nash equilibrium for the two-person investment game as follows. Definition A feasible state (S1, S 2 ) is a Nash equilibrium of the two-person investment game if and only if O 1 (S 1, S 2 ) O 1(S 1, S 2 ) and O 2(S 1, S 2 ) O 2(S 1, S 2) for any feasible states (S 1, S 2 ) and (S 1, S 2). 10

20 1.5 Outline of dissertation In this thesis, we aim to study the conditions under which a pure Nash equilibrium state exists for the two-person investment game. Moreover, we study the complexity of finding a Nash equilibrium when it exists. Algorithms and heuristics will be developed to find such an equilibrium. We are also interested in evaluating the loss of the system due to its deliberate lack of coordination using the concept of the price of anarchy. In this way, we may learn how bad the worst Nash equilibrium could be in a two-person investment game. The dissertation is organized as follows. Literature surveys on related work are given in Chapter 2. In the first section, we review the complexity and algorithms of the background problem - spare allocation problem which helps us understand the underlying structure. Then we move to the game theory literature. We know that pure-strategy Nash equilibria are in general non-existent. It is therefore natural to ask which games have pure-strategy Nash equilibria and if applicable, how difficult it is to find one. A more detailed literature review along these lines is provided in Sections 2 and 3 of Chapter 2. In the last section of this chapter, we review the literature on the price of anarchy. In Chapter 3, we present an existence theorem of pure Nash equilibrium for the two-person investment game. We show that although a two-person investment game can not be treated as a general congestion game, the potential function method can still be used to derive the existing conditions of pure Nash equilibria. Moreover, we extend our model to consider the case where the profit of each project is different for the two players and analyze the existence conditions under this new structure. The problem of finding a pure-strategy Nash equilibrium in a game is in general considered as N P-hard problem. In Chapter 4, we prove that to find a pure Nash equilibrium in a two-person investment game is N P-hard by showing that the 3-set exact covering problem, 11

21 which is known as an N P-complete problem, is polynomially reducible to an investment problem. Then we design a dynamic programming based algorithm to find Nash equilibria for some special cases. At the end of the chapter, we show that the dynamic programming algorithm cannot be generalized to solve more general games because the principle of optimality does not follow. Since the dynamic programming algorithm is not suitable for solving our problem, we propose a genetic-based algorithm for finding Nash equilibrium in Chapter 5. We also conduct numerical experiments and analysis of our genetic algorithm to show its strength. The analysis of price of anarchy is conducted in Chapter 6. We investigate the price of anarchy associated with three simplified two-person investment games, say the conservative investment game, aggressive investment game and mixed investment game. Furthermore, we provide instances for each of the three cases to show that the bounds we derived are tight. In Chapter 7, we conclude the thesis and present some issues for future research. 12

22 Chapter 2 Literature Review In this chapter, we firstly review the complexity and algorithms of the background problem - spare allocation problem which may help us understand the underlying structure. Then we go through the literature of game theory. We mainly focus on the following three aspects: existence of pure Nash equilibrium, algorithms for finding a pure Nash equilibrium and the price of anarchy. For the existence of Nash equilibrium, we review a class of games that are guaranteed to possess pure Nash equilibria. Moreover, we discuss the complexity to decide whether a pure Nash equilibrium exists for a general game. Then we review the complexity, algorithms and heuristics for finding a Nash equilibrium in a game. At the end of this chapter, we introduce the concept of price of anarchy and review the relevant research works on it. 2.1 Spare allocation problem In this section, we review the literature of the spare allocation problem which inspired our work originally. As mentioned in the first chapter, the background of the two-person investment game problem is from the work of Kuo and Fuchs [43] who considered an efficient spare allocation problem in the reconfigurable arrays. With the current trend of technology advancement, a rapidly in- 13

23 creasing number of devices have been placed on some integrated circuits and wafers. It is now possible to build an integrated system on a chip or wafer by interconnecting a large number of identical elements, such as memory cells or processors. Yield degradation is an important problem since the density of VLSI (Very Large Scale Integration) systems increases. One important method to increase the yield is through restructurable VLSI (RVLSI) technology, which employs redundancy that can be used to replace faulty modules. This technique has already been found practical for the manufacturing of large random access memories, where spare rows and columns of memory cells together with their decoders are programmed to repair faulty memories. A number of research works on this issue emerged in the early 1980s. Examples include the fault tolerant procedure in wafer-scale integration, laser programmable redundancy and yield improvement, and the work on defect analysis and repair of redundant memories, cited in [43]. Kuo and Fuchs considered reconfiguration algorithms for the most widely used approach for RVLSI, namely the employment of spare rows and columns for tolerating failures in rectangular arrays of identical computational elements. The computational elements can be as simple as a memory cell or as complex as a processor unit. In the RVLSI design, when a faulty computational element is detected, the entire row or column containing this element is disconnected and a spare is programmed to replace it. However, there are a limited number of spares. Therefore, the goal is to effectively utilize the spares to perfect the system whenever it is repairable. Now we review the algorithms solving this spare allocation problems. Although the problem is important for the practical semiconductor manufacturing, there are few known algorithms published to solve the optimal solution correctly and efficiently. Firstly, Kuo and Fuchs [43] proved that this is an NP-complete problem. They showed the problem of repair analysis can be formulated as a bipartite vertex covering problem. The bipartite graph of Figure 2.1 describes the previous example in Chapter 1. In the figure, an 14

24 edge between R1 and C4 indicates that the cell at the intersection of row 1 and column 4 is faulty. R1 C1 R3 C3 C4 R4 C7 R7 Figure 2.1: C9 Bipartite description of the example. Later they proved a theorem that, Given a bipartite graph BG = (A, B, E) and positive integers x A A, x B B, the problem to determine if there is a vertex cover V A VB, V A A and V B B such that V A x A and V B x B is NP-complete, by transforming the CLIQUE problem into the vertex covering problem. After analyzing the complexity issue of the problem, they made a brief examination of two previously proposed algorithms. The first one is the repair-most algorithm proposed by Boudreau and Murphy in 1984 [72]. This is essentially a greedy approach to the problem. It has two drawbacks: 1. It may not generate a solution for a theoretically repairable device. 2. The solution generated by it may not be optimal in the sense that it may not employ a 15

25 minimal number of the redundant rows and columns. The second algorithm proposed in the work of J. R. Day [16] is an exhaustive search algorithm which generates the entire tree of all possible solutions. This approach is not acceptable when the array size is large with numerous potential failures. Kuo and Fuchs [43] developed a new heuristic algorithm based on a branch and bound with bipartite matching approach in their paper. It is more efficient compared to the exhaustive search algorithm and generates an optimal solution whenever the system is repairable. The flow of the algorithm is shown in Figure 2.2. Start Must-repair Analysis Early-about Analysis Final Analysis Solution Figure 2.2: Flow chart of the repair-analysis algorithm. The must-repair analysis is to find a must-repair row or a must-repair column before the formal repair. A must-repair row is defined to be a row which has more errors than SC and a must-repair column is defined to be a column which has more errors than SR. A new early-abort technique is then presented which efficiently aborts theoretically unrepairable arrays before performing the actual repair-analysis. A bipartite matching algorithm helps to do 16

26 this initial screening. The phase of final analysis performs the actual repair analysis. They employed a branch and bound algorithm because of its ease in implementation and the fact that it will generate an optimal solution whenever the system is repairable. Kuo and Fuchs published another two papers which are related to this spare allocation problem. One is published in 1989 that presented a fault diagnosis algorithm [44], the other published in 1992 [42] focused on the issue of developing efficient algorithm for constructing a flawless subarray from a defective VLSI array. Then Fernau and Niedermeier [24] showed that the branch and bound approach Kuo et al. developed is only efficient for arrays with a moderate number of faulty cells. They also provided a nontrivial fixed parameter algorithm for the spare allocation problem. Their approach always delivers optimal results as well. Furthermore, its complexity basically depends on the number of available spare rows and columns. So far, we have reviewed the literature of the background problem including its structure and algorithms. In the following sections, we will review the literature in the area of algorithmic game theory that cover the topics of the existence, complexity, algorithms and qualification of pure Nash equilibrium. 2.2 Existence of Nash equilibrium Nash s celebrated existence theorem [53] asserts that there exists a mixed-strategy Nash equilibrium for every finite normal-form noncooperative game. However, in many contexts mixed strategies are unappealing and hard to interpret, leading naturally to the question and interest of the existence of pure Nash equilibria. If it is not specially mentioned in this dissertation, a Nash equilibrium solution means a pure Nash equilibrium. In this section, we review the works related to the existence of pure Nash equilibria. For a game, conditions under which there exists at least one pure Nash equilibrium is always critical issue to investigate. In recent years, there has been a growing interest in the study of games in which the existence of pure Nash 17

27 equilibria is guaranteed. Also researchers are interested in the complexity of deciding whether a game possesses a pure Nash equilibrium. In the coming two subsections, firstly, we review the class of congestion games and potential games. More specifically, we introduce their relationship and the conclusions of existence of Nash equilibrium for these two classes of games. Then we go through other games that are guaranteed to have pure Nash equilibria. Moreover, we discuss the complexity to decide whether a general game possesses a pure Nash equilibrium in the end Congestion games and potential games In 1973, Rosenthal [65] introduced a class of noncooperative games in which each player chooses a particular combination of factors out of a common set of primary factors. That is, the pure strategy may be viewed as the selection of a particular subset of the primary factors. The payoff associated with each primary factor is a function of the number of players who include it in their choice. The payoff a player receives is the sum of the payoffs associated with the primary factors he selects. This class of games is called congestion games (or general congestion game). Each game in this class possesses at least one pure-strategy Nash equilibrium. A very important characteristic Rosenthal described in this class of game is all players are equal in a sense that they have the same weight. It does not matter which player chooses a factor, but only how many players choose it. We can see that the class of congestion games is limited, but very important for Economics on the other hand. As Monderer and Shapley indicated in [51], any game where a collection of homogeneous agents have to choose from a finite set of alternatives, and where the payoff of a player depends on the number of players choosing each alternative, is a congestion game. Then every congestion game is guaranteed to have a Nash equilibrium. In the 1990s, with the wide use of computers and the Internet, game theory became an essential tool used in theoretical computer science research to understand multi-agent systems with strategic agents. Papadimitriou has made a good statement in [59] to survey 18

28 the subjects connecting these two areas. Also there are many research topics discussed in the book [54] and related papers. Congestion game and its extensions became an active area of research in computer science because they can model lots of diverse phenomena like processor scheduling, routing, network design and so on. Monderer and Shapley [51] in 1996 introduced a class of games called potential games in which Nash equilibrium is guaranteed as well. A game in game theory is considered as a potential game if the incentive of all players to change their strategies can be expressed in one global function, called the potential function. There are two types of potential functions. One is called ordinal potential function and the other is named as cardinal potential function. An ordinal potential function is a real-valued function over the set of pure strategy-tuples having the property that the potential, or to say the trend, of gain (or loss) of a player shifting to a new strategy is the same as the corresponding increasing (decreasing) trend of the function by this shift. A cardinal potential function, or called exact potential function, is a function with property that the difference in individual payoffs for each player from individual shift of the states must have the same value as the difference in values for the function. A potential game with ordinal (or cardinal) potential functions is called an ordinal (or cardinal) potential game. Monderer and Shapley proved that every finite ordinal potential game possesses a purestrategy Nash equilibrium. To investigate the relationship of congestion game and potential game, Rosenthal showed that any congestion game is a potential game by deducing a potential function. Monderer and shapley proved the converse - every finite exact potential game is isomorphic to a congestion game which has the same potential function. Now we know that for every finite potential game, especially congestion game, it is guaranteed that at least one Nash equilibrium exists. So far this class of games is the most widely known games with the existence of pure Nash equilibrium. After the fundamental work of Rosenthal, Monderer and Shapley on congestion and potential games, many scholars studied 19

29 the extension or generalization of these games. For instance, in 1996, Milchtaich [50] studied the congestion games with player-specific payoff functions. In such a congestion game, the payoff a player receives for playing a particular strategy depends only on the total number of players playing the same strategy. It decreases with that number in a manner which is specific to the particular player. He showed that each game in this class possesses a Nash equilibrium in pure strategies. Then he also considered a generalized class of congestion game - weighted congestion game and proved that a three-player, three-strategy weighted congestion game may not possess a pure-strategy Nash equilibrium. Later, Konishi et al. [38] considered the same class of congestion games as Milchtaich s and showed the existence of a strong Nash equilibrium. Facchini et al. [22] in 1997, studied the weighted potential games and derived a characterization in terms of coordination and dummy games. They also extended Rosenthal s congestion model to an incomplete information setting, in which the related Bayesian games are potential games and therefore have pure Bayesian equilibria. Voorneveld et al. (1999) published a work called congestion games and potential reconsidered [75]. They gave an easy alternative proof of the isomorphism between exact potential games and the set of congestion games. In the mean time, they studied a class of congestion games where the sets of Nash equilibria, strong Nash equilibria and potential-maximizing strategies coincide. In the year of 2006, Kukushkin [41] proposed two tentative concepts about the congestion games and conducted some research on them, one is the generalized congestion game and the other is like the dual of congestion game. For our game, although a general two-person investment game may not be treated as a congestion game because the market profit of each project could contribute differently (not only depending on the number of investors who cover this project) toward individual objective functions, we can still use the idea of potential function to check the existence of Nash equilibrium like what Wang et al. derived for the two-person knapsack game [77]. Furthermore, in case each investor selects only one strategy and the individual objective value decreases as 20

30 the strategy is selected by more investors, Milchtaich [50] showed that a pure Nash equilibrium exists. However, the two-person investment game allows each investor to select multiple strategies and individual objective functions may not decrease in terms of the number of investors who select the strategy. One of our major goals is to investigate the existence conditions for pure Nash equilibrium solutions of the two-person investment game Complexity of determining the existence of Nash equilibrium It is natural to ask the question that, what other games are guaranteed to have at least one pure Nash equilibrium? Vetta identifies in [74] the basic utility games as such a class of games where the Nash dynamics converges. The network creation games [20] are another example. Fabrikant et al. introduced a novel game that models the creation of Internet-like networks by selfish node-agents without central design or coordination. This model takes into account both hardware costs and quality of service costs; however, for the latter it ignores congestions, which makes it different from the focus of other works. They showed that the game does have pure Nash equilibria. Later in [21], Fabrikant et al. considered yet another variant of the congestion game with player-specific delays. By generalizing slightly the results in [26], they presented that this class of games possesses pure Nash euilibria. In 2006, Drabkin et al. studied the connectivity game in [18]. They proved the existence of pure Nash equilibria by showing that better reponse dynamics continuously improve the lexicographic order of the agents utilities and hence converge to a pure Nash equilibrium. Besides the games which are guaranteed to have a pure Nash equilibrium as investigated above, we are interested in learning the complexity of determining the existence of pure Nash equilibria in general games. We found that for most cases, it is N P-hard to decide whether pure Nash equilibria exist. Gottloc et al. published a paper [31] in 2003 to investigate the complexity issues related to pure Nash equilibrium of strategic games. They showed that, even in a very restrictive setting like games in graphic normal form having bounded neighborhood 21

31 or acyclic-graph games, determining whether a game has a pure Nash equilibrium is N P- hard. Dunkel and Schulz [19] proved it is strongly N P-hard to determine the existence of pure Nash equilibrium in weighted network congestion game which is a natural extension of congestion game in They showed the same conclusion on a special class of local-effect games called bidirectional local-effect game. For another generalization of congestion games called the player-specific network congestion game, Ackermann and Skopalik [1] proved similar results for the complexity issues. Recently, Thang [73] proved the N P-hardness of pure Nash equilibrium for some problems of scheduling games and connection games. He also proposed an efficient and standard technique to prove the N P-hardness for deciding the existence of Nash equilibria. Moreover, Conitzer and Sandholm shifted their attention to Bayesian games and repeated and sequential games [13] and proved the N P-hardness of determining pure Nash equilibrium accordingly. There are many other discussions on the complexity issue of pure Nash equilibrium. Please see [23, 32, 4]. 2.3 Algorithms for finding Nash equilibrium In this section, the computational complexity, algorithms and heuristics for finding a pure Nash equilibrium will be reviewed in the following subsections in orders Complexity for finding Nash equilibrium Many research works deal with the complexity for finding a pure Nash equilibrium in congestion or potential games. Maximizing potential functions provides an algorithmic means of reaching pure equilibria in potential games. Unfortunately, it makes no claim regarding its rate of convergence. In some games, the best response dynamics always converge quickly, but in many games, it does not. In some games, potential function can be optimized in polynomial time, but in others it is N P-hard. To get a better handle on the complexity of finding Nash equilibrium in potential games, many researchers cast their attention to find local optima in optimization problems. Now we introduce the so-called PLS class in the following, which is considered as a 22

32 good tool to analyze the complexity of seeking pure Nash equilibrium. The class of polynomial local search problems (PLS) was defined by Johnson [37] as an abstract class of local optimization problems. For a general optimization problem, assume we have a set of instances I, and for each instance x I a set of feasible solutions F (x) and a cost function c x (s) defined on all s F (x). To define a local optimization problem, we must also specify a neighborhood N x (s) F (x) for each instance x I and each solution s F (x). A solution s F (x) is locally optimal if c x (s) c x (s ) for all s N x (s). We say a local optimization problem is in PLS if we have an oracle that, for any instance x I and solution s F (x), decides whether s is locally optimal, and if not, returns s N x (s) with c x (s ) c x (s). A problem in PLS is PLS-complete if every problem in PLS is PLS-reducible to it. More discussion of PLS class and examples of PLS-complete problems can also be found in [40, 60, 56]. A lot of researchers worked on the complexity issue of finding a pure Nash equilibrium for potential and congestion games and most of them, showed that it is a PLS-complete problem. In 2004, Fabrikant et al. [21] showed that finding a Nash equilibrium in potential games is PLS-complete, assuming that the best response of each player can be found in polynomial time. To see that the problem belongs to PLS, we will say that the neighbors of a strategy vector s are all the strategy vectors s that can be obtained from s by a single player changing his or her strategy. By definition, a value of potential function is locally optimal if and only if it is a pure Nash equilibrium, so finding a pure Nash equilibrium is in PLS. They showed it is PLS-complete by polynomially reducing weighted satisfiability problem to it, which is a well known PLS-complete problem proven by Krentel in [40]. In recent years, a number of scholars devoted to proving PLS-complete of finding pure Nash equilibrium in all types of extended congestion games or other games. More literature could be found in [19, 1, 54]. However, our game does not belong to the PLS class because the best response for each 23

33 player cannot be found in polynomial time. We will give a proof of the assertion in Chapter 4 in details. Consequently, we could not follow the traditional way to analyze the complexity of the two-person investment game. Instead, we turned our attention to another paper published by Wang et al. [77] recently. They studied the two-person knapsack game and presented that finding a pure Nash equilibrium for it is an N P-hard problem by showing that even with only one player, the problem which is actually a knapsack problem, is N P-complete. We found this idea of proof is helpful for us to conduct the complexity analysis for our two-person investment game Polynomial-time algorithms for finding Nash equilibrium Though for most potential games or even congestion games, finding a pure Nash equilibrium is a computationally hard problem, we still found papers working on the games with special structures that lead to a polynomial time algorithm for finding a pure Nash equilibrium. For example, Gottlob et al. [31] showed that when the game simultaneously have small neighborhood and bounded hyper-tree width, to find pure Nash equilibrium is in polynomial time. Fotakis et al. [27] studied complexity of Nash equilibria for selfish routing games. They illustrated that a pure Nash equilibrium in their games can be computed by a simple greedy algorithm. Panagopoulou and Spirakis [58, 57] investigated algorithms for pure Nash equilibrium in weighted congestion games. They focused on a potential-based method for finding Nash equilibrium. Their algorithm converts any given non-equilibrium configuration into a pure Nash equilibrium by performing a sequence of greedy selfish steps. A greedy selfish step is a user s change of his current pure strategy to his best pure strategy with respect to the current configuration of all other users. They also provided strong experimental evidence to show the proposed algorithm actually converges to a pure Nash equilibrium in polynomial time. Ieong et al. [34] studied singleton games which is a subclass of congestion games. These games are exponentially more compact that general congestion games. They proposed a dynamic programming algorithm to find a Nash equilibrium in polynomial time. 24

34 Recently, Wang et al. [77] proposed a dynamic programming based algorithm to find a Nash equilibrium of two-person knapsack game named DPKG algorithm in pseudo polynomial time. The DPKG algorithm is an extension of the dynamic programming algorithm used to solve the knapsack problem. With the help of principal of optimality, the enumeration of unnecessary states could be eliminated in the process. However, this dynamic programming approach is not suitable for the general two-person investment game problem because the principal of optimality does not follow anymore in our structure. The separability of the potential function is not guaranteed for the two-person investment game. We will show more details of our analysis in Chapter 4. This issue brings us difficulties to design a dynamic programming algorithm for solving a general two-person investment game problem. Therefore, we seek to design an efficient heuristics to find a Nash equilibrium Heuristics for finding Nash equilibrium In this section, we review the work on tabu search and genetic-based algorithms for finding a pure Nash equilibrium in games. We will derive a genetic-based algorithm for the two-person investment games in Chapter 5. Tabu search is a heuristic algorithm that can be used for solving complex combinatorial optimization problems. Tabu search belongs to the class of local search techniques. It enhances the performance of a local search method by using memory structures: once a potential solution has been determined, it is marked as taboo ( tabu being a different spelling of the same word) so that the algorithm does not visit that possibility repeatedly. In [70], Wurman and Sureka presented an approach, based on best-response dynamics and tabu search, that avoids the requirement for having a complete payoff matrix upfront, and instead computes the payoffs only as they become relevant to the search. The tabu features help break best-response cycles, 25

35 and allow the algorithm to find pure Nash equilibrium in multi-player games where best response would typically fail. Moreover, they tested the algorithm on several classes of standard and random games. Their results showed that the algorithm performs well and provides the designer with the tradeoffs between search time and completeness. Genetic algorithms are another well-known heuristics for solving complicated combinatorial optimization problems. Since the 1960s, genetic algorithms have increased great interests in developing powerful algorithms to solve complex optimization problems. The basic idea of a genetic-based algorithm is to simulate an evolutionary process and obtain better solutions as the algorithm progresses. In every generation, the solutions are altered and combined to generate new solutions. Through the evaluation and selection of the candidate individuals, the fitter ones have better chances of survival. Genetic algorithms have been applied to a variety of problems. A more detailed introduction of genetic algorithm is provided in Chapter 5. Researchers also worked on using the genetic-based algorithms to find Nash equilibrium for game theoretical models. For example, In [69], Sefrioui et al. presented a genetic algorithm based on the concept of a non-cooperative multiple objective algorithm. In Japanya s work [35], a genetic algorithm is proposed to determine the optimal bidding strategy in a competitive auction market. His results indicated that proposed genetic algorithm converges much faster and is more reliable than Monte Carlo Simulation. Another paper we reviewed is to use the genetic algorithm to find a pure Stackelberg-Nash equilibrium. In [46], B. D. Liu designed a genetic algorithm for solving pure Stackelberg-Nash equilibrium of a nonlinear multilevel programming problem with multiple followers in which subject to information. As a byproduct, he also obtained a means for solving classical minimax problems. Finally, he showed some numerical examples to illustrate the effectiveness of the proposed genetic algorithm. We are interested in finding a pure Nash equilibrium solution when it exists in a two-person 26

36 investment game. We will show that though a dynamic programming algorithm can solve some special cases of the two-person investment game, it is not suitable for solving a general twoperson investment game because the principal of optimality does not follow. Hence we design a genetic algorithm for finding a Nash equilibrium in Chapter Price of anarchy It is noticed that economists and computer scientists that, even in very simple setting, the selfish behavior of players can lead to some inefficient outcomes. A well-known example of this phenomenon in economics is provided by the Prisoner s Dilemma [64], in which two players suffer a higher penalty in a Nash equilibrium state without coordination. In recent years, the interface between theoretical computer science and microeconomics, often called the algorithmic game theory, has become an extremely active research area. Researchers have tried to quantify the inefficiency due to the selfish behaviors. In [39], a framework to systematically study this issue is proposed. The price of anarchy [39, 59] is defined as the ratio between the total objective function value of the worst equilibrium and that of the social optimal solution. It is easy to see that for a minimization problem, the price of anarchy is always greater than or equal to 1. But for a maximization problem, it is always less than or equal to 1. If the price of anarchy of a game is 1, then its equilibria are fully efficient. Otherwise, if the price of anarchy is far from 1, implies that the system is less efficient due to its deliberate lack of coordination. More generally, the price of anarchy could help us to learn the worst-possible inefficiency of equilibria in various games. The methods to analyze price of anarchy heavily depend on the structures of games. For the games with a potential function, it is a good idea to utilize the potential function to bound the price of anarchy. More discussions could be found in Roughgarden s survey article [67]. The concept of price of anarchy has been adopted in many papers dealing with the routing games, network design and formation games, scheduling games and so on. We will review them in orders. 27

37 A majority of the research work on the price of anarchy concerns routing games. One reason for its popularity is that it is highly related to practical problems. For example, how to route traffic in a large communication network that has no central authority like the Internet? There are two different models of routing games which are most popular, namely the nonatomic selfish routing and atomic selfish routing. Nonatomic selfish routing problems are a natural generalization of Pigou s example [62], with the assumption that there are a very large number of players, each controlling a negligible fraction of the overall traffic and aiming to minimize its latency (traveling time). The price of anarchy in a nonatomic selfish routing game was first studied by Roughgarden and Tardos [68]. They proved that if the latency of each edge is a linear function of its congestion, the price of anarchy is at most 4 3 (Since this is a minimization latency problem, the price of anarchy is greater than or equal to 1). Then they extended the setting that edge latency functions are assumed only to be continuous and nondecreasing in the edge congestion. Later in [66], Roughgarden analyzed the Braess s Paradox and the computational complexity of detecting it algorithmically. He also described the case of the Stackelberg routing, which improves the price of anarchy using a modest degree of central control. For further details and discussion of selfish routing games, readers could refer to Roughgarden s review article [67] and Nisan s book [54]. In the atomic selfish routing games, each player controls a nonnegligible amount of traffic. The price of anarchy of atomic instances was first studied by Suri et al. [71] in the context of the asymmetric scheduling games. They proved an upper bound of 5 2 on the price of anarchy in such games when each player controls one unit of traffic and all cost functions are affine. Then Awerbuch et al. [5] generalized the results significantly. They proved the price of anarchy of an atomic instance with affine cost functions is Also they extended the case to the atomic instances with cost functions that are polynomial with nonnegative coefficients. More results on atomic selfish routing games are discussed in [5, 9, 55, 54]. Another trend of research on price of anarchy is focused on the network formation games. 28

38 The operations of many large computer networks, such as the Internet, are carried out by a large number of diverse and competitive entities. In these models, entities are seeking a balance between efficiency and stability in the network formation. That is, players want to minimize the expenses they incur when building the network as well as to ensure the quality of service provided by the network they formed. Typical works in this area include the connection games [20, 17, 2, 14] and facility location games [74]. For the connection games, the first bound on price of anarchy was given by Fabrikant et al. [20]. They developed the upper bound and lower bound of price of anarchy depending on different ranges of the parameter α defined in their model. Albers et al. generalized these results in [2]. Then Corbo and Parkes [14] studied bilateral connection games. They developed the lower and upper bounds of price of anarchy in their game and compared the results with the unilateral connection games of Fabrikant et al [20]. Particularly, they showed the worst-case price of anarchy in the bilateral connection game is worse than that in the unilateral connection game. In 2007, Demaine et al. [17] improved the upper bound of the price of anarchy obtained by Fabrikant et al. and also improved the constant upper bound found by Albers et al. Moreover, for bilateral network games, they proved the upper bound derived by Corbo and Parkes is tight under some conditions on α. For the facility location games, Vetta first introduced the game in [74] as one example of the utility games. In the facility location games, we want to locate the facilities, like Web servers, so as to serve a set of clients profitably. In Chapter 19 of [54], it is proved that this kind of games are in fact potential games. And the price of anarchy is at most 2. For more general utility games, the price of anarchy could be very low and near the 1. The price of anarchy of load balancing games or scheduling games is also a hot topic that many researchers are devoted to. They care about what happens to the makespan (the maximum load over all the machines) if selfish users aim at maximizing their individual benefit about the assignment of tasks to the machines. The concept of price of anarchy is introduced by Koutsoupias and Papadimitriou in [39] (called coordination ratio). They studied load bal- 29

39 ancing in form of a special routing game. Therefore, the game theoretical model underlying the load balancing games is also known as the KP model. The upper bound of price of anarchy for pure equilibria in load balancing games with identical machines is obtained by Finn and Horowitz in [25]. The lower bound on the price of anarchy for mixed equilibrium on identical machines is given by Koutsoupias and Papadimitriou in [39]. The analysis of the corresponding upper bound is obtained in Czumaj and Vöcking s work [15]. The upper and lower bounds on the price of anarchy for pure and mixed equilibria in load balancing games with uniformly related machines can be found in [15]. In the past ten years, there are different variants of games for load balancing or routing on parallel links. More works in this area are described in Chapter 20 of Nisan s book [54]. So far, we have reviewed a classic stream of research works on price of anarchy. Most of them are related to the applications of Internet service or other issues in computer science. In the past three or four years, scholars studied the price of anarchy for more games, including those related to Internet technology as well as other areas. For example, researchers studied the price of anarchy in a more general frame of selfish routing model - congestion games in Economics, including Christodoulou and Koutsoupias in[12], and Fotakis et al. in [28]. In addition, Wang et al. studied the two-person knapsack games [77] and the two-group knapsack games [76] inspired by the real-world business problems in investment decision making. They also studied the price of anarchy for these two types of games. Chawla and Niu studied the price of anarchy in Bertrand games [10], which extended the classic Bertrand game in Economics to a network pricing game. Some scholars started to study the strong price of anarchy in recent years [3]. A strong equilibrium, which is proposed by Aumann, is a state when no group of players can coordinate the actions of players in it to increase every player s utility in the group. We should note that every strong equilibrium is necessarily a Nash equilibrium. However, most games do not 30

40 admit strong equilibria. Holzman and Law-Yone studied the strong equilibrium in congestion games [33]. They obtained the conditions for the existence of a strong equilibrium in congestion games. In [3], Andelman, Feldman and Mansour defined Strong Price of Anarchy (SPoA) to be the ratio of the worst strong equilibrium and the social optimum. They studied the strong of price of anarchy in two settings, job scheduling and network creation. They showed that strong equilibria always exist for these two settings except for a subset of network creation games. For the network creation games, SPoA is at most 2. In job scheduling games, it can be bounded as a function of the number of machines and the size of coalition. In [67], Roughgarden surveyed the potential functions and inefficiency of equilibria for several games including routing games and connection games. He argued that potential function technique is by no means the only one known for bounding the price of anarchy. However, so far, it has been the most versatile and powerful. Some network or related games admit wellbehaved potential functions may provide a better approach to analyze the price of anarchy. But, for many games, the bounding and optimization of potential functions is difficult and their applicability is limited. Our two-person investment game belongs to this case. It is hard to analyze the price of anarchy for a general setting. Therefore, in Chapter 6, we limit our analysis of price of anarchy on a special case of the two-person investment game. 31

41 Chapter 3 Existence of Nash Equilibrium From John F. Nash s famous paper [53], we know that there always exists a mixed Nash equilibrium in a noncooperative game. However, this is not true for pure Nash equilibrium. Therefore it is natural to ask under what conditions a pure Nash equilibrium solution exists for the two-person investment game. In this chapter, we explore the existence issue of pure Nash equilibrium for the two-person investment game. More specifically, we follow the idea of potential functions which is defined by Monderer and Shapley [51] to prove the existence of Nash equilibrium in some cases. The concept of potential function is first introduced in [51]. In the following, we give a definition of the potential function of a two-person investment game. Definition For a two-person ivestment game, a potential function of the game is a real valued function over the players feasible states such that its value increases strictly when any player shifts to a new state to improve his objective. As reviewed in Section of Chapter 2, there are two types of potential functions. One is called the ordinal potential function, the other is called the cardinal potential function. It is clear to see that the potential function we defined above is an ordinal one, because we only require that the changing trend of a player s objective by shifting to a new strategy is the 32

42 same as that of the function by the shift. However, the value changed of the objective is not necessarily the same as that of the function. 3.1 Basic model In this section, we derive the existence conditions of pure Nash equilibrium for the basic model of the two-person investment game. Recall that 1 = α 1 β 11 α 2 β 12, 2 = α 2 β 22 α 1 β 21. To show our main theorem which completely characterizes the existence of Nash equilibrium solutions using the product value of 1 and 2, some technical lemmas are presented first. Lemma If 1 = 2 = 0 in the two-person investment game, then Φ 1 (S 1, S 2 ) = p(s 1 ) + p(s 2 ) is a potential function. Proof. Considering 1 = 2 = 0, Equation (1.2) becomes, O 1 (S 1, S 2 ) = β 11 p(s 1 ) β 12 p(s 2 ), O 2 (S 1, S 2 ) = β 22 p(s 1 ) β 21 p(s 2 ). Suppose there is at least one player, say player 1, who can improve his objective by shifting to a new feasible state, say S 1. Then we have O 1(S 1, S 2 ) < O 1 (S 1, S 2). That is, β 11 p(s 1 ) β 12 p(s 2 ) < β 11 p(s 1) β 12 p(s 2 ). Because β 11 0, consequently we have p(s 1 ) < p(s 1 ). Add p(s 2 ) to the both sides of this inequality, then it comes to Φ 1 (S 1, S 2 ) < Φ 1 (S 1, S 2). 33

43 The same conclusion can be derived for player 2. Thus, Φ 1 (S 1, S 2 ) = p(s 1 ) + p(s 2 ) is indeed a potential function in this case. Lemma If 1 = 0 and 2 0 in the two-person investment game, then Φ 2 (S 1, S 2 ) = Mp(S 1 ) + β 22 p(s 2 ) 2 p(s 1 S 2 ) is a potential function with M = k p k. If 1 0 and 2 = 0, then Φ 2 (S 1, S 2 ) = M p(s 2 ) + β 11 p(s 1 ) 1 p(s 1 S 2 ) is a potential function with M = k p k. Proof. We only show that Φ 2 is a potential function. The conclusion for Φ 2 follows similarly. When 1 = 0 and 2 0, substituting them into (1.2) yields, O 1 (S 1, S 2 ) = β 11 p(s 1 ) β 12 p(s 2 ), O 2 (S 1, S 2 ) = β 22 p(s 1 ) β 21 p(s 2 ) 2 p(s 1 S 2 ). If player 1 can improve his objective by shifting to a new state S 1, then O 1(S 1, S 2 ) < O 1 (S 1, S 2). Consequently, we have p(s 1 ) < p(s 1 ) since β 11 > 0, i.e., p(s 1 ) p(s 1) > 0. As we assume that all the projects profits are integers, hence p(s 1 ) p(s 1) 1. Noticing that Φ 2 (S 1, S 2 ) = Mp(S 1 ) + β 22 p(s 2 ) 2 p(s 1 S 2 ), (3.1) Φ 2 (S 1, S 2 ) = Mp(S 1) + β 22 p(s 2 ) 2 p(s 1 S 2 ). (3.2) We subtract (3.1) from (3.2), then Φ 2 (S 1, S 2 ) Φ 2 (S 1, S 2 ) = M[p(S 1) p(s 1 )] 2 [p(s 1 S 2 ) p(s 1 S 2 )]. Because max S 1,S 1 2[p(S,S 1 S 2) p(s 1 S 2 )] = k M 2 p k > 0. k p k 0 = k p k, it is clear that Φ 2 (S 1, S 2) Φ(S 1 S 2 ) 34

44 If player 2 wants to improve his objective by shifting to a state S 2, then O 2(S 1, S 2 ) < O 2 (S 1, S 2 ). This further implies that β 22 p(s 2 ) β 21 p(s 1 ) 2 p(s 1 S 2 ) < β 22 p(s 2) β 21 p(s 1 ) 2 p(s 1 S 2). Subtracting β 21 p(s 1 ) on both sides, we have β 22 p(s 2 ) 2 p(s 1 S 2 ) < β 22 p(s 2) 2 p(s 1 S 2). Therefore, Φ 2 (S 1, S 2) Φ 2 (S 1, S 2 ) = β 22 p(s 2) 2 p(s 1 S 2) (β 22 p(s 2 ) 2 p(s 1 S 2 )) > 0. Hence we have Φ 2 (S 1, S 2 ) > Φ 2(S 1, S 2 ). From Definition 3.0.1, we conclude that Φ 2 (S 1, S 2 ) is indeed a potential function in this case. A similar proof follows for Φ 2 (S 1, S 2 ). Lemma If 1 > 0 and 2 > 0 in the two-person investment game, then Φ 3 (S 1, S 2 ) = 2 β 11 p(s 1 ) + 1 β 22 p(s 2 ) 1 2 p(s 1 S 2 ) is a potential function. Proof. If player 1 changes from S 1 to a new state S 1 to improve his objective value, then O 1 (S 1, S 2 ) < O 1 (S 1, S 2). Equation (1.2) implies that β 11 p(s 1 ) β 12 p(s 2 ) 1 p(s 1 S 2 ) < β 11 p(s 1 ) β 12p(S 2 ) 1 p(s 1 S 2). Using Φ 3 (S 1, S 2) minus Φ 3 (S 1, S 2 ), we have Φ 3 (S 1, S 2 ) Φ 3 (S 1, S 2 ) = 2 β 11 p(s 1) 1 2 p(s 1 S 2 ) 2 β 11 p(s 1 ) p(s 1 S 2 ) = 2 [β 11 p(s 1) 1 p(s 1 S 2 ) (β 11 p(s 1 ) 1 p(s 1 S 2 ))]. 35

45 Since 2 > 0, we see that Φ 3 (S 1, S 2 ) Φ 3 (S 1, S 2 ) > 0. That is, Φ 3 (S 1, S 2 ) > Φ 3 (S 1, S 2 ). If player 2 changes to S 2 to improve his objective, we have O 2(S 1, S 2 ) < O 2 (S 1, S 2 ). Hence β 22 p(s 2 ) 2 p(s 1 S 2 ) < β 22 p(s 2) 2 p(s 1 S 2). Consequently, Φ 3 (S 1, S 2) Φ 3 (S 1, S 2 ) = 1 [β 22 p(s 2) 2 p(s 1 S 2) (β 22 p(s 2 ) 2 p(s 1 S 2 ))]. Since 1 > 0, Φ 3 (S 1, S 2 ) > Φ 3(S 1, S 2 ). Therefore, Φ 3 (S 1, S 2 ) is a potential function. Lemma If 1 < 0 and 2 < 0 in the two-person investment game, then Φ 4 (S 1, S 2 ) = Φ 3 (S 1, S 2 ) is a potential function. Proof. Using the same argument as in Lemma , and by noticing the fact that 1 < 0 and 2 < 0, we can construct a proof. Combining the previous lemmas, we have shown that there always exists a potential function Φ for the two-person investment game when Let us start from any current feasible state of the game, say (S1 k, Sk 2 ). If it is not a Nash equilibrium, then by definition, at least one player can move to a new feasible state, say (S k+1 1, S k+1 2 ), with an improved objective value Φ(S k+1 1, S k+1 2 ) > Φ(S k 1, S k 2 ). 36

46 If (S k+1 1, S k+1 2 ) is a Nash equilibrium, then we are done. Otherwise we can repeat this procedure until a Nash equilibrium is obtained. Since the value of the potential function increases strictly, no feasible state will be visited more than once in this process. As the number of the feasible states is finite, we know after finitely possible different states, two players will reach a Nash equilibrium finally. This leads us to the main existence theorem. Theorem If in the two-person investment game, then there exists at least one Nash equilibrium of the game. Note that the potential function at a Nash equilibrium of the investment game may or may not achieve the maximum value. But a feasible state at which the potential function achieves its maximum must be a Nash equilibrium. Otherwise the above procedure will continue to reach a new feasible state with a potential value strictly bigger than the maximum potential. That s a contradiction. In other words, if in the two-person investment game, a feasible state with the maximum potential value is a Nash equilibrium of the game. Therefore, we have the following corollary: Corollary If in the two-person investment game, then a feasible state with the maximum potential value is a Nash equilibrium of the game. The corollary presents a sufficient condition for identifying a Nash equilibrium, but it is not a necessary condition. Consider a simple example with two players, and four projects {P 1, P 2, P 3, P 4 } with profits p 1 = 3, p 2 = 5, p 3 = 4, and p 4 = 6. Now these projects are packed as four sets which the players can invest. They are: C 1 = {P 1, P 2 } C 2 = {P 3, P 4 } C 3 = {P 1, P 3 } C 4 = {P 2, P 4 } Other parameters of the model in this example are set as: 37

47 b 1 = b 2 = 1, α 1 = 1/3, α 2 = 2/3, β 11 = β 22 = 1, and β 12 = β 21 = 0. Then 1 = 1/3, 2 = 2/3 and the potential function Φ 3 (S 1, S 2 ) = 2 3 p(s 1) p(s 2) 2 9 p(s 1 S 2 ). It can be verified that both (C 2, C 1 ) and (C 4, C 3 ) are Nash equilibria of the game. However, Φ 3 (C 2, C 1 ) = 28/3 and Φ 3 (C 4, C 3 ) = 29/3. Now we turn our attention to the case of 1 2 < 0. Theorem If 1 2 < 0 for the two-person investment game, then there may or may not exist one Nash equilibrium of the game. And there exists at least one instance without Nash equilibrium. Proof. We use two examples to show the situation. Example Assume two investors consider to invest on four project P 1, P 2, P 3, P 4, which are packed to four sets as follows, C 1 = {P 1, P 2 }, C 2 = {P 3, P 4 }, C 3 = {P 1, P 3 }, C 4 = {P 2, P 4 }. Suppose each investor can only choose one set (strategy), i.e., b 1 = b 2 = 1. The expected market profits of projects are given as p 1 = p 2 = 1, p 3 = p 4 = 10. We also assume that α 1 = α 2 = 0.5, β 11 = β 21 = 2, and β 22 = β 12 = 1. 38

48 Consequently, we have 1 = α 1 β11 α 2 β 12 = = 0.5 > 0, 2 = α 2 β22 α 1 β 21 = = 0.5 < 0, O 1 (S 1, S 2 ) = 2p(S 1 ) p(s 2 ) 0.5p(S 1 S 2 ), O 2 (S 1, S 2 ) = p(s 2 ) 2p(S 1 ) + 0.5p(S 1 S 2 ). Now we use the following table to show the objective values of two players at different strategy sets (S 1, S 2 ). Every entry in this table indicates (O 1 (S 1, S 2 ), O 2 (S 1, S 2 )) at state (S 1, S 2 ). Using the result, we can check the existence of Nash equilibrium. Table 3.1: Two players objective values at all the states - Example player 1 player 2 C 1 C 2 C 3 C 4 (0, 0) (-2,2) (-20,20) (-11,11) (-11,11) C 1 (4,-4) (1,-1) (-16,16) (-7.5,7.5) (-7.5,7.5) C 2 (40,-40) (38,-38) (10,-10) (24,-24) (24,-24) C 3 (22,-22) (19.5,-19.5) (-3,3) (5.5,-5.5) (11,-11) C 4 (22,-22) (19.5,-19.5) (-3,3) (11,-11) (5.5,-5.5) It is easy to verify that (C 2, C 2 ) is a Nash equilibrium for this game. Now let us consider another example where 1 2 < 0. Example Suppose we have the same problem, assumptions and coefficients as in example 3.1.1, except that p 3 = 1. Now we use the following table to show the objective values of two players at 39

49 different strategy sets (S 1, S 2 ). Every entry in this table indicates (O 1 (S 1, S 2 ), O 2 (S 1, S 2 )) at state (S 1, S 2 ). Using the result, we can check the existence of Nash equilibrium. Table 3.2: Two players objective values at all the states - Example player 2 player 1 C 1 C 2 C 3 C 4 (0,0) (-2,2) (-11,11) (-2,2) (-11,11) C 1 (4,-4) (1,-1) (-7,7) (1.5,-1.5) (-7.5,7.5) C 2 (22,-22) (20,-20) (5.5,-5.5) (19.5,-19.5) (6,-6) C 3 (4,-4) (1.5,-1.5) (-7.5,7.5) (1,-1) (-7,7) C 4 (22,-22) (19.5,-19.5) (6,-6) (20,-20) (5.5,-5.5) It is easy to verify that there does not exist any pure Nash equilibrium in this game. Through these two examples, we can see that a Nash equilibrium may or may not exist when 1 2 < 0. To conclude our analysis, for the two-person investment game, Nash equilibrium exists if When 1 2 < 0, there exists at least one instance without Nash equilibrium. Notice that i = α i β ii α j β ij for i = 1, 2 and j i. Hence i can be considered as a relative discount on investor i s objective of the projects commonly selected by both investors. When i > 0, any commonly selected projects are disadvantageous to player i, so he tends to improve the objective by not overlapping the opponent s strategy set. On the other hand, when i < 0, any commonly selected projects are favorable to investor i, so he tends to improve the objective by sticking to the commonly selected projects. When i = 0, if there are overlaps between two investors strategy sets or not makes no difference to investor i. As long as both investors hold the same attitude toward the commonly selected project, i.e., 1 2 0, a mutually agreeable steady state will be reached. when 1 2 < 0, one player tends to escape from the opponent s choice while the opponent is chasing for commonly invested projects. Hence there may be no 40

50 equilibrium that can be reached. So far, we have finished exploring the existence condition of Nash equilibrium for the basic model of the two-person investment game. Next, we extend the basic model and explore the existence conditions of Nash equilibrium in a new setting. 3.2 Extensions In this section, we make some extensions of our original model and continue to explore the existence conditions of pure Nash equilibrium. Suppose that the profit of each project is different between two players. For example, when a big-name company opens a new store in an area, it may attract customers more than a small local company. Another example is that, due to the preference of local residents, a seafood restaurant may be more profitable than a vegetarian restaurant even the restaurants are located at the same place. We develop the extended mathematical models as follows. Let p i k indicate the profit of project k of player i, say i = 1, 2. And p1 k p2 k for at least one k when 1 k n. We consider two cases as follows Case 1 - same proportional profits between two players Suppose that p 2 k = mp1 k where m > 0 is a real number for all k. It implies that for all the projects, the ratio of each project s profit for player 2 to that for player 1 remains the same. When m = 1, it becomes our original basic model. Now we have the associated objective 41

51 functions, O 1 (S 1, S 2 ) = β 11 P 1 (S 1, S 2 ) β 12 P 2 (S 1, S 2 ) = β 11 [p(s 1 ) α 1 p 1 (S 1 S 2 )] β 12 [p(s 2 ) α 2 mp 1 (S 1 S 2 )] = β 11 p(s 1 ) β 12 p(s 2 ) (α 1 β 11 α 2 β 12 m)p 1 (S 1 S 2 ) = β 11 p(s 1 ) β 12 p(s 2 ) γ 1 p 1 (S 1 S 2 ), (3.3) O 2 (S 1, S 2 ) = β 22 P 2 (S 1, S 2 ) β 21 P 1 (S 1, S 2 ) = β 22 [p(s 2 ) α 2 mp 1 (S 1 S 2 )] β 21 [p(s 1 ) α 1 p 1 (S 1 S 2 )] = β 22 p(s 2 ) β 21 p(s 1 ) (α 2 β 22 m α 1 β 21 )p 1 (S 1 S 2 ) = β 22 p(s 2 ) β 21 p(s 1 ) γ 2 p 1 (S 1 S 2 ), (3.4) where γ 1 = α 1 β 11 mα 2 β 12 and γ 2 = mα 2 β 22 α 1 β 21. Moreover, p 1 (S 1 S 2 ) is player 1 s profits of all the commonly selected projects. To show our major theorem which completely characterizes the existence of Nash equilibrium solutions using the product value of γ 1 and γ 2, some technical lemmas are presented first. Lemma If γ 1 = γ 2 = 0 in the two-person investment game of the extension case 1, then Φ 1 (S 1, S 2 ) = p(s 1 ) + p(s 2 ) is a potential function for this case. Proof. Following the proof of Lemma , it leads to derive the conclusion accordingly. Lemma If γ 1 = 0 and γ 2 0 in the two-person investment game of the extension case 1, then Φ 5 (S 1, S 2 ) = Np(S 1 ) + β 22 p(s 2 ) γ 2 p 1 (S 1 S 2 ) is a potential function with N = 1 + γ 2 k p 1 k. If γ 1 0 and γ 2 = 0, then Φ 5 (S 1, S 2 ) = N p(s 2 ) + β 11 p(s 1 ) γ 1 p 1 (S 1 S 2 ) 42

52 is a potential function with N = 1 + γ 1 p 1 k. k Proof. We only need to prove the case of γ 1 = 0 and γ 2 0. The second part can be derived similarly. If player 1 can improve his objective by shifting to a new state S 1, then O 1 (S 1, S 2 ) < O 1 (S 1, S 2). Since γ 1 = 0, we have p(s 1 ) < p(s 1 ), i.e., p(s 1 ) p(s 1) 1. Now we subtract Φ 5 (S 1, S 2 ) from Φ 5 (S 1, S 2), it is easy to verify that Φ 5 (S 1, S 2 ) Φ 5 (S 1, S 2 ) = N[p(S 1) P (S 1 )] γ 2 [p 1 (S 1 S 2 ) p 1 (S 1 S 2 )] N γ 2 k p 1 k > 0. If player 2 could improve his objective by shifting to a state S 2, then O 2(S 1, S 2 ) < O 2 (S 1, S 2 ). We can conclude that Φ 5 (S 1, S 2 ) Φ 5(S 1 S 2 ). Therefore, Φ 5 is a potential function. A similar proof follows for Φ 5. Lemma If γ 1 > 0 and γ 2 > 0 in the two-person investment game of the extension case 1, then Φ 6 (S 1, S 2 ) = γ 2 β 11 p(s 1 ) + γ 1 β 22 p(s 2 ) γ 1 γ 2 p 1 (S 1 S 2 ) is a potential function. Proof. Suppose player 1 moves to S 1 to improve his objective. Then we have O 1(S 1, S 2 ) < O 1 (S 1, S 2), i.e., β 11 p(s 1 ) β 12 p(s 2 ) γ 1 p 1 (S 1 S 2 ) < β 11 p(s 1) β 12 p(s 2 ) γ 1 p 1 (S 1 S 2 ). Using Φ 6 (S 1, S 2) minus Φ 6 (S 1, S 2 ), we have, Φ 6 (S 1, S 2 ) Φ 6 (S 1, S 2 ) = γ 2 β 11 p(s 1) γ 1 γ 2 p 1 (S 1 S 2 ) γ 2 β 11 p(s 1 ) + γ 1 γ 2 p 1 (S 1 S 2 ) = γ 2 [β 11 p(s 1) γ 1 p 1 (S 1 S 2 ) (β 11 p(s 1 ) γ 1 p 1 (S 1 S 2 ))]. 43

53 Since γ 2 > 0, we know Φ 6 (S 1, S 2) > Φ 6 (S 1, S 2 ). If player 2 moves to S 2 to improve his objective, then O 2(S 1, S 2 ) < O 2 (S 1, S 2 ). We have β 22 p(s 2 ) γ 2 p 1 (S 1 S 2 ) < β 22 p(s 2) γ 2 p 1 (S 1 S 2). Consequently, Φ 6 (S 1, S 2) Φ 6 (S 1, S 2 ) = γ 1 [β 22 p(s 2) γ 2 p 1 (S 1 S 2) (β 22 p(s 2 ) γ 2 p 1 (S 1 S 2 ))]. Since γ 1 > 0, we have Φ 6 (S 1, S 2 ) > Φ 6(S 1, S 2 ). To conclude, Φ 6 (S 1, S 2 ) is a potential function in this case. Lemma If γ 1 < 0 and γ 2 < 0 in the two-person investment game of the extension case 1, then Φ 7 (S 1, S 2 ) = Φ 6 (S 1, S 2 ) is a potential function. Proof. With the same argument as in Lemma , by noticing the fact that γ 1 < 0 and γ 2 < 0, we can construct a proof. Theorem If γ 1 γ 2 0 in the two-person investment game of the extension case 1, then there exists at least one Nash equilibrium of the game. If γ 1 γ 2 < 0, there exists at least one instance without Nash equilibrium. Proof. From Lemma to Lemma , we conclude that there always exists a potential function for the extension case 1 of two-person investment game when γ 1 γ 2 0. Therefore, at least one Nash equilibrium exists under this condition. Since k = 1 is a special case of this part 1, from Instance 3.1.2, we know that at least there exists one instance without Nash equilibrium when γ 1 γ 2 < 0. 44

54 We are now marching to the case 2 in which we consider a more general case that the ratios of profits between the two players are not the same for different projects. This is a more general and commonly seen practice in the business world that different investors make different profits. However, it is not that easy to develop the existence conditions for the most general frame. We only provide some partial results of a few special situations. 45

55 3.2.2 Case 2 - different proportional profits between two players Consider the ratios of profits between two players are not the same among all the projects. Suppose p 2 k = m kp 1 k where m k [0, Q] for all k and Q is a relatively large number. Let p i (S 1 S 2 ) be the player i s profits of all the commonly selected projects. Then the players objective functions become O 1 (S 1, S 2 ) = β 11 p(s 1 ) β 12 p(s 2 ) (α 1 β 11 p 1 (S 1 S 2 ) α 2 β 12 p 2 (S 1 S 2 )) = β 11 p(s 1 ) β 12 p(s 2 ) (α 1 β 11 p 1 k α 2β 12 m k p 1 k ) {k P k S 1 S 2 } {k P k S 1 S 2 } = β 11 p(s 1 ) β 12 p(s 2 ) (α 1 β 11 α 2 β 12 m k )p 1 k {k P k S 1 S 2 } = β 11 p(s 1 ) β 12 p(s 2 ) γk 1 p1 k, (3.5) {k P k S 1 S 2 } where γ 1 k = α 1β 11 α 2 β 12 m k for all k. O 2 (S 1, S 2 ) = β 22 p(s 2 ) β 21 p(s 1 ) (α 2 β 22 p 2 (S 1 S 2 ) α 1 β 21 p 1 (S 1 S 2 )) = β 22 p(s 2 ) β 21 p(s 1 ) (α 2 β 22 m k p 1 k α 1β 21 p 1 k ) {k P k S 1 S 2 } {k P k S 1 S 2 } = β 22 p(s 2 ) β 21 p(s 1 ) (α 2 β 22 m k α 1 β 21 )p 1 k {k P k S 1 S 2 } = β 22 p(s 2 ) β 21 p(s 1 ) γk 2 p1 k, (3.6) {k P k S 1 S 2 } where γ 2 k = α 2β 22 m k α 1 β 21 for all k. We can see that the analysis of this case 2 is much more complicated than case 1 and the basic model. For the simplicity of analysis, we only present the existence conditions of Nash equilibrium in the following four special cases. 46

56 Lemma If γk 1 = γ2 k = 0 for all k in the two-person investment game of the extension case 2, then Φ 1 (S 1, S 2 ) = p(s 1 ) + p(s 2 ) is a potential function. Proof. It is easy to see that this case becomes the same situation as in Lemma Hence we can follow the proof of Lemma to draw the conclusion. Lemma Assume γ 1 k = 0 for all k and γ2 k 0 for at least one k in the two-person investment game of the extension case 2. Let γmax 2 = max k { γ2 k, 1 k n}, then Φ 8(S 1, S 2 ) = Qp(S 1 ) + β 22 p(s 2 ) is a potential function with Q = 1 + 2γmax 2 p 1 k. γk 2p1 k {k P k S 1 S 2 } Assume γk 2 = 0 for all k and γ1 k 0 for at least one k in the two-person investment game of the extension case 2. Let γmax 1 = max k { γ1 k, 1 k n}, then Φ 8 (S 1, S 2 ) = Q p(s 2 ) + β 11 p(s 1 ) is a potential function with Q = 1 + 2γmax 1 p 1 k. γk 1p1 k {k P k S 1 S 2 } Proof. We only need to prove the case with γk 1 = 0 for all k and γ2 k 0 for at least one k. The second part can be derived similarly. If player 1 can improve his objective by shifting to a new state S 1, then O 1(S 1, S 2 ) < O 1 (S 1, S 2). Since γk 1 = 0 for all k, we have p(s 1) < p(s 1 ), i.e., p(s 1 ) p(s 1) 1. k k Using Φ 8 (S 1, S 2) minus Φ 8 (S 1, S 2 ), we see that Φ 8 (S 1, S 2 ) Φ 8 (S 1, S 2 ) = Q[p(S 1) P (S 1 )] ( γk 2 p1 k γk 2 p1 k ) Q 2γmax 2 p 1 k > 0. k {k P k S 1 S 2} {k P k S 1 S 2 } If player 2 wants to improve his objective by shifting to a state S 2, then O 2(S 1, S 2 ) < O 2 (S 1, S 2 ). Hence we can prove that Φ 8 (S 1, S 2 ) Φ 8(S 1, S 2 ). Therefore, Φ 8 (S 1, S 2 ) = Qp(S 1 ) + β 22 p(s 2 ) is a potential function. γk 2p1 k {k P k S 1 S 2 } 47

57 Lemma If there exists one k n such that γ 1 k n = γ 2 k n = 0, and there exist one k 1 and one k 2 such that γ 1 k 1 > 0 and γ 2 k 2 > 0, then we have k 1 = k 2. That implies the nonzero γ 1 k 1 and γ 2 k 2 must appear in pairs. Proof. We know there exists k n, such that γ 1 k n = γ 2 k n = 0. Then we have, α 1 β 11 α 2 β 12 m kn = 0, α 2 β 22 m kn α 1 β 21 = 0. It is easy to verify that α 1β 11 α 2 β 12 = α 1β 21 α 2 β 22. If there exist one k 1, such that γ 1 k 1 0, then it becomes true that α 2β 12 m k1 1. Therefore, α 2β 22 m k1 1. Consequently, γk 2 α 1 β 11 α 1 β 1 0. Hence the nonzero 21 γk 1 1 and γk 2 2 must appear in pairs. Lemma If there exist only one k 1 and only one k 2 such that γ 1 k 1 > 0 and γ 2 k 2 > 0, and γ 1 k = γ2 k = 0 for all other k in the two-person investment game of the extension case 2, then Φ 9 (S 1, S 2 ) = γ 2 k 1 β 11 p(s 1 ) + γ 1 k 1 β 12 p(s 2 ) γ 1 k 1 γ 2 k 1 p 1 k 1 δ (S1 S 2 )(k 1 ) is a potential function, where δ (S1 S 2 )(k 1 ) = 1 if k 1 S 1 S 2 0 Otherwise Proof. Recall the objective functions are O 1 (S 1, S 2 ) = β 11 p(s 1 ) β 12 p(s 2 ) γk 1 p1 k, {k P k S 1 S 2 } O 2 (S 1, S 2 ) = β 22 p(s 2 ) β 21 p(s 1 ) γk 2 p1 k. {k P k S 1 S 2 } From Lemma , we know that k 1 = k 2. If player 1 improves his objective by shifting to a new state S 1, then O 1(S 1, S 2 ) < O 1 (S 1, S 2). Hence we have Φ(S 1, S 2) Φ(S 1, S 2 ) = 48

58 γ 2 k 1 (β 11 p(s 1 ) β 11p(S 1 ) (γ 1 k 1 p 1 k 1 δ (S 1 S 2) (k 1) γ 1 k 1 p 1 k 1 δ (S1 S 2 )(k 1 ))). Consequently, we have the following four cases: (1) If k 1 / S 1 S 2 and k 1 / S 1 S 2, then Φ(S 1, S 2) Φ(S 1, S 2 ) = γk 2 1 (O 1 (S 1, S 2) O 1 (S 1, S 2 )) > 0. (2) If k 1 S 1 S 2 and k 1 S 1 S 2, then Φ(S 1, S 2) Φ(S 1, S 2 ) = γk 2 1 (β 11 p(s 1 ) β 11p(S 1 )) = γk 2 1 (O 1 (S 1, S 2) O 1 (S 1, S 2 )) > 0. (3) If k 1 / S 1 S 2 and k 1 S 1 S 2, then Φ(S 1, S 2) Φ(S 1, S 2 ) = γk 2 1 (β 11 p(s 1 ) γ1 k 1 p 1 i 1,j 1 β 11 p(s 1 )) = γk 2 1 (O 1 (S 1, S 2) O 1 (S 1, S 2 )) > 0. (4) If k 1 S 1 S 2 and k 1 / S 1 S 2, then Φ(S 1, S 2) Φ(S 1, S 2 ) = γk 2 1 (β 11 p(s 1 ) (β 11 p(s 1 ) γk 1 1 p 1 k 1 )) = γk 2 1 (O 1 (S 1, S 2) O 1 (S 1, S 2 )) > 0. On the other hand, if player 2 improves his objective by shifting to a new state S 2, the symmetric structure of the problem setting takes care of the rest of the proof. Lemma If there exist only one k 1 and only one k 2 such that γ 1 k 1 < 0 and γ 2 k 2 < 0, and γ 1 k = γ2 k = 0 for all other k in the two-person investment game of the extension case 2, then Φ 10 (S 1, S 2 ) = Φ 9 (S 1, S 2 ) is a potential function. Proof. With the same argument as in Lemma , we can construct a proof. To conclude, in the above four cases, we can always find a potential function for each of them using Lemma through Lemma Then there exists at least one Nash equilibrium solution in each of these cases. Other than these cases, it is hard to verify the existence of Nash equilibrium in general. 49

59 3.3 Summary In this chapter, we have described the conditions under which there exists at least one pure Nash equilibrium for the basic model of the two-person investment game. Later by providing two numerical examples, we have illustrated that there may or may not exist a Nash equilibrium if these conditions are not met. Furthermore, we have extended the basic model into two cases to analyze the existence issue of pure Nash equilibrium. For the case 2, we finished the analysis of some basic situations but it is hard to generalize it to more complicated cases. 50

60 Chapter 4 Finding Nash Equilibrium - Dynamic Programming Approach In the last chapter, we showed that the potential function method leads a way to prove that a two-person investment game is guaranteed to possess at least one pure Nash equilibrium, when It is natural to consider the complexity of computing a pure Nash equilibrium when it exists. Will there be a polynomial-time algorithm for finding pure Nash equilibria in the two-person investment game? If not, how should we design a heuristics to find a pure Nash equilibrium in a two-person investment game? The structure of this chapter is organized as follows. In Section 4.1, we prove that computing a pure Nash equilibrium of a two-person investment game is N P-hard. Then we introduce a special class of our game in which a dynamic programming based algorithm (DPKG in [77]) can be implemented for finding a pure Nash equilibrium in pseudo polynomial computing time in Section 4.2. In Section 4.3, we extend the dynamic programming algorithm to solve another special case of the two-person investment game. We also show that this dynamic programming approach does not work for the general two-person investment games due to the absence of principle of optimality in Section 4.4. An example is provided to show this obstacle. Hence 51

61 we need to develop other algorithms to find a pure Nash equilibrium for the general games. 4.1 N P-hardness From the conclusion of Chapter 3, we know that when 1 2 0, the feasible state with the maximum potential value is a Nash equilibrium of the game. However, this makes no claim regarding the complexity of the problem. In some games, the potential function can be optimized in polynomial time, but in others the optimization problem may be N P-hard. To get a good handle on the complexity of finding pure equilibria in a two-person investment game, we consider a closely related problem in which there is only one player in the investment situation. We name it as investment problem and define it as follows. Investment Problem. Suppose one investor is interested in investing on n projects P 1,..., P n. For each project, it has a market profit p i, for i = 1,..., n. These projects are packed into m sets C 1,..., C m for the player s investment consideration. For each project set, its profit is defined as the sum of the profits of all the projects in it. And the profit of a project only counts once towards the total profits even it is chosen multiple times. Suppose the investor has a budget limit on the total number of the sets he can choose, the objective is to maximize the total profits the investor could gain by investing on the project sets under the budget constraint. It is not hard to observe that the investment problem is a combinatorial optimization problem by definition. Now we define the decision version of this optimization problem. Assume the player can only invest on k project sets at most. The decision problem is, given a number K, whether there exist l project sets that the player could choose, such that l k and the total profits gained from these project sets are greater than or equal to K? 52

62 Next, we show that to find an optimal solution for this decision problem is an N P-complete problem. We shall prove that 3-set exact covering problem, which is a well-known N P-complete problem, can polynomially reduce to the decision problem. Hence we know the decision version of the investment problem is also N P-complete. Theorem The decision version of the investment problem is N P-complete. Proof. Firstly, we show that given a candidate solution of this decision problem, a YES answer can be verified in polynomial time. This is because for each project P i, i = 1,..., n, we can always check if it belongs to the given l sets or not. For those projects that belong to the given sets, we calculate the sum of their profits and compare it to K. If the sum is greater than or equal to K, a YES answer is verified. All the processes described above can be finished in polynomial time since we have finite projects and project sets, which implies that a YES answer can be verified in polynomial time. Therefore, the decision problem belongs to the N P class. Secondly, we prove that given an instance of the 3-set exact covering problem, we can construct, in polynomial time, an instance of the decision problem of the investment problem. The 3-set exact covering problem can be described in the following way: for a set S = {u 1, u 2,..., u 3m }, given a collection F of subsets of set S, i.e., F = {S 1, S 2,..., S n } with S i = 3, if there exist m disjointed subsets of F, S i1, S i1,..., S im, so that m j=1 S i j S? Now we construct an instance of the decision version of the investment problem accordingly. We construct an example with 3m projects P 1, P 2,..., P 3m, where P i is corresponding to the element u i in the 3-set exact covering problem, i = 1,..., 3m. Assume for each project, it has a profit of return of 1. In this specific example, we make these projects be packed into n sets {S 1, S 2,..., S n } exactly like F for the investor to select. And we know with each project set, there are 3 projects in it. Now we set K = 3m and k = m. Then the investor is faced the problem that whether he can choose m project sets from {S 1, S 2,..., S n } so that the total profits could be as much as 3m. This is exactly an instance of the decision version of the investment problem and we can 53

63 see all these construction processes can be done in polynomial time. Therefore, we conclude that we can construct an instance of the decision problem in polynomial time given an instance of the 3-set exact covering problem. Thirdly, we prove that a YES solution of the instance of the decision problem is one-to-one corresponding to a YES solution of the 3-set exact covering problem. It is easy to see that in the instance of the decision problem constructed above, the sets that the investor chooses from {S 1, S 2,..., S n } to make 3m profits are precisely the ones which can exactly cover S in the 3-set exact covering problem. On the other hand, an exact cover of set S in the 3-set exact covering problem is actually the collection of the strategy sets that the investor should choose to make maximal profit as much as 3m under the budget constraint. That is, it is a solution of the decision version of the investment problem. To conclude, we proved that 3-set exact covering problem, which is a well-known N P- complete problem, can polynomially reduce to a decision problem of the investment problem. And the decision problem belongs to the N P class. Therefore, the decision version of the investment problem is N P-complete. From Theorem 4.1.1, we know that the decision version of the investment problem is N P- complete. Therefore, the investment problem is an N P-hard problem. We now return to the complexity issue on finding a pure Nash equilibrium in the two-person investment game. Compared to the investment problem, our problem involves two persons and each person has his own budget constraint and objective. The optimization of potential values is more complicated and connects to the interaction of strategies taken by two players. For each player in the twoperson investment game, he actually faces an investment problem when the opponent s decision is fixed. Moreover, verifying if a given state of the two-person investment game is a Nash equilibrium is not easier than solving an investment problem. Therefore, finding a Nash equilibrium 54

64 in the two-person investment game should be at least as hard as the investment problem. Since the investment problem is N P-hard, finding a Nash equilibrium in the two-person investment game should be N P-hard as well. For an N P-hard problem, we may consider investigating the special cases (or simplified cases) and developing the heuristics. A dynamic programming approach will be used to solve some special cases of the two-person investment game in this chapter and a genetic-based algorithm will be developed for the general cases in Chapter A dynamic programming algorithm for a special case In this section, we consider a special case of the two-person investment game. Suppose that two players are interested in investing on a pool of projects. And these projects are packed into sets for investors to select. Particularly, we assume that this collection of sets are mutually disjoint. That is, any two sets in this collection are disjoint. We can see this is a special class of the two-person investment game. And we are interested in finding an algorithm for solving a pure Nash equilibrium solution for this special case. As a matter of fact, if we view every set as a big project, this special class of two-person investment game turns to a two-person knapsack game problem ([77]) which is defined by Wang et al. in A dynamic programming based algorithm called DPKG (Dynamic Programming for Knapsack Game) algorithm was developed by them to compute a pure Nash equilibrium in the two-person knapsack game. Accordingly, we can use their dynamic programming algorithm to solve our special case problem here. Dynamic programming is an algorithm design method that can be used when the solution to a problem may be viewed as the result of a sequence of decisions. Problems to which dynamic 55

65 programming has been applied are usually stated in the terms of stages, states, decisions, transformations and so on. A physical, operational, or conceptual system is considered to progress through a series of consecutive stages. At each stage, the system can be described by a set of parameters called the states. At each stage, and no matter what state the system is in, one or more decisions must be made. And the system undergoes a transformation or transition to the next stage. The art of dynamic programming lies in how to construct these structures. Dynamic programming has been utilized to solve for many combinatorial optimization problems such as knapsack problem and shortest path problem. For a combinatorial optimization problem, we may enumerate all possible solutions and pick the best, but it could be very costly. Dynamic programming often drastically reduces the amount of enumeration by avoiding the enumeration of some solutions that cannot be optimal. In dynamic programming, an optimal decision is reached by making explicit appeal to the principle of optimality. Principle of optimality states that an optimal sequence of decisions has the property that whatever the initial state and decision are, the remaining decisions must constitute an optimal decision sequence with regard to the state resulting from the first decision. Actually, this principle implies that a subsequence of an optimal sequence is also optimal to the subproblem Algorithm Here we introduce the DPKG (Dynamic Programming for Knapsack Game) algorithm and investigate how it can be used to handle our special case problem. In this subsection, the algorithm is presented. Then a numerical example will be conducted to illustrate how the algorithm works in the next subsection. Wang et al. ([77]) designed a dynamic programming based algorithm for finding a pure Nash equilibrium solution of a two-person knapsack game (DPKG algorithm in short), which is an extension of the dynamic programming algorithm used to solve the classic knapsack prob- 56

66 lem presented in [61]. As they proved in their work, the potential functions used to prove the existence of pure Nash equilibrium are additive separable. This separability property of the potential functions guarantees the validity of the principal of optimality. Hence a dynamic programming based algorithm could be developed to find a pure Nash equilibrium. Since our special game can be viewed as a two-person knapsack game, we will adopt their algorithm to handle it. For the two-person investment game, suppose we have n projects which are grouped into m sets for two players to invest. Particularly, these m sets are mutually disjoint. When 1 2 0, we shall introduce the dynamic programming based algorithm (DPKG algorithm in [77]) for finding a Nash equilibrium of this special case as follows. The key elements that one associates with a dynamic programming problem are stages, states, decisions and transformations. We shall now define these terms for our problem correspondingly. In a special two-person investment game problem described above, the stages are projects sets. Hence we have stages 1, 2,..., m. For each stage, the states are pairs of (S 1, S 2 ) given S i as the set of strategies chosen by player i, i = 1, 2 at the current stage. We know that given a feasible state (S 1, S 2 ), it is easy to calculate its potential value Φ(S 1, S 2 ) when As defined in Chapter 1, S i denotes the number of strategies included in strategy set S i, i = 1, 2. To satisfy the investors budget constraints, it is obvious that S i b i. Now we use a quintuple F (S 1, S 2 ) = [S 1, S 2, Φ(S 1, S 2 ), S 1, S 2 ] to record all the information at state (S 1, S 2 ). At each state, the decision is to decide if to keep or discard the quintuple of the current state to the next stage. The transformation function is shown in the Step 2(c) of the dynamic programming algorithm presented in the following. 57

67 DP Algorithm: Step 1. Start with M 0 = {F (, )}, where is the empty set. Step 2. For k = 1, 2,..., m, do (a) Set M k 1 = M k 1. (b) For each F (S 1, S 2 ) M k 1, (i) if S b 1, add F (S 1 k, S 2 ) to M k 1 ; (ii) if S b 2, add F (S 1, S 2 k) to M k 1 ; (iii) if S b 1 and S b 2, add F (S 1 k, S 2 k) to M k 1. (c) Let M k = {F (S1, S 2 ) Φ(S 1, S 2 ) = max {Φ(S 1, S 2 ) S 1 = i, S 2 = j}, F (S 1,S 2 ) M k 1 i I j J F (S 1, S 2 ) M k 1}, where I = [0, min{k, b 1 }] and J = [0, min{k, b 2 }]. Step 3. Check M n to find one F (S 1, S 2 ) with the largest potential value. Output the state (S 1, S 2 ) as the solution. For this algorithm, it is not hard to observe that, with the help of principal of optimality, the unnecessary states are eliminated in Step 2(c) though the enumeration of all feasible states is embedded in Steps 2(a) and (b). Since all parameters in the game are of integer value, we know that this dynamic programming algorithm will eventually find one feasible state with the largest potential value in Step 3. It must be a Nash equilibrium solution of the game. Adopting the complexity analysis in [77], We can see that Step 1 is a trivial step using O(1) computing time. The main computational effort comes from Step 2. Note that there are m stages. For each stage k, k = 1, 2,..., m, since there are at most (b 1 + 1)(b 2 + 1) elements in M k, (a) - (c) 58

68 can be realized in O(b 1 b 2 ) computing time. Consequently, Step 2 needs a total of O(mb 1 b 2 ) computing time. Moreover, Step 3 only needs O(b 1 b 2 ) computing time for comparisons. Therefore, a Nash equilibrium can be found by the proposed dynamic programming algorithm in O(mb 1 b 2 ) computing time A numerical example Now we present an example of the special case of the two-person investment game to help readers understand the proposed dynamic programming algorithm. Example Suppose that there are two players who are interested in investing on 10 projects. For each project, its profit is shown in the following table. Table 4.1: The profits of all the projects - Example P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P And these projects are packed into five sets for investors to choose. C 1 = {P 1, P 3 }, C 2 = {P 2, P 4, P 6 }, C 3 = {P 7 }, C 4 = {P 9, P 10 }. C 5 = {P 5, P 8 }. 59

69 We can see that these sets are mutually disjoint. Suppose b 1 = 2 and b 2 = 1. That means player 1 may choose two strategy sets at most and player 2 can only choose one. Let α 1 = α 2 = 1 2, β 11 = β 22 = 2, and β 12 = β 12 = 1. Then i = 1 2, i = 1, 2. According to the conclusion of Chapter 3, the potential function is Φ(S 1, S 2 ) = p(s 1 ) + p(s 2 ) 1 4 p(s 1 S 2 ). To find a Nash equilibrium of this game, we implement the DP algorithm and document the results of every iteration in the following table. There are five stages at all. The first row of this table indicates all the possible pairs of S i for both investors, i = 1, 2. For example, in this instance, since b 1 = 2 and b 2 = 1, there are 3 2 = 6 possible pairs at each stage. With each pair, the first item shows the number of strategies in the player 1 s current strategy set; and the second item shows the number of player 2 s. For each stage, the first row includes the optimal potential values for all the pairs; and the second row documents the corresponding optimal strategy sets so far at each pair. Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Table 4.2: Dynamic programming algorithm - Example (0, 0) (1, 0) (0, 1) (1, 1) (2, 0) (2, 1) {, } {C 1, } {, C 1 } {C 1, C 1 } {, } {C 2, } {, C 2 } {C 1, C 2 } {(C 1, C 2 ), } {(C 1, C 2 ), C 2 } {, } {C 2, } {, C 2 } {C 1, C 2 } {(C 1, C 2 ), } {(C 1, C 2 ), C 2 } {, } {C 2, } {, C 2 } {C 2, C 4 } {(C 2, C 4 ), } {(C 1, C 2 ), C 4 } {, } {C 5, } {, C 5 } {C 2, C 5 } {(C 2, C 5 ), } {(C 2, C 5 ), C 5 } Finally, it shows that the DP algorithm finds {(C 2, C 5 ), C 5 } as a pure Nash equilibrium with a maximum potential of That is, in this Nash equilibrium state, player 1 chooses set 60

70 2 and set 5, and player 2 chooses set 5. Now we examine the complexity of the DP algorithm for this problem. Step 1 is trivial. For (a) to (c) in Step 2, there are at most 5 9 = 45 calculations of potential values for new states and 5 8 = 40 comparisons. In the last step, we need another 5 comparisons to pick the state with the maximum potential value. Therefore, to solve this problem, we need 45 calculations and 45 comparisons. We can see that this dynamic programming algorithm is valuable for solving the special class of the two-person investment game. It is natural to think that if we can generalize this dynamic programming algorithm to solve more cases. As a matter of fact, it is difficult to extend the proposed dynamic programming algorithm to solve a general two-person investment game. Consider when an investor selects a new strategy into his set, it may intersect with the other player s old strategies. This strategic interaction makes the situations more complicated than in the two-person knapsack game. Actually the intersection leads to the non-separability of the potential functions, which makes the absence of principle of optimality possible. Hence to use this dynamic programming algorithm to solve our general games would be difficult. We shall explain more details in Section 4.4 by providing an example. By knowing that to generalize the DP algorithm for solving general games is hard, we consider if there are some games with some special structures, which we can extend or revise the current dynamic programming algorithm to solve for Nash equilibria. For the extended special case of the two-person investment game presented in the following subsection, we show that a revised dynamic programming algorithm could be developed for finding a pure Nash equilibrium solution. 61

71 4.3 Extension of the dynamic programming algorithm Now we consider another special case of the two-person investment game. Suppose that two players are interested in investing on a pool of projects. And these projects are packed into sets for investors to choose. Assume all the project sets are separated into two groups. In each group, all the sets are mutually disjoint. However, the sets from different groups may have overlaps. We indicate the first group as the row group, denoted as R; and the second group as the column group, denoted as C. Moreover, suppose each investor is allowed to choose only one strategy among the sets in these two groups, i.e., b 1 = b 2 = 1. In the following, we introduce a revised DP algorithm for finding a pure Nash equilibrium in this special game Algorithm To model the case mathematically, suppose that we have n projects which are packed into m sets that two players can invest on. Particularly, these m sets are separated into two large groups, R group and C group. For each set, it can only be included in one group. And for each group, the sets in it are mutually disjoint. Index the sets in group R as 1, 2,..., m 1, and the sets in group C as m 1 + 1, m 1 + 2,..., m 1 + m 2, where m 1 + m 2 = m. When 1 2 0, the following is the revised dynamic programming based algorithm to find a Nash equilibrium for the special two-person investment game described above. Revised DP Algorithm: Step 1. Start with M 0 = {F (, )}, where is the empty set. Step 2. For k = 1, 2,..., m 1, m 1 + 1,..., m, do (a) Set M k 1 = M k 1. (b) For each F (S 1, S 2 ) M k 1, (i) if S , add F (S 1 k, S 2 ) to M k 1 ; (ii) if 62

72 S , add F (S 1, S 2 k) to M k 1 ; (iii) if S and S , add F (S 1 k, S 2 k) to M k 1. (c) Let M k = 0 i 1 0 j 1 F (S 1, S 2 ) M k 1}. {F (S 1, S 2 ) Φ(S 1, S 2 ) = max F (S 1,S 2 ) M k 1 {Φ(S 1, S 2 ) S 1 = i, S 2 = j}, Step 3. Check M n to find one F (S 1, S 2 ) with the largest potential value. Let this state be F (S a 1, Sa 2 ). Step 4. Find the set in the column group with the largest profit throughout the whole group, denote it as C max1. Check F (C max1, S i ) where S i is any set with S i C max1. Find the state with the largest potential value and we call this state as F (S b 1, Sb 2 ). Step 5. Choose F (S 1, S 2 ) = max{f (S a 1, Sa 2 ), F (Sb 1, Sb 2 )}. Output the state (S 1, S 2 ) as the solution. We should notice that this special problem is trivial because b i = 1, i = 1, 2. In the next subsection, we analyze the above algorithm to prove that it can find a pure Nash equilibrium for this special case of the two-person investment game. The complexity of this algorithm is also studied. Compared with the enumeration, we show that the complexity drops from O(m 2 ) to O(m) Proof and complexity Theorem For the special two-person investment game described in Section 4.3, a pure Nash equilibrium can be found by the revised DP algorithm in O(m) computing time. 63

73 Proof. For the simplicity of our analysis, we always assume that player 1 s profit is dominant in the potential functions of the game. We call the sets in the R group as R i and the ones in the C group as C j, for i = 1, 2,..., m 1 and j = m 1 + 1, m 1 + 2,..., m. The sets with the first two largest profits in the row group are named R max1 and R max2. Similarly, let the sets with the first two largest profits in the column group be C max1 and C max2. Through the algorithm, from Steps 1 to 3, it is easy to verify that (S1 a, Sa 2 ) must be one of the elements in the set {(R max1, R max1 ), (R max1, R max2 ), (C max1, C max1 ), (C max1, C max2 ), (R max1, C j )}, where j {m 1 + 1, m 1 + 2,..., m}. In Step 4, we can see that (S1 b, Sb 2 ) could be either (C max1, R i ) or (C max1, C max2 ) where i {1, 2,..., m 1 }. Consequently, in Step 5, we will finally find a state which is the one with the maximal potential value of the following six states: (R max1, R max1 ), (R max1, R max2 ), (C max1, C max1 ), (C max1, C max2 ), (R max1, C j ), and (C max1, R i ). Now we show that this state must have the maximal potential value among all the feasible strategy sets. Consequently, it should be a pure Nash equilibrium solution. We construct our proof by contradiction. If there exists a state (R i, R j ) which has the maximal potential value with i max1 or j / {max1, max2}, then our assumption that R max1 and R max2 are the sets with the first two largest potential values leads to a contradiction. Because the overlap of any two arbitrary different sets in the row group is empty, the maximal potential value has to be achieved at (R max1, R max1 ) or (R max1, R max2 ). A similar analysis can be drawn for the column group. Suppose there exists a state (R i, C j ) with the maximal potential value which is different from any of the six states described above, then we can also draw a contradiction. If p(r i ) p(c max1 ), then the potential value of state (C max1, C j ) must be larger than that of state (R i, C j ). If p(r i ) p(c max1 ), then we have p(r max1 ) p(r i ) p(c max1 ) C j. We can see the potential of state (R i, R max1 ) must be larger than that of state (R i, C j ). Consequently, 64

74 in both cases, the assumption leads to a contradiction. Hence by implementing the revised DP algorithm, we can eventually find a state with the greatest potential value to be a pure Nash equilibrium solution. Now we examine the complexity of the revised DP algorithm. We can see through the algorithm, Step 2(c) avoids a total enumeration. However, it also neglects some states which may be valuable. That is because Step 3(c) always picks the state with the maximal potential value when it compares all the states with the same weight. However, this maximum cannot be accumulated because when a player add a new strategy into his set, it may intersect with the current ones and decrease the potential value. On the other hand, the states which are not picked in Step 3(c) may lead to a Nash equilibrium. Therefore, in Step 4, we handle the states that may be Nash equilibria but deleted through Steps 1 to 3 separately. Now we analyze the complexity step by step. Step 1 is a trivial step using O(1) computing time. Note that there are m stages. For each stage k, k = 1, 2,..., m, through Step 2(a) to 2(c), it can be realized in O(1) computing time. Consequently, Step 2 needs a total of O(m) computing time. Moreover, Step 3 needs O(1) computing time for comparisons. And Step 4 needs O(m) time for the calculation and O(m) + O(m 2 ) for the comparisons. Step 5 also needs O(1) computing time for the comparisons. To summarize, the complexity of this algorithm is O(m). Compared with the enumeration, we can see that the complexity drops from O(m 2 ) to O(m). 4.4 Difficulty of extension for general games In Section 4.2, we introduced a dynamic programming algorithm for finding a pure Nash equilibrium for a special class of the two-person investment games and extended it to solve another special problem in Section 4.3. However, for a general two-person investment game, the proposed dynamic programming based algorithm is not suitable any more for finding a pure Nash equilibrium because the principle of optimality does not hold. This becomes a main obstacle to generalize our DP algorithm to more general cases. Here we present an example to show that 65

75 why the principle of optimality for this algorithm does not follow for the general two-person investment game. Example table: Suppose that there are 7 projects. For each project, its profit is shown in the following Table 4.3: The profits of all the projects - Example P 1 P 2 P 3 P 4 P 5 P 6 P And these projects are grouped into five sets for two investors to choose. C 1 = {P 1, P 4, P 7 }, C 2 = {P 4, P 5 }, C 3 = {P 5, P 6, P 7 }, C 4 = {P 3, P 6 }. C 5 = {P 1, P 2 }. We can see that this problem does not belong to either of the special cases we discussed in sections 4.2 and 4.3. Assume b 1 = 1 and b 2 = 1, implying that both players can only choose one strategy. Let α 1 = α 2 = 1 2, β 11 = β 22 = 2, and β 12 = β 12 = 1. Then i = 1 2, i = 1, 2. Therefore, we know the potential function becomes Φ(S 1, S 2 ) = p(s 1 ) + p(s 2 ) 1 4 p(s 1 S 2 ). If we implement the proposed DP algorithm to solve this problem, the algorithm finds {C 3, C 4 } (or {C 4, C 3 }) as a solution with the potential value of However, this is not 66

76 a Nash equilibrium for our game. Because we can find a state {C 2, C 4 } (or {C 4, C 2 }) with a larger potential value of And it is not hard to verify this state is the only Nash equilibrium of the game. However, it can never be reached by the DP algorithm because C 2 is not an optimal choice for the player which will be replaced by C 3 with a larger potential value in Stage 3 when implementing the DP algorithm. That is, the principle of optimality is not guaranteed in this game. We observe that the optimality can not be accumulated from stage to stage because at an optimal state, when a player adds a new strategy into his set, it may intersect with the other player s current strategies, which is likely to decrease the potential value of the state. And it is also possible for a state which is not selected by the DP algorithm at the current stage becomes optimal in the next. Therefore, for the general two-person investment games, it seems difficult for us to extend this dynamic programming algorithm to find pure Nash equilibria. Instead, we consider to develop a genetic-based algorithm to solve the general two-person investment games. More details of the genetic algorithms will be discussed in the next chapter. 4.5 Summary In this chapter, we first analyzed the complexity of finding a pure Nash equilibrium of the two-person investment game. We have proved that this is an N P-hard problem. Then we introduced a special case for which we can use a dynamic programming algorithm developed by Wang et al. ([77]) to find a pure Nash equilibrium. Also a numerical example showed how the algorithm works. Moreover, we extended this DP algorithm to solve another special two-person investment game and provided the complexity analysis. However, for a general two-person investment game, we used an example to show that it is difficult to generalize the proposed DP algorithm to find a Nash equilibrium because the principle of optimality does not follow. Hence we need to seek other heuristics to solve the general cases. 67

77 Chapter 5 Finding Nash Equilibrium - Genetic-based Algorithm In the last chapter, we showed that the proposed dynamic programming algorithm is not suitable for finding a pure Nash equilibrium for the two-person investment game except for some special cases. In this chapter, we aim to design an efficient heuristics to tackle the problem. Since genetic algorithm is known to be a robust tool for solving difficult or complicated optimization problems, we are interested in adopting the methodology for finding a pure Nash equilibrium for the two-person investment game. This chapter is organized as follows. In the first section, we give a brief introduction of genetic-based algorithms. In Section 5.2, we design the components of our genetic algorithm for the two-person investment game. A genetic algorithm is proposed at the end of the section. Then some numerical experiments are reported to show the strength of the proposed algorithm in Sections 5.3 and 5.4. A comparison of our proposed genetic algorithm to the random search method is presented in the last section. 68

78 5.1 Introduction to genetic algorithms Since the 1960s, genetic algorithms have drawn great interests in developing efficient heuristics to solve complex optimization problems. As we know, combinatorial optimization problems can be characterized as searching the best element among a finite number of potential solutions. Though in principle, this searching process can be carried out by enumeration. However, pure enumeration may not be practical for solving real-life problems because of the complexity. Therefore, heuristics algorithms are often adopted. The basic idea of a genetic-based algorithm is to simulate an evolutionary process for obtaining better solutions as the algorithm progresses. In each generation, the solutions are altered and combined to generate new solutions. Through the evaluation and selection of the candidate individuals, the fitter ones have better chances of survival to produce good solutions. Genetic algorithms have been applied to a variety of problems [29]. As summarized by Michalewicz (1996), in general, a genetic algorithm has five basic components: (i) A genetic representation of solutions to the problem. (ii) A way to create an initial population of solutions. (iii) An evaluation function rating solutions in terms of their fitness and a selection scheme. (iv) Genetic operators that alter the genetic composition of children during the reproduction process. (v) Values for the parameters of the algorithm. A genetic algorithm maintains a population of individuals, say P (t), for generation t. Each of P (t) represents a solution of the problem to be solved, we call it a chromosome. To encode a solution of the problem into a chromosome is the key issue when using genetic algorithms. 69

79 To our knowledge, encoding highly depends on the solutions structures and properties and can be quite different from one problem to another. The basic encoding methods include binary encoding, read-number encoding, integer or literal permutation encoding. After encoding, each individual is evaluated to give some measure of its fitness. Moreover, some individuals are processed by genetic operators to form new generations which are considered as a new population of individual solutions. In this way, new regions of the search space are explored by means of genetic operators. In general, there are two types of genetic operators: one is mutation, which creates new individuals by altering in a single individual; the other is crossover, which creates new individuals by combining genes from two individuals. There are many research work studied these two operators and they show that mutation can sometimes play a vital role in the algorithm. The individuals generated by genetic operators are called offspring C(t). They are evaluated accordingly. A new population is formed by selecting the more fit individuals from the parent population and the offspring population. After several generations, a genetic algorithm may converge to the best individuals (solutions), which hopefully represents an optimal or near optimal solution to the problem. A general structure of the genetic algorithms is illustrated as follows: Procedure: Genetic Algorithms begin t 0; initialize P (t); evaluate P (t); while (termination condition is not met) do 70

80 begin recombine and vary P (t) to generate C(t); evaluate C(t); select P (t + 1) from P (t) and C(t); t t + 1; end end In the next subsection, we present how we approach these five components of genetic algorithms in our problem. Also we propose a genetic-based algorithm to find pure Nash equilibria for the two-person investment game in the end of the next section. 5.2 A genetic algorithm for the two-person investment game Components of our genetic algorithm (i) Problem Representation: Choosing an appropriate representation of the candidate solutions is a key to genetic-based algorithms. The most straightforward approach to encode the two-person investment game problem is the binary encoding. According to our model, suppose there are n projects which are packed into m sets. Player 1 can choose b 1 strategies at most and player 2 can choose up to b 2 sets. In this design, we define a chromosome that contains two segments and each of them has an m-bit string. The first segment represents player 1 s strategy set, that is S 1. The second segment stands for player 2 s, that is S 2. Each m-bit string is composed of 0 and 1. In this binary coding, the position of i th gene in a segment is used to represent i th project set, i = 1,..., m, and the value of the gene is to represent whether to put this set into the player s strategy set. In other words, if a set is selected by the player 1, the corresponding gene in the first m-bit string will be set as one; otherwise, it will be set as zero. The same rules follow for player 2 s strategy 71

81 {}}{ set S 2. Therefore, a chromosome may look like S = {S 1, S 2 } = { , following, we present an example to illustrate the encoding procedure. m m {}}{ }. In the Example There are eight projects {P 1, P 2,..., P 8 } with the profits of {3, 5, 2, 9, 12, 18, 3, 9}. Now these projects are grouped into four sets that the players can invest. They are: C 1 = {P 1, P 5, P 6 } C 2 = {P 5, P 7 } C 3 = {P 2, P 3, P 4, P 8 } C 4 = {P 5, P 6, P 8 } Suppose both players are limited to invest to 2 strategies at most. In our encoding method, S = {S 1, S 2 } = {0101, 1100} represents a possible chromosome. In this chromosome, player 1 chooses project sets 2 and 4 as his strategy set and player 2 selects project sets 1 and 2 accordingly. This binary encoding has the advantage that the chromosomes are of a constant length 2m, which allows for the application of standard genetic operators. Moreover, it is easy to decode from a chromosome to a solution of the original problem. That is, given a chromosome S, we can easily tell the strategies of both players. However, it should be immediately noted that if we randomly generate each bit of chromosomes, not all of them represent feasible candidate solutions for the game. For instance, in the above example 5.1, a chromosome S = {S 1, S 2 } = {1101, 1111} does not represent a feasible state at all because it violates the players budget constraints. Hence we give a definition of chromosome feasibility in the following. Feasibility refers to the situation of whether a solution decoded from a chromosome lies in the feasible region of a given problem. And it is always wise and necessary to check the chromosome 72

82 feasibility after generating any parents or children in the population. Definition For a chromosome, in its segment i, if the sum of the numbers in all positions is less than or equal to b i, i = 1, 2, then we say this chromosome is feasible; otherwise, it is infeasible. (ii) Initial Population: For our knowledge, there are many ways to generate the initial population. How to choose a set of good starting solutions highly depends on the characteristics and structure of the problem. In our problem, the initial population is determined by random generation. We will show, in the next section, this randomly generated initial population is good enough for our numerical examples. (iii) Evaluation and Selection: The fitness of each individual is a value associated with its encoded solution. In our game, giving a chromosome, its fitness is obtained by calculating its value of the potential function of the game. From the conclusion of Chapter 3, we know that the state with the maximum potential value is a pure Nash equilibrium. Therefore, using our genetic algorithm, we are actually trying to find a Nash equilibrium with the maximum potential value. Now we go for the selection process. The principle behind genetic algorithms is essentially the Darwinian natural selection. As a matter of fact, selection provides a driving direction in a genetic algorithm. It directs a genetic algorithm search toward promising regions in the search space. In the literature, many selection methods have been proposed, developed, examined and compared. A good selection seeks the balance between exploring the search space and exploiting the good solutions. In our proposed genetic algorithm, we decide to perform the selection on a enlarged sampling space, implying that both parents and offspring have the same chance of competition for 73

83 survival in the new population. A typical method is the µ + λ selection, which was invented by Bäck and Hoffmeister [6]. With this selection strategy, µ parents and λ children compete for survival and µ best individuals among them are selected as parents of the next generation. (iv) Genetic Operators: There are usually two types of operators in a genetic-based algorithm. One is mutation, which creates new individuals by altering one or multiple genes in a single individual. The other is crossover, which creates new individuals by combining genes from two individuals. There are many research works studying these two operators. However, the performance varies from case to case. We show our two genetic operators as follows. Mutation. Procedure: 1. Randomly generate a position of a given parent, {{}}{. m m.., {}}{... }, mutate the value on this gene from 1 to 0 or 0 to 1 and keep other genes untouched Figure 5.1: Mutation. 74

84 2. Check the feasibility of offspring. Find out which player s strategy set the gene belongs to. If it satisfies the player s budget constraint, implying that it is feasible, then keep it in the set of offspring. Otherwise, go to step Randomly find a gene in the particular player s strategy set with value 1, change it to 0. Crossover. Each individual in the population has a probability P c of being chosen to form a crossover pool of parents. The pairs of parents are selected in order as they are in the pool. Therefore, each individual in the crossover pool is allowed to be parent at most once. Procedure: Given two parents as m {}}{... }, parent 1: {{}}{. m.., parent 2: {{}}{. m.., m {}}{... }. We use the middle point of each chromosome as a cut point to separate each individual into two parts and crossover them separately. 75

85 Parent Parent Figure 5.2: Before crossover. Offspring Offspring Figure 5.3: After crossover. Since both parents are feasible, the offspring generated by this crossover operator must be feasible. And it combines the characteristics of both parents. (v) Parameters: The crossover rate P c is defined as the ratio of the number of offspring produced in each generation by the crossover operation to the population size. A high crossover rate allows exploration of search space and reduces the chance of being trapped in a local optimum; but if the rate is too high, it will waste a lot of computation efforts and decrease the algorithm s efficiency. 76

86 The mutation rate P m we define here is quite different from the traditional definition. Instead of defining P m as the probability of mutating a gene regarding to the population, we defined our P m as the probability of an individual that undergoes the mutation process. It is easy to see that we can always convert our P m to the traditional one. The parameters of our genetic-based algorithm are set as follows: the population size P = 100; the maximum number of generation N = 50; the probability of crossover P c = 0.6 and the probability of mutation P m = 0.8. The terminate condition is to check whether the number of generation hits the maximum number N. In the next section, we will explain the details how to tune these parameters based on our experimental examples Proposed genetic algorithm Genetic Algorithm: For each generation, we have P chromosomes in the population. Use S i to save the population for generation i, i = 0...N. Step 1. Set up the parameters. Set the population size P, the rate of crossover P c, the rate of mutation P m, the maximum generation N,..., and initialize the generation number gen = 0. Set i = 0. Also input the parameters of the two-person investment game. Step 2. Initialize the population. Randomly generate the initial population S 0. Step 3. Evaluate the initial population. Calculate the potential value for each individual. Step 4. Perform the crossover and mutation operations on the current population. (4.1) Apply mutation procedure described above. 77

87 (4.2) Apply crossover procedure described above. Step 5. Evaluation and selection. (5.1) Calculate the potential values of all offsprings. (5.2) Rank all the parents and offsprings and pick P best individuals from them to form the next generation. Set i = i + 1. Update the new generation of population S i. Step 6. Perform the terminating test. Set gen = gen + 1. If the terminating condition is not satisfied, return to step 4 for the next iteration; otherwise, terminate. 5.3 Experimental results - numerical analysis To test the quality of the solutions produced by the proposed genetic-based algorithm, we design some numerical examples to test it. Because our problem structure is very flexible, in this test, we only take into account a class of problems and conduct tests based on them. This class of problems is defined as follows. Suppose there are n 2 projects, which are packed into 2n sets in the following way: the first n sets are generated like {P 1,..., P n }, {P n+1,..., P 2n },..., {P n 2 n,..., P n 2}. The second n sets are generated like {P 1, P n+1..., P (n 1) n+1 }, {P 2, P n+2..., P (n 1) n+2 },..., {P n, P 2n..., P n 2}. We conduct four different-sized experiments generated in the above way, and n = 5, 7, 10, 15. So the four problems are the ones with 25 projects in 10 sets, 49 projects in 14 sets, 100 projects in 20 sets and 225 projects in 30 sets. Ten instances of each experiment are randomly generated. For each of these instances, 50 independent runs of the proposed genetic-based algorithm are conducted. The optimal value for each of these instances is obtained by an enumeration program separately. We code the program using MATLAB version (2008b) on a Thinkpad R61 personal computer. 78

88 The parameters of our game were set as: α 1 = 1/2, α 2 = 1/2, β 11 = 2, β 21 = 1, β 22 = 2, β 12 = 1. Consequently, we have, 1 = α 1 β11 α 2 β 12 = 0.5, 2 = α 2 β22 α 1 β 21 = 0.5. The potential function we use here becomes p(s 1 ) + p(s 2 ) 1 4 p(s 1 S 2 ). Because β ij > 0, for i, j = 1, 2, both players seek to maximize their own profits and suppress the opponent s. It is easy to see that players are always beneficial by choosing as many strategies as they can. Hence the Nash equilibrium must be obtained at the boundary of budget constraints. Therefore, we only need to generate the population under the condition that both players choose exactly b i strategies, i = 1, 2. And the mutation procedure could be simplified as well. 79

89 Example This problem has 25 projects 10 sets. Player 1 can choose at most three sets and player 2 can choose up to two, that is b 1 = 3 and b 2 = 2. If we enumerate to solve the problem, it is needed to go through 5400 potential solutions. We randomly generate 10 instances and run our genetic algorithm 50 times for each instance. For each run of the genetic algorithm, the population size P = 100 and the maximum number of generation N = 50. Consequently, we generate P N = 5000 individuals totally. The following table shows the probabilities of runs hitting optimum, within (0%, 1%) of optimum, within (1%, 2%) of optimum, within (2%, 3%) of optimum and within (3%, 4%) of optimum. Table 5.1: Number of runs to achieve optimality for each instance - Example Instance 0% (0%, 1%) (1%, 2%) (2%, 3%) (3%, 4%) 1 100% % 2% % % 0 4% % 4% % 4% % 8% % 0 8% % %

90 100.00% 80.00% 60.00% 40.00% 20.00% Frequency 0% (0-1)% (1-2)% % 80.00% 60.00% 40.00% 20.00% Frequency 0.00% 0.00% % (0-1)% (1-2)% % 97.40% 80.00% 60.00% 40.00% 20.00% 0.00% 0.00% (2-3)% (3-4)% 0.00% 1.40% 1.20% 0.00% 0.00% 0% (0-1)% (1-2)% (2-3)% (3-4)% Deviation w.r.t. Optimum Figure 5.4: Performance analysis of Example

91 Example This problem has 49 projects in 14 sets. Player 1 can choose at most three sets and player 2 can choose up to two, that is b 1 = 3 and b 2 = 2. If we enumerate to solve the problem, it is needed to go through potential solutions. We randomly generate 10 instances of this size problem and run our genetic algorithm 50 times for each instance. For each run of the genetic algorithm, the population size P = 100 and the maximum number of generation N = 50. Consequently, we generate P N = 5000 individuals totally. The following table shows the probabilities of runs hitting optimum, within (0%, 1%) of optimum, within (1%, 2%) of optimum, within (2%, 3%) of optimum and within (3%, 4%) of optimum. Table 5.2: Number of runs to achieve optimality for each instance - Example Instance 0% (0%, 1%) (1%, 2%) (2%, 3%) (3%, 4%) 1 100% % 12% % 24% % 0 4% % % 6% % % 0 14% % 28% % 0 10%

92 0% (0-1)% (1-2)% % 80.00% 60.00% 40.00% Frequency 0% (0-1)% (1-2)% 20.00% 0.00% 0% (0-1)% (1-2)% % (0-1)% (1-2)% % 90.20% % 80.00% 80.00% 75.60% 60.00% 60.00% 40.00% 40.00% 20.00% 0.00% 7.00% 2.80% 0.00% 0.00% 0% (0-1)% (1-2)% (2-3)% (3-4)% 20.00% 0.00% 0% Deviation w.r.t. Optimum Figure 5.5: Performance analysis of Example

93 Example This problem has 100 projects in 20 sets. Player 1 can choose at most three sets and player 2 can choose up to two, that is b 1 = 3 and b 2 = 2. If we enumerate to solve the problem, it is needed to go through potential solutions. We randomly generate 10 instances of this size problem and run our genetic algorithm 50 times for each instance. For each run of the genetic algorithm, the population size P = 100 and the maximum number of generation N = 50. Consequently, we generate P N = 5000 individuals totally. The following table shows the probabilities of runs hitting optimum, within (0%, 1%) of optimum, within (1%, 2%) of optimum, within (2%, 3%) of optimum and within (3%, 4%) of optimum. Table 5.3: Number of runs to achieve optimality for each instance - Example Instance 0% (0%, 1%) (1%, 2%) (2%, 3%) (3%, 4%) 1 40% 60% % 30% % % 40% % 22% % 14% 14% % % % 48% 2% % 14%

94 % 80.00% 60.00% 40.00% 20.00% 0.00% Frequency 0% (0-1)% (1-2)% 0% (0-1)% (1-2)% % 80.00% 60.00% 40.00% 20.00% 0.00% Frequency % 80.00% 75.60% 60.00% 0.00% 0.00% (2-3)% (3-4)% 40.00% 20.00% 0.00% 22.80% 1.60% 0.00% 0.00% 0% (0-1)% (1-2)% (2-3)% (3-4)% Deviation w.r.t. Optimum Figure 5.6: Performance analysis of Example

95 Example This problem has 225 projects in 30 sets. Player 1 can choose at most three sets and player 2 can choose up to two, that is b 1 = 3 and b 2 = 2. If we enumerate to solve the problem, it is needed to go through potential solutions. We randomly generate 10 instances of this size problem and run our genetic algorithm 50 times for each instance. For each run of the genetic algorithm, the population size P = 100 and the maximum number of generation N = 50. Consequently, we generate P N = 5000 individuals totally. The following table shows the probabilities of runs hitting optimum, within (0%, 1%) of optimum, within (1%, 2%) of optimum, within (2%, 3%) of optimum and within (3%, 4%) of optimum. Table 5.4: Number of runs to achieve optimality for each instance - Example Instance 0% (0%, 1%) (1%, 2%) (2%, 3%) (3%, 4%) 1 46% 54% % 48% 14% % 18% 30% % 2% % 0 30% % 22% % 80% % 44% 2% % 0 10% % 60% 6%

96 0% (0-1)% (1-2)% % 80.00% 60.00% 40.00% Frequency 20.00% 0.00% 0% (0-1)% (1-2)% % 80.00% 60.00% 58.00% 40.00% 32.80% 20.00% 0.00% 9.20% 0.00% 0.00% 0% (0-1)% (1-2)% (2-3)% (3-4)% Deviation w.r.t. Optimum Figure 5.7: Performance analysis of Example

97 Due to the combinatorial nature of the problem, the number of feasible solutions in the search space increases significantly when the problem size enhances. Being able to find the optimum consistently or having very small deviations with respect to it, lead us to conclude that the proposed genetic-based algorithm can be used as a framework for solving variations of the two-person investment game. Because the genetic-based algorithm is merely a probabilistic search, we can derive the required number of runs to achieve optimality, given a certain level of confidence and a certain class of problems. Let p be the probability of finding the optimum in one run and ˆp its estimate. Let K be the number of trials needed to reach the optimum with a confidence level α. From the lemma in the work of Andres L. and Fang [49], we know that, ln(1 α) K = ln(1 p). With the value of ˆp and the formula of K, we constructed Table 5.5 to show the number of runs needed to achieve optimality with α = Table 5.5: Number of runs to achieve optimality size of problems ˆp K As shown in Table 5.5, for the class of problems we generate and for the parameters fixed at P = 100, N = 50, P c = 0.6 and P m = 0.8, the required number of runs to reach the optimality (with at least 99.9% confidence) is 2, 3, 5, and 8 for the four problems respectively, shown in the following table. Therefore, the proposed algorithm required low computational efforts to 88

98 reach very high quality solutions. 5.4 Experimental results - tuning parameters There are four parameters, namely, the population size P, maximum number of generations N, probability of crossover P c, and probability of mutation P m, that have to be fixed in the proposed genetic algorithm for solving the two-person investment game problem. Tuning parameters could help us understand the range of robustness of the proposed algorithm. We conducted an exhaustive experiment on a randomly generated problem with 225 projects in 30 sets. Each test was for a specific choice of P, N, P c and P m. The different levels in which the parameters were tuned is given in Table 5.6. Table 5.6: Tuning experiment: levels of the parameters Parameter Levels P 50, 100, 200 N 30, 50, 100 P c 0.1, 0.2,..., 0.9 P m 0.1, 0.2,...,

99 P Pm Pm Pc Pc Pc Pm Pc P Pc Pm Pc Pm Pc 100 Figure 5.8: Tuning parameters %-1.00% 0.00% 1.00% %-2.00% Pm Pc 0.8 Pc 2.00%-3.00% Pm 3.00%-4.00% Pm Deviation w.r.t Optimization: 4.00%-5.00% Pm N