Optimal Contracts with Imperfect Monitoring

Transcription

1 Optimal Contracts with Imperfect Monitoring Harrison Cheng University of Southern California visiting Kyoto Institute of Economic Research March, 2004 Abstract We use the duality in linear programming to solve the problem of optimal contracts with moral hazards. We show the importance of allowing the partners to throw away outputs under some contingencies. A two-step procedure is used to find the optimal contracts. The first step minimizes the loss from undistributed outputs, and in the second step, second best solution is found near a pure strategy profile. The primal solution gives us the contracts, while the dual solution identifies thepriorityinthe improvement of production technology. Introduction We study the problem of designing optimal contracts when there are moral hazard problems among the players due to unobservable actions. There is a huge literature on the design of optimal contracts under asymmetric information, encompassing many areas of study such as the mechanism design, the theory of auctions, the theory of regulation and procurements, and the incentive theory of public goods. By comparison, the problem of designing optimal contracts while providing incentives to the players are relatively underdeveloped. Part of the reason may be due to complexity with moral hazard problems. As noted in Laffont and Mortimort (2002), "As in the case of adverse selection, asymmetric information also plays a role in the design of the optimal incentive contract under moral hazard. However instead of being an exogenous uncertainty Department of Economics, University of Southern California, Los Angles, Ca , hacheng@usc.edu, fax I am indebted to many over the years for this research. Special thanks to the support from the Institute of Economic Research, Kyoto University. at Kyoto: cheng@kier.kyoto-u.ac.jp

2 for the principal, uncertainty is now endogenous". This endogeneity is asourceofthecomplexity. Most of the literature on optimal contract design take the so-called "differential" approach in which a system of differential equations is solved, and first-order conditions are derived. Ourapproachinthispa- per is the "linear inequality methods". It formulates part of the problem by solving a linear programming problem subject to a system of linear inequality constraints, as surveyed in d Aspremont and Gérard-Varet (998). The "linear inequality methods" was firstusedtohandlead- verse selection in the design of optimal public design mechanisms in d Aspremont and Gérard-Varet (979). It was also used in the design of optimal selling rules for a discriminating monopolist in Cremer and McLean (985). It has also been used to solve moral hazard problems in Legros and Matsushima (99), Fudenberg, Levine, and Maskin (994), Fudenberg and Levine (994), and Cheng (2000). We take one step further by showing how the converted dual of the linear programming problems can be a surprisingly powerful way of finding optimal contracts with moral hazards. Our analysis is performed in a model of partnership with the joint production of a single output subject to production uncertainty. However, the methods are indeed very general and can be applied to any game (or game form) with imperfect monitoring. We provide an exhaustive analysis of the solution of the optimal contracts with two players, two pure strategies, and two public signals (referred to as two-by-two-by-two games). The method can be applied to the case with many players, pure strategies and signals. A general analysis is however beyond the scope of this paper. Because of the complexity, it is necessary that the solution be studied in the important but simpler two-by-two-by-two cases. One interesting phenomenon that comes out of the analysis is that in designing the sharing rules, a general optimal rule is for the partners not to fully distribute all the produced output under all contingencies. A weaker form of the budget constraints should be used. Such a result has been well-known in the models with asymmetric information. For example, in Myerson and Satterthwaite (983), and Laffont and Maskin (979), there is an impossibility result in designing optimal mechanisms that satisfy the incentive constraints, individual rationality and the strict budget balance simultaneously. In the case of moral hazards, Holmstrom (982) also shows that without a budget breaker, we often get inefficiency in partnership games. Given these results, it is important that we allow disposal of outputs under certain contingencies This is of course a source 2

3 of inefficiency for the partners, but in general the inefficiency loss of maintaining strict budget balance under all contingencies is greater. The undistributed outputs should be thrown away (at zero cost) or given away to others in a way not affecting optimal decisions of the partners. For example, an anonymous donation of the undistributed claims to charity institutions can be a good arrangement. The better decisions supported by such incentive schemes can more than compensate the loss from the undistributed outputs for the partners. This can occur quite easily, and an example will be given. The message from the example is the best contract should allow for the possibility of partial distribution of produced outputs among the partners. Although the possibility of using such sharing rules are mentioned in Alchian and H. Demsetz (972), and Holmstrom (982), they did not pursue its implications. We show how important it is to allow such distribution rules and the role it plays in finding an optimal contract for the partners. An optimal contract is one that maximizes the expected distributed outputs(minus costs of efforts) subject to incentive constraints. When the disposal of the outputs is allowed, any strategy profile of the partners is incentive compatible. To find an optimal contract, there are two steps in the procedure. In the first step, given a strategy profile, we find an optimal sharing scheme subject to the condition that the given strategy profile is a Nash equilibrium. In the second step, the strategy profile is varied to find one that is optimal. It can be shown that an optimal contract may not exist, and the best that can be hoped in general is an asymptotically second best contract. Because of the two-step procedure, the analysis of the optimal contracts is substantially more complicated than the case for asymmetric information. In the two-by-two-by-two case, it is shown that the (asymptotically) optimal (or second best) contract is obtained through a strategy profile that is close to a pure strategy profile.weexpectthispropertytohold for an arbitrary number of players, pure strategies and output levels. If confirmed later, it offers a simple computational method for finding the (asymptotically) optimal contract among the partners by solving a finite number of linear programming problems. Thus we regard the pure strategy result to be a very important one for our analysis. Space limitations do not allow us to explore the implications of limited liability. The analysis will also be extended later beyond the 2-by-2- by-2 games of the current paper. A sequel to this paper will be devoted to these objectives. 3

4 Sectiontwostartswiththeanalysisofoptimalcontractsfortheprincipal hiring many agents. It is shown that the principal can extract all the surplus of production from the risk-neutral agents while providing the incentives, a result corresponding to the Cremer and McLean (988) result for asymmetric auction models. Second three deals with case without the principal. The agents are now the partners. A No Surplus Condition is shown to be necessary and sufficient for the incentive compatible equilibrium under strict budget constraints. Second four explains why the budget constraints need to weakened for solving the optimal contract problem. Players do not distribute all the outputs from production under all contingencies. Section 5 converts the linear programming problem of minimizing inefficiency from disposal into a dual problem, and further transforms it into an important tool of analysis. The duality result is then applied in section 6 to analyze the solution of the step one optimization problem in general form in 2-by-2-by-2 games. In section 7, numerical examples are given to illustrate the formula obtained in section 7. In section 8, the formula is used to show that optimal contracts can be computed at or near a pure strategy profile, an important step in making it possible to solve the optimal contract problem with a finite number of linear programming problems. 2 Principal-Agents Contracts Let the principal be denoted by i = 0, and the agents are denoted by i =, 2,...,n. There are only finitely many output levels y k Y, k =, 2,...,K. The principal does not directly participate in production, and delegate the task to the agents. The total output (or revenue) produced by the agents depends on their joint actions or efforts as well as the stochastic factors. There are finitely many actions (pure strategies) for each agent. Let A i be the set of all pure strategies of agent i. The actions of the agents are not observable, but the total revenue is. However, the total revenue y is an imperfect signal of the action profile a =(a,a 2,...,a n ),a i A i. The distribution of revenue on Y depends on a. Let π k (a) be the probability of revenue y k when the action profile a is chosen. The expected total revenue is denoted by E(a) = P K k= π k(a)y k. The output-contingent payment contract offered to the agent i by the principal is denoted by z ik. Thus agent i gets the expected revenue E i (a) = P K k= π k(a)z ik when the joint action is a. The pure strategies of agent i will be indexed by a j i A i. A mixed strategies α i of agent i is the strategy with probability α i (a j i ) of using the pure strategy a j i.letα =(α,α 2,...,α n ) be the mixed strategy profile. We shall let π(α) be the probability distribution on Y with probability 4

5 π k (α,α 2,...,α n ) of revenue y k. The expected total revenue will be denoted by E(α) = P K k= π k(α)y k, and the expected revenue received by agent i is denoted by E i (α) = P K k= π k(α)z ik. The utility function of agent i is given by u i (w i,a i )=w i v i (a i ), where w i is the revenue received by the agent. Let v i (α i )= P j α i(a j i )v i(a j i ) be the expected cost of effort of the mixed strategy α i. Given the strategy profile α i of other agents, each agent will maximize his or her net payoff u i (α) =E i (α i,α i ) v i (α i ) from the chosen strategy α i. Given the quasi-linear utility functions of the agents, it is natural to consider the efficient strategy profiles α which maximizes the total net benefit W (α) =E(α) P n i= v i(α i ). This is the concept of efficiency we will use for the rest of the paper. The measure of efficiency is independent of the payment contracts z ik. Let u i0 be the utility level of the best outside offer of agent i, and the participation constraint of agent i is given by u i (α) u i0. Assume that there exists a strategy profile α satisfying W (α) P i u i0. If the condition is not satisfied, then there is no reason for the principal to hire the agents. The concept of admissible strategies is first introduced in Cheng (2000) for model with a single agent. Here we extend it to the case of many agents. Let α be any mixed strategy profile of the agents. We say that the deviation α 0 i by agent i is non-detectable if π(α 0 i,α i )=π(α) We say that the strategy profile α is inadmissible if for some agent i, there exists a non-detectable deviation strategy α 0 i which has a lower cost for agent i v i (α 0 i) <v i (α i ) If α is not inadmissible, then we say that it is admissible. If α is inadmissible, then there is an agent who findsitprofitable to deviate from α. Such deviation cannot be prevented by a change of the payment (or incentive) contracts. As a result, α cannot be a Nash equilibrium for the agents with any payment contracts z ik. We have proved therefore the following result. Proposition If α is not admissible, then it cannot be an incentive compatible equilibrium for the agents through the design of the outputcontingent payment contracts. We now show that admissibility is sufficient as well as necessary for the incentive compatibility in the principal-agents contracts. 5

6 Theorem 2 The strategy profile α is admissible if and only if it is an incentive compatible individually rational equilibrium for the agents through the design of the output-contingent contracts. Proof: To find an incentive compatible payment contract, we look for the solution of the following system of linear inequalities in z ik X πk (α) π k (a j i,α i) z ik v i (α i ) v i (a j i ) for all aj i k Let the set A i be indexed by a j i. The solution exists if and only if the following is true: X µ j i πk (α) π k (a j i,α i) =0,µ j i 0 implies () k X µ j i vi (α i ) v i (a j i,α i) 0 (2) There is no loss of generality in assuming that P µ j i =. Let the mixed strategy with probability µ j i on action a j i be denoted by α 0 i, then the assumption in () says that π(α) =π(α 0 i,α i ), and the conclusion (2) is v i (α 0 i) v i (α i ).Thisisexactlythestatementthatprofitable deviations by player i are detectable. It is quite easy to satisfy the individual rationality condition. Let z ik be an incentive compatible payment contract, and choose t i such that u i (α i )+t i u i0 for all i Let zik 0 = z ik + t i. The payment t i is independent of k. It is a noncontingent payment to agent i. The introduction of t i does not change the incentive property of z ik, and at the same time makes the contract individually rational Note that an efficient strategy profile must be admissible. We have the following full surplus extraction result under moral hazards. It corresponds to the Cremer and McLean (988) full surplus extraction result for the asymmetric information models. Theorem 3 Let α be an efficient admissible strategy profile. Then the principal can find an incentive compatible contract so that α is a Nash equilibrium and u i (α) =u i0 for all i. All the surplus W (α) P i u i0 from production goes to the principal. Proof. At the end of the previous proof, just choose t i = u i0 u i (α i ). 6

7 3 Partnership Contracts Since the individual rationality condition can be easily handled in our model. We shall omit it from our discussions from now on. Consider now the case when there is no principal. The agents agree among themselves the payments contracts. In this case, we refer to the agents as partners and the payment contracts as share contracts. The partnership incentive problem is to find conditions under which there are revenue dependent share contracts s ik so that the partners have the incentive to choose the profile α with the highest possible total payoff. One form of budget constraints usually used in the literature is X s ik = y k (3) i Let α be an admissible strategy profile. Not all admissible strategy profile can be an incentive compatible equilibrium through the design of the payment contract satisfying the exact budget constraints (3). In fact, we shall give an example of a partnership game in which the efficient strategy profile is not an incentive compatible equilibrium whatever payment contracts are agreed upon by the partners. Although such examples are known in the literature, we find it useful to use one simple example for illustrating our ideas in the paper. Example 4 There are two possible outputs y =24,y 2 =2. When both partners work, the probability of high output is 2. If only one works, the 3 probability is, and if neither works, it is 0. The disutility (cost) of work 6 is 3.5 for both, and shirking has 0 cost. E(w, w) =20 7=3 E(w, s) 3.5 =E(s, w) 3.5 =0.5 E(s, s) 0=2 Hence it should be clear that (work, work) is the efficient strategy profile of the game. It is easily seen to be admissible. We want to show that this is not a Nash equilibrium for any choice of the contracts s ik satisfying (3). 7

8 Given any choice of payment contracts of the partners s ik, the payoffs to the partners are as follows When both work: ( s + 3 s 2, s 2+ 3 s 22) When only player one works: ( s s 2, 6 s s 22) When only player two works: ( 6 s s 2, s s 22) When no one works: (s 2,s 22 ) Suppose (work, work) is a Nash equilibrium, we want to derive a contradiction. The Nash equilibrium property for player one requires that s + 3 s 2 6 s s 2 which yields 2 s 2 s or s s 2 7 Similarly, the equilibrium property for player two implies that s 2 s 22 7 Summing the two requirements together, and use (3), we get (s + s 2 ) (s 2 + s 22 ) 7 or y y 2 4 which is contradictory, as y =24,y 2 =2. This shows that there is no way to design the payment contracts satisfying (3) and achieve the incentive compatible efficient outcome (work, work). There is a necessary and sufficient condition in the literature for a strategy profile α to be an incentive compatible equilibrium with the proper choice of the contracts s ik satisfying (3). In Legros and Matsushima (99, Theorem 2), the condition is that a certain measure of the likelihood of deviation is bounded above. In other words, the "degree" of the incentive to deviate is not infinite. Although this is a good condition with some intuition,it is not the condition that we find useful for our purpose. Instead, our condition is more in the spirit of the mathematical conditions in Lemma A of their paper, but with a further mathematical manipulation to obtain an economic interpretation. There are several versions of similar conditions in section 3 of d Aspremont 8

9 and Gérard-Varet (998). However, none of those in the literature is stated in the way useful for us here. Instead, I shall state the conditions through the concept of unidentifiable deviations and the "surplus" from such deviations. Our condition is that the "surplus" from unidentifiable deviations should be negative. We refer to it as the "No Surplus Condition". Such "surplus" concept is useful when we consider the problem of optimal contracts later. Consider the collection of deviations α 0 i,i=, 2...n,We say that the collection of deviations α 0 = {α 0,α 0 2,...,α 0 n} are unidentifiable deviations if π(α 0 i,α i ),i=, 2,...,nare all identical. If α 0 = {α 0,α 0 2,...,α 0 n} are unidentifiable deviations, we let E 0 = E(α 0 )=E(α 0 i,α i ) be the common expected revenue after deviations. Let the total surplus from the deviations be defined as Ã! Ã! nx nx S(α 0 )= E0 v i (α 0 i) E(α) v i (α i ) i= = E0 E(α) nx [v i (α 0 i) v i (α i )] i= We say that the No Surplus Condition is satisfied at α, if for all unidentifiable deviations α 0 = {α 0 i} n i=, wehaves(α 0 ) 0. Note that if the "pairwise identifiability" conditions in Fudenberg, Levine and Maskin (994) are satisfied, then the No Surplus Condition is also satisfied. Note also that the admissibility condition is automatically satisfied when the No Surplus Condition holds. Theorem 5 The mixed strategy profile α is an incentive compatible equilibrium through the design of the share contracts satisfying the exact budget constraints (3) if and only if it satisfies the No Surplus Condition. Proof. It is quite easy to show that the no surplus condition is a necessary condition. Let {α 0 i} n i=be unidentifiable deviations. When partner i deviates to α 0 i, the change in payoff to partner i is E(α 0 i,α i ) v i (α 0 i) (E(α) v i (α i )). Let s ik be the sharing rule satisfying the incentive conditions and (3). We have the following incentive inequalities KX [π k (α 0 i,α i ) π k (α)]s ik v i (α 0 i) v i (α i ) (4) k= Summing over i, we have nx i= KX [π k (α 0 i,α i ) π k (α)]s ik k= 9 i= nx [v i (α 0 i) v i (α i )] (5) i=

10 Since π k (α 0 i,α i ) is independent of i, we have à KX nx! nx [π k (α 0,α ) π k (α)] s ik [v i (α 0 i) v i (α i )] (6) or or k= i= KX [π k (α 0,α ) π k (α)]y k k= E 0 E(α) i= nx [v i (α 0 i) v i (α i )] (7) i= nx [v i (α 0 i) v i (α i )] (8) i= hence the no surplus condition holds at α. To prove sufficiency, we look for the solutions in s ik of the following inequalities v i (α 0 i)+ KX π k (α 0 i,α i )s ik v i (α i )+ k= KX π k (α)s ik for all α i (9) k= nx s ik = y k for all k (0) i= We can simplify the problem by looking at the pure strategy deviations only. That is, it is sufficient to consider the following incentive inequalities for all pure strategy deviations a j i K v i (a j i )+ X π k (a j i,α i)s ik v i (α i )+ k= KX π k (α)s ik for all a j i () k= nx s ik = y k for all k (2) i= The above inequalities can be rewritten as KX [π k (α) π k (a j i,α i)]s ik v i (α i ) v i (a j i ) for all aj i (3) k= nx s ik = y k for all k (4) i= Thesolutionexistsifforallµ j i 0,η k satisfying X [π k (α) π k (a j i,α i)]µ j i + η k =0for all k and i (5) j 0

11 we must have X X [v i (α i ) v i (a j i )]µj i + X i j k η k y k 0 (6) Let µ j i 0,η k satisfy (5). We want to show (6) if α satisfies the P no surplus condition. Choose a large enough constant c>0 such that j µj i /c for all i. Let α i be the mixed strategy of the partner i with probability of choosing a j i given by α i (a j i )=µj i c + α(aj i )( X j 0 µ j0 i c ) (7) We have v i (α i ) v i (α 0 i)= X j µ j i c v i(α i ) X j µ j i c v i(a j i ) (8) = X j [v i (α i ) v i (a j i )]µj i c and π k (α) π k (α 0 i,α i )= X j µ j i c π k(α) X j µ j i c π k(a j i,α i) (9) = X j [π k (α) π k (a j i,α i)] µj i c Conditions (5) and (9) then imply that π k (α) π k (α 0 i,α i )= η k for all i and k (20) c Hence the deviations α 0 i are not identifiable.. By the no surplus condition, we have X [v i (α i ) v i (α 0 i)] + X i k Using condition (8), we then get X X [v i (α i ) v i (a j i )]µj i c + X i j k η k c y k 0 η k c y k 0 which is exactly the condition (6) we want to prove. At the efficient profile α, the following result says that there is no need to consider the case when the expected output after deviation is larger than that of the efficient expected output.

12 Lemma 6 Let E 0 = E(α 0 i,α i) be the common expected output for the unidentifiable deviations α 0 i. The no surplus condition is automatically satisfied if E 0 E(α ) Proof. Let α 0 i be unidentifiable deviations. Let E 0 = E(α 0 i,α i) be the common expected output after deviations. By the efficiency property of the profile α, no unilateral deviation by any partner can increase the total net benefit, hence we have E 0 v i (α 0 i) E(α ) v i (α i ) (2) Sum (2) over i, we get nx ne 0 v i (α 0 i) ne(α ) Hencewehave nx v i (α i ) i= i= nx v i (α i ) (22) i= nx v i (α 0 i) n[e(α ) E 0 ] E(α ) E 0 (23) i= and the no surplus condition holds. An immediate corollary of the lemma is Corollary 7 Let α be an efficient strategy profile. Then α is an incentive compatible equilibrium through the design of the share contracts satisfying the exact budget constraints (3) if and only if it satisfies the No Surplus Condition for all unidentifiable deviations with E 0 <E(α ). We can use the no surplus condition to argue that it is more difficult to construct efficient incentives in large partnerships than in small ones. Let there be two partners with A = A 2. Assume that the two partners are symmetric in the sense that π(a,a 2 )=π(a 2,a ) for all a,a 2. Let a =(a,a 2),a = a 2 be a symmetric efficient action profile. Assume that there are only two revenue levels y =0or T. Let p be the probability of the high output T at a. When a partner exerts less effort a 0 i <a i, let p(a 0 i) be the probability of the high output at (a 0 i,a i). The efficiency condition at a requires that (p p(a 0 i))t v i (a i ) v i (a 0 i). The no surplus condition, hence the existence of efficient incentives, requires that (p p(a 0 i))t 2(v i (a i ) v i (a 0 i)). If there are n such symmetric partners, then the no surplus condition requires that (p p(a 0 i))t n(v i (a i ) v i (a 0 i)) or (p p(a 0 i)) T n v i(a i ) v i (a 0 i) (24) 2

13 Assume that the per capita revenue T involved is more or less fixed as n n gets larger. With the larger number of partners, the individual impact on the success probabilities is likely to be smaller. Thecostishowever fixed. Hence the condition becomes more stringent as n becomes larger. With a larger number of partners, it then becomes clear that the no surplus condition is often violated, and there is no efficient incentives for such partnerships. 4 Weaker Budget Constraints From example 4, we know that in general the best we can hope for is to find the second best equilibrium, one that is best possible in the presence of moral hazards and imperfect monitoring. For the example 4, we want to show that (shirk, shirk) is a second best equilibrium when the share contracts are required to satisfy the exact constraints (3). Thepurestrategyprofiles (work, shirk), (shirk, work) are less efficient than (shirk, shirk), hence any outcome α which is more efficient must have some completely mixed strategy α i. We will show that both players in fact have to use a completely mixed strategy. Suppose one of the player, say player two, uses a pure strategy in equilibrium, and player one s strategy α is completely mixed. If player two s strategy is work, then the equilibrium condition for player one implies that s + 3 s 2 = 6 s s 2 which is equivalent to s s 2 =7 By (3), we must have s 2 s 22 =5 (25) But we can show that with this contract, work is not optimal for player two. Let (p, p) be the mixed strategy of player one with probability p of working. If player two chooses w, the payoff is [ 2 3 p + 6 ( p)]s 2 +[ 2 3 p 6 ( p)]s =( p)s 2 +( p)s while if he chooses s, the payoff is 6 ps 2 +( 6 p)s 22 3

14 If w were optimal for player two, we must have ( p)s 2 +( p)s ps 2 +( 6 p)s 22 which translates into ( p)(s 2 s 22 ) 3.5 and from (25), we have p 3.5 However this is a contradiction. The highest value for p is, hence the left-hand side is bounded above by < 3.5. Hence we have 6 3 shown that w is not optimal for player two, and it must be the case that player two s strategy is shirk. In this case, the equilibrium is only as efficient as (shirk, shirk). Hencewehaveshownthatboth playersmust use completely mixed strategies to be more efficient than (work, work). We now show that it is impossible to have a completely mixed Nash equilibrium. If there is a completely mixed Nash equilibrium under the payment contract s ik, then (shirk, shirk) is no longer a dominant strategy for either player. For player one, this means that either or Hence, either we have or we have s + 3 s 2 > 6 s s s s 2 >s 2 s s 2 > 7 s s 2 > 27 In either case, we have s s 2 > 7. Using the same argument for player two, we also conclude that s 2 s 22 > 7. Taken together, we must have y + y 2 > 4 and again we get a contradiction. Hence there exists no completely mixed Nash equilibrium under any payment contract s ik. We have proved that (shirk, shirk) is a second best equilibrium through the use of share contracts satisfying the exact constraints (3). The second best contracts we just established is subject to questioning. It is in fact possible to do better than the solution (shirk, shirk) if 4

15 the partners are willing to distribute less than the full amount of outputs and throw the undistributed outputs away when the performance is bad. If the outputs can be disposed freely, then any admissible strategy can be incentive compatible (see Theorem 8). For instance, in example 4, consider the strategy pair (work, work). We have shown that it is not incentive compatible when the budget constraints are satisfied exactly. Now consider the case when the outputs can be disposed freely, so that the following weaker version of the budget constraints are imposed X s ik y k (26) i We can show that (work, work) is now incentive compatible with the weaker budget constraints. Let s = s 2 =2,s 2 = s 22 =5 When the output is 2, only 0 units are distributed to the players, and 2 units are disposed off. With such contracts in place, (work, work) is now a Nash equilibrium, as and s + 3 s 2 = 6 = 6 s s s 2+ 3 s 22 = 6 = 6 s s 22 Note that even though the efficient strategy (work, work) is now incentive compatible, it does not mean that contract is efficient. The total amount of 2 units of output is wasted when the outcome is bad. This loss of efficiency can be computed as 3 2=2 3 By including this measure of inefficiency into our consideration, the efficiency measure of the contract is given by the expected total output minus the sum of the expected costs and the expected waste from disposal. Therefore in such an equilibrium, the efficiency measure is given by =2 3 We have shown earlier that we cannot do better than (shirk, shirk) with the efficiency measure 2, if we insist on the strict budget constraints. Allowing for the weaker budget constraints (thus throwing away resources 5

16 occasionally) actually makes sense. The contract arrangement is better than the equilibrium (shirk, shirk). We have the following general result when the weaker budget constraints are used. Theorem 8 If α is admissible, then it is incentive compatible with the weaker budget constraints (26). Furthermore, admissibility is equivalent to incentive compatibility with the weaker budget constraints (26) Proof. Let z ik be the incentive compatible payment contracts obtained in Theorem 5. Let x k = P i z ik. We can choose t 0 large enough so that x k t y k for all k and let s ik = z ik t, we have X s ik = t + X z ik = t + x k y k i i and (26) is satisfied, while all the incentive properties are unchanged. Borrowing the terminology from the implementation literature, we can α implementable, or we can refer to it also as sustainable. The problem of finding optimal contracts can be broken down into steps. The firststepistotakeanadmissibleα as given, and find the optimal incentive compatible s ik to implement α. It is equivalent to minimizing thelossfromdisposal.letl(α) be the the minimum cost of implementing α. This step is a linear programming problem. The second step is to find α to maximize m(α) =W (α) l(α). The function W (α) is wellunderstood. To find l(α), we need to convert the linear programming problem into its dual, and through further transformation, obtain a formula to compute m(α) =W (α) l(α). This is the duality result to be developed in the next section. 5 Duality Given an incentive compatible (or sustainable) α, we want to choose payment contracts so that the incentive constraints are satisfied, and at the same time the expected loss from disposal is minimized. Minimizing the expected loss from disposal is the same as maximizing the expected total payoff of the players from the distributed outputs minus costs. 6

17 Before we can find the optimal α for the second best solution, we will first determine the best payment contract for any given admissible strategy profile α. Let m(α) be the efficiency measure of this best contract. In this section, we determine m(α) for each admissible α. This is needed later to find the overall second best contract. We shall extend the concept of deviation strategies α 0 i to include any vectors p j i = α0 i(a j i ) negative or positive with the only condition P j pj i =. For such possibly infeasible deviation strategies π(α 0 i,α i ),v i (α 0 i) can be mathematically defined. This is done for easy mathematical expressions. It is possible to express our results without using such extended deviation strategies. However, the mathematical expressions will not be as elegant as the one given here. The simpler expressions facilitate computations when the result is applied to the analysis of economic games. Let denote the simplex of the probability distributions over the outcome space Y. We have the following result. Theorem 9 Let α be a completely mixed admissible strategy profile, and α 0 i an extended deviation strategy of player i. Then we have m(α) =minηy X i v i (α 0 i) subject to π(α 0 i,α i )=η for all i and η Proof. The function m(α) is the value of the following maximization problem max X v i (α i )+ X X π k (α)s ik i i k subject to X s ik y k for all k (27) i v i (α i )+ X k π k (α)s ik v i (a i )+ X k π k (a i,α i )s ik for all i and a i Denote the generic element of the set A i by a j i. The maximization problem can be written as max X v i (α i )+ X X π k (α)s ik (28) i i 7 k

18 subject to X s ik y k for all k (29) i X πk (a j i,α i) π k (α) s ik v i (a j i ) v i(α i ) for all a j i A i k We shall refer to this as the primary problem. Because of the admissibility of α, the system of constraints has a solution. The objective function is bounded above by E(α) P i v i(α i ), hence there exists a maximum for the objective function. Let e kk 0 =0when k 6= k 0, and e kk =for all k. To convert to the dual, use the following "switch box" for player i the coefficient π k 0(α) of s ik 0 in the objective function dual var. the coefficient of s ik 0 in the constraints right hand side µ j i π k 0(a j i,α i) π k 0(α) v i (a j i ) v i(α i ) η k e kk 0 y k Hence the dual problem is the minimization problem min X v i (α i )+ X η k y k + X X µ j i [v i(a j i ) v i(α i )] (30) i k i j subject to µ j i 0,η k 0 for all i, j, k and X µ j i [π k(a j i,α i) π k (α)] + η k = π k (α) for all i, k (3) j Summing (3) over k, we have X η k = (32) k We have the following minimization problem equivalent to the dual problem min X v i (α i )+ X η k y k + X X µ j i [v i(a j i ) v i(α i )] (33) i k i j 8

19 subject to µ j i 0,η k 0 for all i, j, k, P k η k =, and X µ j i [π k(a j i,α i) π k (α)] + η k = π k (α) for all i, k (34) j We now make use of the dual variables µ j i,η k to construct the strategy variables of the players. Define the extended deviation strategy α 0 i with α 0 i(a j i )=α i(a j i ) µj i +(X j µ j i )α i(a j i ) then we have π(α 0 i,α i ) π(α) = X j µ j i [π(α) π(aj i,α i)] and v i (α 0 i) v i (α i )= X j µ j i [v i(α i ) v i (a j i )] Hence the expression in (33) can be written as X i v i (α i )+ X k η k y k + X i [v i (α i ) v i (α 0 i)] = X k η k y k X i v i (α 0 i) (35) and the constraint (34) can be written as π(α) π(α 0 i,α i )+η = π(α) for all i or π(α 0 i,α i )=η for all i (36) Hence the value of the objective function (33) at µ j i,η k is the value of (35) at a collection of deviation strategies satisfying (36), and hence is larger than the minimum value of the following optimization problem min ηy X i v i (α 0 i) (37) subject to π(α 0 i,α i )=η for all i (38) where η is the vector with the components η k, and is the probability simplex of the outcome space. Conversely, given any collection of extended deviations strategies α 0 i, and η satisfying the constraint (38), define α 00 i = α i + ε(α i α 0 i)=(+ε)α i εα 0 i (39) 9

20 Since α i (a j i ) > 0 for all i, j, we must have pj i = α 00 i (a j i ) > 0 when ε is chosen to be small enough. We have π(α 00 i,α i ) π(α) =ε[π(α) π(α 0 i,α i )] or ε [π(α00 i,α i ) π(α)] = π(α) π(α 0 i,α i ) (40) Similarly, we have We now define µ j i = pj i ε X j ε [v i(α 00 i ) v i (α i )] = v i (α i ) v i (α 0 i) and η k the k-th component of η, then we have µ j i [v i(a j i ) v i(α i )] = ε [v i(α 00 i ) v i (α i )] = v i (α i ) v i (α 0 i) or v i (α 0 i)= v i (α i )+ X µ j i [v i(a j i ) v i(α i )] j Hence the objective function in (37) can be written as ηy X i v i (α 0 i)=ηy + X Ã v i (α i )+ X! µ j i [v i(a j i ) v i(α i )] (4) i j = X v i (α i )+ X η k y k + X X µ j i [v i(a j i ) v i(α)] i k i j which is the same as in (33). The constraint equation in (38) can be written as and from (40), it becomes π(α) π(α 0 i,α i )=π(α) η with η ε [π(α00 i,α i ) π(α)] = π(α) η with η By the definition of µ j i,η k, it is transformed further into X µ j i [π(aj i,α i) π(α)] = π(α) η with η j 20

21 or X µ j i [π k(a j i,α i) π k (α)] + η k = π k (α) for all i, k and η j whichisthesameonein(34). Therefore, we know that given any collection of extended deviation strategies α 0 i and η satisfying the constraint (38), the value of the objective function (37) is one of the value of the objective function in (33) at µ j i,η k we just defined. This means that the minimum value of the objectivefunction(37)isaslargeastheminimumvalueoftheobjective function (33). We have just shown that the two minimization problems areequivalentandhavethesameminimumvalue. Theproofisnow complete. For boundary strategy profiles, we have the following result. Theorem 0 Let α be an admissible strategy profile and α 0 i an extended deviation strategy for player i. We have m(α) =minηy X i v i (α 0 i) subject to α 0 i(a j i ) whenever α i(a j i )= (42) α 0 i(a j i ) 0 whenever α i(a j i )=0 (43) π(α 0 i,α i )=η for all i and η (44) If there exist no non-trivial collection of extended deviation strategies α 0 i satisfying all the constraints in (42) (44), then we must have m(α) = E(α) P i v i(α i ). ProofoftheTheorem: The proof is a refinement of the arguments used in Theorem 9. In the arguments for Theorem 9, given µ j i,η k,wedefine the extended deviation strategies as follows α 0 i(a j i )=α i(a j i ) µj i +(X j µ j i )α i(a j i ) Follow the same arguments as in the proof of Theorem 9, we get a collection of deviation strategies satisfying the constraints (44). If α i (a j i )=, we have α 0 i(a j i )=α i(a j i ) µj i + X µ j i α i(a j i )= j 2

22 If α i (a j i )=0, we have α 0 i(a j i )=α i(a j i ) µj i α i(a j i )=0 Therefore constraints (42) and (43) are satisfied in this choice of deviation strategies. Follow the same step as in the proof of Theorem 9, and we obtain a collection of extended deviation strategies satisfying all the required constraints. In the construction in the other direction, we are given a collection of deviation strategies α 0 i satisfying (42) (44). We defined the following deviation strategies in (39) α 00 i =(+ε)α i εα 0 i and used the property of completely mixed strategy profile α to insure that p j i = α00 i (a j i ) 0 andthenusep j i to define the non-negative µj i. From the constraints, we know that when α i (a j i )=, we have α0 i(a j i ) α i(a j i ). Hence we have α 00 i (a j i )=α i(a j i ) ε[α i(a j i ) α i(a j i )] α i(a j i ) and when ε>0issmall enough, we also have α 00 i (a j i ) > 0. When α i(a j i )= 0, we have the constraint property α 0 i(a j i ) 0. Hence we have α 00 i (a j i )= εα0 i(a j i ) 0 andagainwecaninsurethatp j i = α00 i (a j i ) 0 from the constraint properties without using the completely mixed property. Hence our arguments used in the proof of the Theorem 9 also insure that µ j i 0,η k 0. This completes the proof that m(α) can be written as stated in the theorem. If there exists no collection of non-trivial extended deviation strategies satisfying all the constraints in the theorem, then the minimum occurs at the trivial one α 0 i = α i, and we must have m(α) =ηy P i v i(α i ). End of Proof. We now give an example of the application of the theorems to determine the efficiency measure m(α). The same method will be used in the next section to derive a general formula for m(α) in the 2-by-2-by-2 games. Knowing the behavior of m(α) is essential for finding the optimal contracts. 22

23 Example We shall use the same partnership in example 4. Let α =(work, work). HenceweneedtoapplyTheorem0. Theextended deviation strategy α 0 should be of the form (x, x) with x. To get the minimum possible value for the expression in that theorem, x should satisfy the following equation µ 0 µ 2 (x, x) 3 6 = 0 6 or 2 3 x + ( x) = 6 and we get x = 5. The two extended deviations strategies 3 α0 =( 5, 2)= 3 3 α 0 2 are not identifiable, and satisfy all the constraints in Theorem 0. We have m(α) =y v (α 0 ) v 2 (α 0 2)= =2 3 We have constructed the payment contracts earlier with the same total payoff for the players. Hence the payment contract we constructed is an optimal one for the partners, conditional on the implementation of α = (work, work). To know whether it is an optimal (second best) contract, we need to know how m(α) varies with α. 6 Two-by-Two-by-Two games Let y >y 2 be the two possible signals. Let a,a 2 ; b,b 2 be the two pure strategies of player one and two respectively. Let π ij be the probability of signal y occurring when the chosen strategy profile is (a i,b j ).Letthe costs of choosing a i,b j by players, 2 be c i = v i (a i ),d j = v j (b j ). In our analysis, by changing the label if necessary, we can assume and we can also assume c >c 2,d >d 2 (45) (π,π 2 ) 6= (π 2,π 22 ), (π,π 2 ) 6= (π 2,π 22 ) (46) to avoid the trivial cases. It is useful to think of a,b as higher efforts (or consuming more resources in adopting the strategies), therefore higher costs for the respective players. Let α =(p,p 2 ) be the player one strategy with probabilities p,p 2 of using the pure strategies a,a 2 respectively. Similarly let α 2 =(q,q 2 ) 23

24 be the player two strategy. We will use the simplified notation (p,q ) for the strategy profile ((p, p ), (q, q )) when the context is clear. At the strategy profile α =(α,α 2 ), the change of probability of y occurring due to player one s "unitary" deviation in the direction of more effort is denoted by r and is given by µ µ π π r =(, ) 2 q =(π π 2 π 22 q π 2 )q +(π 2 π 22 )q 2 2 Hencewehave r = π 2 π 22 +(π + π 22 π 2 π 2 )q (47) Similarly, when player two deviates in the same "unitary" direction (, ), the change of probability of y is given by µ µ π π r 2 =(p,p 2 ) 2 =(π π 2 π 22 π 2 )p +(π 2 π 22 )p 2 hence r 2 = π 2 π 22 +(π + π 22 π 2 π 2 )p (48) When r 6=0,r 2 6=0, let T = y y 2 c c 2 r d d 2 r 2 To compute m(α), we need to (i) adopt individual directions of deviations (more or less effort in the strategy space) so that ±r, ±r 2 have the same signs: (ii) adjust individual intensity of deviations (probability of deviations in the strategy space) so that ±k r, ±k 2 r 2 have the same magnitude; (iii) choose common direction of deviations in the probability of outcomes so that T becomes negative; (iv) choose common scale of deviations in the probability space of outcomes so that these probabilities do not become negative. In the first two steps, we obtain unidentifiable deviations. In the last two steps, we compute the total surplus from such deviations. In thelaststep,althoughtherequirementseemstobeintuitive,whenitis translated into the probability space of strategies, it may require intensity of deviations leading to negative probabilities. This is the reason we extend the notions of deviation strategies (just for the computation in Theorems 9 and 0). A clear distinction between the three spaces: α in the strategy space, π in the outcome space, and T in the utility space is essential for the understanding of the following analysis. It should be noted that in general r i,t are functions of the strategy profile and will be denoted by r i (α),t(α) when such dependence needs 24

25 to be spelled out. Let Π = π + π 22 π 2 π 2. When Π =0,r = π 2 π 22,r 2 = π 2 π 22,T are constants. In this case, the analysis becomes easier. When Π 6= 0, there will be more complications. First we shall assume that Π =0, and take α to be an admissible completely mixed strategy profile. Notethatwemusthaver 6= 0, otherwisewehaveπ 2 π 22 =0. Since Π = π + π 22 π 2 π 2 =0, we must have π π 2 =0as well, violating the assumption in (46). Similarly, we must have r 2 6=0as well. The computations will depend on the sign of T. If T 0, the value of m(α) will computed at η =(, 0); if T 0, it will be computed at η =(0, ). Theorem 2 Assume that Π =0and T = y y 2 c c 2 π 2 π 22 d d 2 π 2 π For a completely mixed admissible strategy profile α =(p,q ), we have m(α)=y c 2 d 2 (c c 2 )( π 22 π 2 π 22 π 2 π 22 π 2 π 22 q ) (d d 2 )( π 22 π 2 π 22 π 2 π 22 π 2 π 22 p ) Proof: Let k = π (α),k 2 = π (α) r r 2 Define the extended deviation strategies as follows (49) α 0 = α + k (sgn(r ), sgn(r )) = (p + sgn(r )k, p sgn(r )k ) α 0 2 = α 2 + k 2 (sgn(r 2 ), sgn(r 2 )) = (q + sgn(r 2 )k 2, q sgn(r 2 )k 2 ) We have π (α 0,α 2 )=π (α)+sgn(r )k r = π (α)+ π (α) = hence we have π(α 0,α 2 )=(, 0) (50) and similarly π(α,α 0 2)=(, 0) Thus α 0,α 0 2 are unidentifiable deviations. The total surplus from these deviations is 25

26 ( π(α))(y y 2 ) sgn(r )k (c c 2 ) sgn(r )k 2 (d d 2 ) µ =( π(α)) y y 2 c c 2 sgn(r ) r d d 2 sgn(r 2 ) r 2 µ =( π(α)) y y 2 c c 2 d d 2 =( π(α))t 0 r r 2 and therefore we have m(α) =y c 2 d 2 (c c 2 )(p +sgn(r )k ) (d d 2 )(q +sgn(r 2 )k 2 ) (5) We want to compute m(α). First we relate π (α) to r, µ µ π π π (α)=(p, p ) 2 q π 2 π 22 q 2 =(0, ) µ µ π π 2 q + p π 2 π 22 q (, ) 2 µ µ π π 2 q π 2 π 22 q 2 Hencewehave π (α) =π 2 q + π 22 q 2 + p r (52) From the definition for k, we have k = π (α) r = π 2q π 22 q 2 r sgn(r )p and hence sgn(r )p + k = π 2q π 22 q 2 r Multiply by sgn(r ), we have p + sgn(r )k = π 2q π 22 q 2 r We can substitute q for q 2, and we have or p + sgn(r )k = π 22 (π 2 π 22 )q r (53) p + sgn(r )k = π 22 π 2 π 22 π 2 π 22 π 2 π 22 q (54) 26

27 Similarly, for player two we have or q + sgn(r 2 )k 2 = π 2p π 22 p 2 r 2 = π 22 (π 2 π 22 )p r 2 (55) q + sgn(r 2 )k 2 = π 22 π 2 π 22 π 2 π 22 π 2 π 22 p (56) Substituting (54) and (56) into (5), we have End of Proof. m(α)=y c 2 d 2 (c c 2 )( π 22 π 2 π 22 π 2 π 22 π 2 π 22 q ) (d d 2 )( π 22 π 2 π 22 π 2 π 22 π 2 π 22 p ) Now we want to consider the case when T = y y 2 c c 2 0. We have the following result. d d 2 π 2 π 22 π 2 π 22 Theorem 3 Assume that Π =0and T = y y 2 c c 2 π 2 π 22 d d 2 π 2 π For a completely mixed admissible strategy profile α =(p,q ), we have m(α)=y 2 c 2 d 2 + (c c 2 )π 22 π 2 π 22 + (c c 2 )(π 2 π 22 ) π 2 π 22 q + (d d 2 )π 22 π 2 π 22 + (d d 2 )(π 2 π 22 ) π 2 π 22 p Proofofthetheorem: Let k = π (α),k 2 = π (α) r r 2 Define the extended deviation strategies as follows (57) α 0 = α k (sgn(r ), sgn(r )) = (p sgn(r )k, p + sgn(r )k ) α 0 2 = α 2 k 2 (sgn(r 2 ), sgn(r 2 )) = (q sgn(r 2 )k 2, q + sgn(r 2 )k 2 ) We have π (α 0,α 2 )=π (α) sgn(r )k r = π (α) π (α) =0 hence we have π(α 0,α 2 )=(0, ) (58) 27

28 and similarly π(α,α 0 2)=(0, ) Thus α 0,α 0 2 are unidentifiable deviations. these deviations is The total surplus from π (α)(y y 2 )+sgn(r )k (c c 2 )+sgn(r )k 2 (d d 2 ) µ = π (α) y y 2 c c 2 sgn(r ) r d d 2 sgn(r 2 ) r 2 µ = π (α) y y 2 c c 2 d d 2 = π(α)t 0 r r 2 and therefore we have m(α) =y 2 c 2 d 2 (c c 2 )(p sgn(r )k ) (d d 2 )(q sgn(r 2 )k 2 ) (59) From the definition for k, and (52), we have k = π (α) r = π 2q + π 22 q 2 r + sgn(r )p and hence sgn(r )p + k = π 2q + π 22 q 2 r Multiply by sgn(r ), we have p sgn(r )k = π 2q π 22 q 2 r We can substitute q for q 2, and we have or p sgn(r )k = π 22 (π 2 π 22 )q r p sgn(r )k = π 22 π 2 π 22 π 2 π 22 π 2 π 22 q (60) Similarly, for player two we have q sgn(r 2 )k 2 = π 2p π 22 p 2 r 2 = π 22 (π 2 π 22 )p r 2 28

29 or q sgn(r 2 )k 2 = π 22 π 2 π 22 π 2 π 22 π 2 π 22 p (6) Substituting (60) and (6) into (59), we have or π 22 m(α)=y 2 c 2 d 2 (c c 2 )( π 2 π 22 q ) π 2 π 22 π 2 π 22 π 22 (d d 2 )( π 2 π 22 p ) π 2 π 22 π 2 π 22 m(α)=y 2 c 2 d 2 + (c c 2 )π 22 π 2 π 22 + (c c 2 )(π 2 π 22 ) π 2 π 22 q End of Proof. + (d d 2 )π 22 π 2 π 22 + (d d 2 )(π 2 π 22 ) π 2 π 22 p Now, we consider the more complex case when Π 6= 0. If α is admissible, and r =0, then player one is indifferent between the two strategies a,a 2. The unidentifiable deviations must satisfy π(α 0,α 2 ) π(α) =π(α,α 0 2) π(α) =0 By the admissibility condition, we must have v i (α 0 i) v i (α i ) 0 for i =, 2 and hence m(α) =W (α) in this case. Similarly, if r 2 =0, we also have m(α) =W (α). We will now assume that r 6=0,r 2 6=0. Theorem 4 Suppose Π 6= 0,r 6=0,r 2 6=0. For an admissible completely mixed strategy profile α =(p,q ) with T = y y 2 c c 2 r d d 2 r 2 0, we have m(α)=y c 2 d 2 (c c 2 )(π 22 π 2 ) (d d 2 )(π 22 π 2 ) π + π 22 π 2 π 2 π + π 22 π 2 π 2 (c c 2 )( + π 2π 2 π π 22 π +π 22 π 2 π 2 ) π 2 π 22 +(π + π 22 π 2 π 2 )q (d d 2 )( + π 2π 2 π π 22 π +π 22 π 2 π 2 ) π 2 π 22 +(π + π 22 π 2 π 2 )p 29

30 Proof: If T 0, from (53), we have p + sgn(r )k = π 22 (π 2 π 22 )q r π 22 (π 2 π 22 )q = π 2 π 22 +(π + π 22 π 2 π 2 )q Reduce the numerator into a constant term, making use of the equation, π 22 + (π 22 π 2 )(π 22 π 2 ) π + π 22 π 2 π 2 = we get the following expression for p + sgn(r )k π 2 π 2 π π 22 π + π 22 π 2 π 2 (62) π 22 π 2 + π 2π 2 π π 22 π + +π 22 π 2 π 2 (63) π + π 22 π 2 π 2 π 2 π 22 +(π + π 22 π 2 π 2 )q Similarly, q + sgn(r 2 )k 2 is given by π 22 π 2 + π 2π 2 π π 22 π + +π 22 π 2 π 2 (64) π + π 22 π 2 π 2 π 2 π 22 +(π + π 22 π 2 π 2 )p Substitute (63) and (64) into (5), we have m(α)=y c 2 d 2 (c c 2 )(π 22 π 2 ) (d d 2 )(π 22 π 2 ) π + π 22 π 2 π 2 π + π 22 π 2 π 2 (c c 2 )( + π 2π 2 π π 22 π +π 22 π 2 π 2 ) π 2 π 22 +(π + π 22 π 2 π 2 )q (d d 2 )( + π 2π 2 π π 22 π +π 22 π 2 π 2 ) π 2 π 22 +(π + π 22 π 2 π 2 )p End of Proof. Similarly, for Π 6= 0,T 0, we have the following result. 30

31 Theorem 5 Suppose Π 6= 0,r 6=0,r 2 6=0. For an admissible completely mixed strategy profile α =(p,q ) with T = y y 2 c c 2 r d d 2 r 2 0, we have m(α)=y c 2 d 2 (c c 2 )(π 22 π 2 ) (d d 2 )(π 22 π 2 ) π + π 22 π 2 π 2 π + π 22 π 2 π 2 (c c 2 ) π 2π 2 π π 22 π +π 22 π 2 π 2 π 2 π 22 +(π + π 22 π 2 π 2 )q (d d 2 ) π 2π 2 π π 22 π +π 22 π 2 π 2 ) π 2 π 22 +(π + π 22 π 2 π 2 )p To find sup α m(α), it is important that we consider boundary strategy profiles. We will see that it occurs often at pure strategy profiles. The above arguments can also be applied to "boundary" strategy profiles aswell,aslongastheunidentifiable deviations α 0,α 0 2 with a negative surplus also satisfy the boundary conditions in Theorem 0. This means that there is a feasible direction of unidentifiable deviations with a positive surplus at the strategy profile. In particular, if r < 0,T 0, deviations in the direction of deviation (, ) is feasible and yields positive total surplus when p i > 0,q i > 0. Henceforthestrategyprofile with p =, 0 <q <, Theorem 2 applies only if r > 0,T 0; in other cases the total surplus from unidentifiable and feasible deviations is always 0, hence we must have m(α) =W (α). Similarly, for the strategy profile with 0 <p <,q =, Theorem 2 applies only if r 2 > 0,T 0; in other cases we must have m(α) =W (α). For the strategy profile with p = 0, 0 < q <, Theorem 2 applies only if r < 0,T 0; in other cases, we must have m(α) =W (α). For the strategy profile with 0 <p <,q =0, Theorem 2 applies only if r 2 < 0,T 0. For the profile p ==q, Theorem 2 applies only if r > 0,r 2 > 0,T 0; in other cases we must have m(α) = W (α). For the strategy profile p =,q = 0, Theorem 2 applies only if r > 0,r 2 < 0,T 0; in other cases we must have m(α) = W (α). For the strategy profile p =0,q =, Theorem 2 applies only if r < 0,r 2 > 0,T 0; in other caseswemusthavem(α) =W (α). For the strategy profile p =0=q, Theorem 2 applies only if r < 0,r 2 < 0,T 0; in other cases we must have m(α) =W (α). Thus we have the following result for all strategy profiles. Theorem 6 Assume that Π =0and T = y y 2 c c 2 π 2 π 22 d d 2 π 2 π For any mixed admissible strategy profile α =(p,q ), we have the same formula given in Theorem 2 if and only if all of the following four 3