On the Existence of Pure Nash Equilibria in Weighted Congestion Games

Transcription

1 MATHEMATICS OF OPERATIONS RESEARCH Vol. 37, No. 3, August 2012, pp ISSN X (print) ISSN (online) INFORMS On the Existence of Pure Nash Equilibria in Weighted Congestion Gaes Tobias Harks School of Business and Econoics, Maastricht University, Maastricht, The Netherlands, Max Kli Institut für Matheatik, Technische Universität Berlin, Berlin, Gerany, We study the existence of pure Nash equilibria in weighted congestion gaes. Let denote a set of cost functions. We say that is consistent if every weighted congestion gae with cost functions in possesses a pure Nash equilibriu. Our ain contribution is a coplete characterization of consistency of continuous cost functions. We prove that a set of continuous functions is consistent for two-player gaes if and only if contains only onotonic functions and for all nonconstant functions c 1 c 2, there are constants a b such that c 1 x = a c 2 x + b for all x 0. For gaes with at least three players, we prove that is consistent if and only if exactly one of the following cases holds: (a) contains only affine functions; (b) contains only exponential functions such that c x = a c e x + b c for soe a c b c, where a c and b c ay depend on c, while ust be equal for every c. The latter characterization is even valid for three-player gaes. Finally, we derive several characterizations of consistency of cost functions for gaes with restricted strategy spaces, such as weighted network congestion gaes or weighted congestion gaes with singleton strategies. Key words: weighted congestion gaes; pure Nash equilibria; cost functions. MSC2000 subject classification: Priary: 91A10; secondary: 91A43, 90B10, 68W01 OR/MS subject classification: Priary: gaes, noncooperative; secondary: atheatics, functions. History: Received Deceber 7, 2010; revised August 1, 2011, February 1, Published online in Articles in Advance May 17, 2012, and updated June 19, Introduction. In any situations, the state of a syste is deterined by a finite nuber of independent players, each optiizing an individual objective function. A natural fraework for analyzing such decentralized systes are noncooperative gaes. While it is well known that for finite noncooperative gaes a Nash equilibriu in ixed strategies always exists, this need not be true for Nash equilibria in pure strategies (PNE for short). One of the fundaental goals in gae theory is to characterize conditions under which a Nash equilibriu in pure strategies exists. In this paper, we study this question for weighted congestion gaes. Congestion gaes, as introduced by Rosenthal [32], odel the interaction of a finite set of players that copete over a finite set of facilities. A pure strategy of each player is a set of facilities. The cost of facility f is given by a real-valued cost function c f that depends on the nuber of players using f, and the private cost of every player equals the su of the costs of the facilities in the strategy that she chooses. Rosenthal [32] proved in a seinal paper that such congestion gaes always adit a PNE by showing that these gaes possess an exact potential function. In a weighted congestion gae, every player has a deand d i >0 that she places on the chosen facilities. The cost of a facility is then a function of the total load on the facility. An iportant subclass of weighted congestion gaes are weighted network congestion gaes. Here, every player is associated with a positive deand that she wants to route fro her origin to her destination on a path of iniu cost. In contrast to unweighted congestion gaes, weighted congestion gaes do not always adit a PNE. Fotakis et al. [16] and Liban and Orda [24] each constructed a single-coodity network instance with two players having deands one and two, respectively, and showed that these gaes do not have a PNE. Their instances use different nondecreasing cost values per edge that are defined at the three possible loads, Goeans et al. [19] constructed a two-player single-coodity instance without a PNE that uses different polynoial cost functions with nonnegative coefficients and degree of at ost two. Interestingly, Anshelevich et al. [5] showed that for cost functions of the for c f x = c f /x c f 0, every two-player gae possesses a PNE. For gaes with affine cost functions, Fotakis et al. [16, 17] proved that every weighted congestion gae possesses a PNE. Later, Panagopoulou and Spirakis [30] proved that PNE always exist for instances with unifor exponential cost functions c f x = e x. Harks et al. [22] generalized this existence result to nonunifor exponential cost functions of the for c f x = a f e x +b f for soe a f b f, where a f and b f ay depend on the facility f, while ust be equal for all facilities. It is worth noting that the positive results of Fotakis et al. [16, 17], Harks et al. [22], and Panagopoulou and Spirakis [30] are particularly iportant as they establish existence of PNE for the respective sets of cost functions independent of the underlying gae structure, that is, independent of the underlying strategy set, deand vector, and nuber of players, respectively. 419

2 420 Matheatics of Operations Research 37(3), pp , 2012 INFORMS In this paper, we further explore the equilibriu existence proble in weighted congestion gaes. Our goal is to precisely characterize which types of cost functions actually guarantee the existence of PNE. To forally capture this issue, we introduce the notion of PNE-consistency or siply consistency of a set of cost functions. Let be a set of cost functions and let be the set of all weighted congestion gaes with cost functions in. We say that is consistent if every gae in possesses a PNE. Using this terinology, the results of Fotakis et al. [16, 17], Harks et al. [22], and Panagopoulou and Spirakis [30] yield that is consistent if contains either affine functions or certain exponential functions. A natural open question is to decide whether there are further consistent functions, that is, functions guaranteeing the existence of a PNE. We thus investigate the following question: How large is the set of consistent cost functions? We also introduce a stricter notion of consistency which we ter FIP-consistency. Forally, we say that a set of cost functions is FIP-consistent, if every gae in possesses the finite iproveent property, that is, every sequence of unilateral iproveents is finite; see Monderer and Shapley [28] Our results. To obtain a coplete characterization of the equilibriu existence proble in weighted congestion gaes, we first derive necessary conditions. Let be a set of continuous functions. We show that if is consistent, then ay only contain onotonic functions. We here use onotonic in the literal sense; i.e., every function c is either nondecreasing or nonincreasing. We further show that onotonicity of cost functions is necessary for consistency even in singleton gaes with only two players, two facilities, identical cost, functions, and syetric strategies. As our first ain result we show that a set of continuous cost functions is consistent for two-player gaes if and only if contains only onotonic functions and for all nonconstant c 1 c 2, there are constants a b such that c 1 x = a c 2 x + b for all x 0. This characterization precisely explains the seeing dichotoy between the positive result of Anshelevich et al. [5] for two-player gaes and the two-player instances without PNE given by Fotakis et al. [16], Goeans et al. [19], and Liban and Orda [24]. Our second ain result establishes a characterization for the general case. We prove that a set of continuous functions is consistent for gaes with at least three players if and only if exactly one of the following cases holds: (a) contains only affine functions; (b) contains only exponential functions such that c x = a c e x + b c for soe a c b c, where a c and b c ay depend on c, while ust be equal for every c. This characterization is even valid for three-player gaes. We further show that for both two player gaes and gaes with at least three players, consistency of is equivalent to FIP-consistency. While the above characterizations hold for arbitrary strategy spaces, we also study consistency of cost functions for restricted strategy spaces. First, we consider weighted network congestion gaes. Assuing strictly positive costs, we show that essentially all results translate to directed network congestion gaes. For gaes on undirected networks, we give respective characterizations for gaes with two players and at least four players leaving a gap for three-player gaes. For singleton weighted congestion gaes with two players we show that is consistent if and only if contains only onotonic functions. This characterization does not extend to gaes with three players. We give an instance with three players and onotonic cost functions without a PNE. For syetric singleton weighted congestion gaes, however, we prove that is consistent if and only if contains only onotonic functions. In contrast to the characterizations for arbitrary strategy spaces, both characterizations do not carry over to FIP-consistency. We provide corresponding instances with iproveent cycles Techniques and outline of the paper. The proofs of our ain results essentially rely on two ingredients. First, we derive in 3 for continuous and consistent cost functions two necessary conditions (Monotonicity Lea (Lea 3.3) and Extended Monotonicity Lea (Lea 3.4)). The Monotonicity Lea states that any continuous and consistent cost function ust be onotonic. The lea is proved by constructing a generic two-player weighted congestion gae in which we identify a unique 4-cycle of deviations of two players. Then, we show that for any nononotonic cost function, there is a weighted congestion gae with a unique iproveent cycle. By adding additional players and carefully choosing the players weights and strategy spaces, we then derive the Extended Monotonicity Lea, which ensures that the set of cost functions contained in a certain finite integer linear hull of the considered cost functions ust be onotonic. By analyzing functions contained in the finite integer linear hull corresponding to two-player gaes, we prove in 4 that a set of continuous cost functions is consistent for two-player gaes if and only if all cost functions are onotone and every two nonconstant cost functions are affine transforations of each other. In 5, we consider gaes with at least three players. We show that the Extended Monotonicity Lea for gaes with at least three players iplies that consistent and continuous cost functions ust be either affine or exponential. In 6 and 7, we derive characterizations of consistency and FIP-consistency of cost functions for gaes with restricted strategy spaces, such as weighted network congestion gaes and weighted singleton congestion gaes, respectively.

3 Matheatics of Operations Research 37(3), pp , 2012 INFORMS Significance. Weighted congestion gaes are aong the core topics in the gae theory, operations research, coputer science, and econoics literature. This class of gaes has several applications such as scheduling gaes, routing gaes, facility location gaes, network design gaes, etc; see Ackerann et al. [1], Anshelevich et al. [5], Chen and Roughgarden [11], Gairing et al. [18], Ieong et al. [23], and Milchtaich [27]. In all of the above applications there are two fundaental goals fro a syste design perspective: (i) the syste ust be stabilizable, that is, there ust exist a stable point (PNE) fro which no player wants to unilaterally deviate; (ii) yopic play of the players should guide the syste to a stable state. Because the nuber of players and their types (expressed by the deands and the strategy spaces) are only known to the players and not available to the syste designer, it is very natural to study the above two issues with respect to the used cost functions. In fact, in ost of the above-entioned applications, the cost functions are under control of the syste designer since they represent the technology associated with the resources, e.g., queuing discipline at routers, latency function in transportation networks, etc. Therefore, our results ay help to predict and explain unstable traffic distributions in telecounication networks and road networks. For instance in telecounication networks, relevant cost functions are the so-called M/M/1-delay functions (see also Roughgarden and Tardos [35]). These functions are of the for c a x = 1/ u a x, where u a represents the capacity of arc a. In road networks, for instance, the ost frequently used functions are onoials of degree 4 put forward by the U.S. Bureau of Public Roads [10]. Our results iply that, for these special types of cost functions, there is always a ulti-coodity instance (with three players and identical cost functions) that is unstable in the sense that a PNE does not exist. On the other hand, our characterizations can be used to design a stable syste: for instance, unifor M/M/1-delay functions are consistent for two-player gaes. Our results are also relevant for the large body of work quantifying the worst-case efficiency loss of PNE for different sets of cost functions; see Awerbuch et al. [6], Christodoulou and Koutsoupias [12], and Aland et al. [3]. While ixed Nash equilibria are guaranteed to exist, their use is often unrealistic in practice. On the other hand, our work reveals that for ost classes of cost functions pure Nash equilibria as the stronger solution concept ay fail to exist in weighted congestion gaes. Thus, our work provides additional justification to study the worst-case efficiency loss for different solution concepts, such as sink equilibria (Goeans et al. [19]), correlated and coarse correlated equilibria (Bhawalkar et al. [9], Roughgarden [34]) Related work. In contrast to ordinary congestion gaes as introduced by Rosenthal [32], gaes with weighted players and/or player-specific cost functions need not possess a PNE. As for weighted players, even two-player gaes ay fail to adit a PNE; see the exaples given by Fotakis et al. [16], Goeans et al. [19] and Liban and Orda [24]. Also related is the early work of Rosenthal [33] who showed that in weighted congestion gaes where players are allowed to split their deand integrally, a PNE need not exist. On the positive side, Fotakis et al. [16] and Panagopoulou and Spirakis [30] proved the existence of a PNE in gaes with affine and exponential costs, respectively. Dunkel and Schulz [13] showed that it is strongly NP-hard to decide whether or not a weighted congestion gae with nonlinear cost functions possesses a PNE. If the strategy of every player contains a single facility only (singleton gaes), Fotakis et al. [15] showed the existence of PNE for linear cost functions (without a constant). Even-Dar et al. [14] derived the existence of PNE for load balancing gaes on parallel unrelated achines. Andelan et al. [4] proved even the existence of a strong Nash equilibriu a strengthening of the pure Nash equilibriu to resilience against coalitional deviations in scheduling gaes on unrelated achines. In fact, strong Nash equilibria exist in all singleton weighted congestion gaes with nondecreasing costs; see Harks et al. [21]. This also holds for nonincreasing cost functions as proven by Rozenfeld and Tennenholtz [36]. Allowing for player-specific cost functions, Milchtaich [25] showed that unweighted singleton congestion gaes with player-specific cost functions possess at least one PNE. He also presented an instance with weighted players and player-specific cost functions without a PNE. Gairing et al. [18] showed that best response dynaics do not cycle if the player-specific cost functions are linear without a constant ter. Milchtaich [27] further showed that general network gaes with player-specific costs need not adit a PNE in general. In fact, the corresponding decision proble turns out to be NP-coplete, as shown by Ackerann and Skopalik [2]. Ieong et al. [23] proved that in congestion gaes with singleton strategies and nondecreasing cost functions, best response dynaics converge in polynoial tie to a PNE. Ackerann et al. [1] extended this result to weighted congestion gaes with a so-called atroid property, that is, the strategy of every player fors a basis of a atroid. In the sae paper, they showed that the atroid property is the axial property that gives rise to a PNE for all nondecreasing cost functions, that is, for any strategy space not satisfying the atroid property, there is an instance of a weighted congestion gae not having a PNE. The consistency approach that we pursue in this paper is orthogonal to that of Ackerann et al. [1]. While they characterize the structure of the strategy space guaranteeing the existence of a PNE assuing arbitrary

4 422 Matheatics of Operations Research 37(3), pp , 2012 INFORMS positive and nondecreasing costs, we characterize the structure of cost functions guaranteeing the existence of a PNE assuing arbitrary strategy spaces. Orda et al. [29] study the issue of uniqueness of PNE in weighted network congestion gaes with splittable deands (see also Bhaskar et al. [8], Milchtaich [26], Richan and Shikin [31], and Yang and Zhang [37]). They give sufficient conditions for uniqueness of PNE for several classes of cost functions. Interestingly, in the final section of their paper, the authors raise the issue of the existence of pure Nash equilibria in such gaes (depending on the cost functions) under the assuption that the flow is unsplittable. The results in this paper give a coplete answer to their question. An extended abstract of parts of this paper appeared in the Proceedings of the 37th International Colloquiu on Autoata, Languages, and Prograing, 2010; see Harks and Kli [20]. 2. Preliinaries. We consider finite strategic gaes G = N S, where N = 1 n is the nonepty and finite set of players, S = i N S i is the nonepty strategy space, and S n is the cobined private cost function that assigns a private cost vector s to each strategy profile s S. We consider cost iniization gaes, and (unless specified otherwise) we allow private cost functions to be negative or positive. We call an eleent s S a strategy profile. For i N, we write S i = j i S j and s = s i s i eaning that s i S i and s i S i. A strategy profile s is a pure Nash equilibriu (PNE) if i s i t i s i for all i N and t i S i. A pair s t i s i S S is called an iproving ove (or profitable deviation) of player i if i s i s i > i t i s i. We call a sequence of strategy profiles = s 1 s 2 an iproveent path if for every k the tuple s k s k+1 is an iproving ove for soe player i. A closed path s 1 s l s 1 is referred to as an l-iproveent cycle. A gae has the finite iproveent property (FIP) if no such cycle exists. A function P S with P s > P t for all iproving oves s t is called a potential function. As noticed by Monderer and Shapley [28], every gae that adits a potential function has the FIP and every finite gae with the FIP possesses a PNE. A tuple M = N F S = i N S i c f f F is called a congestion odel, where N is the set of players, F is a nonepty, finite set of facilities, and for each player i N, her collection of pure strategies S i is a nonepty, finite set of subsets of F. A cost function c f 0 is associated with every facility f F. In contrast to ost previous works, we neither assue onotonicity nor positivity of costs. In the following, we define weighted congestion gaes siilar to Goeans et al. [19]. Definition 2.1 (Weighted Congestion Gae). Let M = N F S c f f F be a congestion odel and d i i N be a vector of deands with d i >0. The corresponding weighted congestion gae is the strategic gae G M = N S, where is defined as = i N i, i s = f s i d i c f l f s and l f s = j N f s j d j. We soeties write G instead of G M. Let be a set of cost functions and let be the set of all weighted congestion gaes with cost functions in. Then, we say that is consistent if every G adits a PNE; we call FIP-consistent if every G has the FIP. If the set is restricted, for instance to two player gaes etc., we say that is consistent for if every G possesses a PNE. 3. Necessary conditions on the existence of a PNE. As a first result, we prove that if is consistent, then every function c is onotonic. We first need a technical lea. Lea 3.1. Let c 0 be a continuous function. Then, the following two stateents are equivalent: (i) c is onotonic on 0. (ii) The following two conditions hold: (a) For all x > 0 with c x > c 0 there is > 0 such that c y c x for all y x x +. (b) For all x > 0 with c x < c 0 there is > 0 such that c y c x for all y x x +. Proof. (i) (ii): Trivial. For proving (ii) (i), we first derive a useful property of functions satisfying (ii). Let c 0 be a continuous function satisfying (ii). Moreover, assue that there is an open interval with c x c 0 for all x. We clai that c is nondecreasing on if c x > c 0 for all x and that c is nonincreasing on if c x < c 0 for all x. We prove only the first case because the second follows by the sae arguents. Let c x > c 0 for all x. For a contradiction, assue that there are p 1 p 2 with p 1 < p 2 and c p 1 > c p 2. We define p 1 = ax x p 1 p 2 c x c p 1. Note that the set x p 1 p 2 c x c p 1 is nonepty because it contains p 1 and closed because c is continuous. Using (ii), there is = p 1 > 0 such that c y c p 1 c p 1 for all y p 1 p 1 +, contradicting the axiality of p 1. Now we prove (ii) (i). Let = inf x > 0 c x c 0. If =, we are done as c is constant. Otherwise, we clai that c x c 0 for all x >. For a contradiction, let = in x > c x = c 0 and

5 Matheatics of Operations Research 37(3), pp , 2012 INFORMS 423 = c + /2 (the iniu is attained because c is continuous). By construction, c x c 0 for all x. If c x > c 0 for all x, we have c x > c 0 for all x + /2 and thus c 0 = c = li x c x > c 0, a contradiction. If, on the other hand, c x < c 0 for all x, we get c 0 = c = li x c x < c 0, again a contradiction. We conclude that c x c 0 for all x >. Thus, for every >, the function c is onotonic on the open interval and thus, c is onotonic on 0. The following existence result for continuous, nononotonic functions can be derived directly fro Lea 3.1 and will be very useful in the reainder of this paper. Lea 3.2. Let c 0 be a continuous, nononotonic function. Then, there are x y >0 with y > x such that either c y x < c y < c x or c y x > c y > c x. Proof. Using the characterization of onotonic functions of Lea 3.1, for every continuous nononotonic function c, there is x > 0 such that one of the following holds: c x > c 0 and for every > 0 there is y = y x x + such that c y < c x ; or c x < c 0 and for every > 0 there is y = y x x + such that c y > c x. Fix such x. Because of the continuity of c, we have c y x c 0 and c y c x for 0. For sufficiently sall, x and y have the desired property. The two cases occurring in Lea 3.2 are depicted in Figure 1. Now consider a facility f with a nononotonic cost function and two players with deands d 1 = y x and d 2 = x, where x and y are as in Lea 3.2. Clearly, in case c y x < c y < c x player 1 prefers to be alone on f while player 2 would like to share the facility with player 1. If c y x > c y > c x, the arguentation works the other way around. This observation is the key to construct a two-player weighted congestion gae with singleton strategies that does not adit a PNE. Lea 3.3 (Monotonicity Lea). Let be a set of continuous functions. If is consistent, then every c C is onotonic. Proof. For a contradiction, suppose that c is a nononotonic function and consider the congestion odel M = N F S c f f F with N = 1 2, F = f g, S 1 = S 2 = f g, c f = c g = c. Since c is nononotonic, by Lea 3.2 we can find x y >0 with y > x such that either c y x < c y < c x or c y x > c y > c x. Regard the gae G M with d 1 = y x and d 2 = x. Calculating the differences of the deviating players private costs along the 4-cycle = f f g f g g f g f f, we obtain 1 g f 1 f f = y x c y x c y 1 f g 1 g g = y x ( c y x c x ) 2 g g 2 g f = x c y c x 2 f f 2 f g = x c y c x (2) If c y x < c y < c x, the differences (1) (2) are negative and is an iproveent cycle. If c y x > c y > c x, we can reverse the direction of and still get an iproveent cycle. Using that every strategy cobination is contained in, the claied result follows. Besides the continuity of the functions in, the proof of Lea 3.3 relies on rather ild assuptions and thus, this result can be strengthened in the following way. (a) c (b) c (1) y x x y x y x x y x Figure 1. As shown in Lea 3.2, for every continuous nononotonic function there are x y >0 with y > x such that one of the following cases holds: (a) c y x < c y < c x ; (b) c y x > c y > c x.

6 424 Matheatics of Operations Research 37(3), pp , 2012 INFORMS Corollary 3.1. Let be a set of continuous functions. Let be the set of weighted congestion gaes with cost functions in satisfying one or ore of the following properties: (i) Each gae G has two players; (ii) Each gae G has two facilities; (iii) For each gae G and each player i N, the set of her strategies S i contains a single facility only; (iv) Each gae G has syetric strategies; (v) Cost functions are identical, that is, c f = c g for all f g F. If is consistent for, then each c ust be onotonic. We now extend the Monotonicity Lea to obtain an even stronger result by regarding ore players and ore coplex strategies. To this end, for K we consider those functions that can be written as the integral linear cobination of K functions in, possibly with an offset. Forally, we define the finite integer linear hull of as { L = c 0 c x = K } a k c k x + b k K a k b k 0 c k (3) and show that consistency of iplies that L contains only onotonic functions. Lea 3.4 (Extended Monotonicity Lea). Let be a set of continuous functions. If is consistent, then L contains only onotonic functions. Proof. Assue by contradiction that c L is not onotonic. By allowing c k = c l for k l, we can oit the integer coefficients a k and rewrite c as c x = + c k x + b k c k x + b k for soe c k c k +, and b k b k 0. We define the congestion odel M = N F S c f f F, where N = N p N + N and F = F 1 F 2 F 3 F 4. The set of players N + contains for each c k, 1 k +, a player with deand b k and the set of players N contains for each c k, 1 k, a player with deand b k. We call the players in N N + offset players. The set N p = 1 2 contains two additional (nontrivial) players. Offset players with deand equal to 0 can be reoved fro the gae. For ease of exposition, we assue that all offsets b k, k = 1 +, and b k, k = 1, are strictly positive. We now explain the strategy spaces and the sets F 1 F 2 F 3 F 4. For each function c k, 1 k +, we introduce two facilities fk 2 f k 3 with cost function c k. For each function c k, 1 k, we introduce two facilities fk 1 f k 4 with cost function c k. To odel the offsets b k in (3), for each offset player k N +, we define S k = fk 2 f k 3. Siilarly, for each offset player k N, we set S k = fk 1 f k 4. The nontrivial players in N p have strategies S 1 = F 1 F 2 F 3 F 4 and S 2 = F 1 F 3 F 2 F 4 where F 1 = f1 1 f 1, F 2 = f1 2 f 2 +, F 3 = f1 3 f 3 +, and F 4 = f1 4 f 4. As c is assued to be nononotonic, by Lea 3.2, there are x y >0 with y > x such that either c y x < c y < c x or c y x > c y > c x. We consider the weighted congestion gae G M with d 1 = y x and d 2 = x for 1 2 N p. For the 4-cycle = F 1 F 2 F 1 F 3 F 3 F 4 F 1 F 3 F 3 F 4 F 2 F 4 it is straightforward to calculate that F 1 F 2 F 2 F 4 F 1 F 2 F 1 F 3 1 F 3 F 4 F 1 F 3 1 F 1 F 2 F 1 F 3 = y x ( + c k d 1 +d 2 +b k c k d 1 +d 2 + b k + + c k d 1 + b k 2 F 3 F 4 F 2 F 4 2 F 3 F 4 F 1 F 3 ( + + = x c k d 1 + d 2 + b k + c k d 1 + d 2 + b k + c k d 2 + b k 1 F 1 F 2 F 2 F 4 1 F 3 F 4 F 2 F 4 = y x ( + c k d 1 +d 2 +b k c k d 1 +d 2 + b k + + c k d 1 + b k ) c k d 1 +b k = y x c y c y x ) c k d 2 + b k = x c x c y ) c k d 1 +b k = y x c y c y x 2 F 1 F 2 F 1 F 3 2 F 1 F 2 F 2 F 4 ( + + ) = x c k d 1 + d 2 + b k + c k d 1 + d 2 + b k c k d 2 + b k + c k d 2 + b k = x c x c y

7 Matheatics of Operations Research 37(3), pp , 2012 INFORMS 425 If c y x > c y > c x, all private cost differences are negative and is an iproveent cycle; if on the other hand c y x < c y < c x, the 4-cycle in the other direction is an iproveent cycle. Because every strategy cobination is contained in we get the claied result. 4. A characterization for two-player gaes. We analyze iplications of the Extended Monotonicity Lea (Lea 3.4) for two-player weighted congestion gaes. First, for ease of exposition, we restrict ourselves to the case K = 2; that is, we only regard those functions that can be written as an integral linear cobination of two functions in without offset. We define the following set of functions L 2 = c 0 c x = a 1 c 1 x + a 2 c 2 x a 1 a 2 c 1 c 2 L We reark that by setting all offsets b k in (3) equal to zero, the construction in the proof of Lea 3.4 only involves two players. Thus, we iediately obtain the following result. L 2 Proposition 4.1. Let be a set of continuous functions. If is consistent for two-player gaes, then contains only onotonic functions. We proceed to investigate sets of functions that guarantee that L 2 contains only onotonic functions. Lea 4.1. Let c 1 c 2 0 be two continuous, onotonic, and nonconstant functions. Then, the following are equivalent. (i) For all a 1 a 2 the function a 1 c 1 + a 2 c 2 is onotonic on 0. (ii) There are a b such that c 2 x = a c 1 x + b for all x 0. Proof. (ii) (i): Calculus. (i) (ii): Without loss of generality, we ay assue that c 1 and c 2 are nondecreasing because ultiplying functions with 1 has no ipact on either stateent of the lea. As c 1 is nonconstant and nondecreasing, there is x 1 0 with c 1 x 1 = c 1 0 and c 1 x > c 1 0 for all x > x 1. Fix such x > x 1. For all a 1 a 2, the function a 1 c 1 + a 2 c 2 is onotonic. This iplies that for every y > x 1 and every the expressions c 1 x + c 2 x c 1 0 c 2 0 and c 1 y + c 2 y c 1 0 c 2 0 have identical signs. Thus, for all y > x 1 and all at least one of the following two cases holds: (i) c 2 x c 2 0 c 1 x c 1 0 and c 2 y c 2 0 c 1 y c 1 0 ; (ii) c 2 x c 2 0 c 1 x c 1 0 and c 2 y c 2 0 c 1 y c 1 0. This clearly iplies for all y > x 1, because otherwise any rational c 2 y c 2 0 c 1 y c 1 0 = c 2 x c 2 0 c 1 x c 1 0 ( { in c } { 2 y c 2 0 c 1 y c 1 0 c 2 x c 2 0 ax c }) 2 y c 2 0 c 1 x c 1 0 c 1 y c 1 0 c 2 x c 2 0 c 1 x c 1 0 would violate both constraints. Fro (4), we obtain (4) c 2 y = c 2 x c 2 0 c 1 x c 1 0 c 1 y c 2 x c 2 0 c 1 x c 1 0 c c 2 0 for all y > x 1. This shows the existence of a b with c 2 x = a c 1 x + b for all x > x 1. Exchanging the roles of c 1 and c 2, we ay also derive the existence of a b such that c 1 x = a c 2 x + b for all x > x 2, where x 2 is such that c 2 x 2 = c 2 0 and c 2 x > c 2 0 for all x > x 2. This iplies x 1 = x 2. Using the fact that c 1 and c 2 are continuous and constant on 0 x 1 finishes the proof. We are now ready to prove our first ain result.

8 426 Matheatics of Operations Research 37(3), pp , 2012 INFORMS Theore 4.1. Let be a set of continuous functions. Let 2 be the set of two-player gaes such that cost functions are in. Then, the following conditions are equivalent. (i) is consistent for 2. (ii) is FIP-consistent for 2. (iii) contains only onotonic functions and for all nonconstant c 1 c 2, there are constants a b such that c 1 x = a c 2 x + b for all x 0. Proof. (ii) (i) is trivial. (i) (iii): Using Proposition 4.1 we get that L 2 contains only onotonic functions. As L2, this iplies in particular that contains only onotonic functions. For all nonconstant functions c 1 c 2 and all a 1 a 2, the function a 1 c 1 + a 2 c 2 L 2 is onotonic. Applying Lea 4.1 then yields the result. (iii) (ii): Let be as specified in (iii). Trivially, the claied result holds if contains only constant functions. If contains a nonconstant function c, consider the set = a c x + b a b. We show that is consistent for 2. To this end, consider an arbitrary two-player gae with costs in and deands d 1 < d 2. We distinguish the following three cases. First case: c d 1 < c d 2 < c d 1 + d 2, or c d 1 > c d 2 > c d 1 + d 2. Since c is strictly onotonic with respect to the points d 1, d 2, and d 1 + d 2, there is a strictly onotonic function c with c d 1 = c d 1, c d 2 = c d 2, and c d 1 + d 2 = c d 1 + d 2. Consequently, we can replace every cost function c C = a c x + b a b by a cost function c = a c x +b a b without changing the players private costs. As shown by Harks et al. [22], for any strictly onotonic function c, every weighted congestion gae G with two players and cost functions in = a c x + b a b adits a potential function and, thus, has the FIP. Second case: c d 1 = c d 2. We set d 1 = d 2 = 1 and choose for every facility f F a new cost function c f with c f 1 = c f d 1 = c f d 2 and c f 2 = c f d 1 + d 2. By construction, the unweighted congestion gae: with deands d 1, d 2 and costs c f f F has the sae private costs as the original gae. Rosenthal [32] showed the existence of a potential function in all unweighted congestion gaes; thus, the gae has the FIP. Third case: c d 2 = c d 1 + d 2. We have c d 2 = c d 1 + d 2 for all c and thus player 2 is always indifferent whether player 1 shares a facility with her or not. For the FIP and the existence of a PNE, we argue as follows: Consider the vector-valued function S, s 2 s 1 s which assigns to every strategy profile the vector which has the private cost of players 2 and 1 in first and second coponent respectively. We clai that decreases lexicographically along any iproveent path. This is trivial for iproveent oves of player 2. Since player 2 is indifferent whether player 1 shares with her a facility or not, every iproveent ove of player 1 does not affect the private cost of player 2 but decreases the private cost of player 1. This iplies that the lexicographical order of s decreases along any iproveent path; thus, every such path is finite. 5. A characterization for the general case. We now consider the case n 3; that is, we consider weighted congestion gaes with at least three players. We will show that a set of continuous cost functions is consistent if and only if this set contains either only linear or only certain exponential functions. Our ain tool for proving this result is to analyze iplications of the Extended Monotonicity Lea (Lea 3.4) for three-player weighted congestion gaes. Forally, define L 3 = c 0 c x = a 1 c 1 x + a 2 c 1 x + a 1 a 2 c 1 >0 L Note that L 3 involves a single offset > 0, which requires only three players in the construction of the proof of the Extended Monotonicity Lea. However, regarding three-player gaes in which the strategy available to the third player does not intersect with the strategies of the first two players we still get as a necessary condition that L 2 ay only contain onotonic functions. We, thus, obtain the following result. Proposition 5.1. Let be a set of continuous functions. If is consistent for three-player gaes, then both L 2 and L3 contain only onotonic functions. We proceed to characterize the set of cost functions for which L 3 contains only onotonic functions. Lea 5.1. Let be a set of continuous functions. Then, the following two are equivalent: (i) L 3 contains only onotonic functions. (ii) For every c either c x = a e x + b for soe a b, or c x = a x + b for soe a b.

9 Matheatics of Operations Research 37(3), pp , 2012 INFORMS 427 Proof. (ii) (i): Let c be an exponential or an affine function. By siple calculus one can verify that every function c x = a 1 c x + a 2 c x + with a 1 a 2, >0 is exponential if c is exponential and affine if c is affine. (i) (ii): By contradiction, assue that c is neither affine nor exponential. As L 3 contains only onotonic functions, for all > 0 and all a 1 a 2 the function c 0 x a 1 c x + a 2 c x + is onotonic. Referring to Lea 4.1, this iplies that for all > 0 there are a b such that for all x 0: c x + = a c x + b (5) As c is neither affine nor exponential on 0, there ust exist four points 0 p 1 < p 2 < p 3 < p 4 following neither an exponential nor an affine law; i.e. there are neither such that c p i = e p i + for all i nor are there such that c p i = p i + for all i As c is continuous, we ay assue without loss of generality that p 1 p 2 p 3 p 4 are rational and we write the as p 1 = 2 1 / 2k p 4 = 2 4 / 2k for soe k. For = 1/k we derive fro (5) that there are a b such that for all n : Subtracting (6) fro (7) and rearranging ters, we obtain for all n : c n + 1 /k = a c n/k + b (6) c n + 2 /k = a c n + 1 /k + b (7) c n + 2 /k a + 1 c n + 1 /k + a c n/k = 0 (8) This defines a second-order linear hoogeneous recurrence relation on the sequence c n/k n. Such recurrence relations can be solved with the ethod of characteristic equations; see Balakrishnan [7, 3.2] for ore details. The characteristic equation of the recurrence relation equals x 2 a + 1 x + a = x 1 x a. If a 1, then the characteristic equation has two distinct roots and we obtain for even that c /k = 1 + a = + a = exp ln a + for soe constants. If on the other hand a = 1, we can evaluate c /k as c /k = = + for soe constants. We are now ready to state our second ain theore. Theore 5.1. Let be a set of continuous functions. Then, the following three are equivalent: (i) is consistent. (ii) is FIP-consistent. (iii) contains only affine functions or contains only functions of type c x = a c e x +b c, where a c b c ay depend on c while is independent of c. Proof. (ii) (i) is trivial. (iii) (ii) follows because every weighted congestion gaes with such cost functions possesses a weighted potential; see Fotakis et al. [16], Harks et al. [22], and Panagopoulou and Spirakis [30]. (i) (iii): By Proposition 5.1 both L 2 and L3 ay only contain onotonic functions. Applying Lea 5.1 we obtain that every c is either affine or exponential. In addition, as shown in Lea 4.1 for each two nonconstant functions c 1 c 2, there are a b such that c 2 x = a c 1 x + b for all x 0. Both results together iply (iii). We conclude this section by giving an exaple that illustrates the ain ideas presented so far. Recall, that Theore 5.1 establishes that for each continuous, nonaffine and nonexponential cost function c, there is a weighted congestion gae G with unifor cost function c on all facilities that does not adit a PNE. In the following exaple, we show how such a gae for c x = x 3 is constructed. Exaple 5.1. As the function c x = x 3 is neither affine nor exponential, there are a 1 a 2 and >0 such that c x = a 1 c x + a 2 c x + has a strict local extreu. In fact, we can choose a 1 = 2 a 2 = 1, and = 1, that is, the function c x = 2c x c x + 1 = 2x 3 x has a strict local iniu at x 0 = In particular, we can choose d 1 = 1 and d 2 = 2 such that c d 1 = 6 > c d 2 = 11 < c d 1 + d 2 = 10 The weighted congestion gae without PNE is now constructed as follows: We introduce 2 a 1 + a 2 facilities f 1 f 6 and the following strategies s1 a = f 1 f 2 f 3, s1 b = f 4 f 5 f 6, s2 a = f 1 f 2 f 4, s2 b = f 3 f 5 f 6, and

10 428 Matheatics of Operations Research 37(3), pp , 2012 INFORMS (a) Strategies (b) Iproveent cycle s 2 a f 1 f 4 f 2 s 3 (s 1 a, s 2 a, s 3 ) = (62, 2 81, 35) (s 1 b, s 2 a, s 3 ) = (66, 2 80, 65) s 2 b f 3 s 1 a f 5 f 6 s 1 b (s 1 a, s 2 a, s 3 ) = (66, 2 80, 65) (s 1 b, s 2 b, s 3 ) = (62, 2 81, 35) Figure 2. (a) The players strategies and (b) the iproveent cycle of the gae constructed in Exaple 5.1 that does not adit a PNE. s 3 = f 3 f 4. We then set S 1 = s a 1 sb 1, S 2 = s a 2 sb 2, and S 3 = s 3 ; see Figure 2(a) for an illustration of the strategies. The so-defined gae has four strategy profiles, naely s a 1 sa 2 s 3, s a 1 sb 2 s 3, s b 1 sa 2 s 3, s b 1 sb 2 s 3. As Player 3 is an offset player, she has a single strategy only; thus, the players private costs depend only on the choice of players 1 and 2. We derive that the 4-cycle shown in Figure 2(b) is a best-reply cycle in G. As there are no strategy profiles outside we conclude that G has no PNE. 6. Weighted network congestion gaes. In this section, we discuss the iplications of our characterizations to the iportant subclass of weighted network congestion gaes. In these gaes, the facilities correspond to edges of a directed or undirected graph. Every player is associated with a positive deand that she wants to route fro her origin to her destination on a path of iniu cost. We consider directed and undirected networks separately, starting with directed networks Directed networks. We first give a version of the Extended Monotonicity Lea for directed networks with two players and strictly positive costs. Lea 6.1 (Extended Monotonicity Lea for Two-Player Gaes on Directed Networks). Let be a set of strictly positive and continuous functions. If is consistent for two-player directed network congestion gaes, then L 2 contains only onotonic functions. Proof. Because singleton congestion gaes are a subclass of directed network congestion gaes, by Corollary 3.1 every set of consistent functions contains only onotonic functions. For a contradiction, assue that there are a 1 a 2 and onotonic functions c 1 c 2 such that the function c 0 defined as c x = a 1 c 1 x + a 2 c 2 x is not onotonic. By Lea 3.2 there are x y >0 with y > x such that either c y x < c y < c x or c y x > c y > c x. We choose the deands equal to d 1 = y x and d 2 = x. Note that c is onotonic if and only if c is onotonic; thus we ay assue w.l.o.g. that a 2 > 0. To define the players strategies we distinguish the following two cases. (a) Case a 1 < 0 s 2 P 1 P 2 v 1 v 2 v 3 v 4 (b) Case a 1 > 0 P 1 s 1 t 1 s t P 3 P 4 v 5 v 6 v 7 v 8 P 2 t 2 Figure 3. Directed network congestion gaes used in the proof of the Extended Monotonicity Lea for Two-Player Directed Networks (Lea 6.1).

11 Matheatics of Operations Research 37(3), pp , 2012 INFORMS 429 First case: a 1 < 0: We use a construction siilar to the proof of Lea 3.4. To define the players strategy spaces, consider the network in Figure 3. The two players are represented by the two source-terinal pairs s i t i i = 1 2. The set of strategies available to player i equals the set of directed s i t i -paths. The dashed edges in Figure 3(a) correspond to directed paths P 1 P 4, which we choose as follows: the directed path P 1 fro v 1 to v 2 contains a 1 edges with cost function c 1, the directed path P 2 fro v 3 to v 4 contains a 2 edges with cost function c 2, the directed path P 3 fro v 5 to v 6 contains a 2 edges with cost function c 2, and the directed path P 4 fro v 7 to v 8 contains a 1 edges with cost function c 1. All other edges have an arbitrary cost function in, say c 1. Because all costs are strictly positive, for player 1 all strategies except the upper path P u = s 1 v 1 P 1 v 2 v 3 P 2 v 4 t 1 and the lower path P d = s 1 v 5 P 3 v 6 v 7 P 4 v 8 t 1 are strictly doinated in the sense that they have strictly higher costs than either P u or P d regardless of the strategy played by player 2. For player 2, all strategies except the left path P l = s 2 v 1 P 1 v 2 v 5 P 3 v 6 t 2 and the right path P r = s 2 v 3 P 2 v 4 v 7 P 4 v 8 t 2 are strictly doinated. We consider the 4-cycle = ( P u P l P d P l P d P r P u P r P u P l ), and calculate that 1 P d P l 1 P u P l = y x c 1 y x + a 2 c 2 y + c 1 y x a 1 c 1 y x + c 1 y x c 1 y x + a 1 c 1 y c 1 y x a 2 c 2 y x c 1 y x = y x a 1 c 1 y + a 2 c 2 y a 1 c 1 y x a 2 c 2 y x = y x c y c y x In the sae fashion, we obtain 2 P d P r 2 P d P l = x c x c y, 1 P u P r 1 P d P r = y x c y c y x, and 2 P u P l 2 P u P r = x c x c y. If c y x > c y > c x, then is an iproveent cycle which gives that none of the strategy profiles contained in is a PNE. If, on the other hand, c y x < c y < c x, we can reverse the direction of and get an iproveent cycle. Because every strategy profile that uses only nondoinated strategies is contained in, the constructed directed network congestion gae does not adit a PNE. Second case: a 1 > 0: Consider the network shown in Figure 3(b). Here, both players want to route fro s to t; that is, S 1 = S 2 = P 1 P 2. The directed paths P 1 and P 2 each contain a 1 edges with cost function c 1 and a 2 edges with cost function c 2. If c y x < c y < c x, player 1 prefers to be alone on an s t -path while player 2 wants to share the path with player 1. If c y x > c y > c x, the arguentation works the other way round. We conclude that the gae does not adit a PNE. Together with Lea 4.1 and Theore 4.1, we obtain the following characterization of consistency for two-player network congestion gaes on directed networks. Theore 6.1. Let be a set of strictly positive and continuous functions and let 2 dn be the set of twoplayer directed network gaes such that cost functions are in. Then, the following conditions are equivalent. (i) is consistent for 2 dn. (ii) is FIP-consistent for 2 dn. (iii) contains only onotonic functions and for all nonconstant c 1 c 2, there are constants a b with c 1 x = a c 2 x + b for all x 0. Using siilar ideas as in the case of two players, we can also prove a version of the Extended Monotonicity Lea for directed network gaes with three or ore players. Lea 6.2 (Extended Monotonicity Lea for Directed Networks). Let be a set of strictly positive and continuous functions. If is consistent for three-player directed network congestion gaes, then L 3 contains only onotonic functions. Proof. Assue by contradiction that there are a 1 a 2, >0 and a onotonic function c 1 such that the function c 0 defined as c x = a 1 c 1 x + a 2 c 1 x + is not onotonic. We ay again assue w.l.o.g. that c 1 is onotonic and that a 2 > 0. Note that because c 1 is onotonic, this iplies a 1 < 0. Consider the network in Figure 4 where again the directed paths P 1 and P 4 contain a 1 edges each, and the directed paths P 2 and P 3 contains a 2 edges each. In addition to the players i = 1 2 corresponding to the pairs s i t i i = 1 2, we now have a third player corresponding to the pair s 3 t 3 with a single strategy P Q = P 3 Q P 2 and deand d 3 =. Moreover, we set d 1 = y x, d 2 = x for x y >0 with y > x such that either c y x < c y < c x or c y x > c y > c x holds (by Lea 3.2 such values exist). We design the directed path Q fro v 6 to v 3 so as to contain a sufficiently large nuber of edges, such that for players 1 and 2, all s i t i -paths not containing Q are strictly less costly than every path that contains Q. As every

12 430 Matheatics of Operations Research 37(3), pp , 2012 INFORMS s 2 P v 1 P 2 1 v 2 v 3 t 3 Q s 1 t 1 P s 3 P 3 v 6 v 4 7 v 8 Figure 4. Multi-coodity directed network instance used in the proof of the Extended Monotonicity Lea for Directed Networks (Lea 6.2). s i t i -path that does not contain Q has costs less than 2 a 2 a c 1 y + and every edge in Q has cost at least c 1, it is sufficient to let Q contain 2 a 2 a c 1 y + /c edges. By construction of Q, for player 1, all strategies except the upper path P u = s 1 v 1 P 1 v 2 v 3 P 2 t 3 t 1 and the lower path P d = s 1 s 3 P 3 v 6 v 7 P 4 v 8 t 1 are strictly doinated in the sense that they have strictly higher costs than either P u or P d regardless of the strategies played by players 2 and 3. For player 2, all strategies except the left path P l = s 2 v 1 P 1 v 2 s 3 P 3 v 6 t 2 and the right path P r = s 2 v 3 P 2 t 3 v 7 P 4 v 8 t 2 are strictly doinated. With the sae calculations as in Lea 6.1 one can show that the 4-cycle = P u P l P Q P d P l P Q P d P r P Q P u P r P Q P u P l P Q is an iproveent cycle when traversed in the right direction. Because every strategy profile that uses only nondoinated strategies is contained in, we conclude that the thus constructed network congestion gae does not adit a PNE. Using Lea 5.1, we obtain the following characterization of cost functions that are consistent for weighted directed network congestion gaes. Theore 6.2. Let be a set of strictly positive and continuous functions and let dn be the set of directed network congestion gaes such that cost functions are in. Then, the following are equivalent: (i) is consistent for dn. (ii) is FIP-consistent for dn. (iii) contains only affine functions or contains only functions of type c x = a c e x +b c where a c b c ay depend on c while is independent of c. This characterization is even valid for three-player gaes. Reark 6.1. In gaes with negative costs the players strive to establish long paths. In this case, our construction does not work since, e.g., player 2 prefers to take the detour v 6 v 7 v 8 t 2 instead of the edge v 6 t Undirected networks. We first show that a version of the Extended Monotonicity Lea holds also for two-player gaes on undirected networks. In such a gae, we are given an undirected graph and for each player i, two designated vertices s i and t i. Facilities correspond to the edges of the graph and the strategy set of each player i contains all siple s i t i -paths. Each edge can be traversed in any direction and its cost depends on the aggregated flow. Lea 6.3 (Extended Monotonicity Lea for Two-Player Gaes on Undirected Networks). Let be a set of strictly positive and continuous functions. If is consistent for two-player undirected network congestion gaes, then L 2 contains only onotonic functions. Proof. For a contradiction, let a 1 a 2 and c 1 c 2 be such that the function c 0 defined as c x = a 1 c 1 x + a 2 c 2 x is not onotonic and w.l.o.g. a 2 > 0. Moreover, let x y >0 with y > x be such that either c y x < c y < c x or c y x > c y > c x holds. We set d 1 = y x, d 2 = x and distinguish the following two cases. If a 1 < 0 we consider the network in Figure 5 where the paths P 1 and P 4 each contain a 1 edges with cost function c 1 and the paths P 2 and P 3 each contain a 2 edges with cost function c 2. With siilar calculations as in the proof of Lea 6.1 one can verify that the 4-cycle = P 1 P 2 P 1 P 3 t 2