Load Balancing Congestion Games and their Asymptotic Behavior

Transcription

1 Load Balancing Congestion Games and their Asymptotic Behavior Eitan Altman, Corinne Touati Inria CRS, LIG, Univ. Grenole Alpes, LIG, F Grenole, France {eitan.altman, arxiv: v1 [cs.gt] 31 Dec 2015 Astract A central question in routing games has een to estalish conditions for the uniqueness of the equilirium, either in terms of network topology or in terms of costs. This question is well understood in two classes of routing games. The first is the non-atomic routing introduced y Wardrop on 1952 in the context of road traffic in which each player (car) is infinitesimally small; a single car has a negligile impact on the congestion. Each car wishes to minimize its expected delay. Under aritrary topology, such games are known to have a convex potential and thus a unique equilirium. The second framework is splitale atomic games: there are finitely many players, each controlling the route of a population of individuals (let them e cars in road traffic or packets in the communication networks). In this paper, we study two other frameworks of routing games in which each of several players has an integer numer of connections (which are population of packets) to route and where there is a constraint that a connection cannot e split. Through a particular game with a simple three link topology, we identify various novel and surprising properties of games within these frameworks. We show in particular that equiliria are non unique even in the potential game setting of Rosenthal with strictly convex link costs. We further show that non-symmetric equiliria arise in symmetric networks. I. ITRODUCTIO A central question in routing games has een to estalish conditions for the uniqueness of the equiliria, either in terms of the network topology or in terms of the costs. A survey on these issues is given in [1]. The question of uniqueness of equiliria has een studied in two different frameworks. The first, which we call F1, is the non-atomic routing introduced y Wardrop on 1952 in the context of road traffic in which each player (car) is infinitesimally small; a single car has a negligile impact on the congestion. Each car wishes to minimize its expected delay. Under aritrary topology, such games are known to have a convex potential and thus have a unique equilirium [2]. The second framework, denoted y F2, is splitale atomic games. There are finitely many players, each controlling the route of a population of individuals. This type of games have already een studied in the context of road traffic y Haurie and Marcotte [3] ut have ecome central in the telecom community to model routing decisions of Internet Service Providers that can decide how to split the traffic of their suscriers among various routes so as to minimize network congestion [4]. In this paper we study properties of equiliria in two other frameworks of routing games which exhiit surprising ehavior. The first, which we call F3, known as congestion games [5], consists of atomic players with non splitale traffic: each player has to decide on the path to e followed y for its traffic and cannot split the traffic among various paths. This is a non-splitale framework. We further introduce a new semisplitale framework, denoted y F4, in which each of several players has an integer numer of connections to route. It can choose different routes for different connections ut there is a constraint that the traffic of a connection cannot e split. In the case where each player controls the route of a single connection and all connections have the same size, this reduces to the congestion game of Rosenthal [5]. We consider in this paper routing games with additive costs (i.e. the cost of a path equals to the sum of costs of the links over the path) and the cost of a link is assumed to e convex increasing in the total flow in the link. The main goal of this paper is to study a particular symmetric game of this type in a simple topology consisting of three nodes and three links. We focus oth on the uniqueness issue as well as on other properties of the equiliria. This game has already een studied within the two frameworks F1-F2 that we mentioned aove. In oth frameworks it was shown [6] to have a unique equilirium. Our first finding is that in frameworks F3 and F4 there is a multitude of equiliria. The price of staility is thus different than the price of anarchy and we compute oth. We show the uniqueness of the equilirium in the limit as the numer of players grows to infinity extending known results [3] from framework F2 to the new frameworks. In framework F2 uniqueness is in fact achieved not only for the limiting games ut also for all large enough. We show that this is not the case for F3-F4: for any finite there may e several equiliria. We finally show a surprising property of F4 that exhiits non symmetric equiliria in our symmetric network example while under F1, F2 and F3 there are no asymmetric equiliria. The structure of the paper is as follows. We first introduce the model and the notations used in the while study, we then move on to the properties of frameworks F3 (Section III) and F4 (Section IV) efore concluding the paper. For completeness, we also include in the Appendix the proofs of the theorems and propositions of the paper although they will e removed from the final manuscript so as to comply with the conference regulations for final manuscript ut will e made availale on ArXiv.

2 II. MODEL AD OTATIOS We shall use throughout the term atomic game to denote situations in which decisions of a player have an impact on other players utility. It is non-atomic when players are infinitesimally small and are viewed like a fluid of players, such that a single player has a negligile impact on the utility of other players. We consider a system of three nodes (A, B and C) with two incoming traffic sources (respectively from node A and B) and an exit node C. There are a total of connections originating from each one of the sources. Each connection can either e sent directly to node C or rerouted via the remaining node. The system is illustrated in Figure 1. A Fig. 1. C B Physical System This model has een used to model load alancing issues in computer networks, see [6] and references therein. Jos arrive to two computing centers represented y nodes A and B. A jo can e processed locally at the node where it arrives or it may e forwarded to the other node incurring further communication delay. The costs of links [AC] and [BC] represent the processing delays of jos processed at nodes A and B respectively. Once processed, the jos leave the system. A connection is a collection of jos with similar characteristics (e.g. elonging to the same application). We introduce the following notations: A link etween two nodes, say A and B, is denoted y [AB]. Our considered system has three links [AB], [BC] and [AC]. A route is simply referred y a sequence of nodes. Hence, the system has four connections: two originating from node A (route AC and ABC) and two originating from node B (route BC and BAC). Further, in the following, n AC, n BC, n ABC and n BAC will refer to the numer of connections routed via the different routes while n[ac], n[bc] and n[ab] will refer to the numer of connections on each susequent link. By conservation law, we have: n AC +n ABC = n BC +n BAC = n[ac] = n AC +n BAC, and n[bc] = n ABC +n BC, n[ab] = n BAC +n ABC. For each route r, we also define the fraction (among ) of flow using it, i.e. f r = n r /. The conservation law ecomes f AC +f ABC = f BC +f BAC = 1. Finally, the performance measure considered in this work is the cost (delay) of connections experienced on their route. We consider a simple model in which the cost is additive (i.e. the cost of a connection on a route is simply taken as the sum of delays experienced y the connection over the links that constitute this route). We further assume that the costs on each link are linear with coefficient a/ on link [AB] and coefficient / on link [AC] and [BC], i.e. C [AB] = a n[ab] = a(f BAC +f ABC ), C [AC] = n[ac] = (f BAC +f AC ), C [BC] = n[bc] = (f BC +f ABC ). and then: C AB = C [AB], C ABC = C [AB] +C [BC], C BC = C [BC], C BAC = C [AB] +C [AC]. We restrict our study to the (pure) ash equiliria and give the equiliria in terms of the corresponding flows marked y a star. By conservation law, the equiliria is uniquely determined y the specification of f ABC and f BAC (or equivalently n ABC and n BAC ). We recall that in this paper, we consider two types of decision models. In the first (F3), the decision is taken at the connection level (Section III), i.e. each connection has its own decision maker that seeks to minimize the connection s cost, and the connection cannot e split into different routes. In the second (F4), (Section IV) each one of the two source nodes decides on the routing of all the connections originating there. Each connection of a given source node (either A or B) can e routed independently ut a connection cannot e split into different route. We hence refer to F4 this semi-splitale framework. ote that the two-approaches (F3 and F4) coincide when there is only = 1 connection at each source, which we also detail later. III. ATOMIC O-SPLITABLE CASE AD ITS O-ATOMIC LIMIT (F3 FRAMEWORK) We consider here the case where each connection elongs to an individual user acting selfishly. We first show that for fixed parameters, the game may have several equiliria, all of which are symmetric for any numer of players. The numer of distinct equiliria can e made aritrary large y an appropriate choice of the parameters a and, and for any choice of a and, there exists 0 such that the numer of equiliria remain constant for all 0. We then show properties of the limiting game otained as the numer of of players increases to infinity. A. on-uniqueness of the equilirium Theorem 1. The set of pure ash equiliria of the game are the points satisfying n BAC = n ABC 2a.

3 Proof: Consider an equilirium(n ABC,n BAC ). Then, we have the following conditions: C [AC] = C AC (C [AB] +a/)+(c [BC] +/) C [BC] = C BC (C [AB] +a/)+(c [AC] +/) (1) C [AB] +C [BC] = C ABC C [AC] +/ C [AB] +C [AC] = C BAC C [BC] +/ ote that the last two equations lead to: { C[AB] C [BC] +C [AC] +/ C [AB] C [AC] +C [BC] +/ One can check that (n ABC,n BAC ) = (0,0) is a solution. If the equilirium is not the trivial null solution, then either n ABC 0 or n BAC 0. Either way leads to C [AB] > 0 and thus / < C [AC] C [BC] < / which implies that C [AC] = C [BC]. Equation 1 ecomes: { 0 a(n ABC +1+n BAC )+ a(n ABC +n BAC ) a(n ABC +n BAC) But then: C [AC] = C [BC] (n AC + n BAC ) = (n BC + n ABC ) n ABC + n BAC = n BAC + n ABC n ABC = n BAC. Therefore the equilirium is symmetrical. Jointly with a(n ABC +n BAC ), this leads to the conclusion. Corollary 2. For 0 = 2a, there exists exactly /2a+ 1 ash equiliria in pure strategies. B. The potential and asymptotic uniqueness When the numer of players grows to infinity, the limiting game ecomes a non-atomic game with a potential [7] F (f ABC,f BAC ) = (f ABC f BAC ) 2 + a 2 (f ABC +f BAC ) 2. Indeed, recall that the potential g is unique up to an additive constant and that it satisfies g def = C AC = (f AC +f BAC ) f AC g def g def f BC g def = C ABC = a(f ABC +f BAC )+(f ABC +f BC ) = C BC = (f BC +f ABC ) = C BAC = a(f ABC +f BAC )+(f BAC +f AC ). One can check that the function g(f AC,f ABC,f BC,f BAC ) = a 2 (f ABC +f BAC ) ((f AC +f BAC ) 2 +(f BC +f ABC ) 2 ) readily satisfies these conditions. Then g can e rewritten as g(f ABC,f BAC ) = a 2 (f ABC +f BAC ) (1+(f ABC f BAC ) 2 ). As the potential is unique up to an additive constant, we consider F = g.id/2. Proposition 3. The non-atomic game has a unique ash equilirium, which is f ABC = f BAC = 0. Proof: ote that: F = a(f ABC +f BAC )+2(f ABC f BAC )) F = a(f ABC +f BAC )+2(f BAC f ABC )) Hence, the potential is twice differentiale with Hessian matrix ( ) a+2 a 2. a 2 a+2 This Hessian is definite positive and hence the potential is (strictly) convex. Therefore it has a unique minimum, which is the only ash equilirium of the game. Finally, note that f ABC (0,1), f BAC (0,1), F (f ABC,f BAC ) 0 and that F (0,0) = 0, which concludes the proof. To show the uniqueness of the equilirium in the limiting game, we made use of the fact that the limiting game has a potential which is convex. Yet, not only the limiting game has a convex potential, ut also the original one, as we conclude from next theorem, whose proof is a direct application of [5]. Theorem 4. For any finite numer of players, the game is a potential game [8] with the potential function: F(f ABC,f BAC ) = (f ABC f BAC ) 2 + a 2 (f ABC +f BAC )(f ABC +f BAC +1/). (2) Proof: Consider a connection following route ABC. Its cost is a(f ABC + f BAC ) + (f ABC + 1 f BAC ). If this connection switches its strategy to route AC, then its cost ecomes(1 f ABC +f BAC +1/). Therefore the associated change of cost is = a(f ABC +f BAC )+(f ABC +1 f BAC ) (1 f ABC +f BAC +1/) = a(f ABC +f BAC )+(2f ABC 2f BAC 1/). ow: 1 (F(f ABC,f BAC ) F(f ABC 1/,f BAC )) = [ (f ABC f BAC ) 2 (f ABC 1/ f BAC ) 2] + a 2 [(f ABC +f BAC )(f ABC +f BAC +1/) (f ABC +f BAC 1/)(f ABC +f BAC )] = (2f ABC 2f BAC 1/)+ a (f ABC +f BAC ) = /. By symmetry, the same argument holds for a connection originating from source B. ote that unlike the framework of non-atomic games, the fact that the game has a convex potential does not imply uniqueness. The reason for that is that in congestion games, the action space over which the potential is minimized is not a convex set (due to the non-splitale nature) so that it may have several local minima, each corresponding to another equilirium, whereas a for a convex function over the Euclidean space, there is a unique local minimum which is also a gloal minimum of the function (and thus an equilirium of the game).

4 C. Efficiency Theorem 5. In the non-atomic setting, the only ash equilirium is also the social optimum (i.e. the point minimizing the sum of costs of all players) of the system. Proof: The sum of costs of all players is f ABC C ABC +f AC C AC +f BAC C BAC +f BC C BC = a(f ABC +f BAC ) 2 +((f BC +f ABC ) 2 +(f AC +f BAC ) 2 ) = a(f ABC +f BAC ) 2 +2(1+(f ABC f BAC ) 2 ). The minimum is hence otained for (f ABC,f BAC ) = (0,0). Since the game possesses several equiliria, we can expect the PoA (Price of Anarchy - the largest ratio etween the sum of costs at an equilirium and the sum of costs at the social optimum) and PoS (Price of Staility - the smallest corresponding ratio) to e different. Theorem 6. The price of staility of the game is 1 and the price of anarchy is 1+ 2a 2. Proof: From Eq. 3 the price of anarchy (resp. staility) is y definition the maximum (resp. minimum) value over the ash equiliria of: a(f ABC +f BAC )2 +2(1+(f ABC f BAC )2 ) 2 Then, from Theorem 1: and (2p/) ap 2 / 2 + PoA = max = max p /2a 2 p /2a = 2a(/2a)2 + = 2a2 4a = 2a 2 +1 a(2p/) ap 2 / 2 PoS = min = min +1 = 1. p /2a 2 p /2a We make the following oservations: (i) In the splitale atomic games studied in [6] the PoA was shown to e greater than one for sufficiently small numer of players (smaller than some threshold), and was 1 for all large enough numer of players (larger than the same threshold). Here for any numer of players, the PoS is 1 and the PoA is greater than 1. (ii) The PoA decreases in and tends to 1 as tends to infinity, the case of splitale games. (iii) We have shown that the PoA is unounded: for any real value K and any numer of players one can choose the cost parameters a and so that the PoA exceeds K. This corresponds to what was oserved in splitale games [6] and contrast with the non-atomic setting of single commodity flows (i.e. when there is only one source node instead of two), and aritrary topology networks where the PoA equals 4/3 [9]. (3) IV. ATOMIC SEMI-SPLITABLE CASE AD ITS SPLITABLE LIMIT (F4 FRAMEWORK) The game can e expressed as a 2-player matrix game where each player (i.e. each source node A and B) has +1 possile actions, for each of the +1 possile values of f ABC and f BAC respectively. The utility for player A is U A (f ABC,f BAC ) = f AC C AC +f ABC C ABC = f ABC +f BAC +(a 2)f ABC f BAC +(a+2)f 2 ABC Similarly, for player B: ote that and Hence U B (f ABC,f BAC ) = f BC C BC +f BAC C BAC = f BAC +f ABC +(a 2)f BAC f ABC +(a+2)f 2 BAC U A = +(a 2)f BAC +2(a+2)f ABC U B = +(a 2)f ABC +2(a+2)f BAC. 2 U A f 2 ABC = 2(a + 2) = 2 U B f 2 BAC (4) (5). Therefore, oth u A : f ABC U A (f ABC,f BAC ) and u B : f BAC U B (f ABC,f BAC ) are (strictly) convex functions. This means that for each action of one player, there would e a unique est response to the second player if its action space was the interval (0,1). Hence, for the limit case (when ), the est response is unique. In contrast, for any finite value of, there are either 1 or 2 possile est responses which are the discrete optima of functions u A : f ABC U A (f ABC,f BAC ) and u B : f BAC U B (f ABC,f BAC ). We will however show that in the finite case, there may e up to 2 2 = 4 ash equiliria while in the limit case the equilirium is always unique. A. Efficiency ote that the total cost of the players is Σ(f ABC,f BAC ) = U A (f ABC,f BAC )+U B (f ABC,f BAC ) = 2+2(a 2)f ABC f BAC +(a+2)(f 2 ABC +f2 BAC ) = 2+a(f ABC +f BAC ) 2 +2(f ABC f BAC ) 2 2. Further, note that Σ = 2(F + ). Hence Σ is strictly convex. Also Σ(0,0) = 2. Therefore (0,0) is the (unique) social optimum of the system. Yet, for sufficiently large (that is, as soon as we add enough flexiility in the players strategies), this is not a ash equilirium, as stated in the following theorem: Theorem 7. The point (f ABC,f BAC ) = (0,0) is a ash equilirium if and only if a +2. Proof: By symmetry and as u A : f ABC U A (f ABC,f BAC ) is convex, then (0,0) is a ash equilirium

5 iff U A (0,0) U A (1/,0) = / +(a+2)/ 2 which leads to the conclusion. Also, we can ound the total cost y: Σ(f ABC,f BAC ) = = 2+2(a 2)f ABC f BAC +(a+2)(f 2 ABC +f2 BAC ) 2+(a 2)(f 2 ABC +f2 BAC )+(a+2)(f2 ABC +f2 BAC ) 2+2a(f 2 ABC +f2 BAC ) 2+4a This ound is attained at Σ(1,1) = 2+2(a 2)+2(a+ 2) = 4a+2. Yet, it is not otained at the ash equilirium for sufficiently large values of : Theorem 8. (1,1) is a ash equilirium if and only if 2+a 3a+. Proof: We have U A (1,1) = +2a and U A (1 1/,1) = 2a+ 3a/ / +2/ 2 +a/ 2. Therefore U A (1 1/,1) U A (1,1) 2+a (3a+). The conclusion follows y convexity. Therefore, for max( a 2+a +2, 3a+ ) the ash equiliria are neither optimal nor worse-case strategies of the game. B. Case of = 1 In case of = 1 (one flow arrives at each source node and there are thus two players) the two approach coincides: the atomic non-splitale case (F3) is also a semi-splitale atomic game (F4). f ABC and f BAC take values in {{0},{1}}. From Eq. 4 and Eq. 5, the matrix game can e written ( ) (,) (2,a+2) (a+2,2) (2a+,2a+) and the potential of Eq. 2 ecomes ( 0 a+ a+ 3a ). Then, assuming that either a or is non null, we get that (0,0) is always a ash equilirium and that (1,1) is a ash equilirium if and only if 3a a+, i.e. 2a <. We next consider any integer and identify another surprising feature of the equilirium. We show that depending on the sign of a 2, non-symmetric equiliria arise in our symmetric game. In all frameworks other than the semi-splitale games there are only symmetric equiliria in this game. We shall show however that in the limit (as grows to infinity), the limiting game has a single equilirium. C. Case a 2 < 0 In this case, there may e multiple equiliria, as shown in the following example. Example 9. Consider a = 1, = 3 and = 4, then the cost matrices are given elow, with the two ash equiliria of the game represented in old letters: U A = 1 16 U B = , and. ote that due to the shape of U A and U B the cost matrices of the game are transpose of each other. Therefore in the following, we shall only give matrix U A. We have the following theorem: Theorem 10. All ash equiliria are symmetrical, i.e. f ABC = f BAC. The proof is given in Appendix A. D. Case a = 2 (with a > 0) When a = 2, we shall show that some non-symmetrical equiliria exists. Theorem 11. If a = 2, there are exactly either 1 or 4 ash equiliria. For any, let = 8. If mod 8 = 4, there are 4 equiliria (n ABC,n BAC ), which are (,), ( + 1,), (, + 1) and ( + 1, +1). Otherwise, there is a unique equilirium, which is(, ) if mod 8 < 4 or ( +1, +1) if mod 8 > 4. Proof: The ash equiliria are the optimal points for oth u A and u B. They are therefore either interior or oundary points (i.e. either f ABC or f BAC are in 0,1). We detail the interior point cases in Appendix B. The rest of the proof derives directly from the definition of Indeed: U A and U B. U A = (a 2)f BAC +2(2+a)f ABC = 8f ABC U B = (a 2)f ABC +2(a+2)f BAC = 8f BAC. Both are minimum for 1/8. Therefore, it is attained if is a multiple of 8. Otherwise, the est response of each player is either 1 8 if mod 8 3 or 1 8 if mod 8 5. If mod 8 = 4, then each player has 2 est responses which are and Then, one can check that the oundary points follow the law of Theorem 15 when = 8 = 0.

6 E. Case a 2 > 0 Theorem 12. If a 2 > 0, there are exactly either 1, 2 or 3 ash equiliria. Let α = a+2 3a+2, β = 2a 3a+2 and γ = 3a+2. Define further Ñ = γ and z() = γ Ñ. The equiliria are of the form Either (Ñ,Ñ), (Ñ +1,Ñ), (Ñ,Ñ +1) if is such that z() = α (mode 3-A in Figure 2) Or (Ñ +1,Ñ +1), (Ñ +1,Ñ), (Ñ,Ñ +1) if is such that z() = β (mode 3-B) Or (Ñ,Ñ +1), (Ñ +1,Ñ) if is such that α < z() < β (mode 2) Or (Ñ,Ñ) if is such that β < z() < α+1 (mode 1). Mode 3-A Mode 3-B Mode 3-A Mode 3-B Mode 2 Mode 1 Mode 2 α β α+1 β +1 Fig. 2. Different modes according to different values of. We illustrate the different modes in the following example. Example 13. Suppose that a = 10 and = 3 (we represent only the part of the matrices corresponding to 1/ f ABC,f BAC 4/). If = 24, there are 3 ash equiliria: If = 26, there are 2 ash equiliria: If = 27, there are 3 ash equiliria: If = 28, there is a single ash equilirium: F. Limit Case: Perfectly Splitale Sessions We focus here in the limit case where +. Theorem 14. There exists a unique ash equilirium and it is such that fbac = f ABC = 3a+2. U A U B Proof: ote that (1) > 0 and (1) > 0. If f ABC = 0 then f BAC = 2a+4 which implies that + (a 2) 2a+4 0, which further implies that a 6 > 0 which is impossile. Hence f ABC > 0. Similarly f BAC > 0 which concludes the proof. Recall that the optimum sum (social optimum) is given y (0,0) and that the worse case is given y (1,1). Hence, regardless of the values ofaand, at the limit case, we oserve that there is a unique ash equilirium, that is symmetrical, and is neither optimal (as opposed to F3), nor the worst case scenario. The price of anarchy is then: PoA = PoS = 2+2f 2 ABC a a = 1+ 2 (3a+2) 2. V. COCLUSIOS We revisited in this paper a load alancing prolem within a non-cooperative routing game framework. This model had already received much attention in the past within some classical frameworks (the Wardrop equilirium analysis and the atomic splitale routing game framework). We studied this game under other frameworks - the non splitale atomic game (known as congestion game) as well as a the semi-splitale framework. We have identified many surprising features of equiliria in oth frameworks. We showed that unlike the previously studied frameworks, there is no uniqueness of equilirium, and non-symmetric equiliria may appear (depending on the parameters). For each of the frameworks we identified the different equiliria and provided some of their properties. We also provided an efficiency analysis in terms of price of anarchy and price of staility. In the future we plan to investigate more general cost structures and topologies. REFERECES [1]. Shimkin, A survey of uniquenes results for selfish routing, in Proc. of the International Conference on etwork Control and Optimization (etcoop), L.. in Computer Science 4465, Ed., 2007, pp. pp [2] M. Beckmann, C. McGuire, and C. Winsten, Studies in the Economics of Transportation. ew Haven: Yale University Press, [3] A. Haurie and P. Marcotte, On the relationship etween ash-cournot and Wardrop equiliria, etworks, vol. 15, no. 3, [4] A. Orda, R. Rom, and. Shimkin, Competitive routing in multiuser communication networks, IEEE/ACM Trans. etw., vol. 1, no. 5, pp , Oct [5] R. W. Rosenthal, A class of games possessing pure-strategy ash equiliria, International Journal of Game Theory, vol. 2, pp , [6] E. Altman, H. Kameda,, and Y. Hosokawa, ash equiliria in load alancing in distriuted computer systems, International Game Theory Review (IGTR), vol. 4, no. 2, pp , June [7] W. H. Sandholm, Potential games with continuous player sets, Journal of Economic Theory, vol. 97, no. 1, pp , [8] D. Monderer and L. S. Shapley, Potential games, Games and economic ehavior, vol. 14, no. 1, pp , 1996.

7 [9] T. Roughgarden, Selfish routing and the price of anarchy. MIT Press, APPEDIX A. Proof of Theorem 10. Suppose that (fabc,f BAC ) is a ash equilirium with fabc f BAC. Then, y definition: U A (fabc,f BAC ) U A(fBAC,f BAC ) and U B (fabc,f BAC ) U B(fABC,f ABC ), which gives, after some manipulations, (a 2)fABC f BAC 2afBAC 2 +f ABC f BAC (a+2)fabc 2 (a 2)fABC f BAC 2afABC 2 +f BAC f ABC (a+2)f 2 BAC. Therefore 2(a 2)fABCf BAC (a 2)(fABC+f 2 BAC) 2 and hence 0 (a 2)(fABC f BAC )2 which is impossile. B. Boundary equiliria when a = 2. Theorem 15. If a = 2, there exists a single ash equilirium of the form (0,fBAC ) and (f BAC,0) with f BAC non null. It is otained for = 4 and fbac = 1/4. The points (0,0) are ash equiliria if and only if 4. Further, there are no equilirium of the form (f ABC,1) or (1,f BAC ). Proof: We first study the equiliria of the form(0,f ABC ). (0,γ) is a ash equilirium iff ( ) 1 U A (0,γ) U A,γ 2+a ( U B (0,γ) U B 0,γ + 1 ) ( U B (0,γ) U B 0,γ 1 ) (a+2)(2γ + 1 ) (a+2)(2γ 1 ) (2γ + 1 ) 1 4(2γ 1/) { 4 /8 1/2 γ /8+1/2 If 3 then /8 + 1/2 7/8 < 1 which cannot e otained y the player otherwise than in 0. For = 4, the second inequality ecomes 0 γ 1 4 which hence leads to the only non null ash equilirium. We next study the potential equiliria of the form (f ABC,1). Let (γ,1) e a ash equilirium. Then U B (γ,1) U B (γ,1 1/). Then γ +a+2 (1 1/)+γ +(a+2)(1 1/) 2 a+2 / +(a+2)(1+1/ 2 2/) 0 +(a+2)(1/ 2) 2a+3 (a+2)/ 1/4. C. Boundary equiliria when a 2 > 0. Theorem 16. (0,α) and (α,0) are ash equiliria iff: a 2 1 a+2 a 2 α 2(a+2) Further, there are no ash equilirium of the form (A,1). Proof: We first focus on the ash equiliria of the form (0,A). SinceU A (.,f BAC ) andu B (f ABC,.) are convex,(0,γ) is a ash equilirium iff ( ) 1 U A (0,γ) U A (,γ U B (0,γ) U B 0,γ + 1 ) ( U B (0,γ) U B 0,γ 1 ) (a 2)γ + 2+a 2 a γ (a 2) (a+2)(2γ + 1 a 2 ) γ 2(a+2) (a+2)(2γ 1 ) +a+2 γ 2(a+2) 2 a But (a 2) a 2 2(a+2) which concludes the proof. and hence a 2 2(a+2) γ +a+2 2(a+2) We now study the potential equiliria of the form(a, 1). Let (A,1) e a ash equilirium. Then U B (A,1) U B (A,1 1/). Then +(a 2)A+(a+2) (1 1/) +(a 2)A(1 1/)+(a+2)(1 1/) 2 0 (a 2)A+(a+2)( 2+1/) (a 2)A 2a 3+(a+2)/ 2a+3 (a 2)A+2a+3 (a+2)/ But 2a+3 (a+2)/ a+2 2a+3 < 1. D. Proof of Theorem 12. We first start y showing that there are at most 4 interior ash equiliria and that they are of the form: (A,A),(A + 1,A),(A,A+1),(A+1,A+1). Proof: Let f ABC,f BAC e a ash equilirium in the interior (i.e. 0 < f ABC < 1 and 0 < f BAC < 1). Then f ABC and f BAC are the (discrete) minimizers of x U A (x,f BAC ) and x U B (f ABC,x) respectively. Further: U A = +(a 2)f BAC +2(2+a)f ABC U B = +(a 2)f ABC +2(a+2)f BAC The optimum values are therefore respectively: x A = θf BAC and x B = θf ABC

8 with = 2(2+a) and θ = a 2. Therefore: x A 1 2 f ABC x A x B 1 2 f BAC x B Hence θ ( θ f ABC + 1 ) f ABC θ ( θ f ABC 1 ) 2 Then +θ Then 2 ( θ) f ABC +θ = 2+3a, 2 ( θ) = 2(a+2), which gives 2 (6+a) 2 ( θ) + +θ 2 ( θ) = 4+2a 2 (6+a) and 2+3a a+2 (6+a) f ABC 2+a (6+a) + 2+3a. Similarly, we have 2+3a (2+a) (6+a) f BAC 2+3a + 2+a (6+a). 1 ote that 6+a < 1. Therefore there are either 1 or 2 possile values, which are identical for f ABC and f BAC. There are therefore 4 possile equiliria. ow, the potential equiliria are of the form(a, A), (A, A+ 1), (A + 1,A) and (A + 1,A + 1). By symmetry, note that if (A,A+1) is a ash equilirium, then (A+1,A) also is. The following lemma reduces the numer of cominations of equiliria: 2 < 2+a Lemma 17. If(A, A) is a ash equilirium then(a+1, A+1) is not a ash equilirium. Proof: Suppose that (A,A) and (A+1,A+1) are two ash equiliria. Then U A (A,A) U A (A+1,A) and U A (A+ 1,A+1) U A (A,A+1), which implies A +(a 2)A 2 +(2+a)A 2 (A+1) +(a 2)A(A+1)+(2+a)(A+1) 2 (A+1) +(a 2)(A+1) 2 +(2+a)(A+1) 2 A +(a 2)A(A+1)+(2+a)A 2 { (a 2)A+(2+a)(2A+1) (a 2)(A+1)+(2+a)(2A+1) (a 2)(A+1) (2+a)(2A+1) (a 2)A Hence (a 2)(A+1) (a 2)A and therefore a 2 0 which is impossile. Therefore the different possile cominations are mode 1, mode 2, mode 3-A and mode 3-B in Figure 2). We first start y the occurrence of mode 3-A: Lemma 18. Suppose that a 2 > 0. Suppose that (A,A) and (A+1,A) are two ash equiliria. Then A = 2 a. 3a+2 Proof: Suppose that (A,A) and (A+1,A) are two ash equiliria. Then necessarilyu A (A,A) = U A (A+1,A). Hence i.e. A +(a 2)A 2 +(2+a)A 2 = (A+1) +(a 2)A(A+1)+(2+a)(A+1) 2 = (a 2)A+(2+a)(2A+1) 2 a = (3a+2)A which leads to the conclusion. Hence, the system is in mode 3-A iff 2 a is divisile y 3a+2 or in other words, if is of the form (3a+2)K+2a for some integer K. We then move on to Mode 3-B: Lemma 19. Suppose that a 2 > 0. Suppose that (A + 1,A+1) and (A+1,A) are two ash equiliria. Then A = 2a 3a+2. Proof: Suppose that (A+1,A+1) and (A,A +1) are two ash equiliria, then U 1 (A+1,A+1) = U 1 (A,A+1). This implies (A+1)+(a 2)(A+1) 2 +(2+a)(A+1) 2 = A+(a 2)A(A+1)+(2+a)A 2 (a 2)(A+1)+(2+a)(2A+1) = (3a+2)A = 2a which concludes the proof. Hence, the system is in mode 3-B iff 2a is divisile y 3a + 2 or in other words, if is of the form (3a+2)K +2+a for some integer K. Finally, for Mode 2: Lemma 20. Suppose that a 2 > 0. Suppose that (A,A+1) and (A+1,A) are only two ash equiliria. Then (3a+2)A+2+a < < (3a+2)A+2a. Proof: Suppose that (A,A+1) and (A+1,A) are two ash equiliria, then: ie U A (A,A+1) U A (A+1,A+1) and U A (A+1,A) U A (A,A) { (3a+2)A+2a (3a+2)A+2+a The conclusion comes from Lemma 18 and 19, since neither (A,A) nor (A+1,A+1) are ash equiliria. Finally the system is in mode 1 if it is not in any over modes. One can then check that the oundary cases found in Theorem 16 corresponds to the case where A = 0 which concludes the proof.