Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence

Dynamics of Profit-Sharing

John Augustine, Ning Chen, Edith Elkind, Angelo Fanelli, Nick Gravin, Dmitry Shiryaev School of Physical and Mathematical Sciences Nanyang Technological University, Singapore

Abstract information about other players. Such agents may simply re- spond to their current environment without worrying about An important task in the analysis of multiagent sys- the subsequent reaction of other agents; such behavior is said tems is to understand how groups of selfish players to be myopic. Now, coalition formation by computationally can form coalitions, i.e., work together in teams. In limited agents has been studied by a number of researchers in this paper, we study the dynamics of coalition for- multi-agent systems, starting with the work of [Shehory and mation under . We consider set- Kraus, 1999] and [Sandholm and Lesser, 1997]. However, tings where each team’s profit is given by a concave myopic behavior in coalition formation received relatively lit- function, and propose three profit-sharing schemes, tle attention in the literature (for some exceptions, see [Dieck- each of which is based on the concept of marginal mann and Schwalbe, 2002; Chalkiadakis and Boutilier, 2004; . The agents are assumed to be myopic, i.e., Airiau and Sen, 2009]). In contrast, myopic dynamics of they keep changing teams as long as they can in- non-cooperative games is the subject of a growing body of crease their payoff by doing so. We study the prop- research (see, e.g. [Fabrikant et al., 2004; Awerbuch et al., erties (such as closeness to or to- 2008; Fanelli et al., 2008]). tal profit) of the states that result after a polyno- mial number of such moves, and prove bounds on In this paper, we merge these streams of research and ap- the of anarchy and the price of stability of the ply techniques developed in the context of analyzing the dy- corresponding games. namics of non-cooperative games to coalition formation set- tings. In doing so, we depart from the standard model of games with , which allows the players in 1 Introduction a team to share the payoff arbitrarily: indeed, such flexibility and collaborative task execution are fundamen- will necessitate a complicated negotiation process whenever a tally important both for human societies and for multiagent player wants to switch teams. Instead, we consider three pay- systems. Indeed, many tasks are too complicated or resource- off models that are based on the concept of , consuming to be executed by a single agent, and a collective i.e., the contribution that the player makes to his current team. effort is needed. Such settings are usually modeled using the Each of the payoff schemes, when combined with a coopera- framework of cooperative games, which specify the amount tive , induces a non-cooperative game, whose dynamics of payoff that each subset of agents can achieve: when the can then be studied using the rich set of tools developed for game is played the agents split into teams (coalitions), and such games in recent years. the payoff of each team is divided among its members. We will now describe our payment schemes in more detail. The standard framework of cooperative is We assume that we are given a concave cooperative game, static, i.e., it does not explain how the players arrive at a i.e., the values of the teams are given by a submodular func- particular set of teams and a payoff . However, tion; the submodularity property means that a player is more understanding the dynamics of coalition formation is impor- useful when he joins a smaller team, and plays an important tant for many applications of cooperative games, and game role in our analysis. In our first scheme, the payment to each theorists have studied bargaining and coalition formation in agent is given by his marginal utility for his current team; by cooperative environments (see, e.g. [Chatterjee et al., 1993; submodularity, the total payment to the team members never Moldovanu and Winter, 1995; Yan, 2003]). Much of this exceeds the team’s value. This payment scheme rewards each research assumes that the agents are fully rational, i.e., can agent according to the value he creates; we will therefore call predict the consequences of their actions and maximize their these games Fair Value games. Our second scheme takes into (expected) utility based on these predictions. However, full account the history of the interaction: we keep track of the or- rationality is a strong assumption that is unlikely to hold in der in which the players have joined their teams, and pay each many real-life scenarios: first, the agents may not have the agent his marginal contribution to the coalition formed by the computational resources to infer their optimal strategies, and players who joined his current team before him. This ensures second, they may not be sophisticated enough to do so, or lack that the entire payoff of each team is fully distributed among

37  its members. Moreover, due to the submodularity property S by changing the of player i from si to si, i.e.,   a player’s payoff never goes down as long as he stays with (S−i,si)=(s1,s2,...,si−1,si,si+1,...,sn). the same team. This payoff scheme is somewhat reminiscent Nash Equilibria and Dynamics Given a strategy profile of the reward schemes employed in industries with strong la-  S =(s1,s2,...,sn), a strategy si ∈ Σi is an improvement bor unions; we will therefore refer to these games as Labor   move for player i if ui(S−i,si) >ui(S); further, si is called Union games. Our third scheme can be viewed as a hybrid  α i ui(S−i,s ) > (1 + α)ui(S) of the first two: it distributes the team’s payoff according to an -improvement move for if i , α> sb ∈ the players’ Shapley values, i.e., it pays each player his ex- where 0. A strategy i Σi is a for player i S pected marginal contribution to a coalition formed by its pre- in state if it yields the maximum possible payoff given u S ,sb ≥ decessors when players are reordered randomly; the resulting the strategy choices of the other players, i.e., i( −i i ) u S ,s s ∈ α games are called Shapley games. i( −i i) for any i Σi.An -best response move is both an α-improvement and a best response move. Our Contributions We study the equilibria and dynamics A (pure) Nash equilibrium is a strategy profile in which of the three games described above. We establish that all our every player plays her best response. Formally, S = games admit a Nash equilibrium. Further, we argue that in (s1,s2,...,sn) is a Nash equilibrium if for all i ∈ N and all these games the total profit of the system in a Nash equi-   for any strategy s ∈ Σi we have ui(S) ≥ ui(S−i,s ).We librium, i.e., the sum of teams’ values, is within a factor of 2 i i denote the set of all (pure) Nash equilibria of a game G by from optimal. In addition, for the first two classes of games, NE(G). A profile S =(s ,...,sn) is called an α-Nash equi- a natural dynamic process quickly converges to a state that is 1 librium if no player can improve his payoff by more than a almost as good as a Nash equilibrium, and the best Nash equi-  factor of (1+α) by deviating, i.e., (1+α)ui(S) ≥ ui(S−i,si) librium is in fact an optimal state. We also show that Labor  for any i ∈ N and any ui ∈ Σi. The set of all α-Nash equilib- Union games have other desirable properties. α ria of G is denoted by NE (G).Inastrong Nash equilibrium, Related Work The games studied in this paper belong to no group of players can improve their payoffs by deviating, the class of potential games, introduced by Monderer and i.e., S =(s1,...,sn) is a if for all Shapley [1996]. In potential games, any sequence of im- I ⊆ N S s ,...,s and any strategy vector =(1 n) such that   provements by players converges to a pure Nash equilibrium. si = si for i ∈ N \ I,ifui(S ) >ui(S) for some i ∈ I, then  However, the number of steps can be exponential in the de- uj(S )

38 Potential Games A non-cooperative game G is called a 4 Profit-Sharing Games if there is a function Φ:Σ→ N such that  We will now describe three non-cooperative games that for any state S and any improvement move si of a player i in  can be constructed from any monotone concave coopera- S we have Φ(S−i,s ) − Φ(S) > 0; the function Φ is called i tive game. Each of our games can be described by a triple the potential function of G. The game G is called an exact   G =(N,v,M), where (N,v) is a monotone concave coop- potential game if we have Φ(S−i,s )−Φ(S)=ui(S−i,s )− i i erative game with N = {1,...,n}, and M = {1,...,m} is ui(S). It is known that any potential game has a pure Nash a set of m parties; we require m ≤ n. All three games con- equilibrium [Monderer and Shapley, 1996; Rosenthal, 1973]. sidered in this section model the setting where the players in Cooperative Games A cooperative game G =(N,v) is N form a coalition structure over N that consists of m coali- given by a set of players N and a characteristic function v : tions. Thus, each player needs to choose exactly one party N + 2 → R ∪{0} that for each set I ⊆ N specifies the profit from M, i.e., for each i ∈ N we have Σi = M. In some that the players in I can earn by working together. We assume cases (see Section 4.2), we also allow players to be unaffili- that v(∅)=0.Acoalition structure over N is a partition of ated. To model this, we expand the set of strategies by setting players in N, i.e., a collection of sets I1,...,Ik such that (i) Σi = M ∪{0}. Intuitively, the parties correspond to different Ii ⊆ N for i =1,...,k; (ii) Ii∩Ij = ∅ for all i0 the basic Nash dynamic converges to a state S with 4.1 Fair Value Games G   f(SF ) ≥ OPTf ( ) (1 − ) n ln 1 β in at most β  steps, starting In our first model, the utility ui(S) of a player i in a state from any initial state. S =(Q1,...,Qm) is given by i’s marginal contribution to

39 the coalition he belongs to, i.e., if i ∈ Qj, we set ui(S)= in party j, ordered according to their arrival time. As be- vj(S) − vj(S \{i}). As this payment scheme rewards each fore, the profit of party j is given by the function vj; note player according to the value he creates, we will refer to that the value of vj does not depend on the order in which this type of games as Fair Value games. Observe that since the players join j. The payoff of each player, however, is the functions vj are assumed to be submodular, we have dependent on their position in the affiliation order. Specifi- u S ≤ v Q j ∈ M i ∈ P P i i∈Qj i( ) j( j) for all , i.e., the total pay- cally, for a player j, let j( ) be the set of players that ment to the employees of a company never exceeds the profit appear in Pj before i. Player i’s payoff is then defined as of the company. Moreover, it may be the case that the profit ui(P)=vj(Pj(i) ∪{i}) − vj(Pj(i)). of a company is strictly greater than the amount it pays to its We remark that, technically speaking, Labor Union games employees; we can think of the difference between the two are not non-cooperative games. Rather, each state of a La- quantities as the owner’s/shareholders’ value. Consequently, bor Union game induces a non-cooperative game as described in these games the total profit of all parties may differ from above; after any player makes a move, the induced non- the social welfare, as defined in Section 2. cooperative game changes. Abusing terminology, we will say We will now argue that Fair Value games have a number of that a state P of a Labor Union game G is a Nash equilibrium desirable properties. In particular, any such game is a poten- if for each player i ∈ N staying with his current party is a tial game, and therefore has a pure Nash equilibrium. best response in the induced game; all other notions that were G defined for non-cooperative games in Section 2, as well as the Theorem 2. Every Fair Value game is a perfect 2-nice ex- results in Section 3, can be extended to Labor Union games act potential game w.r.t. the total profit function. in a similar manner. Combining Theorem 2, Lemmas 1 and 2 and Theorem 1, We now state two fundamental properties of our model.  we obtain the following corollaries. • Guaranteed payoff: Consider two players i and i in Pj. G α ≥ Suppose i moves to another party. The payoff of player Corollary 1. For every Fair Value game and every 0  we have PoAα(G) ≤ 2+α. In particular, PoA(G) ≤ 2. i will not decrease. Indeed, if i succeeds i in the se- quence Pj, then by definition, i’s payoff is unchanged. If G >0  Corollary 2. For every Fair Value game and any , the i precedes i in Pj, then, since vj is non-decreasing and SF basic Nash dynamic converges to a state  with total profit submodular, i’s payoff will not decrease; it may, how- SF ≥ OPT(G) −  n 1 tp( ) 2 (1 ) in at most 2 ln  steps, ever, increase. Corollary 2 generalizes to basic α-Nash dynamic; we omit • Full payoff distribution: The sum of the payoffs of the exact statement of this result due to space constraints. players within a party j is a telescopic sum that eval- Since every Fair Value game is an exact potential game uates to vj(Pj). Therefore, the total profit tp(P)= with the potential function given by the total profit, any profit- j∈M vj(Pj) in a state P equals to the social wel- maximizing state is necessarily a Nash equilibrium. This im- fare in this state. In other words, in Labor Union plies the following proposition. games, the profit of each enterprise is distributed among Proposition 1. PoS(G)=1for any Fair Value game G. its employees, without creating any value for the own- ers/shareholders. 4.2 Labor Union Games The guaranteed payoff property distinguishes the Labor In Fair Value games, the player’s payoff only depends on his Union games from the Fair Value games, where a player who current marginal value to the enterprise, i.e., one’s salary may maintains his affiliation to a party might not be rewarded, but go down as the company expands. However, in many real-life may rather see a reduction in his payoff as other players move settings, this is not the case. For instance, in many industries, to join his party. This, of course, may incentivize him to shift especially ones that are highly unionized, an employee that his affiliation as well, leading to a vicious cycle of moves. In has spent many years working for the company typically re- contrast, in Labor Union games, a player is guaranteed that ceives a higher salary than a new hire with the same set of his payoff will not decrease if he maintains his affiliation to a skills. Our second class of games, which we will refer to party. This suggests that in Labor Union games stability may as Labor Union games, aims to model this type of settings. be easier to achieve. In what follows, we will see that this is Specifically, in this class of games, we modify the notion of indeed the case. state so as to take into account the order in which the play- We will first show that Labor Union games are perfect 2- ers have joined their respective parties; the payment to each nice with respect to the total profit (or, equivalently, social welfare); this will allow us to apply the machinery devel- player is then determined by his marginal utility for the coali- P tion formed by his predecessors. The submodularity property oped in Section 3. Abusing notation, let Δi( ) denote the improvement in the payoff of player i if he performs a best guarantees than a player’s payoff never goes down as long as P P P he stays with the same party. response move from , and let Δ( )= i∈N Δi( ). Formally, in a Labor Union game G that corresponds to a Proposition 2. Any Labor Union game G is a perfect 2-nice tuple (N,v1,...,vm,M), we allow the players to be unaffil- game with respect to the total profit function. iated, i.e., for each i ∈ N we set Σi = M ∪{0}. If player As in the case of Fair Value games, Proposition 2 allows us to i plays strategy 0, we set his payoff to be 0 irrespective of bound the price of anarchy of any Labor Union game, and the the other players’ strategies. A state of G is given by a tu- time needed to converge to a state with “good” total profit; ple P =(P1,...,Pm), where Pj is the sequence of players again, Corollary 4 generalizes to basic α-Nash dynamic.

40  Corollary 3. For every Labor Union game G and every α ≥ the party. This immediately implies i∈Q ui(S)=vj(Qj). α j 0 we have PoA (G) ≤ 2+α. In particular, PoA(G) ≤ 2. Thus, Shapley games share features with both the Fair Value Corollary 4. For every Labor Union game G and any >0, games and the Labor Union games. Like Fair Value games, the basic Nash dynamic converges to a state SF with total the order in which the players join the party is unimportant. G   profit tp(SF ) ≥ OPT( ) (1 − ) in at most n ln 1 steps, Moreover, if all payoff functions are additive, i.e., we have 2 2  u S ∪{j} − u S u {j} i ∈ N from any initial state. i( ) i( )= i( ) for any and any S ⊆ N \{i}, then the respective Shapley game coincides with O G O ,...,O Let ( )=( 1 m) be a state that maximizes the the Fair Value game that corresponds to (N,v ,...,vm,M). G G O G 1 total profit in a game , and let OPT( )=tp( ( )).Asin On the other hand, similarly to the Labor Union games, the O G the case of Fair Value games, it is not hard to see that ( ) entire profit of each party is distributed among its members. G is a Nash equilibrium, i.e., PoS( )=1. In fact, in Labor We will first show that any Shapley game is an exact potential Union games, no coalition of players can profitably deviate game and hence admits a Nash equilibrium in pure strategies. from O(G) regardless of the order in which they deviate: by G N,v ,...,v ,M the guaranteed payoff property, any such deviation would not Theorem 3. Any Shapley game =( 1 m ),is harm the non-deviators and thus would lead to a state whose an exact potential game with the potential function given by O G   total profit exceeds that of ( ), a contradiction. This im- (|Q|−1)!(|Qj|−|Q|)! plies the following result. Φ(S)= vj(Q). |Qj|! Proposition 3. In any Labor Union game G, O(G) is a strong j∈M Q⊆Qj Nash equilibrium, i.e., the strong price of stability is 1. Just like in other profit-sharing games, the price of anarchy Moreover, for Labor Union games under certain dynamics in Shapley games is bounded by 2. and certain initial states one can guarantee convergence to α- Theorem 4. In any Shapley game G =(N,v1,...,vm,M) Nash equilibrium or even Nash equilibrium. 1 with |N| = n, we have PoA(G) ≤ 2 − n . G Proposition 4. Consider any Labor Union game = The following claim shows that the bound given in Theo- N,v ,...,v ,M v I ≥ j ∈ M ( 1 m ) such that j( ) 1 for any rem 4 is almost tight. and any I ∈ 2N \ {∅}. For any such G, the α-Nash dy- Proposition 6. For any n ≥ 3, there exists a Shapley game namic starting from any state in which all players are affili- G N,v ,v ,M |N| n |M| α =( 1 2 ) with = and =2such that ated with some party converges to an -Nash equilibrium in G − 2 G − 4 O n log W W PoA( )=2 n+1 and PoS( )=2 n+1 . ( log(1+α) ) steps, where is the maximum payoff that any player can achieve. 5 Cut Games and Profit Sharing Games Proof. After each move in the α-Nash dynamic, a player im- We will now describe a family of succinctly representable proves her payoff by a factor of 1+α, and the guaranteed profit-sharing games that can be described in terms of undi- payoff property ensures that payoffs of other players are un- rected weighted graphs. It turns out that while two well- affected. So, if a player starts with a payoff of at least 1, she studied classes of games on such graphs do not induce profit- W O log W will reach a payoff of after ( log(1+α) ) steps. Therefore, sharing games, a “hybrid” approach does. n W in O( log ) steps, we are guaranteed to reach an α-Nash In the classic cut games [Schaffer¨ and Yannakakis, 1991], log(1+α) players are the vertices of a weighted graph G =(N,E). equilibrium. The state of the game is a partition of players into two par- Proposition 5. Suppose a Labor Union game G with n play- ties, and the payoff of each player is the sum of the weights ers starts at a state in which every player is unaffiliated. Then, of cut edges that are incident on him. A cut game naturally in exactly n steps of the Nash dynamic, the system will reach corresponds to a coalitional game with the set of players N, a Nash equilibrium. where the value of a coalition S ⊆ N equals to the weight of the cut induced by S and N \ S. However, this game is 4.3 Shapley Games not monotone, so it does not induce a profit-sharing game, as In our third class of games, which we call Shapley games, defined in Section 4. In induced subgraph games [Deng and the players’ payoffs are determined in a way that is inspired Papadimitriou, 1994], the value of a coalition S equals to the by the definition of the [Shapley, 1953]. Like total weight of all edges that have both endpoints in S; while in Fair Value games, a state of a Shapley game is fully de- these games are monotone, they are not concave. scribed by the partition of the players into parties. Given a Finally, consider a game where the value of a coalition S ⊆ N state S =(Q1,...,Qm) and a player i ∈ Qj, we define equals the total weight of all edges incident on ver- player i’s payoff as tices in S, i.e., both internal edges of S (as in induced sub-  graph games) and the edges leaving S (as in cut games). It is |Q|!(|Qj|−|Q|−1)! ui(S)= (vj(Q∪{i})−vj(Q)). not hard to see that this game is both monotone and concave, |Qj|! Q⊆Q \{i} and hence induces a profit-sharing game as described in Sec- j tion 4. We will now explain how to compute players’ payoffs Intuitively, the payment to each player can be viewed as his in the corresponding Fair Value games, Labor Union games average payment in the Labor Union model, where the av- and Shapley games. Consider a state of the game with two erage is taken over all possible orderings of the players in parties S and N \ S and a player i ∈ S. Let A (resp., B)

41 denote the total weight of edges incident on i that connect i [Awerbuch et al., 2008] Awerbuch, B., Azar, Y., Epstein, A., Mir- to a predecessor (resp., successor) within S, and let C be the rokni, V. S., and Skopalik, A. (2008). Fast convergence to nearly total weight of the cut edges incident on i. Then i’s payoff optimal solutions in potential games. In EC, pages 264–273. can be computed as follows: [Bhalgat et al., 2010] Bhalgat, A., Chakraborty, T., and Khanna, S. A B (2010). Approximating pure Nash equilibrium in cut, party affil- i + + C Fair Value Games: The payoff of is given by 2 . iation, and satisfiability games. In EC, pages 73–82. Intuitively, an edge between two players represents a shared skill, and i’s unique skills within a coalition are [Chalkiadakis and Boutilier, 2004] Chalkiadakis, G. and Boutilier, C. (2004). Bayesian reinforcement learning for coalition forma- weighted more toward his payoff than his shared skills. tion under uncertainty. In AAMAS, pages 1090–1097. i B C Labor Union Games: The payoff of is given by + . [Chatterjee et al., 1993] Chatterjee, K., Dutta, B., Ray, D., and Sen- Intuitively, i’s payoff reflects the unique skills that i pos- gupta, K. (1993). A noncooperative theory of coalitional bargain- sessed when he joined the party. Players who share skills ing. Review of Economic Studies, 60(2):463–77. i i with , but join after , will not get any payoff for those [Chien and Sinclair, 2007] Chien, S. and Sinclair, A. (2007). Con- shared skills. vergence to approximate Nash equilibria in congestion games. In i A+B C SODA, pages 169–178. Shapley Games: The payoff of is given by 2 + , just as in Fair Value games. [Deng and Papadimitriou, 1994] Deng, X. and Papadimitriou, C. H. (1994). On the complexity of cooperative solution concepts. One can see that this interpretation easily extends to multiple Math. Oper. Res., 19:257–266. parties and hyperedges. We also note that many of the notions [ ] that we have discussed are naturally meaningful in this variant Dieckmann and Schwalbe, 2002 Dieckmann, T. and Schwalbe, m U. (2002). Dynamic coalition formation and the . Journal of the cut game: for instance, an optimal state for =2is a of Economic Behavior and Organization, 49:363–380. configuration in which the weighted cut size is maximized. [Fabrikant et al., 2004] Fabrikant, A., Papadimitriou, C. H., and Talwar, K. (2004). The complexity of pure Nash equilibria. In 6 Conclusions and Future Work STOC, pages 604–612. We have studied the dynamics of coalition formation un- [Fanelli et al., 2008] Fanelli, A., Flammini, M., and Moscardelli, L. der marginal contribution-based profit division schemes. We (2008). The speed of convergence in congestion games under have introduced three classes of non-cooperative games that best-response dynamics. In ICALP (1), pages 796–807. can be constructed from any concave cooperative game. We [Gairing and Savani, 2010] Gairing, M. and Savani, R. (2010). have shown that all three profit distribution schemes consid- Computing stable outcomes in hedonic games. In SAGT, pages ered in this paper have desirable properties: all three games 174–185. admit a Nash equilibrium, and even the worst Nash equilib- [Gairing and Savani, 2011] Gairing, M. and Savani, R. (2011). rium is within a factor of 2 from the optimal configuration. Computing stable outcomes in hedonic games with voting-based In addition, for Fair Value games and Labor Union games a deviations. In AAMAS, to appear. natural dynamic process quickly converges to a state with a [Moldovanu and Winter, 1995] Moldovanu, B. and Winter, E. fairly high total profit. Thus, when rules for sharing the pay- (1995). Order independent equilibria. Games and Economic Be- off are fixed in advance, we can expect a system composed havior, 9(1):21–34. of bounded-rational selfish players to quickly converge to an [Monderer and Shapley, 1996] Monderer, D. and Shapley, L. S. acceptable set of teams. (1996). Potential games. Games and Economic Behavior, Of course, the picture given by our results is far from com- 14(1):124 – 143. plete; rather, our work should be seen as a first step towards [Rosenthal, 1973] Rosenthal, R. W. (1973). A class of games pos- understanding the behavior of myopic selfish agents in coali- sessing pure-strategy Nash equilibria. International Journal of tion formation settings. In particular, our results seem to sug- Game Theory, 2:65–67. gest that keeping track of the history of the game and dis- [Sandholm and Lesser, 1997] Sandholm, T. and Lesser, V. (1997). tributing payoffs in a way that respects players “seniority” Coalitions among computationally bounded agents. Artif. Intell., leads to better stability properties; it would be interesting to 94:99–137. see if this observation is true in practice, and whether it gen- [Schaffer¨ and Yannakakis, 1991] Schaffer,¨ A. A. and Yannakakis, eralizes to other settings, such as congestion games. M. (1991). Simple local search problems that are hard to solve. Acknowledgements: this work was supported by a NAP SIAM J. Comput., 20(1):56–87. grant, NRF (Singapore) under grant NRF RF2009-08, and A- [Shapley, 1953] Shapley, L. S. (1953). A value for n-person games. STAR SINGA graduate fellowship. Contributions to the theory of games, 2:307–317. [Shehory and Kraus, 1999] Shehory, O. and Kraus, S. (1999). Fea- References sible formation of coalitions among autonomous agents in non- superadditve environments. Comp. Intell., 15:218–251. [Ackermann et al., 2008] Ackermann, H., Roglin,¨ H., and Vocking,¨ [Skopalik and Vocking,¨ 2008] Skopalik, A. and Vocking,¨ B. B. (2008). On the impact of combinatorial structure on conges- (2008). Inapproximability of pure Nash equilibria. In STOC, tion games. J. ACM, 55(6). pages 355–364. [ ] Airiau and Sen, 2009 Airiau, S. and Sen, S. (2009). A fair payoff [Yan, 2003] Yan, H. (2003). Noncooperative selection of the core. distribution for myopic rational agents. In AAMAS, pages 1305– International Journal of Game Theory, 31:527–540. 1306.

42