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Coalition Formation and Price of Anarchy in Cournot Oligopolies Joint work with: Nicole Immorlica (Northwestern University) Georgios Piliouras (Georgia Tech) Vangelis Markakis Athens University of Economics and Business

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2 Motivation and goals Some degree of cooperation is often allowed or even encouraged in various games Price of anarchy can be reduced if players are allowed to form coalition structures [Hayrapetyan et al ’06, Fotakis et al. ’06]: Static models for congestion games (coalition structure exogenously forced) Dynamic models? Inefficiency of stable partitions w.r.t the dynamics?

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3 Outline Cournot games Nash equilibria and price of anarchy Coalition Formation in Cournot games A model for dynamic coalition formation Stable partitions Quantifying inefficiency of stable partitions

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4 Cournot Oligopolies [Cournot 1838] Games among firms producing/offering the same (or a similar) product

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5 Linear and symmetric Cournot games n firms producing the same product Strategy space: R + (quantity that the firm will produce) Cost of producing per unit: c Given a strategy profile q = (q 1, q 2,…,q n ): Price of the product: depends linearly on Q = Σq i p(Q) = a – b Q Payoff to agent i: u i = q i p(Q) - cq i

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6 Linear and symmetric Cournot games Cournot games have a unique Nash equilibrium where: q i = q * = (a - c)/b(n+1) p(Q) = (a + nc)/(n+1) u i = (a – c) 2 /b(n+1) 2 Total welfare of the agents can be very low: [Harberger ’54] (empirical observations) [Guo, Yang ’05, Kluberg, Perakis ’08] (theoretical analysis) PoA = (n)

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7 Cooperation in Cournot games In practice, competition among firms is not exactly a non-cooperative game Suppose firms are allowed to partition themselves into coalition structures S1S1 S2S2 S3S3 S4S4 S5S5 S6S6

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8 Cooperation in Cournot games Definition (the static case): Given a fixed partitioning Π = (S 1,…,S k ), the Cournot super-game consists of k super-players Strategy space of superplayer: product space of its players Utility of superplayer: sum of utilities of its players Lemma: In all Nash equilibria of the super-game: Social welfare is the same Payoff of a superplayer is the payoff of a firm in a k-player Cournot game

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9 Cooperation in Cournot games Are all partitions equally likely to arise? What if players are allowed to join/abandon existing coalitions? Inefficiency of stable partitions? (stable w.r.t. allowed moves)

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10 A coalition formation game Given a current partition Π = (S 1,…,S k ) At an equilibrium of the super-game, a player j S i considers his current payoff to be u(S i )/| S i | We allow 3 types of moves from Π Type 1: A group of existing coalitions merge

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11 A coalition formation game Type 2: A subset S of an existing coalition S i, abandons S i and forms a separate coalition. Left over coalition S i \S dissolves SiSi S

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12 A coalition formation game Type 3: A strict subset S of an existing coalition S i can leave and join another existing coalition S j. Left over coalition S i \S dissolves SiSi SjSj S

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13 Inefficiency of stable partitions Definition: A partition is stable if there is no move that strictly increases the payoff of all deviators PoA := max. inefficiency of a stable partition Theorem: PoA = Θ(n 2/5 ) Note: constants independent of supply-demand curves (i.e. of game parameters, a, b, c)

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14 Proof sketch of upper bound Lemma 1: For stable partitions with k coalitions PoA = O(k) Because equilibria of super-game have same welfare as the equilibium of a k-player Cournot game Need upper bound on size of stable partitions For Π = (S 1,…,S k ), let k 1 = # singleton coalitions k 2 = # non-singleton coalitions S1S1 S2S2 S3S3 S4S4 S5S5

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15 Proof sketch of upper bound Proposition (characterization): A partition Π = (S 1,…,S k ) is stable iff k 1 (k 2 +1) 2 For each non-singleton S i, |S i | k 2 suffices to solve a non-linear program

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16 Proof sketch of upper bound PoA = Solving PoA n 2/5

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17 Proof of lower bound By (almost) tightening the inequalities of the math. program For any integer N, let n:= 4N 4/5 N 1/5 + N 2/5 We need k 1 = N 2/5 singletons And k 2 = N 1/5 coalitions of size 4N 4/5 k = k 1 + k 2 = Ω (n 2/5 ) Lemma 2: The resulting partition is stable

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18 Other behavioral assumptions So far we assumed partitions reach a Nash equilibrium of the super-game Theorem: Same result holds when super-players of a partition employ no-regret algorithms. No-regret converges to Nash utility of each superplayer

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19 Future work Apply the same to other classes of games Routing games, socially concave games Need to ensure the super-game has a well-defined payoff for the super- players Need to define how players split the superplayer’s payoff Other models of coalition formation Thank you!

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