Coalition Formation and Price of Anarchy in Cournot Oligopolies

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

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?

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

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

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 = (q1, q2,…,qn): • Price of the product: depends linearly on Q = Σqi p(Q) = a – b Q • Payoff to agent i: ui = qi p(Q) - cqi

6. Linear and symmetric Cournot games • Cournot games have a unique Nash equilibrium where: • qi = q* = (a - c)/b(n+1) • p(Q) = (a + nc)/(n+1) • ui = (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)

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 S4 S5 S6 S1 S2 S3

8. Cooperation in Cournot games • Definition (the static case): Given a fixed partitioning Π = (S1,…,Sk), 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

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)

10. A coalition formation game Given a current partition Π = (S1,…,Sk) • At an equilibrium of the super-game, a player jSi considers his current payoff to be u(Si)/| Si| • We allow 3 types of moves from Π • Type 1: A group of existing coalitions merge 

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

12. A coalition formation game • Type 3: A strict subset S of an existing coalition Si can leave and join another existing coalition Sj. Left over coalition Si\S dissolves S Sj Si 

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(n2/5) Note: constants independent of supply-demand curves (i.e. of game parameters, a, b, c)

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 = (S1,…,Sk), let k1 = # singleton coalitions • k2 = # non-singleton coalitions S4 S5 S1 S2 S3

15. Proof sketch of upper bound • Proposition (characterization): A partition Π = (S1,…,Sk) is stable iff • k1 (k2 +1)2 • For each non-singleton Si, |Si|  k2 •  suffices to solve a non-linear program

16. Proof sketch of upper bound PoA = Solving  PoA  n2/5

17. Proof of lower bound • By (almost) tightening the inequalities of the math. program • For any integer N, let n:= 4N4/5 N1/5 + N2/5 • We need k1 = N2/5 singletons • And k2 = N1/5 coalitions of size 4N4/5 • k = k1 + k2 = Ω (n2/5) • Lemma 2: The resulting partition is stable

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

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!