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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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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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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Cournot Oligopolies [Cournot 1838]

Games among firms producing/offering the same (or a similar) product

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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

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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)

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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

S1 S2 S3S4 S5 S6

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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

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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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A coalition formation game

Given a current partition Π = (S1,…,Sk)

At an equilibrium of the super-game, a player jSi 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

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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

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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

Si Sj

S

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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)

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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

S1 S2 S3S4 S5

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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

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Proof sketch of upper bound

PoA =

Solving PoA n2/5

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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

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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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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!