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)
Motivation and goals
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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
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Cournot games
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Coalition Formation in Cournot games
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Nash equilibria and price of anarchy
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
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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
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Cournot games have a unique Nash equilibrium
where:
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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:
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[Harberger ’54] (empirical observations)
[Guo, Yang ’05, Kluberg, Perakis ’08] (theoretical
analysis)
PoA = (n)
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Cooperation in Cournot games
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In practice, competition among firms is not exactly a
non-cooperative game
Suppose firms are allowed to partition themselves
into coalition structures
S1
S2
S3
S4 S5 S6
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Cooperation in Cournot games
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Definition (the static case): Given a fixed partitioning
Π = (S1,…,Sk), the Cournot super-game consists of
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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:
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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
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Are all partitions equally likely to arise?
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What if players are allowed to join/abandon
existing coalitions?
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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)
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At an equilibrium of the super-game, a player jSi
considers his current payoff to be u(Si)/| Si|
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We allow 3 types of moves from Π
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Type 1: A group of existing coalitions merge

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A coalition formation game
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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
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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

Si
Sj
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Inefficiency of stable partitions
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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
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Lemma 1: For stable partitions with k coalitions PoA =
O(k)
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S1
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
S2
S3
S4 S5
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Proof sketch of upper bound
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Proposition (characterization): A partition Π =
(S1,…,Sk) is stable iff
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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
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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
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So far we assumed partitions reach a Nash
equilibrium of the super-game
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Theorem: Same result holds when super-players of
a partition employ no-regret algorithms.
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No-regret converges to Nash utility of each superplayer
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Future work
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Apply the same to other classes of games
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Routing games, socially concave games
Need to ensure the super-game has a well-defined payoff for the superplayers
Need to define how players split the superplayer’s payoff
Other models of coalition formation
Thank you!
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