Coalition Formation and Price of Anarchy in Cournot Oligopolies
-
Upload
letitia-dominick -
Category
Documents
-
view
25 -
download
0
description
Transcript of Coalition Formation and Price of Anarchy in Cournot Oligopolies
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
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
S1 S2 S3S4 S5 S6
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 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
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
Si Sj
S
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
S1 S2 S3S4 S5
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!