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The concept of correlated equilibria in game theory, including its definition, relationship with nash equilibria, and existence in every game. It also explores the development of a polynomial runtime algorithm for computing correlated equilibria in a wide range of games using linear programming and markov chains. Additionally, the paper introduces succinct games and their role in efficiently calculating correlated equilibria.
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Tools of the Trade The Exponential Curse
Andreas Hoenselaar
CS 244
January 11, 2006
Tools of the Trade The Exponential Curse
(^1) Introduction Basic definitions The Mission
(^2) Tools of the Trade Linear Programming Markov Chains
(^3) The Exponential Curse Succinct Games
Tools of the Trade The Exponential Curse
Basic definitions The Mission
Definition A distribution x ∈ ∆ is a correlated equilibrium if the following is true for all players p and pairs of strategies (i, j): After drawing a strategy profile from x where player p’s component is i, there is no incentive to play another strategy j and it holds that ∑
s∈S−p
(upis − upjs)xis ≥ 0
Tools of the Trade The Exponential Curse
Basic definitions The Mission
Nash equilibria are special cases of correlated equilibria.
Nash equilibria Distribution x on S is a product distribution: xs =
∏n p= 1 x
p sp ∀s^ ∈^ S ⇒ x is completely defined by its marginal distributions xp
Correlated Equilibria As long as x is a valid distribution, it can have any form All |S| entries are required for a full description
Tools of the Trade The Exponential Curse
Linear Programming Markov Chains
Primal
min c>x s.t. Ax ≥ b x ≥ 0
Dual
max π>b s.t. A>π ≤ c π ≥ 0
Strong duality For optimal x∗^ and π∗^ it holds that c>x∗^ = π∗>b.
Tools of the Trade The Exponential Curse
Linear Programming Markov Chains
Optimization results for linear programs An LP can have an optimal feasible solution x∗ be infeasible (contradicting constraints) be unbounded (ie there are feasible points with arbitrarily large objective value)
Tools of the Trade The Exponential Curse
Linear Programming Markov Chains
Find x = (x 1 ,... , x|S|) s.t. Ux ≥ 0 ∑
s∈S
xs = 1
0 ≤ x ≤ 1
max
s∈S
xs
s.t. Ux ≥ 0 x ≥ 0
Lemma Problem (P) is unbounded iff (CE) has a solution.
Tools of the Trade The Exponential Curse
Linear Programming Markov Chains
Lemma For every y ≥ 0 there is a product distribution x such that xU>y = 0.
∑
s∈S−p
(upis − upjs)xis ≥ 0
q 6 =p
xqsq
xpi
j∈Sp
ypji −
j∈Sp
xpj ypij
Tools of the Trade The Exponential Curse
Linear Programming Markov Chains
The Markov property For every state sequence (Xt)t= 0 , 1 ,... (Xt^ ∈ Q) it holds that
P(Xt+^1 = qm|X^0 ,... , Xt) = P(Xt+^1 = qm|Xt) ∀t ∈ N^0 , qm ∈ Q.
This implies that the state at any given point in time only depends on the previous state and the transition matrix.
Example
Q = {q 1 , q 2 , q 3 }, T =
(^) , p^0 =
3
1 3
1 3
Tools of the Trade The Exponential Curse
Linear Programming Markov Chains
Motivation What can we say about the probabilty to be in a specific state qm after n steps?
Computation of the steady-state distribution Given an initial state distribution p^0 we can calculate the state distribution after n steps: pn^ = p^0 · (T)n
Surprisingly, this sequence converges towards an equilbirium distribution p∗^ in the limit of infinitely many steps: lim n→∞
p^0 · (T)n
= p∗
Tools of the Trade The Exponential Curse Succinct Games
Exponential size of distributions A distribution x over the set of strategy profiles S has an exponential number of entries:
|x| = |S| =
∏^ n
i= 1
|Si| ∈ O(sn) where s = max (|S 1 |,... , |Sn|)
A first insight If a correlated equilibrium is to be calculated in polynomial time, the game and thus the set of strategy profiles must have a more succinct representation.
Tools of the Trade The Exponential Curse Succinct Games
Definition A succinct game G = (I, T, U) is defined by a set of inputs I an algorithm T that returns the number of players n the cardinalities of the strategy sets (t 1 ,... , tn) in polynomial time an algorithm U that returns the utility ui(s) for player i under a distribution s = (s 1 ,... , sn) The game is of polynomial type if n and all ti’s are polynomially bounded in their argument.
Tools of the Trade The Exponential Curse Succinct Games
Symmetric Games In symmetric games, players cannot be distinguished and have the same set of strategies: The payoff depends on a player’s choice and the distribution of the other players’ choices over the strategy set Requires O(ns) space instead of O(sn) for its description
Tools of the Trade The Exponential Curse Succinct Games
in graphical games, players are identified with nodes in a graph and games are played only along the (undirected) edges. Description length: O(n · sk) for maximum node degree k plus explicit representation of gp, the game with all players in the neighborhood Correlated equilibria can be calculated in polynomial time in graphical games
Polynomial expectation property Calculate expectation by iterating over all stratagy profiles in gp, ignore players not in the neighborhood