Maximum Entropy Correlated Equilibria by L. Ortiz, R. Schapire and S. Kakade. Course: Applications of Information Theory to Computer Science CSG195 , Fall 2008 CCIS Department, Northeastern University Dimitrios Kanoulas. Maximum Entropy Correlated Equilibria.
Applications of Information Theory to Computer Science
CSG195, Fall 2008
CCIS Department, Northeastern University
Algorithmic Game Theory
Studies the behavior of players in competitive and collaborative situations
[Christos Papadimitriou in SODA 2001]
Two cars, a red and a white one [players of the game] get to a road intersection without traffic light, at the same time.
Each driverdecides to stop (S) or go (G) [two pure strategies of the game]
Payoffs for red/white car are defined from the matrix:
GOAL for each player: Maximize his payoff
Equilibrium in a Game:
Each player picks a strategy such that:
no one wants to unilaterally deviate from this.
White car stops && Red car goes (pure NE)
Red car stops and White car goes (pure NE)
Both cars with 1/2 go and 1/2 stop (mixed NE)
Movie: Beautiful Mind
There always exists a mixed strategy Nash Equilibrium.
There is a traffic light that suggest individually to the cars:
The general problem of equilibrium computation
is fundamental in Computer Science
Every player is “happy” by playing a [pure or mixed] strategy, which means that he cannot increase his payoff by unilaterally deviate from his strategy.
Correlated equilibrium (CE):
A joint probability distribution P(a1, . . . , an) such that:
• Every player individually receives “suggestion” from P
• Knowing P, players are happy with this “suggestion” and don’t want to deviate from this.
Nash Equilibria (NE):
Is a special case of CE: P a product distribution -> P = ΠP(ai)
NE always exists but the problem of finding a NE is hard even for a 2-players game.
[Chen & Deng]
Is the equilibrium “good” or “bad” ?
What if I want to add some properties to my equilibrium ?
A player is willing to negotiate and agree to some form of “joint” strategy
with the other players.
At the same time, the player wants to try to hide as much as he can his own behavior, by making it difficult to predict.
We want to suggest a joint strategy that satisfies all the players but complicates their prediction of each others’ individual strategies
The conditional entropy in information theory provides
a measure of the predictability of a random process from another
The larger the conditional entropy, the harder the prediction.
[Cover and Thomas]
Ai : the strategy of playeri(random variable)
A−i: the strategy of the rest of the players (random variable)
P(ai|a−i): the conditional mixed strategy where:
player i picks aigiven that the rest of the players pick a−I
HAi|A−i (P) = − Σa−i in A−iP(a−i) Σai in AiP(ai|a−i) logP(ai|a−i)
the conditional entropy of the strategy of player i
given the strategies of the rest of the players
the larger the conditional entropy,
the harder the prediction.
is the joint mixed strategy P* = argmaxP in CEHAi|A−i (P)
[The probability distribution over the strategies which give a CE such that maximizes its entropy]
MaxEnt CE satisfies all the players and maximizes the hardness of predictions.
A mathematician is a device
for turning coffee into theorems.