Outline for today. Stat155 Game Theory Lecture 17: Correlated equilibria and the price of anarchy. Correlated equilibrium. A driving example.

Transcription

1 Outline for today Stat55 Game Theory Lecture 7: Correlated equilibria and the price of anarchy Peter Bartlett s Example: October 5, 06 A driving example / 7 / 7 Payoff Go (-00,-00) (,-) (-,) (-,-) Nash equilibria (Go, ), (, Go), (( 0, 99 ) ( 0, 0, 0)) 99. (because we want indifference: 00p + p = p ( p)) For a two player game with strategy sets S = {,..., m} and S = {,..., n}, a correlated strategy pair is a pair of random variables (R, C) with some joint probability distribution over pairs of actions (i, j) S S. Example In the traffic signal example, Pr(, Go) = Pr(Go, ) = /. Better solution? A traffic signal: Pr((Red, Green)) = Pr((Green, Red)) = /, and both players agree: Red means, Green means Go. After they both see the traffic signal, the players have no incentive to deviate from the agreed actions. c.f. a pair of mixed strategies If we have x Sm and y Sn, then choosing the two actions (R, C) independently gives Pr(R = i, C = j) = x i y j. In the traffic signal example, we cannot have Pr(, Go) > 0 and Pr(Go, ) > 0 without Pr(Go, Go) > 0. 3 / 7 4 / 7

2 A correlated strategy pair for a two-player game with payoff matrices A and B is a correlated equilibrium if for all i, i S, Pr(R = i) > 0 E [a i,c R = i] E [ a i,c R = i ]. for all j, j S, Pr(C = j) > 0 E [b R,j C = j] E [ b R,j C = j ]. c.f. a Nash equilibrium A strategy profile (x, y) Sm Sn is a Nash equilibrium iff for all i, i S, Pr(R = i) > 0 E [a i,c ] E [ a i,c ]. for all j, j S, Pr(C = j) > 0 E [b R,j ] E [ b R,j ]. When R and C are independent, these expectations and the conditional expectations are identical. Thus, a Nash equilibrium is a correlated equilibrium. A correlated strategy pair for a two-player game with payoff matrices A and B is a correlated equilibrium if for all i, i S, Pr(R = i) > 0 E [a i,c R = i] E [ a i,c R = i ]. for all j, j S, Pr(C = j) > 0 E [b R,j C = j] E [ b R,j C = j ]. Traffic signal example Pr(C = Go R = ) =. Pr(C = R = ) = 0. E [a,c R = ] =, E [a Go,C R = ] = / 7 6 / 7 Payoff Go (-00,-00) (,-) (-,) (-,-) A correlated equilibrium Value: (0,0). 0 Nash equilibria (Go, ), value (,-). (, Go), value (-,). (( ) (, 0, 99 0 value (-,-). 0, 0)) 99, A correlated strategy pair Value: (-/3,-/3). Pr(C = Go R = ) =. A correlated strategy pair E [a,c R = ] =, Pr(C = Go R = Go) = E [a Go,C R = ] = 99/. E [a Go,C R = Go] =, E [a,c R = Go] =. 7 / 7 8 / 7

3 Correlated equilibria Correlated equilibria Nash s Theorem implies there is always a correlated equilibrium. They are easy to find, via linear programming. It is not unusual for correlated equilibria to achieve better solutions for both players than Nash equilibria. Combinations of correlated equilibria Implementation We can think of a correlated equilibrium being implemented in two equivalent ways: There is a random draw of a correlated strategy pair with a known distribution, and the players see their strategy only. There is a draw of a random variable ( external event ) with a known probability distribution, and a private signal is communicated to the players about the value of the random variable. Each player chooses a mixed strategy that depends on this private signal (and the dependence is common knowledge). Given any two correlated equilibria, you can combine them to obtain another: Imagine a public random variable that determines which of the correlated equilibria will be played. Knowing which correlated equilibrium is being played, the players have no incentive to deviate. The payoffs are convex combinations of the payoffs of the two CEs. 0 / 7 9 / 7 Outline 009: New York City closed Broadway at Times Square.... with the aim of reducing traffic congestion. It was successful. Before After s Example: / 7 (NYC DOT) / 7

4 Example: before Other examples 005: Cheonggyecheon highway removed, speeding up traffic in downtown Seoul, South Korea. 990: 4nd Street in NYC closed for Earth Day. Traffic improved. 969: Congestion decreased in Stuttgart, West Germany, after closing a major road. Example: after Explanation? Drivers, acting rationally, seek the fastest route, which can lead to bigger delays (on average, and even for everyone). 3 / 7 4 / 7 (Karlin and Peres, 06) Mechanical and electrical versions (Nature, 99) 5 / 7 (Nature, 99) 6 / 7

5 Outline s Example: 7 / 7