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Algorithmic game theory • What if we have a large number of strategies? – Or large number of players?
intractability How computationally hard is it to find a Nash equilibrium(even in a two-player game)? -Optimal one:NP-hard.[Gilboa-Zemel'89] -Any one:PPAD-hard. [Daskalakis-Goldberg-Papadimitriou'06,Chen- Deng'06] 。Computationally Unfriendly！ Even worse,inefficient in game-theoretic sense!
intractability • How computationally hard is it to find a Nash equilibrium (even in a two-player game)? – Optimal one: NP-hard. [Gilboa-Zemel’89] – Any one: PPAD-hard. [Daskalakis-Goldberg-Papadimitriou’06, ChenDeng’06] • Computationally Unfriendly! • Even worse, inefficient in game-theoretic sense!
correlated equilibrium:game 2 pure NE:go to the same game. Payoff:(2,4)or (4,2) Battle of the Sexes -Bad:unfair. 1 mixed NE:each goes to city Baseball Softball preferred w.p.2/3. -Good:Fair -Bad:Low payoff:both 4/3 2 0 Baseball Worse:+ve chance of separate 0 CE:(Baseball,Softball)w.p., 0 4 Softball (Softball,Baseball)w.p. 0 2 Fair,high payoff,0 chance of separate
correlated equilibrium: game city Baseball Softball Baseball 2 4 0 0 Softball 0 0 4 2 • 2 pure NE: go to the same game. Payoff: (2,4) or (4,2) – Bad: unfair. • 1 mixed NE: each goes to preferred w.p. 2/3. – Good: Fair – Bad: Low payoff: both = 4/3 – Worse: +ve chance of separate • CE: (Baseball,Softball) w.p. ½ , (Softball,Baseball) w.p. ½ – Fair, high payoff, 0 chance of separate. Battle of the Sexes