lect1207

An Intro to Game Theory 15-451 Avrim Blum 12/07/04

Plan for Today ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],test material not test material

2-Player Zero-Sum games ,[object Object],[object Object],[object Object],[object Object],shooter goalie shooter loses and goalie wins (-1,1) (1,-1) (1,-1) (-1,1) Left Right Left Right

Minimax-optimal strategies ,[object Object],[object Object],[object Object],[object Object]

Minimax-optimal strategies ,[object Object],Minimax optimal for both players is 50/50. Gives expected gain of 0. Any other is worse. (-1,1) (1,-1) (1,-1) (-1,1) Left Right Left Right

Minimax-optimal strategies ,[object Object],Minimax optimal for both players is (2/3,1/3). Gives expected gain 1/3. Any other is worse. (0,0) (1,-1) (1,-1) (-1,1) Left Right Left Right

Matrix games and Algorithms ,[object Object],[object Object],[object Object],Of course matrix may be HUGE. But helpful conceptually.

An simple example ,[object Object],[object Object],2 3 3 3 2 3 3 3 2 (A,B) first (A,C) first (B,C) first (payoff to adversary) adversary largest is: C B A

Matrix games and Algs ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Alg player Adversary

E.g., hashing ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Alg player Adversary

Minimax Theorem (von Neumann 1928) ,[object Object],[object Object],[object Object],Counterintuitive: Means it doesn’t hurt to publish your strategy if both players are optimal. (Borel had proved for symmetric 5x5 but thought was false for larger games)

Nice proof of minimax thm (sketch) ,[object Object],[object Object],[object Object],[object Object],[object Object],V C V R

Proof sketch, contd ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

General-sum games ,[object Object],[object Object],person driving towards you you (1,1) (-1,-1) (-1,-1) (1,1) Left Right Left Right

General-sum games ,[object Object],[object Object],Spongebob Oceans-12 Spongebob Oceans-12 No longer a unique “value” to the game. (8,2) (0,0) (0,0) (2,8)

Nash Equilibrium ,[object Object],[object Object],[object Object],NE are: both left, both right, or both 50/50. (1,1) (-1,-1) (-1,-1) (1,1) Left Right Left Right

Nash Equilibrium ,[object Object],[object Object],[object Object],NE are: both Sb, both Oc, or (80/20,20/80) Spongebob Oceans12 Spongebob Oceans-12 (8,2) (0,0) (0,0) (2,8)

Uses ,[object Object],[object Object],[object Object],don’t pollute pollute don’t pollute pollute Need to add extra incentives to get good overall behavior. (2,2) (-1,3) (3,-1) (0,0)

NE can do strange things ,[object Object],[object Object],[object Object],Fine. NE is 50/50. Travel time = 1.5 travel time = 1, indep of traffic travel time t(x)=x . s x 1 1 t x

NE can do strange things ,[object Object],[object Object],[object Object],Add new superhighway. NE: everyone uses zig-zag path. Travel time = 2. s x 1 1 t x travel time = 1, indep of traffic travel time t(x)=x . 0

Existence of NE ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Existence of NE ,[object Object],[object Object],[object Object],[object Object],[object Object]

Proof ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Proof (cont) ,[object Object],[object Object],[object Object],[object Object],[object Object]

Try #1 ,[object Object],[object Object],[object Object],(-1,1) (1,-1) (1,-1) (-1,1) Left Right Left Right

Instead we will use... ,[object Object],[object Object],[object Object],p p’ Note: quadratic + linear = quadratic.

Instead we will use... ,[object Object],[object Object],[object Object],p Note: quadratic + linear = quadratic. p’

Instead we will use... ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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