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Into. Game theory

Explore the concept of Nash Equilibrium in game theory and learn how to solve zero-sum games using linear programming. Understand the connections between these concepts.

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Into. Game theory

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  1. Into. Game theory

  2. Background: Game theory

  3. Background: Prison’s Dilemma Given you’re Dave, what’s the best choice?

  4. Nash Equilibrium A set of strategies is a Nash equilibrium if no player can do better by changing his or her strategy

  5. Nash Equilibrium (Cont’) • Nash showed (1950), that Nash equilibria (in mixed strategies) must exist for all finite games with any number of players. • Before Nash's work, this had been proven for two-player zero-sum games (by John von Neumann and Oskar Morgenstern in 1947). • Today, we’re going to find such Nash equilibria using Linear Programming for zero-sum game

  6. Zero-Sum Game • A strictly competitive or zero-sum game is a 2-player strategic game such that for each action aA, we have u1(a) + u2(a) = 0. (u represents for utility) • What is good for me, is bad for my opponent and vice versa

  7. Zero-Sum Game • Mixed strategy • Making choice randomly obeying some kind of probability distribution • Why mixed strategy? (Nash Equilibrium) • E.g. : P(1) = P(2) = 0.5; P(A) = P(B)=P(C)=1/3

  8. Solving Zero-Sum Games • Let A1 = {a11, …, a1n }, A2 = {a21, …, a2m } • Player 1 looks for a mixed strategy p • ∑i p(a1i ) = 1 • p(a1i ) ≥ 0 • ∑i p(a1i ) · u1(a1i, a2j) ≥ r for all j {1, …, m} • Maximize r! • Similarly for player 2.

  9. Solve using Linear Programming • What are the unknowns? • Strategy (or probability distribution): p • p(a11 ), p(a12 ),..., p(a1n-1 ), p(a1n )~n numbers • Denoted as p1,p2,...,pn-1,pn • Optimum Utility or Reward: r • Stack all unknowns into a column vector • Goal: maximize: where

  10. Solve Zero-Sum Game using LP • What are the constraints?

  11. Solve Zero-Sum Game using LP Let: Ae=(0,1,…,1) Aie=(1;-UT) f=(1,0,…,0)T The LP is: maximize fTx subject to: Aiex<=0 Aex=1

  12. Summary • Zero-Sum Game  Mixed Strategy  NE • Connections with Linear Programming

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