Lecture 16 Linear Program Duality Duality

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Lecture 16 Linear Program Duality Duality

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Lecture 16 Linear Program Duality Duality Two-Player Zero-sum Games Game played with two competing players, when one player wins, the other player loses. Goal: Find the best strategy in the game Game as a matrix Can represent the game using

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Lecture 16 Linear Program Duality

Duality

Two-Player Zero-sum Games Game played with two competing players, when one player wins, the other player loses.
Goal: Find the best strategy in the game

Game as a matrix Can represent the game using a 2-d array

A[i, j] = if row player uses strategy i, column player uses strategy j, the payoff for the row player
Recall: payoff for the column player is - A[i, j]

Pure Strategy vs. Mixed Strategy Pure strategy: use a single strategy (correspond to a single row/column of the matrix)
Obviously not a good idea for Rock-Paper-Scissors.

Mixed strategy: Play Rock with probability p1…

Payoff of the game.

Solving two player games by LP Try to use LP to find a good strategy for Duke.

What is a good strategy for Duke? Solution: (9,6,4,19)/19.

Duality: what would UNC do? Solution: (1,1,1,3)/3.

Comparing the Solution to two LPs

Min-Max Theorem Theorem [Von Neumann] For any two-player, zero-sum game, there is always a pair of optimal strategies and a single value V.
If the row player plays its optimal strategy, then it can guarantee a payoff of at least V.
If the column player plays its optimal strategy, then it can guarantee a payoff of at most V.

Corollary: The solution to the two LP must be equal.(x4=y4)

Duality for Linear Programs

Dual LP Strong Duality: The two LP has the same optimal value.

Lecture 16 Linear Program Duality Duality

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