Linear Programming and zerosum twoplayer games

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Linear Programming and zero-sum two-player games

• By Warren Schudy

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Outline

• Intro to Linear Programs (LPs)
• LP Duality
• Corollaries
• Minimax theorem via duality
• Solving two-player zero-sum games in polynomial
  time via LPs
• For more on LPs, take CS 149 Intro to
  Combinatorial Optimization

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A Production Problem

• Town dump has 4 bike frames, 5 unicycle frames
  and 11 wheels
• Bicycles sell for 1, Unicycles for 2

y (unic.)
OPT
2
0
0
2
x (bikes)
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Alternate view competitive equilibrium

• Suppose you want to pay the dump enough for the
  parts that your little brother wont have an
  incentive to compete with you
• Rival has incentive to make
• bicycles if price(bike frame)2price(wheel)lt1
• unicycles if price(unic. frame)price(wheel)lt2
• Will show via duality preventing competition
  forces you to pay the dump all of your revenue!

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Primal and Dual
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Weak duality

• Intuition competition will keep you from making
  money
• Theorem For feasible we have
• Proof

Uses
Uses
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Upper Bounds
Also implies UB 20 because


Perpendicular to objective function and valid
-gt Upper Bound
Best Upper Bound of this form
2
0
0
2
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Equality Constraints

• Suppose the dump manager hates dealing with bike
  frames (too big), so he wants you to take all of
  them, even if he has to pay you

Equality
Price can go negative
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LP for games and its dual

• Proves minimax theorem using LP duality!

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Game from LP

• Prove LP duality from games
• A bit unintuitive see notes for details

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Conclusions

• Two-player zero-sum games are poly-time solvable
  using linear programming