Zerosum Games

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Zero-sum Games

• Steffen Hoernig
• Game Theory
• FEUNL, Fall 2006

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Why important?

• History first games fully understood
• Simple structure of payoffs
• Each players win is the other players loss
• no scope for cooperation or coordination
• Equilibrium concept precedes Nash equilibrium
• Value, maxmin, minimax
• actually coincides with Nash equilibrium

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(Two-player) Zero-sum?

• Also called
• strictly competitive games
• more generally constant-sum games
• Examples
• Sport games chess etc
• Seller/buyer relationships
• War (again..)

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Maxmin Strategy

• Assume finite zero-sum normal form game
• For each strategy s1, player 2 has best response
  b2(s1)
• which minimizes player is payoffu1(s1,b2(s1))
  mins2 u1(s1,s2)
• The same is true for any mixed strategy p
  u1(p,b2(p)) mins2 Epu1(s1,s2)

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Maxmin Strategy (2)

• Limit losses by choosing largest minimum maxp
  mins2 Epu1(s1,s2)
• gt maxmin or security strategy
• Resulting payoff m1 is maxmin payoff
• will get at least this payoff in game!
• unilateral concept!
• Player 2 m2 maxq mins1 Eq-u1(s1,s2)

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Minmax strategy

• Assume that rival assumes that player 1 plays
  best responses, and maximizes his payoff
• Player 1 chooses best response with lowest payoff
  (this WILL be a best response!)
• M1 minq maxs1 Equ1(s1,s2)
• Result Minmax payoff and strategy

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Observations

• Concepts applicable to all games
• Also true in all games M1 ? m1
• In zero-sum games m1 - M2 M1 - m2

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Nash Equilibrium

• If (p,q) is a Nash equilibrium of a zero-sum
  game, then p and q are security strategies and
  the maxmin and minmax payoffs are equal.
• If the maxmin and minmax payoffs are equal then
  the security strategies constitute a Nash
  equilibrium of the game.
• All Nash equilibria have the same payoffs!

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Readings and Exercises

• Readings Dutta ch. 11
• Exercises Dutta 10.9, 10.20