minimax

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minimax
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Outline

• zero-sum games
• pure strategy minimax
• saddle (equilibrium) points
• mixed strategy minimax
• minimax theorem
• non-zero sum games
• minimax as optimization

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Minimax Beginning of Game Theory

• Long before Nash
• "As far as I can see, there could be no theory of
  games without that theorem I thought there was
  nothing worth publishing until the 'Minimax
  Theorem' was proved," remarked von Neumann.

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Zero-Sum Games

• 2 players
• R_1(i,j) M(i,j)
• R_2(i,j) -M(i,j)
• player 1 is maximizer
• player 2 is minimizer
• Matching Pennies
• Rock-Paper-Scissors

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Strategies

• row player what is the best payoff I can
  guarantee if column player knows/guesses my move
• maxi (minj M(i,j)) - maximin value
• column player what is the best payoff I can
  guarantee if row player knows/guesses my move
• - minj (maxi M(i,j))
• minj (maxi M(i,j)) - minimax value

opponent chooses the worst move for me
I move first
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Example
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Min and Max

• maxi (minj M(i,j)) minj (maxi M(i,j))
• better to go second can react
• Proof
• for any i
• M(i,j) maxi M(i,j) for all j
• minj M(i,j) minj (maxi M(i,j))
• i arg maxi (minj M(i,j))
• maxi (minj M(i,j)) minj (maxi M(i,j))

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minimax gt maximinMatching Pennies example

• maximin -1
• minimax 1

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Pure Strategy Saddle Points
(2,2) is a saddle point

• Def (i,j) is a saddle point if
• it is
• 1. minimum of the row i
• M(i,j) M(i,j)
• 2. maximum of the column j
• M(i,j) M(i,j)
• Equivalent to pure strategy Nash Equilibria in
  2-player zero-sum games

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Property of Saddle Points

• Let (i1,j1), (i2,j2) be two saddle points. Then
• 1. M(i1,j1) M(i2,j2)
• 2. M(i1,j2) M(i2,j1)
• same value at all saddle points (i,j)
• v M(i,j) value of the game

(1,1), (2,1), (1,3), (2,3)
saddle points
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Pure Strategy Saddle Points and Minimax

• In games with pure strategies, saddle point
  exists if and only if
• maxi (minj M(i,j)) minj (maxi M(i,j))
• Row player is guaranteed to get at least the
  maximin value
• Column player is guaranteed to pay at most the
  minimax value
• Proof (?) a saddle point (i,j) exists if
• maxi minj M(i,j) minj maxi M(i,j)

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• i arg maxi minj M(i,j)
• maxi minj M(i,j) minj M(i,j) M(i,j) for
  all j
• j arg minj maxi M(i,j)
• minj maxi M(i,j) maxi M(i,j) M(i,j)
• But maxi minj M(i,j) minj maxi M(i,j) gt
• minj M(i,j) M(i,j) maxi M(i,j)
• Or equivalently
• M(i,j) minj M(i,j) M(i,j)
• M(i,j) maxi M(i,j) M(i,j)
• Prove the other direction

(i,,j) minimum of row i
(i,,j) maximum of column j
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Pure Strategy Games with Saddle Points

• called strictly determined games
• the outcome is known
• can tell the opponent what you are going to do
• what is there are no pure strategy saddle points?

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Mixed Strategies

• do not know what you are going to play
• p (p1,..,pm)
• q (q1,..,qm)
• ?ipi 1, ?jqj 1
• M(p,q) ?i?jM(i,j) piqj pRqT
• Def (p,q) is a saddle point if
• M(p,q) M(p,q) M(p,q) for all p,q

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Minimax Theorem

• Every m-by-n 2-person zero-sum game has a
  solution. More precisely, there is a unique
  number v, called the value of the game, and there
  are optimal (pure or mixed) strategies p,q such
  that
• maxpminq M(p,q) minqmaxp M(p,q) v M(p,q)
• i.e., we know whats rational

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Non-Zero Sum Games

• minimax (maximin) strategies are not optimal
• Battle of the Sexes (modified)
• 3 Nash equilibria

husband
wife
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Nash vs Minimax

• Minimax rational in 2-player zero-sum games,
  i.e. players know what to do
• Same value at all saddle points
• Nash applies to all n-player games
• Not clear how to reach Nash (multiple equilibria)
• Nash outcome may not be the best one (Prisoners
  Dilemma)
• players do not know what to do

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Maximin Optimization Problem

• maxpminq M(p,q)
• maxpv s.t.
• v minq M(p,q)
• ?ipi 1, ?jqj 1
• pi 0 for all i
• qj 0 for all j

• Mj column j, Mi row i
• maxp,v v s.t.
• pTMj v for all j
• ?ipi 1
• pi 0 for all i

when column player knows the strategy of the row
player, there is no need to randomize
player 2 cannot make the payoff lower than v
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Maximin Optimization Problem

• maxpminq M(p,q)
• Mj column j, Mi row i
• maxp,v v s.t.
• pTMj v for all j
• ?ipi 1
• pi 0 for all i

player 2 cannot make the payoff lower than v