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SCIT1003Chapter 4: Minimax Equilibrium Chapter 4: Minimax Equilibrium

in Zero Sum Gamein Zero Sum Game

Prof. Tsang

Maximin & Minimax Equilibrium in a zero-sum game

• Minimax - minimizing the maximum loss (loss-ceiling, defensive)

• Maximin - maximizing the minimum gain (gain-floor, offensive)

• Minimax = Maximin

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The Minimax Theorem

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“Every finite, two-person, zero-sum game has a rational solution in the form of a pure or mixed strategy.”

John Von Neumann, 1926

For every two-person, zero-sum game with finite strategies, there exists a value V and a mixed strategy for each player, such that (a) Given player 2's strategy, the best payoff possible for player 1 is V, and (b) Given player 1's strategy, the best payoff possible for player 2 is −V.

Pure strategy game: Saddle pointSaddle point

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A zero-sum game with a saddle point.

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Is this a Nash Equilibrium?

MaxiMin

MiniMax

Pure & mixed strategiesPure & mixed strategies

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A pure strategy provides a complete definition of how a player will play a game. It determines the move a player will make for any situation they could face.

A mixed strategy is an assignment of a probability to each pure strategy. This allows for a player to randomly select a pure strategy.In a pure strategy a player chooses an action for sure, whereas in a mixed strategy, he chooses a probability distribution over the set of actions available to him.

All you need to know about All you need to know about Probability Probability

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If E is an outcome of action, then P(E) denotes the probability that E will occur, with the following properties:

1. 0 P(E) 1 such that:If E can never occur, then P(E) = 0If E is certain to occur, then P(E) = 1

2. The probabilities of all the possible outcomes must sum to 1

Mixed strategy

• In some zero-sum game, there is no pure strategy solution (no Saddle point)

• Play’s best way to win is mixing all possible moves together in a random (unpredictable) fashion.

• E.g. Rock-Paper-Scissors

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Mixed strategiesMixed strategies

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Some games, such as Rock-Paper-Scissors, do not have a pure strategy equilibrium. In this game, if Player 1 chooses R, Player 2 should choose p, but if Player 2 chooses p, Player 1 should choose S. This continues with Player 2 choosing r in response to the choice S by Player 1, and so forth.

In games like Rock-Paper-Scissors, a player will want to randomize over several actions, e.g. he/she can choose R, P & S in equal probabilities.

A soccer penalty shot at 12-yardleft or right?

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Left Right

Left 42 558 95

Right 7 3093 70

Goalie

Kicker

p.145 payoffs are winning probability

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A penalty shot at 12-yardleft or right?

If you are the kicker, which side you use?The best chance you have is 95%. So you kick left.But the goalie anticipates that because he knows that’s your best chance. So his anticipation reduces your chance to 58%.What if you anticipate that he anticipates … so you kick right & that increase your chance to 93%.What if he anticipates that you anticipate that he anticipates …If you use a pure strategy, he always has a way to reduce you chance to win.

• To end this circular reasoning, you do something that the goalie cannot anticipate.

• What if you mix the 2 choices randomly with 50-50 chance?

• Your chance of winning is(58+93)/2 if the goalie moves to left(93+70)/2 if the goalie moves to right

Is this better?

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A penalty shot at 12-yardleft or right?

12p.166 graphical solution

Kicker’s mixture

13p.168 graphical solution

Goalie’s mixture

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If the goalie improves his skill at saving kicks to the Right side

A Parking meter game (p.164)

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If you pay for the parking, it cause you $1.

If you don’t pay for the parking and you are caught by the enforcer, the penalty is $50.

Should you take the risk of not paying for the parking?How often the enforcer should patrol to keep the car drivers honest (to pay the parking fee)?

Parking meter game

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Pay Not pay

Enforce -1 -501 50

Not enforce

-1 01 0

Car driver

Enforcer

p.164

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nono

No Nash equilibrium for pure strategy

x y 1-x-y

x=probability to take action R

y=probability to take action S

1-x-y=probability to take action P

Mixed strategiesMixed strategies

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They have to be equal if expected payoff independent of action of player 2

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Janken step game (Japanese RSP)

p.171

Two-Person, Zero-Sum Games: Summary

• Represent outcomes as payoffs to row player• Find any dominating equilibrium• Evaluate row minima and column maxima• If maximin=minimax, players adopt pure strategy

corresponding to saddle point; choices are in stable equilibrium -- secrecy not required

• If maximin minimax, find optimal mixed strategy; secrecy essential

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Summary: Ch. 4

• Look for any equilibrium• Dominating Equilibrium• Minimax Equilibrium• Nash Equilibrium

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Assignment 4.1

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Assignment 4.1