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1 Chapter 4: Minimax Equilibrium in Zero Sum Game SCIT1003 Chapter 4: Minimax Equilibrium in Zero Sum Game Prof. Tsang

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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 2

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The Minimax Theorem 3 “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.

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Saddle point Pure strategy game: Saddle point 4 A zero-sum game with a saddle point. 1 3 4 3 Is this a Nash Equilibrium? MaxiMin MiniMax

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Pure & mixed strategies 5 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.

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All you need to know about Probability 6 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) = 0 If E is certain to occur, then P(E) = 1 2.The probabilities of all the possible outcomes must sum to 1

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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 7

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Mixed strategies 8 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.

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A soccer penalty shot at 12-yard left or right? 9 LeftRight Left425 5895 Right730 9370 Goalie Kicker p.145 payoffs are winning probability

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10 A penalty shot at 12-yard left 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.

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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? 11 A penalty shot at 12-yard left or right?

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12 p.166 graphical solution Kicker’s mixture

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13 p.168 graphical solution Goalie’s mixture

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

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A Parking meter game (p.164) 15 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)?

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Parking meter game 16 PayNot pay Enforce-50 150 Not enforce 0 10 Car driver Enforcer p.164

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17 no No Nash equilibrium for pure strategy x y1-x-y x=probability to take action R y=probability to take action S 1-x-y=probability to take action P Mixed strategies

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

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19 Janken step game (Japanese RSP) p.171

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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 20

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Summary: Ch. 4 Look for any equilibrium Dominating Equilibrium Minimax Equilibrium Nash Equilibrium 21

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Assignment 4.1 22

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23 Assignment 4.1

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