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A very little Game Theory Math 20 Linear Algebra and Multivariable Calculus October 13, 2004

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A Game of Chance You and I each have a six-sided die We roll and the loser pays the winner the difference in the numbers shown If we play this a number of times, who’s going to win?

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The Payoff Matrix Lists one player’s (call him/her R) possible outcomes versus another player’s (call him/her C) outcomes Each a ij represents the payoff from C to R if outcomes i for R and j for C occur (a zero-sum game). C’s outcomes 123456 R ’s outcomes 10-2-3-4-5 210-2-3-4 3210-2-3 43210-2 543210 6543210

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Expected Value Let the probabilities of R’s outcomes and C’s outcomes be given by probability vectors

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Expected Value The probability of R having outcome i and C having outcome j is therefore p i q j. The expected value of R’s payoff is

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Expected Value of this Game A “fair game” if the dice are fair.

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Expected value with an unfair die Suppose Then

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Strategies What if we could choose a die to be as biased as we wanted? In other words, what if we could choose a strategy p for this game? Clearly, we’d want to get a 6 all the time! C’s outcomes 123456 R ’s outcomes 10-2-3-4-5 210-2-3-4 3210-2-3 43210-2 543210 6543210

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Flu Vaccination Suppose there are two flu strains, and we have two flu vaccines to combat them. We don’t know distribution of strains Neither pure strategy is the clear favorite Is there a combination of vaccines that maximizes immunity? Strain 12 Vaccine 10.850.70 20.600.90

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Fundamental Theorem of Zero-Sum Games There exist optimal strategies p* for R and q* for C such that for all strategies p and q: E(p*,q) ≥ E(p*,q*) ≥ E(p,q*) E(p*,q*) is called the value v of the game In other words, R can guarantee a lower bound on his/her payoff and C can guarantee an upper bound on how much he/she loses This value could be negative in which case C has the advantage

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Fundamental Problem of Zero-Sum games Find the p* and q*! In general, this requires linear programming. Next week! There are some games in which we can find optimal strategies now: Strictly-determined games 2 2 non-strictly-determined games

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Network Programming Suppose we have two networks, NBC and CBS Each chooses which program to show in a certain time slot Viewer share varies depending on these combinations How can NBC get the most viewers?

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Payoff Matrix CBS shows 60 Minutes Survivor CSI Everybody Loves Raymond NBC Shows Friends60203055 Dateline50754560 Law & Order 70453530

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NBC’s Strategy NBC wants to maximize NBC’s minimum share In airing Dateline, NBC’s share is at least 45 This is a good strategy for NBC 60 M Surv CSI ELR F60203055 DL50754560 L&O70453530

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CBS’s Strategy CBS wants to minimize NBC’s maximum share In airing CSI, CBS keeps NBC’s share no bigger than 45 This is a good strategy for CBS 60 M Surv CSI ELR F60203055 DL50754560 L&O70453530

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Equilibrium (Dateline,CSI) is an equilibrium pair of strategies Assuming NBC airs Dateline, CBS’s best choice is to air CSI, and vice versa 60 M Surv CSI ELR F60203055 DL50754560 L&O70453530

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Characteristics of an Equlibrium Let A be a payoff matrix. A saddle point is an entry a rs which is the minimum entry in its row and the maximum entry in its column. A game whose payoff matrix has a saddle point is called strictly determined Payoff matrices can have multiple saddle points

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Pure Strategies are optimal in Strictly-Determined Games If a rs is a saddle point, then e r T is an optimal strategy for R and e s is an optimal strategy for C.

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Proof

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So for all strategies p and q: E(e r T,q) ≥ E(e r T,e s ) ≥ E(p,e s ) Therefore we have found the optimal strategies

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2x2 non-strictly determined In this case we can compute E(p,q) by hand in terms of p 1 and q 1

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Optimal Strategy for 2x2 non-SD Let This is between 0 and 1 if A has no saddle points Then

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Optimal set of strategies We have

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Flu Vaccination Strain 12 Vaccine 10.850.70 20.600.90

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Flu Vaccination Strain 12 Vaccine 10.850.70 20.600.90 So we should give 2/3 of the population vaccine 1 and 1/3 vaccine 2 The worst that could happen is a 4:5 distribution of strains In this case we cover 76.7% of pop

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Other Applications of GT War Battle of Bismarck Sea Business Product Introduction Pricing Dating

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Two-Player Zero-Sum Games

Two-Player Zero-Sum Games

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