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4. Mixed strategies. In this section we shall learn How to not lose a game when it appears your opponent has a counter to all your moves. How to use unpredictability to your advantage. Mixed strategies. Anyone for tennis?
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4. Mixed strategies • In this section we shall learn • How to not lose a game when it appears your opponent has a counter to all your moves. • How to use unpredictability to your advantage. Games People Play.
Mixed strategies Anyone for tennis? You are playing tennis against some bald bloke called Agassi, you both have two possible strategies to play the ball cross-court or down-the-line. The probabilities of your winning the point (and therefore of Andre losing it) are What do you do? Games People Play.
Mixed strategies Anyone for tennis? The problem you are encountering here is that if you become predictable you will lose. The answer is to become unpredictable. This is the idea behind mixed strategies. Games People Play.
Mixed strategies • In the tennis game there is no equilibrium in pure strategies, you need to become more unpredictable. • You must play down-the-line and cross-court with given probabilities. Perhaps spin your racket and if the makers logo comes out on top play DL otherwise play CC. Now you are playing the two possibilities with probabilities of ½ each. • But is 50/50 the best mix of your two possibilities, or can you improve on it? Games People Play.
Mixed strategiesComputing your best mixed strategy • The intuition behind an best mixed strategy is strange but also obvious. • You must choose the probabilities attached to each of you options such that the opponent is indifferent between his options! • Why? Because if he weren’t indifferent it means he can gain an advantage by doing a particular thing. • But his gain is your loss, so make him indifferent. Games People Play.
Mixed strategiesEquilibria • If you opponent and yourself are both indifferent then we have a mixed strategy equilibrium. • A mixed strategy Nash equilibriumalwaysexists, which means that even if you don’t know what to do in terms of pure strategies you can always work out what to do in mixed. Games People Play.
Mixed strategiesCalculating your Equilibrium mix. • If you choose DL with probability p and CC with probability (1-p). • Then if Agassi chooses DL he gets • p(50)+(1-p)(10) • If Agassi chooses CC he gets • p(20)+(1-p)(80) • To make him indifferent thus requires • p(50)+(1-p)(10)= p(20)+(1-p)(80) • Which we solve to get • p=0.7 • Which is the optimal frequency for you to play DL Games People Play.
Mixed strategiesCalculating your Equilibrium mix. • Similarly Agassi chooses DL with probability q and CC with probability (1-q). • Then if you chooses DL you get • q(50)+(1-q)(80) • If you choose CC you get • q(10)+(1-q)(20) • To make you indifferent thus requires • q(50)+(1-q)(80) = q(90)+(1-q)(20) • Which we solve to get • q=0.6 • Which is the optimal frequency for Agassi to play DL Games People Play.
Mixed strategiesCalculating your Equilibrium mix. • So what are your chances of winning? • You play DL with probability 0.7, and Agassi plays DL with probability 0.6. • So your probability of winning is • pq(50) + p(1-q)80 + (1-p)q(90) + (1-p)(1-q)20 • = (0.42)(50) + (0.28)80 + (0.18)(90) + (0.12)20 • = 62 or 62% • World #1 indeed!!! Games People Play.
