games people play. 4. mixed strategies in this section we shall learn how to not lose a game when it...
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Games People Play.Games People Play.
4. Mixed strategies4. Mixed strategies
In this section we shall learnIn this section we shall learnHow 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.Games People Play.
Games People Play.Games People Play.
Mixed strategies Mixed strategies Anyone for tennis?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 are
What do you do?
AgassiAgassi
DLDL CCCC
YouYou
DLDL 5050 8080
CCCC 9090 2020
Games People Play.Games People Play.
Mixed strategiesMixed strategies
Anyone for tennis?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.
AgassiAgassi
DLDL CCCC
YouYou
DLDL 5050 8080
CCCC 9090 2020
Games People Play.Games People Play.
Mixed strategiesMixed strategies
In the tennis game there is no equilibrium in pure In the tennis game there is no equilibrium in pure strategies, you need to become more unpredictable.strategies, you need to become more unpredictable.
You must play down-the-line and cross-court with given You must play down-the-line and cross-court with given probabilities. Perhaps spin your racket and if the makers probabilities. Perhaps spin your racket and if the makers logo comes out on top play DL otherwise play CC. Now logo comes out on top play DL otherwise play CC. Now you are playing the two possibilities with probabilities of you are playing the two possibilities with probabilities of ½ each.½ each.
But is 50/50 the best mix of your two possibilities, or can But is 50/50 the best mix of your two possibilities, or can you improve on it?you improve on it?
Games People Play.Games People Play.
Mixed strategiesMixed strategiesComputing your best mixed strategyComputing your best mixed strategy
The intuition behind an best mixed strategy is strange but The intuition behind an best mixed strategy is strange but also obvious.also obvious.You must choose the probabilities attached to each of You must choose the probabilities attached to each of you options such that the opponent is indifferent between you options such that the opponent is indifferent between his options!his options!Why? Because if he weren’t indifferent it means he can Why? Because if he weren’t indifferent it means he can gain an advantage by doing a particular thing.gain an advantage by doing a particular thing.But his gain is your loss, so make him indifferent.But his gain is your loss, so make him indifferent.
Games People Play.Games People Play.
Mixed strategiesMixed strategiesEquilibriaEquilibria
If you opponent and yourself are both indifferent then we If you opponent and yourself are both indifferent then we have a mixed strategy equilibrium.have a mixed strategy equilibrium.A mixed strategy Nash equilibriumA mixed strategy Nash equilibrium alwaysalways exists, which exists, which means that even if you don’t know what to do in terms of 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 pure strategies you can always work out what to do in mixed.mixed.
Games People Play.Games People Play.
Mixed strategiesMixed strategiesCalculating your Equilibrium mix.Calculating your Equilibrium mix.
AgassiAgassi
DLDL CCCC
YouYou
DLDL 5050 8080
CCCC 9090 2020
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.Games People Play.
Mixed strategiesMixed strategiesCalculating your Equilibrium mix.Calculating your Equilibrium mix.
AgassiAgassi
DLDL CCCC
YouYou
DLDL 5050 8080
CCCC 9090 2020
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(90)+(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.Games People Play.
Mixed strategiesMixed strategiesCalculating your Equilibrium mix.Calculating 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!!!World #1 indeed!!!