Game Theory

Louis Oliphant

oliphant@cs.wisc.edu

CS540-2

How To Calculate the Optimal Mixed Strategy (for 2x2 Matrix)
Provided Opponent plays optimally

Using the following probabilities:

1 Finger for Even – p

2 Fingers for Even– (1-p)

1 Finger for Odd – q

2 Fingers for Odd – (1-q)

Now assume that Odd will set q=1 and Even knows this. Calculate the Expected Outcome

Now assume that Odd will set q=0 and Even knows this. Calculate the Expected Outcome

Find where the two expected outcomes are equal

This is the point where it doesn't matter what q is set to, the expected outcome will be the same either way

How To Calculate the Optimal Mixed Strategy (for 2x2 Matrix)Provided Opponent plays optimally

Now assume that Odd will set q=1 and Even knows this. Calculate the Expected Outcome

Expected Outcome=2p-3(1-p)

Expected Outcome=5p-3

Now assume that Odd will set q=0 and Even knows this. Calculate the Expected Outcome

Expected Outcome=-3p+4(1-p)

Expected Outcome=-7p+4

Find where the two expected outcomes are equal

5p-3=-7p+4

12p=7

p=7/12

How To Calculate the Optimal Mixed Strategy (for 2x2 Matrix)Provided Opponent plays optimally

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What if q is something between q=0 and q=0? say q=1/2.

Expected Outcome=p(1/2*2+1/2*-3)+(1-p)(1/2*-3+1/2*4)

Expected Outcome=1/2-p

How To Calculate the Optimal Mixed Strategy (for 2x2 Matrix)Provided Opponent plays optimally

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What if q is something between q=0 and q=0? say q=1/2.

Expected Outcome=p(1/2*2+1/2*-3)+(1-p)(1/2*-3+1/2*4)

Expected Outcome=1/2-p

How To Calculate the Optimal Mixed Strategy (for 2x2 Matrix)Provided Opponent plays optimally

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What if q is something else? say q=3/4.

Expected Outcome=p(3/4*2+1/4*-3)+(1-p)(3/4*-3+1/4*4)

Expected Outcome=2p-5/4

How To Calculate the Optimal Mixed Strategy (for 2x2 Matrix)Provided Opponent plays optimally

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What if q is something else? say q=3/4.

Expected Outcome=p(3/4*2+1/4*-3)+(1-p)(3/4*-3+1/4*4)

Expected Outcome=2p-5/4

How To Calculate the Optimal Mixed Strategy (for 2x2 Matrix)Provided Opponent plays optimally

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What if q is something else? say q=3/4.

Expected Outcome=p(3/4*2+1/4*-3)+(1-p)(3/4*-3+1/4*4)

Expected Outcome=2p-5/4

At This Point it Doesn't matter what q is.

This is the best I can do, if q is playing optimally.

How To Calculate the Optimal Mixed Strategy (for 2x2 Matrix)
Provided Opponent plays optimally

So what is the Expected Outcome when p=7/12

Just plug 7/12 back into one of the Expected Outcome formulas:

2*7/12-3*5/12 = -1/12

So if Even follows this Mixed policy he can expect to loose 1/12 of a dollar every game

That is the bad news. The good news is Odd can't do anything about it. No matter what the strategy (from q=0 to q=1) the expected outcome will be the same – PROVIDED Even doesn't know Odd's strategy

What if you know the others strategy (and its not optimal)

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2005-09-29T09:22:47