- #1
kidsmoker
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Homework Statement
Quite a long intro to the question so I thought it easier to include it as an image:
http://img96.imageshack.us/img96/7264/78941753.jpg
http://img686.imageshack.us/img686/7780/39557949.jpg
The Attempt at a Solution
I can do Q2.3 and get the payoff matrix given when V=4 and C=6.
For Q2.4a I get
[tex]E_{H,x}=-x_{H}+4x_{D}+x_{B}[/tex]
[tex]E_{D,x}=2x_{D}+x_{B}[/tex]
[tex]E_{B,x}=-0.5x_{H}+3x_{D}+2x_{B}[/tex].
For Q2.4b I normalize the payoff matrix to get
[tex]\[ \left( \begin{array}{ccc}
0 & 2 & -0.5 \\
1 & 0 & -1 \\
0.5 & 1 & 0 \end{array} \right)\][/tex]
Now comes the problems.
For an ESS we must have
[tex]E_{H,x}=E_{D,x}=E_{B,x}[/tex] (*)
By using the normalized matrix we can rewrite these as
[tex]E_{H,x}=2x_{D}-0.5x_{B}[/tex]
[tex]E_{D,x}=x_{H}-x_{B}[/tex]
[tex]E_{B,x}=0.5x_{H}+x_{D}[/tex].
Let x=(h,d,b) be our interior ESS, then by (*) we have
2d - 0.5b = 0.5h + d and h - b = 0.5h + d .
The first of these can be rearranged to give h=2d-b while the second can be rearranged to give h=2d+2b. Clearly these can only both be satisfied when b=0. But this contradicts the fact that x=(h,d,b) is an interior ESS. Hence there can be no interior ESS's.
Now that seemed correct to me, but it doesn't tie-in with Q2.4c. This question claims that the only ESS is the pure strategy B. By considering the H-D subgame I get an ESS at (2/3,1/3,0).
Assuming the question is written correctly, where am I going wrong?
Thanks for any help!
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