Evolutionary Game Theory question

[kidsmoker]

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

$$E_{H,x}=-x_{H}+4x_{D}+x_{B}$$
$$E_{D,x}=2x_{D}+x_{B}$$
$$E_{B,x}=-0.5x_{H}+3x_{D}+2x_{B}$$.

For Q2.4b I normalize the payoff matrix to get

$$\[ \left( \begin{array}{ccc}<br /> 0 & 2 & -0.5 \\<br /> 1 & 0 & -1 \\<br /> 0.5 & 1 & 0 \end{array} \right)\]$$

Now comes the problems.

For an ESS we must have

$$E_{H,x}=E_{D,x}=E_{B,x}$$ (*)

By using the normalized matrix we can rewrite these as

$$E_{H,x}=2x_{D}-0.5x_{B}$$
$$E_{D,x}=x_{H}-x_{B}$$
$$E_{B,x}=0.5x_{H}+x_{D}$$.

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!

 

[kidsmoker]

Anyone? :-(