(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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 [Broken]

http://img686.imageshack.us/img686/7780/39557949.jpg [Broken]

3. 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!!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Evolutionary Game Theory question

**Physics Forums | Science Articles, Homework Help, Discussion**