Where Did the ESS Calculation Go Wrong in Evolutionary Game Theory?

[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

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

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!

 

[nate808]

Thank you for sharing your thoughts and approach to solving this problem. It seems like you have a good understanding of the concepts involved. However, I would suggest double-checking your calculations and assumptions to see if there may be any errors or inconsistencies that could be causing the discrepancy between your solution and the answer provided.

For Q2.4c, it is possible that the pure strategy B is the only ESS in this scenario. It is important to note that an ESS does not necessarily have to be an interior solution, as it can also be a pure strategy. In this case, it is possible that the pure strategy B is the only stable solution because it has a higher payoff than the other two strategies in all possible scenarios.

I would also recommend considering the possibility of mixed strategies in this scenario, as they may also play a role in determining the ESS. It is important to consider all possible strategies and their corresponding payoffs to fully understand the dynamics of this game.

Overall, your approach and calculations seem logical, so I would suggest double-checking your work and considering all possible solutions before concluding that there are no interior ESS's in this scenario.

Best of luck with your studies!

 

[Blueshift5]

I would first commend the student for their thorough analysis and attempt at finding an ESS (evolutionarily stable strategy) for the given payoff matrix. However, I would also suggest that the student double check their calculations and assumptions to ensure accuracy.

One potential issue could be in the normalization of the payoff matrix. It is possible that there was a mistake made in this step, which could lead to incorrect conclusions about the existence of an interior ESS.

Additionally, it is important to note that the conditions for an ESS are not just that the expected payoffs are equal, but also that the strategy is resistant to invasion by other strategies. This may require further analysis and consideration of different scenarios and potential strategies.

I would also suggest consulting with a colleague or professor for further guidance and discussion on this topic. Evolutionary game theory can be complex and it is always beneficial to have multiple perspectives and insights when tackling a problem.