Reference Materials to the Bishop-Cannings Proof

Here are the rules we learned earlier (on the introduction to game theory page):

Summary of rules for Finding a Pure ESS in a Two Strategy Game: Assumption -- one strategy is very rare compared to the other. In this example, let A be the common strategy and we will determine whether or not it is immune from invasion by B. Rule #1: E(A,A)> E(B,A) or if: Rule #2: E(A,A) = E(B, A) and E(A,B) > E(B,B) There is a method that uses rule #1 to find an ESS by inspection of a payoff matrix. Click here to see how to use this method.

For our purposes, assume that mixed strategy is the common strategy (since it is inclusive) and a is the less common one (since it is in support of the mixed strategy var). As long as the payoffs to the common interactions favor mix -- i.e.,:

E(mix,mix) > E(a, mix)

then mix cannot be displaced by one of its supporting strategies.

Again, remember that E(mix,mix) is the payoff to a mix strategist who is playing some strategy in support of mix (including a) when vs. any other mix strategist (also playing some strategy in support of the mix).

Review of why expression 2a is false. Recall eq. 2a:

2a. E(a, mix) > E(mix, mix)

It is easy to dismiss statement 2a: if mix is an ESS (as we stipulated at the start) then according to rule #1 for finding an ESS, then statement 2a cannot be true (press here to review rule #1 in this context).