General Equations for fitness of a strategy in a two-strategy game. Here the strategies are C and S at frequencies c and s where s = 1 - c
| eq. 7b: W(C) = E(C,C) * c + E(C, S) * (1 - c) |
A similar calculation can be made for the fitness of satellite:
| eq. 7c: W(S) = E(S,C) * c + E(S,S) * (1 - c) |
Reference: Matrix Needed for Solution to Problem:
? Using the expression for B vs. A that you just wrote and the matrix below, explain whether or not B is stable against invasion by A. (ANS)
Opponent's Strategy |
||
| Focal Strategy | A |
B |
A |
E(A,A) = 0 | E(A,B) = 1 |
B |
E(B,A) = - 0.5 | E(B,B) = 0.5 |
? REVIEW OF PROBLEM (from earlier in text) DEALING WITH FREQUENCY OF INTERACTIONS: This simple problem is to illustrate the assumptions we made about the frequency of various contests in our population mainly composed of A strategists. Assume that the frequency of strategy A is 0.9999. ....
Will the proportion of the total number of payoffs to A when vs. B be any different than the proportion of the total number of payoffs to B when vs. A?
ANS: From the last two calculations above, you can see that the total frequency of those payoffs in the entire population is equal. But that is not what matters when considering whether or not A or B are pure strategies. We need to know how common each particular interaction is. And that is simply given by the frequency of the strategy with which the focal strategy interacts. OK, let's see what this means:
For payoffs to A: 99.99% of them will be with other A strategists and 0.01% will be with B strategists. Thus, 9999X more interactions will occur against A; the B interactions would not seem to be very important.
For payoffs to B: once again, 99.99% of them will be with other A strategists and 0.01% will be with B strategists.
Thus, the important payoffs for calculating the fitness of A and B respectively when B is rare are E(A,A) and E(B,A) which account for 99.99% of the interactions for both strategists!
Opponent |
||
| Focal Strategy | Call |
Satellite |
Call |
E(C,C) | E(C,S) |
Satellite |
E(S,C) | E(S,S) |