We have seen that rule #1 (the equilibrium property) is a good way to determine whether or not a strategy is a pure ESS (for an extended discussion of when rule #1 works, press here).
Maynard Smith (1982) outlines a simple way to look for a pure ESS according to rule #1. Simply inspect each column of payoffs (labeled A, B, C and D in the four-strategy matrix below). If the payoff to the strategy vs. itself is greater than any other in the column, it is likely that that particular strategy is a pure ESS. Notice that these reference payoffs for a particular strategy are on a diagonal from upper left to lower right; they are shown as darkened cells in the matrix below:
Opponent's Strategy |
||||
| Focal Strat. | A |
B |
C |
D |
A |
E(A,A) = 0 |
E(A,B) = 0 |
E(A,C) = 0.5 | E(A,D) = 1.0 |
B |
E(B,A) = - 0.5 |
E(B,B) = 0.5 |
E(B,C) = 0 | E(B,D) = 0.5 |
C |
E(C,A) = -0.1 | E(C,B) = 1.0 | E(C,C) = 0.8 | E(C,D) = 0.5 |
D |
E(D,A) = 0.1 | E(D,B) = -0.1 | E(D,C) = 0 | E(D,D) = 0.8 |
Let's check this totally arbitrary set of payoffs (I simply put numbers in each cell with no real strategies in mind) for a pure ESS.
The blue column (strategy A) gives the payoffs to every strategy when vs. A. Comparing all of these with E(A,A) we quickly see that strategy D receives a greater payoff. Thus, according to rule #1, strategy A is not a pure ESS.
The yellow column (strategy B) gives the payoffs to every strategy when vs. B. Here strategy C does better against B than B does against itself (darkened cell). B is not a pure ESS.
The reddish column (strategy C) -- here we have a pure ESS since E(C,C) is better than any of the other common payoffs if they are invaders of a population of C.
Finally, we can see that in the green column, strategy D is not a pure ESS (comforting since we already had evidence that strategy C was).
! To recapitulate, the easy way to implement rule #1 is to look down each column and see if any of the values exceed the payoff for the strategy vs. itself; these are located on a diagonal from upper left to lower right. One caveat -- there may be rare situations, especially where a large number of strategies are involved where mixed ESSs or no ESS solutions are possible even though this method predicts that a strategy is a pure ESS. We will not need to worry about these. |
Questions: 1. When using this method to find a pure ESS, what is the hypothesized situation with respect to the frequencies of each strategy? ANS 2. In a three or more strategy game, will failure to find any pure ESS strategy mean that the remaining strategies form a mixed ESS? ANS |
1. When using this method to find a pure ESS, what is the hypothesized situation with respect to the frequencies of each strategy?
The strategy that heads the column (the one that everyone is playing against) is assumed to be very common and all of the alternatives are rare invaders. Thus, for the first column of the matrix above, A is common and B, C, and D are all assumed to be rare. Thus, A vs. A is the most important interaction to determine the fitness of strategy A. E(A,A) is used to see if A is stable against the invaders whose most important payoffs are either E(B,A), E(C,A) or E(D,A). When moving to the next column (headed by "B"), we now assume that B is the most common and the others are all invaders etc.
2. In a three or more strategy game, will failure to find any pure ESS strategy mean that the remaining strategies form a mixed ESS?
No. You will learn later in the hypertext materials that if there are more than two strategies it is quite possible that no mixed or pure ESS exists. You will have a chance to demonstrate this to yourself with a simulation later on.
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Copyright © 1999 by Kenneth N. Prestwich
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