We have seen that rule#1 (the equilibrium property) is a good way to determine whether ornot a strategy is a pure ESS (for an extended discussion of when rule #1works, press here).
Maynard Smith (1982)outlines a simple way to look for a pure ESS according to rule #1. Simplyinspect each column of payoffs (labeled A, B, C and D in the four-strategymatrix below). If the payoff to the strategy vs. itself is greater thanany other in the column, it is likely that that particular strategy is apure ESS. Notice that these reference payoffs for a particular strategyare on a diagonal from upper left to lower right; they are shown as darkenedcells 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 numbersin each cell with no real strategies in mind) for a pure ESS.
The blue column (strategy A) gives the payoffs to every strategy whenvs. A. Comparing all of these with E(A,A) we quickly see that strategy Dreceives 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 whenvs. 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 apopulation of C.
Finally, we can see that in the green column, strategy D is not a pureESS (comforting since we already had evidence that strategy C was).
! To recapitulate, the easy way to implement rule #1 isto look down each column and see if any of the values exceed the payofffor the strategy vs. itself; these are located on a diagonal from upperleft to lower right. One caveat -- there may be rare situations, especiallywhere a large number of strategies are involved where mixed ESSs or no ESSsolutions are possible even though this method predicts that a strategyis 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 hypothesizedsituation with respect to the frequencies of each strategy? ANS 2. In a three or more strategy game, will failure to find any pure ESSstrategy mean that the remaining strategies form a mixed ESS? ANS |
1. When using this method to find a pure ESS, whatis the hypothesized situation with respect to the frequencies of each strategy?
The strategy that heads the column (the one that everyone is playingagainst) is assumed to be very common and all of the alternativesare rare invaders. Thus, for the first column of the matrixabove, 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 strategyA. E(A,A) is used to see if A is stable against the invaders whose mostimportant payoffs are either E(B,A), E(C,A) or E(D,A). When moving to thenext column (headed by "B"), we now assume that B is the mostcommon and the others are all invaders etc.
2. In a three or more strategy game, will failureto find any pure ESS strategy mean that the remaining strategies form amixed ESS?
No. You will learn later in the hypertext materials that if there aremore than two strategies it is quite possible that no mixed or pure ESSexists. You will have a chance to demonstrate this to yourself with a simulationlater on.
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Copyright © 1999 by Kenneth N. Prestwich About FairUse of these materials Last modified 2 - 19 - 99 |