Question: Is it correct to talk about the fitness of either strategy in isolation -- that is, if only one strategy is present in the population, do the "fitness values" calculated by eqs above have any real meaning?
ANS: Recall that fitness is a relative measure -- these games have to do with competing strategies and so it makes no sense to even consider a situation where both are not at least potentially present. Remember that fitness is a relative measure because over the long run, individuals (strategies, genes) that leave more offspring (copies, whatever) of themselves come to dominate the population. If everyone has the same strategy, then, with respect to the evolution of this strategy, they enjoy equal fitness benefits or decrements and so there is no evolution with respect to this strategy. Thus, the fitnesses calculated from the payoff matrix and frequency of different strategies only have meaning in the context of competition. If you have problems with this, review the sections of fitness and competition.
One additional point however. It would be correct to assume that the smaller the fitness value calculated by either of these equations, the smaller the number of offspring to that strategy. Likewise, if the fitness calculation yielded a negative number, that would mean that the strategy would be declining from one generation to the next. While negative fitnesses could not go on indefinitely, if one strategy's fitness was less negative than the other, it would increase relative to the other even though overall, numbers are dropping!
ANS: The payoff to an individual playing B against one playing strategy A.
Question: If two alternative strategies make up a mixed ESS at frequencies of 0.8 for strategy A and 0.2 for strategy B, and if all individuals practice both A and B describe each individual's behavior.
ANS: Each individual plays A 80% of the time and B 20% of the time
Question: Explain the differences between a pure strategy and pure ESS. Between a mixed strategy and a mixed ESS.
ANS: A pure strategy is a set of behaviors that an individual will employ
in a given set of circumstances. A pure ESS is a single strategy that cannot
be invaded by any other known strategy.
A mixed strategy is one composed of several pure strategy components. Maynard Smith (1982) states that there is a random component in the organism's behavior (in terms of which behavioral component it will employ in a given situation). We saw examples of mixed strategies in two-strategy games that had no pure ESS and in the "war of attrition". By contrast, a mixed ESS involves more than one behavior making up an equilibrium. This could a mixed strategy or an equilibrium between individuals of different strategies.
Question: Write the conditions for strategy "B" being a pure ESS.
ANS: E(B,B)> E(A,B)
or if E(B,B) = E(A,B)
E(B,A) > E(A,A)
Question:Using the expression for B vs. A that you just wrote and the data in the payoff matrix:
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 |
explain whether or not B is stable against invasion by A.
ANS: First review the expressions above. From the payoff matrix, E(B,B) = 0.5 and E(A,B) = 1.0. Thus, B is not stable to invasion by A.
ANS: Yes provided that E(A,A) = E(B,A).
Question: If a strategy is not a pure ESS, does that mean that the opposing strategy is a pure ESS ?
ANS: Absolutely not -- remember that a pure ESS is not inevitable. In a two strategy game, if one strategy is not a pure ESS then you must test to see if the other is as well. If it isn't, then the solution is a mixed ESS. We will later see in three strategy games that if no pure ESS is found using the rules we have just learned then either a mixed ESS or no ESS at all are the possible solutions.
Question: -- Does it seem reasonable that hawks pay no cost in winning?
Discussion: Probably not. No costs would seem to imply a very brief contest with injuries only going to the loser. That implies a contest that is probably very asymmetrical -- the winner is able to quickly impress its superiority on the loser. Yet hawk vs. hawk contests are supposed to be symmetrical. It has frequently been documented (in many game theory based studies) that animals that are evenly matched tend to fight longer and injury is more likely to occur. In such cases, at least minor injuries would be expected even to the winner. Then there are energy, time and perhaps even predation costs that would be expected to be incurred. For instance in my lab, I have found that the costs of struggles in spiders are very high, especially when compared to walking or other more routine activities that might closely approximate displays. Moreover, these struggles involve anaerobic metabolism (which takes spiders a long time to recover from) and depletion of stores of compounds very important to rapid motion. And sometimes struggles can last for a considerable period of time.
Question: -- See if the Hawk Dove game:
Opponent |
||
Focal Strategy |
Hawk |
Dove |
Hawk |
-25 | +50 |
Dove |
0 | +15 |
meets the criteria for a pure ESS.
ANS: Define: Hawk A and Dove B.
E(A,A) is E(H,H) = -25,
E(B,A) is E(D,H) = 0
E(H,H) < E(D,H), therefore H is not an ESS
for D
E(D,D) = +15 and E(H,D) = +50
therefore D is not an ESS since E(D,D) < E(H,D)
NO PURE ESS
Question: In the list of cost and benefits:
Action |
Benefit or Cost (arbitrary units) |
| Gain Resource | + 50 |
| Lose Resource | 0 |
| Injury to Self | - 100 |
| Cost of Display to Self | - 10 |
it is assumed that injury costs are large compared to the payoff for gaining the resource. Give a situation where this relative weighing might accurately reflect the forces acting on an animal.
ANS: If the animal has a reasonable expectation of continued reproduction if it passes by the present fight for this particular resource, and if injuries are likely to be severe and lower significantly its future fitness, then C would be large compared to benefit. A young male elephant seal with little prospect of actually holding a section of beach occupied by females and great chance of injury against a larger experienced male would be a decent example of this sort of situation where C > B.
Question : Calculate the mixed ESS frequencies of Hawk and Dove using the payoff matrix .
ANS: h (freq of Hawk) = 0.58
thus:
freq of Dove = (1 - h) = 0.42.
Go to:
Copyright © 1999 by Kenneth N. Prestwich
About Fair Use of these materials Last modified 2 - 22- 99 |