During the previous lecture we began thinking about Game Theory. We were thinking in terms of two strategies, A and B.

Transcription

1 During the previous lecture we began thinking about Game Theory. We were thinking in terms of two strategies, A and B. One way to organize the information is to put it into a payoff matrix Payoff to A B A E(A,A) E(A,B) B E(B,A) E(B,B) Where, for example, E(A,A) gives the expected payoff to A when interacting with A. 1

2 We could rewrite the matrix to read Payoff to A B A a b B c d Note that 1) if a>c, then A is and ESS 2) if d>b, then B is and ESS 3) Either A or B could be an ESS. The outcome of selection would then depend on initial conditions. Hence Natural selection does not necessarily result in the best strategy winning. Show this graphically. In addition, no claim is made that either A or B represent the optimal strategy. 2

3 4) If neither A nor B is an ESS, then selection is frequency-dependent. Both strategies will increase when rare under the action of natural selection, leading to a stable polymorphism. Since by definition, there is no Evolutionary Stable Strategy, we call this an Evolutionarily Stable State. It is a state of the population that is stable. We would like to know the frequency of A at the stable equilibrium point. How do we solve for that? Let q be the frequency of strategy A. Then 1-q is the frequency of strategy B. In the space below, show how you would solve for q at equilibrium, ˆq. Hint at equilibrium, the fitness of strategy A must be equal to the fitness of strategy B. 3

4 Now consider a mixed strategy such that a single individual could play either A or B. Let the name of the mixed strategy be I. (play RPS) Strategy I: play A with probability q and play B with probability (1-q) The Bishop-Cannings theorem: If I is an ESS, then at the ESS, the following is true (note: you don t need to know the math here, just the basic ideas as discussed in class.) E(A, I) = E(B, I) = E(I, I). Why is that true?* In addition, if I is an ESS, then E(I, A) > E(A, A), and E(I, B) > E(B, B). (i.e., condition b from the previous lecture is met. Why is this true? What does it mean?? *Hint: write out E(I, A) > E(A, A). Compare to previous page 4

5 Some definitions. Strategy: specification of what an organism will do in a given situation. Pure strategy: conditional (e.g. Bourgeois) or unconditional (e.g. Hawk) fixed response. The strategy is pure in the sense that the behavior can be specified in advance. For example, the conditional strategy, Bourgeois, can be specified in advance: play hawk if owner, play dove if intruder. Mixed strategy: stochastic strategy (e.g., play Hawk with probability q and Dove with probability 1- q. The strategy is mixed in the sense that only the probability of playing hawk can be specified; the exact behavior cannot be exactly predicted. The best example of a mixed strategy might be the Rock- Paper-Scissors game, where rock is played one third of the time; paper is played one third of the time, and scissors in played one third of the time. 5

6 THE HAWK-DOVE GAME We assume that two individuals pair up at random in a large population. Hawks will always fight for the contested resource. Doves will retreat from Hawks, and split the resource with other doves. Let V = the Value of the resource. We assume V>0. Let C = the Cost of fighting Here is the payoff matrix Payoff to Hawk Dove Hawk E(H,H) E(H, D) Dove E(D,H) E(D, D) Payoff to Hawk Dove Hawk (V-C)/2 V Dove 0 V/2 6

7 Okay, now answer the following questions: 1. When is Hawk an ESS? 2. When is Dove an ESS? 3. When would the population be polymorphic at equilibrium? What would be the frequency of hawks at equilibrium? 4. When would a mixed strategy be favored by selection? What would be the probability that an individual plays Hawk in this mixed strategy at equilibrium? Now consider a new strategy, Bourgeois. Bourgeois behaves like a hawk if it is the owner of the territory (e.g. sunspot), and it behaves like a dove, if it is the intruder on the territory. To get the simplest payoff matrix, we assume that Bourgeois is the territory owner ½ of the time, and B is the intruder ½ of the time. Payoff to Hawk Dove Bourgeois Hawk (V-C)/2 V 0.5[((V C) / 2) + V ] Dove 0 V/2 V/4 Bourgeois (V-C)/4 3 4 V V/2 Convince yourself that the new gray entries are correct. Then show that Bourgeois is the only ESS if C>V. 7

8 Calcs for B when Calcs for B when Payoff to Hawk Dove Bourgeois B is owner B is intruder Hawk (V-C)/2 V 0.5[((V C) / 2) + V ] ½E(H, H B ) ½E(H, D B ) Dove 0 V/2 V/4 ½E(D, H B ) ½E(D, D B ) Bourgeois (V-C)/4 3 4 V V/2 ½E(H B, D B ) ½E(D B, H B ) calcs for B when B is owner calcs for B when B is intruder ½E(H B, H) ½E(H B, D) ½E(D B, H) ½E(D B, D) Payoff to Hawk Dove Bourgeois Hawk (V-C)/2 V ½E(H, H B ) + ½E(H, D B ) Dove 0 V/2 ½E(D, H B ) + ½E(D, D B ) Bourgeois ½E(H B, H) +½E(D B, H) ½E(H B, D) +½E(D B, D) ½E(H B, D B ) + ½E(D B, H B ) Calculations for B are in this blue color. Note: H B means Bourgeois play hawk. D B means Bourgeois play dove. 8

9 à Class discussion of Davies (1978) paper on Resident always wins. (Lecture 8.) The logic of animal conflict: Davies 1978 Asymmetric contests: Larger animal wins Bigger payoff to owner Uncorrelated Asymmetries (Maynard Smith and Parker) Contest settled by cues unrelated to fighting ability Escalation occurs with cues are imperfect Observations 1:1 sex ratio Males move less, and stay in sunspots Short spiral flights with conspecific males 62% of males are in sunspots on average Males sometimes drop down from canopy to spots. 9

10 Hypothesis: Spiral flights are a form of territory defense Exp 1: Remove males from sunspots. Result: New males rapidly colonize Question: why do males want sunspots? Observations from 7 m up in trees. Question: what is the probability of a canopy male getting a sunspot during the day? Question: What is the effect of sunspot size? A. More males in bigger sunspots B. Profitability of larger sunspots per male is the same as small sunspots. Question: What happens if you remove a male? Question: What happens if two males think they are the owner? Should we accept this paper for publication? 10