Social Behavior & Game Theory

1. Social Behavior & Game Theory Social Behavior: Define by Economic Interaction Individuals Affect Each Other’s Fitness, Predict by Modeling Fitness Currency Interactions Competitive Game Theory, ESS

2. Social Behavior & Game Theory Example: Hawk-Dove Game Evolution of Diversity in Aggressive Behavior Common Strategy, Rare Mutant (Invade?) Pure ESS, Mixed ESS

3. Social Behavior & Game Theory Suppose 2 Randomly Selected Individuals Encounter a Resource Item Resource Benefits Survival or Reproduction Potential Mate, Food, … How Will They Interact?

4. Hawk-Dove Game Hawk: Always Fights for Resource Chance for Benefit May Lose, May Be Injured Cost of Aggression

5. Hawk-Dove Game Dove: Never Fights for Resource Withdraws from Hawk Divides Resource with Dove No (Cost of) Aggression

6. Hawk-Dove Game Among-Species Diversity in Aggression Within-Species Diversity in Aggression Why? Functional Significance When is Aggression Adaptive?

7. Hawk-Dove Game Basic Model: Action Set = {Hawk, Dove} Payoffs: Benefit of resource > 0, Cost of aggression > 0 “Alleles” H, D p = Freq(H), 1- p = Freq(D)

8. Hawk-Dove Game

9. Hawk-Dove Game Suppose Hawk Common: Evolutionarily Stable? E(H, H) > E(D,H)  ESS p* = 1 E(H, H) > E(D,H)  (B/2) – C > 0 Cost Aggression < Benefit/2, Pure Hawk is ESS

10. Hawk-Dove Game Suppose Dove Common: Evolutionarily Stable? E(D,D) > E(H,D)  ESS p* = 0 E(D,D) > E(H,D)  (B/2) > B Impossible Pure Dove, p* = 0, Never ESS

11. Hawk-Dove Game Suppose C > B/2 E(H,H) < E(D,H) and Dove invades Hawk Know B > B/2 so Hawk Invades Dove Mixed ESS, 0 < p* < 1 ESS: Nash Solution with Equal Payoffs

12. Hawk-Dove: Frequency-Dependence Payoff to H = p E(H,H) + (1 - p) E(H,D) p[(B/2) – C] + (1 – p) B Payoff to D = p E(D,H) + (1 – p) E(D,D) p(0) + (1 – p) B/2

13. Hawk-Dove Mixed ESS p[(B/2) – C] + (1 – p) B = p(0) + (1 – p) B/2 p B + B/2 – p (B + C) = 0 p* = B/2C when C > B/2

14. Mixed ESS Behavioral Diversity: 2 Variants 1. Between-Phenotype Variation p* Hawks, (1 – p*) Doves 2. Within-Phenotype Variation Each Individual Has Same Mixed Strategy Play H with Probability = p*