# Hawk-Dove Game

```Social Behavior &amp; Game Theory
Social Behavior:
Define by Economic Interaction
Individuals Affect Each Other’s Fitness,
Predict by Modeling Fitness Currency
Interactions Competitive
Game Theory, ESS
Social Behavior &amp; Game Theory
Example: Hawk-Dove Game
Evolution of Diversity in Aggressive Behavior
Pure ESS, Mixed ESS
Social Behavior &amp; Game Theory
Suppose 2 Randomly Selected
Individuals Encounter a Resource Item
Resource Benefits Survival or Reproduction
Potential Mate, Food, …
How Will They Interact?
Hawk-Dove Game
Hawk:
Always Fights for Resource
Chance for Benefit
May Lose, May Be Injured
Cost of Aggression
Hawk-Dove Game
Dove:
Never Fights for Resource
Withdraws from Hawk
Divides Resource with Dove
No (Cost of) Aggression
Hawk-Dove Game
Among-Species Diversity in Aggression
Within-Species Diversity in Aggression
Why?
Functional Significance
Hawk-Dove Game
Basic Model: Action Set = {Hawk, Dove}
Payoffs: Benefit of resource &gt; 0,
Cost of aggression &gt; 0
“Alleles” H, D
p = Freq(H), 1- p = Freq(D)
Hawk-Dove Game
Payoff
Matrix
H
D
H
(B/2) - C
B
D
0
B/2
Hawk-Dove Game
Suppose Hawk Common:
Evolutionarily Stable?
E(H, H) &gt; E(D,H)  ESS p* = 1
E(H, H) &gt; E(D,H)  (B/2) – C &gt; 0
Cost Aggression &lt; Benefit/2, Pure Hawk is ESS
Hawk-Dove Game
Suppose Dove Common:
Evolutionarily Stable?
E(D,D) &gt; E(H,D)  ESS p* = 0
E(D,D) &gt; E(H,D)  (B/2) &gt; B Impossible
Pure Dove, p* = 0, Never ESS
Hawk-Dove Game
Suppose C &gt; B/2
E(H,H) &lt; E(D,H) and Dove invades Hawk
Know B &gt; B/2 so Hawk Invades Dove
Mixed ESS, 0 &lt; p* &lt; 1
ESS: Nash Solution with Equal Payoffs
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
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 &gt; B/2
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*
```