Social Behavior: Evolutionary Game Theory
Matrix (Discrete) Games
General Rules for Solving
Example: Hawk-Dove Game
Hypothesis: Fitness Increases with Payoff
Solve: Evolutionarily Stable Strategy (ESS)
Game Theory
Economic Interaction
2 or More (N) “Players”
Each Has Behavioral Strategy
Assume Each Player’s Behavior
Affects Own and Other
Player’s Fitness
Game Theory
Model for Competition, Mutualism,
Reciprocity, Cooperation
Evolutionarily Stable Strategy
If Common, Repels All Rare
Mutants (Other Strategies)
ESS Theory
Population
Behavior = 2 Alleles
A Common, B Rare
Can B Invade A?
If Not, A is an ESS
ESS Theory
A Common, B Rare
B Does Not Invade A
Pure A:
Evolutionarily Stable
(Against B)
ESS Theory
A Common, B Rare
B Invades and
Excludes A
A Does Not Advance
When Rare
Pure B is an ESS
ESS Theory
A Common, B Rare
B Invades; A Persists
Equilibrium System
Mixed ESS
Polymorphism
Individuals Mix
ESS Theory
Payoff Matrix
Payoff to Player
Controlling Rows
Discrete Game,
Identical Players
(Symmetric)
ESS Theory
Evolutionarily Stable Strategy
Payoff Matrix: Symmetric Game
Payoff matrix: Player 1
Player 2 Action
A
B
Player 1
A
Player 1
B
Finding ESS
Finding ESS
ESS: Find p*
6
𝑀=
3
5
4
2
𝑀=
8
1
4
ESS: Find p*
6
𝑀=
3
4
5
Bistable: 2 ESS frequencies,
p* = 0 AND p* = 1
Diversity Among Populations
ESS: Find p*
3
𝑀=
5
5
2
No Pure ESS; 𝑝∗ ≠ 0, 1
Mixed ESS
𝑝∗ 3 + 1 − 𝑝∗ 5 = 𝑝∗ 5 + 1 − 𝑝∗ 2
3𝑝∗ − 5𝑝∗ + 5 = 5𝑝∗ − 2𝑝∗ + 2
−2𝑝∗ − 3 𝑝∗ = −3
𝑝∗ = 3/5
Diversity Within Populations