Evolutionary Games. Econ 171. The Hawk-Dove Game. Animals meet encounter each other in the woods and must decide how to share a resource. There are two possible strategies. Hawk: Demand the entire resource and be prepared to fight for it. Dove: Be willing to share or retreat.
Does this Game have a Pure Strategy Nash Equilibrium?
Both play Hawk is the only pure strategy Nash Equilibrium.
Both play Dove is the only pure strategy Nash Equilibrium.
There are two pure strategy Nash Equilibria.
There is no pure strategy Nash equilibrium .
Fraction of Hawks
When fraction of hawks is smaller than equilibrium, Hawks reproduce faster than
Doves. Fraction of Hawks grows.
When fraction of hawks is larger than equilibrium, Doves reproduce faster than Hawks,
Fraction of Doves grows.
Notion 1: Equilibrium is dynamically stable in the sense that a small number of mutants does worse than the equilibrium population and so mutation cannot invade.
Notion 2: Equilibrium is a symmetric Nash equilibrium (possibly in mixed strategies) such that no possible mutation does better against the equilibrium strategy and if it does as well as the equilibrium strategy against the equilibrium strategy, it does worse against itself.
This game has two pure strategy Nash equilibria.
This game has one pure strategy Nash equilibria where both use the
one minute strategy.
C) This game has no pure strategy Nash equilibria and no mixed strategy equilibria.
D) This game has no pure strategy Nash equilibria and one symmetric mixed
Strategy Nash equilibrium.
2=4p+1. That is: p=1/4.
Payoff to two-minute flies
Payoff to one-minute flies
Fraction of One-minute flies
This game has only one pure strategy Nash equilibrium: Both Defect
This game has only one pure strategy Nash equilibrium: Both Cooperate
This game has two pure strategy Nash equilibria and one mixed strategy
D) This game has two pure strategy Nash equilibriaand no mixed strategy
E) This game has no pure strategy Nash equilibria, but one mixed strategy
Payoff to Defect
Payoff to Cooperate
Probability of Cooperate
What are the evolutionary stable states?
Why do male zebras not help raise their babies? Why don’t female zebras desert?
Why do male birds usually cooperate in child care?
There are no pure strategy Nash equilibria. There is a unique symmetric mixed
Strategy equilibrium with probabilities (1/3,1/3,1/3).
All strategies do equally well against this strategy. But the strategy stone does better against itself than the strategy (1/3,1/3,1/3) does against stone. So a mutant stone population could invade.