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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.

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### 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

Strategic Form

Creature 2

Creature 1

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 .

Another interpretation

- There is a population of animals in the woods. Some are hardwired to play hawk. Some are hardwired to play dove.
- The number of babies any animal has is determined by its payoff in the games it plays.
- When babies grow up, they play as their parent did. (True story is a little more complex. ) Remember your h.s. sex education classes. )

Expected Payoffs

- Suppose that the fraction of Hawks in the woods is p and the fraction of Doves is 1-p.
- Who does better?
- Expected payoff of a hawk is
-Lp+V(1-p)=V-p(L+V).

- Expected payoff of a dove is
0p+(V/2)(1-p)=V/2-pV/2.

- Expected payoff of a hawk is
- Two types do equally well when
V-p(L+V)=V/2-pV/2

- This implies p=V/(2L+V).
- That’s a symmetric mixed strategy equilibrium

A graphical view

Expected Payoff

V

Hawk’s Payoff

V/2

Dove’s Payoff

Fraction of Hawks

1

0

Equilibrium

-L

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.

Evolutionary stable strategy in a symmetric two-player game with 2 strategies

- Two interpretations of mixed strategy equilibrium.
- All individuals use same mixed strategy with same probability p of doing strategy 1.
- All individuals use pure strategies. The fraction p use strategy 1. The fraction 1-p use strategy 2.
- Both interpretations lead to actions being taken in the same proportions.

Two equivalent notions of ESS with 2 strategies

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.

Strategic Form: Cowpat game with 2 strategies

Fly 2

Fly 1

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.

Expected payoffs with 2 strategies

- Let p be the probability that other fly is a one minute fly.
- Expected payoff to one minute strategy is
2p+2(1-p)=2.

- Expected payoff to two minute strategy is 5p+(1-p)=4p+1.
- Symmetric mixed strategy equilibrium if
2=4p+1. That is: p=1/4.

Dung Fly Evolutionary Dynamics with 2 strategies

5

Payoff to two-minute flies

Payoff to one-minute flies

2

1

0

1

1/4

Fraction of One-minute flies

Evolutionary Dynamics of with 2 strategies Cooperative Hunting

Strategic Form with 2 strategies

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

Nash equilbrium.

D) This game has two pure strategy Nash equilibriaand no mixed strategy

Nash equilibrium.

E) This game has no pure strategy Nash equilibria, but one mixed strategy

Nash equilibrium.

Finding with 2 strategies equilibria.

- There are two pure strategy Nash equilibria.
- What about mixed strategy equilibrium?
- Suppose that each cooperates with probability p.
- Expected payoff to cooperating is 4p+(1-p)=1+3p.
- Expected payoff to defecting is 3.
- Mixed strategy N.E. has 1+3p=3, which implies
P=2/3.

Dynamics of Hunting Game with 2 strategies

4

Payoff to Defect

3

Payoff to Cooperate

1

1

2/3

0

Probability of Cooperate

What are the evolutionary stable states?

The Child Care Game with 2 strategies

Parental Roles in Different Species with 2 strategies

- Both care for offspring.
- Most but not all species of birds
- Males also help build nest
- Some Primates
- Baboons
- Humans
- Not Chimps

- Wolves

Male deserts, Female stays with 2 strategies

- Most vegetarian mammals
- Horses, cows, goats, sheep,
- Deer
- Elephants,

- Some Birds
- Chickens, turkeys, some ducks

- Cat Family
- Lions, Tigers, House cats

- Bears
- Pigs

Female deserts, Male stays with 2 strategies

- Sea horses
- Penguins
- Emus

Both parents desert with 2 strategies

- Most reptiles
- Cuckoos…
- Most fish

Parental Care: An asymmetric Game with 2 strategies

Female

Male

Why do male zebras not help raise their babies? Why don’t female zebras desert?

Why do male birds usually cooperate in child care?

Equilibria with 2 strategies depend on payoffs.

- Both cooperate is an equilibrium.
- R>TM and R>TF

- Male desert, female stay is an equilibrium
- TM>R and R>TF

- Female desert, male stay is an equilibrium
- TF>R and R>TM

- Both desert
- PM>SM and PF>SF

A game with no ESS: with 2 strategies Stone, paper scissors with a premium for ties

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.

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