Evolutionary Games - PowerPoint Presentation

Slide1

Evolutionary Games

Econ 171Slide2

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 retreatSlide3

Strategic Form

Hawk

Dove

Hawk

-L, -L

V, 0

Dove0, VV/2, V/2

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 .Slide4

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.

)Slide5

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.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 equilibriumSlide6

A graphical view

V

V/2

1

0

-L

Hawk’s Payoff

Dove’s Payoff

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.

Equilibrium

Expected PayoffSlide7

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.Slide8

Two equivalent notions of ESS

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.Slide9

Dung Fly Games

One Minute

Two Minutes

Two daysSlide10

Strategic Form: Cowpat game

One Minute

Two

Minutes

One Minute

2,2

2,5 Two Minutes 5,2

1,1

Fly 1

Fly 2

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.Slide11

Expected payoffs

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.Slide12

Dung Fly Evolutionary Dynamics

1

2

5

0

Fraction of One-minute flies

1

1/4

Payoff to two-minute flies

Payoff to one-minute fliesSlide13

Evolutionary Dynamics of

Cooperative HuntingSlide14

Strategic Form

Cooperate

Defect.

Cooperate

4,4

1,3

Defect3,1

3,3

This game has only one pure strategy Nash equilibrium: Both DefectThis 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

equilibria

and no mixed strategy

Nash

equilibrium.

E) This game has no pure strategy Nash

equilibria

, but one mixed strategy

Nash equilibrium.Slide15

Finding 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 impliesP=2/3.Slide16

Dynamics of Hunting Game

0

1

1

4

3

Probability of Cooperate

2/3

Payoff to Defect

Payoff to Cooperate

What are the evolutionary stable states?Slide17

The Child Care Game

Slide18

Parental Roles in Different Species

Both care for offspring.Most but not all species of birds

Males also help build nest

Some Primates

Baboons

Humans

Not ChimpsWolvesSlide19

Male deserts, Female stays

Most vegetarian mammalsHorses, cows, goats, sheep,

Deer

Elephants,

Some Birds

Chickens, turkeys, some ducks

Cat FamilyLions, Tigers, House catsBearsPigsSlide20

Female deserts, Male stays

Sea horsesPenguinsEmusSlide21

Both parents desert

Most reptilesCuckoos…Most fishSlide22

Parental Care: An asymmetric Game

Stay

Desert

Stay

R,R

S

M,TF

Desert TM

, SF

P

M

,P

F

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

Why do male birds usually cooperate in child care?

Male

FemaleSlide23

Equilibria depend on payoffs.

Both cooperate is an equilibrium.R>T

M

and R>T

F

Male desert, female stay is an equilibrium

TM>R and R>TFFemale desert, male stay is an equilibriumTF>R and R>TM Both desertPM>SM and PF>SFSlide24

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

Stone

Paper

Scissors

Stone

1,1

-2,22,-2

Paper2,-21,1

-2,2Scissors

-2,2

2,-2

1,1

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.