Econ 171 The HawkDove 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 ID: 270586
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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.