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. • 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 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.
Dung Fly Games Two Minutes One Minute Two days
Strategic Form: Cowpat game 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 • 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 5 Payoff to two-minute flies Payoff to one-minute flies 2 1 0 1 1/4 Fraction of One-minute flies
Strategic Form 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 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 4 Payoff to Defect 3 Payoff to Cooperate 1 1 2/3 0 Probability of Cooperate What are the evolutionary stable states?
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 Chimps • Wolves
Male deserts, Female stays • 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 • Sea horses • Penguins • Emus
Both parents desert • Most reptiles • Cuckoos… • Most fish
Parental Care: An asymmetric Game 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 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: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.