Game DynamicsEcon 171

Two Population Games

A Predator

Why Does He Need to be so fast?

Here’s Why

The Prey

Why does he need to be so fast?

Speed is costly

Predator-Prey Arms Race

Fast Slow

Fast 1,1 2,0Slow 0,2 3,3

Cheetah

Gazelle

What are the symmetric Nash equilibria of this game?

ESS and the Payoffs to speed

• Let p be the fraction of gazelles who are fast and 1-p be the fraction who are slow. Let q be the fraction of cheetahs who are fast and 1-q the fraction who are slow.

• Expected payoffs to gazelles– Fast: q1+(1-q)2=2-q– Slow: q0+(1-q)3=3-q– Fast is better if q>1/2. Slow is better if q<1/2.

• Expected payoffs to cheetahs– Fast: p1+(1-p)2=2-p– Slow: p0+(1-p)3=3-p– Fast is better if p>1/2. Slow is better if p<1/2.

Two-population games

• In last lesson, we had a single population and just two strategies. There was just one variable that evolved over time—the fraction p of population doing strategy A. (The fraction doing strategy B would be just 1-p.)

• This time we have two populations, say predator and prey. Even if each population has only two possible strategies there are two variables to solve for.—fraction of population 1 doing strategy A and fraction 1 of population 2 doing strategy A.

• This requires a new kind of diagram—a two dimensional phase diagram.

Phase diagram of dynamics

p0 1

1

0.5

0.5

So what is an ESS for this game? What does it predict about predators and prey?

Handedness in baseball

• Batters are on average more successful against pitchers who are opposite-handed from themselves.

• In the major leagues of the US, fractions of both pitchers and batters who are left-handed grew from about 15% to about 35% between 1875 and 1985.

• Think of an evolutionary process in which only the most successful are selected by managers to play in the major leagues.

Payoff matrix

Right Handed Left HandedRight Handed 30,30 44,24Left Handed 36,21 30,30

Pitcher Type

Batter Type

Are there any symmetric pure strategy Nash equilibria? How about symmetric mixed strategy equilibrium?

Performance of Left and Right

• Let p be the fraction of pitchers who are right-handed and b the fraction of batters who are right-handed.

• Expected payoff to batters– Right-handed batters: p30+(1-p)44=44-14p– Left-handed batters: p36+(1-p)30=30+6p– Right handers do better if p<7/10. Right handers do better if

p>7/10.• Expected payoff to pitchers

– Right-handed pitchers: 30b+21(1-b)=21+9b– Left-handed pitchers: 24b+30(1-b)=30-6b– Right handers do better if b>3/5. Left handers do better if b<3/5.

Dynamic Phase diagram

b

p

0

1

1.6

.7

A winding path

b

p

0

1

1.6

.7

Maybe it winds in

b

p

0

1

1.6

.7

Or maybe it winds out

b

p

0

1

1.6

.7

Depends on the details

• If it winds in, it eventually gets and stays close to mixed equilibrium

• Textbook example winds out. (Author does numeric simulation to show this.)

• This means that there is a never ending cycle with fluctuations of the two types.

• General conditions that determine what happens are studied in theory of differential equations.

What is missing from the model?

• Recent history suggests that proportions of left and right handed batters and pitchers has settled down.

• Textbook example predicts continuing large fluctuations.

• What real world features are missing from the model that might settle this down?

Final Exam

• Will cover entire course.• Read over text. Look at problems. Look at

lecture notes.• You probably will not be as rushed as you

were on the second midterm.

So long…

Enjoy the game of life!