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##### Game Dynamics

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**Game Dynamics**Econ 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?**Predator-Prey Arms Race**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**1 0.5 1 0.5 0 p 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**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**1 .7 p 1 0 .6 b**A winding path**1 .7 p 1 0 .6 b**Maybe it winds in**1 .7 p 1 0 .6 b**Or maybe it winds out**1 .7 p 0 1 .6 b**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!