Principles of Game Theory

1. Principles of Game Theory Lecture 9: examples

2. Administrative • Homework 3 is up (you should have already started) and due at the beginning of class on Sunday. • We’ll (i.e., you’ll) be going through the problems in class • Exam 1: Tuesday • Open note, open book. Nothing electronic but feel free to bring a calculator

3. Last time • Mixed strategies • Two ways to think about mixed strategies; use whatever works for you: • Probabilities as beliefs about what the player will do • Randomization device

4. Mixed Strategies • Mixed strategies are just a probability distribution over the player’s strategy space • It can be any well formed probability distribution • Hence all pure strategies are mixed strategies played with probability 1 • The support of a mixed strategy σi is just the subset of the strategy space Si to which σiassigns positive support. • Support does not need to be the full Si • Nor does the support only need to include 2 strategies

5. Existence (Nash 1950) Every finite strategic form game has a mixed strategy equilibrium. • Finite means finite strategy space • Recall all extensive form games have a normal form representation. (Wilson 1971) Almost all finite games have an odd number of equilibria. • “Almost all” does have a technical meaning but in practice it means “all” • Use this theorem as a hint if you’re looking for all equilibria.

6. Finding all mixed strategy equilibria • How do you find mixed strategy equilibria? • 2 strategies: easy (we did it last time) • 3+ strategies: harder… • No general algorithm to find them other than brut force (without relying on other properties of the game at hand) • What do I mean by brut force? • What if the strategy space isn’t finite? • Maybe. Finite strategy space is a sufficient condition, not a necessary one, for existence.

7. Another example: car problems • Imagine your cousin has a problem with her car • Two types of problems: major and minor • Probability of a major problem is r, with 0 < r < 1 • An expert can tell what kind of problem it is • He can recommend either a major or minor repair • Major repairs cost more to fix. • He can also either be honest or not. • Your cousin doesn’t know what kind of problem or if he is being honest, only r • Your cousin can either accept the recommendation or seek another remedy • A major repair fixes both minor and major problems.

8. Car problems • Assume that your cousin always accepts the experts advice to obtain a minor repair • No reason to doubt a suggestion of a ‘minor repair’ • The same isn’t true for advice to repair a major repair. Why? • The expert might be honest • A major repair is needed for a major problem • Or the expert might be dishonest • She actually has a minor problem, doesn’t know it, and the extra money for the major repair is pocketed by the expert.

9. Car problems • Payoff assumptions • Expert’s profit is the same for minor and major repairs, if he is being honest: π • If he dishonest and sells a major repair to her for a minor problem, he gets π’ with π’ > π • Cousin must pay E for a major repair and I for a minor one • If she chooses some other remedy, she will pay E’> E if it’s a major problem and I’ > I if it’s a minor problem.

10. Car Problems • Model this as a game: • Expert: (Honest, Dishonest) • Cousin: (Accept, Reject) • <Honest, Accept> • With probability r cousin’s problem is major, so she pays E. With prob(1-r) it’s minor, so she pays I. Cousin’s expected payoff is -rE – (1-r)I. • Expert’s payoff is π

11. Car Problems • Model this as a game: • Expert: (Honest, Dishonest) • Cousin: (Accept, Reject) • <Dishonest, Accept> • The cousin’s payoff is –E. • Expert: the cousin’s problem is major with probr, yielding the expert π, and minor with prob (1-r) yielding a payoff of π’. Expected payoff is rπ +(1-r)π’

12. Car Problems • Model this as a game: • Expert: (Honest, Dishonest) • Cousin: (Accept, Reject) • <Honest, Reject> • Cousin: cost is E’ if problem is major and I if it’s minor; so expected payoff is r(–E’) + (1-r)(-I) • Expert: the expert only gets a payoff if it’s minor so his expect payoff is (1-r)π

13. Car Problems • Model this as a game: • Expert: (Honest, Dishonest) • Cousin: (Accept, Reject) • <Dishonest, Reject> • Cousin: will never accept his advice and faces a cost E’ if problem is major and I’ if it’s minor; so expected payoff is r(–E’) + (1-r)(-I') • Expert: doesn’t get any business and gets a payoff of 0.

14. Car problems • Putting those together: • Solve for the mixed strategy eq:

15. Car problems • After a fair amount of algebra… • Find a q to make the Expert indifferent between being Honest and Dishonest: Or • Find p to make Cousin indifferent Or

16. Comparative Statics • What happens to p* and q* as we change parameters of the problem? • First note when we have an eq • What happens if cars become more reliable (major problems are less likely)? • Rewrite s.t.

17. Comparative Statics • What happens if cars become more reliable (major problems are less likely)? • p* increases as r decreases • As major problems are less likely, experts are more honest • But your cousin is just as likely to follow his advice • Suppose major repairs become less expensive relative to minor ones (E decreases as E’ and I’ are held constant)? • p* decreases  when major repairs are less costly, experts are less honest. • Suppose profit π’ from fixing a minor problem while claiming a major problem decreases. • q* increases  cousin becomes less wary.