A very little Game Theory

1. A very little Game Theory Math 20 Linear Algebra and Multivariable Calculus October 13, 2004

2. A Game of Chance • You and I each have a six-sided die • We roll and the loser pays the winner the difference in the numbers shown • If we play this a number of times, who’s going to win?

3. The Payoff Matrix • Lists one player’s (call him/her R) possible outcomes versus another player’s (call him/her C) outcomes • Each aij represents the payoff from C to R if outcomes i for R and j for C occur (a zero-sum game).

4. Expected Value • Let the probabilities of R’s outcomes and C’s outcomes be given by probability vectors

5. Expected Value • The probability of R having outcome i and C having outcome j is therefore piqj. • The expected value of R’s payoff is

6. Expected Value of this Game • A “fair game” if the dice are fair.

7. Expected value with an unfair die • Suppose • Then

8. Strategies • What if we could choose a die to be as biased as we wanted? • In other words, what if we could choose a strategyp for this game? • Clearly, we’d want to get a 6 all the time!

9. Flu Vaccination • Suppose there are two flu strains, and we have two flu vaccines to combat them. • We don’t know distribution of strains • Neither pure strategy is the clear favorite • Is there a combination of vaccines that maximizes immunity?

10. Fundamental Theorem of Zero-Sum Games • There exist optimal strategies p* for R and q* for C such that for all strategies p and q: E(p*,q) ≥ E(p*,q*) ≥ E(p,q*) • E(p*,q*) is called the value v of the game • In other words, R can guarantee a lower bound on his/her payoff and C can guarantee an upper bound on how much he/she loses • This value could be negative in which case C has the advantage

11. Fundamental Problem of Zero-Sum games • Find the p* and q*! • In general, this requires linear programming. Next week! • There are some games in which we can find optimal strategies now: • Strictly-determined games • 22 non-strictly-determined games

12. Network Programming • Suppose we have two networks, NBC and CBS • Each chooses which program to show in a certain time slot • Viewer share varies depending on these combinations • How can NBC get the most viewers?

13. Payoff Matrix

14. NBC wants to maximize NBC’s minimum share In airing Dateline, NBC’s share is at least 45 This is a good strategy for NBC NBC’s Strategy

15. CBS wants to minimize NBC’s maximum share In airing CSI, CBS keeps NBC’s share no bigger than 45 This is a good strategy for CBS CBS’s Strategy

16. (Dateline,CSI) is an equilibrium pair of strategies Assuming NBC airs Dateline, CBS’s best choice is to air CSI, and vice versa Equilibrium

17. Characteristics of an Equlibrium • Let A be a payoff matrix. A saddle point is an entry ars which is the minimum entry in its row and the maximum entry in its column. • A game whose payoff matrix has a saddle point is called strictly determined • Payoff matrices can have multiple saddle points

18. Pure Strategies are optimal in Strictly-Determined Games • If ars is a saddle point, then erT is an optimal strategy for R and es is an optimal strategy for C.

19. Proof

20. Proof

21. Proof • So for all strategies p and q: E(erT,q) ≥ E(erT,es) ≥ E(p,es) • Therefore we have found the optimal strategies QED

22. 2x2 non-strictly determined • In this case we can compute E(p,q) by hand in terms of p1 and q1

23. Optimal Strategy for 2x2 non-SD • Let • This is between 0 and 1 if A has no saddle points • Then

24. Optimal set of strategies • We have

25. Flu Vaccination

26. So we should give 2/3 of the population vaccine 1 and 1/3 vaccine 2 The worst that could happen is a 4:5 distribution of strains In this case we cover 76.7% of pop Flu Vaccination

27. Other Applications of GT • War • Battle of Bismarck Sea • Business • Product Introduction • Pricing • Dating