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

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?

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

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

Expected Value  The probability of R having outcome i and C having outcome j is therefore p i q j.  The expected value of R’s payoff is

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

Expected value with an unfair die  Suppose  Then

Strategies  What if we could choose a die to be as biased as we wanted?  In other words, what if we could choose a strategy p for this game?  Clearly, we’d want to get a 6 all the time! C’s outcomes R ’s outcomes

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? Strain 12 Vaccine

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

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

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?

Payoff Matrix CBS shows 60 Minutes Survivor CSI Everybody Loves Raymond NBC Shows Friends Dateline Law & Order

NBC’s Strategy  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 60 M Surv CSI ELR F DL L&O

CBS’s Strategy  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 60 M Surv CSI ELR F DL L&O

Equilibrium  (Dateline,CSI) is an equilibrium pair of strategies  Assuming NBC airs Dateline, CBS’s best choice is to air CSI, and vice versa 60 M Surv CSI ELR F DL L&O

Characteristics of an Equlibrium  Let A be a payoff matrix. A saddle point is an entry a rs 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

Pure Strategies are optimal in Strictly-Determined Games  If a rs is a saddle point, then e r T is an optimal strategy for R and e s is an optimal strategy for C.

Proof

 So for all strategies p and q: E(e r T,q) ≥ E(e r T,e s ) ≥ E(p,e s )  Therefore we have found the optimal strategies

2x2 non-strictly determined  In this case we can compute E(p,q) by hand in terms of p 1 and q 1

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

Optimal set of strategies  We have

Flu Vaccination Strain 12 Vaccine

Flu Vaccination Strain 12 Vaccine  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

Other Applications of GT  War  Battle of Bismarck Sea  Business  Product Introduction  Pricing  Dating