Linear Programming Notes IX Two Person Zero Sum Game Theory 1 Introduction Economists use the word rational in a narrow way To an economist a rational actor is someone who makes decisions that maximize her or his preferences subject to constraints imposed by the environment So this actor knows her preferences and knows how to go about optimizing It is a powerful approach but it probably is only distantly related to what you mean when you think of yourself as rational Decision theory describes the behavior of a rational actor when her actions do not influence the behavior of the people around her Game theory describes the behavior of a rational actor in a strategic situation Here decisions of other actors determine how well you do Deciding where to go to dinner can be thought of as a decision problem if all you care about is what you eat and where you eat it It is a strategic problem if you also want to meet a friend at the restaurant In the first case you go to the restaurant that serves the food you like best In the second case the restaurant that you prefer depends not only on the food served but also on the where your friend goes 2 Zero Sum Games These notes describe a simple class of games called two player zero sum games You can probably figure out what a two player game is Zero sum games refer to games of pure conflict The payoff of one player is the negative of the payoff of the other player This formulation is probably appropriate for most parlor games where the outcomes are either win lose or draw and there is at most one winner or loser Maybe it describes war It is a restrictive assumption and is not appropriate to most economic applications where there is a strong component of common interests mixed with the conflict For example in a bargaining situation the conflict is clear the buyer wants to pay a low price and the seller wants to receive a high price The cooperative element arises because it is frequently the case that making a transaction at an intermediate price is better for both sides than a failure to reach an agreement Concretely if something is worth 10 to the seller and 15 to the potential buyer then making a sale at the price 12 or any price between 10 and 15 is better for both buyer and seller than making no sale at all Problems that describe aspects of firm competition models of Cournot duopoly that you may have seen in a micro class have non zero sum aspects Why limit attention to zero sum games They are simpler There is a beautiful theory that is more compelling than the general theory of games Predicting outcomes in these games uses linear programming in ways that do 1 not generalize to other kinds of game The general structure of a game involves a list of players a set of strategies for each of the players a payoff for each vector of strategies I will assume that the game has only two players 3 Strategies The intuition behind a strategy is that it tells you how you are going to play the game In examples it will be just a choice from one of a finite list of possible things you can do This story might help you understand the notion of a strategy You made an arrangement to talk to a friend about what you were going to do together but you unexpectedly cannot be home when the friend is supposed to call Your roommate will be home and promises to talk to your friend You want to give your roommate instructions about what kind of arrangements to make You would like to walk on the beach but not if it is going to rain You would like to go to the Belly Up but only if you can dance You would like to see a movie but only if Leonardo DiCaprio isn t in it Most of all you would like to do something that your friend also wants to do What kind of instructions do you give your roommate Complete instructions will account for all possible contingencies You won t say Tell my friend that I ll do whatever he or she wants to do Instead you ll say something like If she wants to go to a movie find out if DiCaprio is in it If he isn t tell her OK If he is tell her no And so on In game theory a strategy is a complete set of instructions It allows your roommate to negotiate for you no matter what your friend on the phone says When you specify a strategy for each player you determine the outcome of the game Payoffs associate to each outcome a number for each player You can therefore describe two player games using a payoff matrix The rows of the matrix represent the strategies of one player The columns of the matrix represent the strategies of the other player The cells of the matrix represent outcomes In these cells you place payoff numbers In general each cell should have a payoff for each player in it In zero sum games you need only have one number in each cell This number represents the payoff to the player who picks rows The negative of this number is the payoff to the player who picks columns Take the game of matching pennies Two players simultaneously place a penny on the table If the pennies match both heads up or both heads down then the Row player wins the Column player s penny If the pennies do not match exactly one head then the Column player wins the Row player s penny The payoff matrix is below Heads Tails Heads 1 1 2 Tails 1 1 In matching pennies each player has two strategies The player can either play heads or play tails Now consider a variant of matching pennies that I play with my son First I decide whether to play heads or tails Next he looks at what I did Finally he decides whether to play heads or tails I win if the coins match He wins if they do not In this game both players must decide whether to play heads or tails So you might think that we both have two strategies This is not correct I have two strategies but my son can make his decision based on what I did He therefore has four strategies HH Play heads no matter what I do T T Play tails no matter what I do HT Play heads if I play heads and tails if I play tails match T H Play tails if I play heads and heads if I play tails mismatch Therefore the payoff matrix for this version of matching pennies is Heads Tails HH 1 1 TT 1 1 HT 1 1 TH 1 1 Naturally my son plays T H and I always lose The point is that even though my son ends up either playing heads or …