“If you don’t think the math matters, then you don’t know the right math.” Chris Ferguson 2002 World Series of Poker Champion
Prisoners You and I have been arrested for a bank robbery (of which we are guilty!), and they have evidence to convict us, but not enough for the maximum sentence. They decide to separate us and try to get each of us to confess and implicate the other one.
A Dilemma If we both stonewall (i.e. keep our mouths shut) then we will get 2 years in prison. If we both confess, we’ll each get 4 years in prison. But if only one person finks (confesses), that person will get 1 year in prison, while the person who stayed quiet gets 5 years!
Summary of Your Outcomes I Stonewall I Fink You Stonewall 2 years prison 5 years prison You Fink 1 year prison 4 years prison
A Dominant Strategy No matter what I choose, you’ll be better off having finked. But the same reasoning applies for me! So even though we’d both be better off had we both stonewalled (2 years prison each instead of 4), unless we can cooperate (i.e. make binding agreements), we are both forced to fink.
I Stonewall I Fink You Stonewall (-2,-2)(-5,-1) You Fink (-1,-5)(-4,-4) Summary of Both of Our Outcomes Equilibrium Outcome
A ‘game’ is any situation in which: There are at least two players Each player has a number of possible strategies The strategies chosen by each player determine the outcome of the game Associated to each outcome is a collection of numerical payoffs to each player
Payoffs in the Prisoner’s Dilemma: Me You The first number in each ordered pair indicates the payoff to the first player, whose strategy options correspond to the rows of the matrix. (-2,-2)(-5,-1) (-1,-5)(-4,-4) SFSF SFSF
Domination A strategy A dominates a strategy B if every outcome in A is at least as good (for that player) as the corresponding outcome in strategy B. Dominance Principle: A rational player should never play a dominated strategy.
A game of total conflict: The following table indicates payoffs for a game. Notice that the payoffs are the negative of each other, so what is good for you is bad for me, and vice versa! Me You (5,-5)(2,-2) (0,0)(-4,4) ABAB ABAB
Strategy A is dominant for You. Strategy B is dominant for Me. By the Dominance Principle, the outcome of this game should be AB for rational players. Me You (5,-5)(2,-2) (0,0)(-4,4) ABAB ABAB
Zero-Sum Games In the previous game, since we know that the payoffs for the second player are negatives of the payoffs for the first player, we could have described the outcomes with a matrix where each entry has payoffs for the first player only. Me You 52 0-4 ABAB ABAB
For the game below, what is your rational strategy? 137 646 621 ABCABC Me ABCABC
137 646 621 A B C Me ABCABC You The rational outcome is BB.
A Game without domination 39 648 82-2 A B C ABCABC Me You Goal: Guarantee yourself the least-worst result. Minimum 4 -2 maximin 8 4 9 maximum minimax
Saddle Points An outcome in a matrix game is called a saddle point if the entry at that outcome is both less than or equal to any entry in its row, and greater than or equal to any entry in its column. If the minimax column and maximin row are equal in payoff, the corresponding outcome is a saddle point. That payoff is called the value of the game.
Expected value This game does not have a dominant strategy or saddle point. But you happen to know that I am going to play strategy A half the time and Strategy B the other half of the time. What should you do? Me You 13 42 ABAB A B
Expected Value If you choose strategy A, your expected payoff will be If you choose strategy B, your expected payoff will be
Mixed Strategy What happens if I play strategy A one fourth of the time and strategy B the remaining three fourths of the time? Me You 13 42 ABAB A B
Mixed Strategy Your expected payoff for playing strategy A will be: Your expected payoff for playing strategy B will be:
Mixed Strategy By playing randomly with the correct weights for each strategy, I can guarantee that your strategy choice makes no difference. This is called a mixed-strategy on my part. You can even know exactly what my mixed strategy is, and it gives you no advantage!
Computing a Mixed Strategy Equilibrium I want to play strategy A with some weight p, and strategy B with weight 1-p. I want the expected payoffs to you to be independent of your strategy. So I need to solve the equation: The solution is p=1/4.
Utility Theory Payoffs (also called utilities by economists) can be selected by ranking outcomes in order of preference. The magnitude of these rankings need not matter, just the order. Thus the payoffs are called ordinal utilities.
Utility Theory If the magnitudes can be compared, then the payoffs are called cardinal utilities. The most common situation with cardinal utilities is when the payoffs correspond to money. Only cardinal utilities make sense with mixed strategies!
Multiple-move games It is X’s turn to act in Tic-Tac-Toe: What are X’s strategies? OOA BXC DXO
Bibliography Game Theory and Strategy, Philip D. Straffin For All Practical Purposes, COMAP The Mathematics of Poker, Bill Chen and Jerrod Ankenman Microeconomic Theory, Walter Nicholson Theory of Games and Economic Behavior, John von Neumann, et. al.