Rationality, Choice and Values

1. Rationality, Choice and Values • In Game Theory we assume that players make rational choices based on the payoffs in the game. The assumption is that choices of rational players are predictable. • Real players don’t always make the choices that game theory predicts for a variety of reasons • Payoffs may not accurately reflect all values that players associated to a certain outcome (eg money, equity, pride etc) • Players may not understand the implications of their choices, or do not play logically (they play irrationally) • The game may require a mixed strategy that we are not able to accurately compute without dice. • The rational strategy may lead to a sub-optimal choice which we do not like! • Two different rational arguments may conflict!

2. Newcomb’s Problem and Freewill • Game theory requires us to make predictions about what other players will do. To what extend are our choices really predictable? Do we have free will? • Newcomb’s Problem: • Suppose a being (God, an alien, or an intelligent psychologist) claims that they can predict what you will do and proposes the following game: • There are two boxes: • Box A is either empty or contains $1000 depending on what the being predicts you will do. • Box B contains $100. • Your choices: • pick box A • pick both boxes. • The being: • If she predicts you will pick both she puts $0 in box A. • If she predicts you will pick only box A she puts $1000 in box A

3. Payoffs for Newcomb’s problem Payoff matrix, with your payoffs What should you pick? Argument #1: When I make my choice the money is already in the boxes. Regardless of what she predicted I will get more if I pick both then if I pick A. Pick both. Argument #2: If the being has any predictive powers she will know if I choose to pick both, so I will get $100. But if she does have predictive powers, then if I chose A she would predict that and I would get $1000. $1000 is better than $100 . Pick A.

4. Solution Both arguments are rational arguments. Are they in conflict? How accurately does the being have to be able to predict in order for you to follow argument #2? Suppose the being is able to predict accurately with probability p. Expected payoff with choice of A is: 1000p + 0 (1-p) = 1000p Expected payoff with choice of both is: 100p + 1100(1-p) = 1100-1000p Expected payoff are equal when: 1000p = 1100 – 1000p which means 2000p=1100 so that p=1100/2000=0.55 If the being can predict with better than 55% accuracy, you should pick A If the being cannot predict with better than 55% accuracy, pick both.

5. A second look at the Prisoner’s Dilemma • You are alone in a room participating in a psychology experiment for pay. • You are told you will be partnered with a participant you do not know who is in another room. • There are two buttons in each room marked A and B. • You are each given the following instructions: • If you push A the researcher will give you $1. • If you push B the researcher will give your partner $3. • What button would you choose if you will meet your partner later? B A

6. Rational Argument Rational Argument #1 Regardless of what my partner chooses I will always do better by pressing A. Press A. Rational Argument #2 If my partner and I reason logically we must come to the same conclusion about what the right choice is. We will either both decide that A is the right choice or we will both decide that B is the right choice. The outcome BB is better than AA, so as rational players we must both choose B. What is the correct rational argument?

7. Changing the Payoff’s In the Prisoner’s Dilemma Suppose $10 is given to you if you press A and $12 is given to your partner if you press B? Now suppose instead that the original payments are multiplied by 100? What button would you press?

8. Correcting The Payoffs We see different results in these games because individuals attach a value to cooperative behavior. Suppose in the original game you and your partner assign the value of $2 to being viewed as a cooperator. Then the game is changed and the cooperative strategy B becomes an equilibrium. The same change in the second game results in the following matrix, which is still a prisoner’s dilemma

9. Assigning Values to Outcomes • Utility: The value we associate with a particular outcome. • The utility should reflect all values (both market and social) • Changing the utility can change the nature of the game How do we determine utility? An ordinal utility (rank) may be enough to determine the nature of the game. A cardinal utility is necessary when strategies are mixed, or when we wish to examine solutions to social dilemmas. (mixed strategies depend on the difference between payoffs, not just the order.)