Econ 431: David Reiley
Due: Monday, 26 March 2007

## Problem Set #6

1. (10 points) DS, question 10.2.
2. (10 points) DS, question 10.3.
3. (10 points) DS, question 10.5.
4. (20 points) DS, question 10.7. To clarify part (a), what I want you to do is change the game so that the children now prefer being Good to being Mischievous. They still care even more about the nanny being Nice instead of Harsh, and the nanny's payoffs should stay the same. If you change the payoffs in this way, you should see that the kids' statement is now technically a threat rather than a promise.
5. (25 points) DS, question 10.8.
6. (25 points) DS, question 10.9. Please give concrete examples. For example, "The US should promise to be nice to Cuba" is not a specific enough answer. Better, and more concrete, is, "The US could promise to restore full trade relations with Cuba if Cuba allows investment by US companies."
7. (15 points) DS, question 11.2.
8. (15 points) DS, question 11.4.
9. (15 points) DS, question 11.6.
10. (25 points) DS, question 11.8.
11. (15 points) DS, question 11.9. For part (c), since the proposed \$120/\$10 split will not support a subgame-perfect equilibrium to the infinitely repeated game, you are asked to find the most favorable split you can get and still be supported by a subgame-perfect equlibrium.
12. (25 points) DS, question 11.10
13. (15 points) DS, question 11.11.
14. (Extra credit -20 points) We might expect that the grim strategy will provide better enforcement of cooperation in the infinitely repeated prisoners' dilemma than the tit-for-tat (TFT) strategy does, because the grim strategy punishes cheaters more harshly. By contrast, an advantage of TFT is that it is mkore forgiving of accidental errors. In this problem, we will show that for some infinitely repeated prisoners' dilemma games, grim indeed provides more incentive for cooperation than TFT does, but for other games, they provide exactly the same enforcement incentives.

Consider the general model of the symmetric prisoners' dilemma found in section 11.2.D. The game can be described by the parameters C, D, H, and L. If the grim strategy can sustain cooperation in subgame-perfect equilibrium for a larger set of interest rates (or effective rates of return) than TFT can, then we can say the grim strategy provides more incentive for cooperation. If the set of interest rates that can sustain cooperation is the same for the grim strategy as for the TFT strategy, then we can say they have the same enforcement incentive.

There are two extreme cases of deviation from cooperation: defecting once or defecting always. Which type of defection is optimal depends on the value of the interest rate r and the strategy of the opponent.
1. Suppose you are facing a TFT opponent. For what values of r will a "defect once" deviation provide a higher payoff than a "defect forever" deviation from your cooperative strategy? (You will need to compare the present discounted value of the gains and losses of each deviation, relative to complete cooperation. Your answer should be in terms of C, D, H, and L.)
2. Use the previous result to figure out, overall, for what values of r you would prefer to deviate from cooperation against a TFT opponent. (Again, your answer should be in terms of C, D, H, and L.)
3. Suppose you are facing a grim opponent. For what values of r will a "defect once" deviation provide a higher payoff than a "defect forever" deviation from your cooperative strategy?
4. Use the previous result to figure out, overall, for what values of r you would prefer to devi ate from cooperation against a grim opponent.
5. Compare your answers to (b) and (d), and use them to conclude for what values of C, D, H, and L a player would be able to support more cooperation from his opponent using grim than using TFT. If you've done the work correctly, you should be able to interpret your result as saying that grim provides more enforcement incentives than TFT if and only if "the gain to defecting defensively is larger than the gain to defecting offensively."