Quiz 5. Prisoner’s dilemma in repeated games.

1. You have to decide whether to invest $100 in your friend’s enterprise, where in a year’s time the money will increase to $140. You have agreed that your friend will then repay you $130, keeping $10 for himself. But instead he may choose to run away with the whole $140. Any of your money that you don’t invest in your friend’s venture you can invest elsewhere safely at the prevailing rate of interest r, and get $100(1+r) next year.

1. Draw the game tree for this situation and show the rollback equilibrium. [3]

Next, suppose this game is played repeatedly infinitely often. That is, each year you have the opportunity to invest another $100 in your friend’s enterprise, and the agreement is to split the resulting $140 in the manner already described. From the second year onward, you get to make your decision of whether to invest with your friend in the light of whether he made the agreed payment the preceding year. The rate of interest between any two successive periods is r, the same as the outside rate of interest and the same for you and your friend.

2. For what values of r can there be an equilibrium outcome of the repeated game, in which each period you invest with your friend and he repays as agreed? [5]

3. If he rate of interest is 10% per year, can there be an alternative profit-splitting agreement that is an equilibrium outcome of the infinitely repeated game, where each period you can invest with your friend and he repays as agreed?[4]

2. What are the pros [4] of Tit For Tat strategy?

Answers

1. Modification is needed in game tree. There should be 130 instead of 120 and 140 instead of 130

(a) The game tree shows payoffs in the order (You, Friend). The rollback equilibrium is shown by making the chosen branches thicker at each node; the equilibrium is (Don’t invest, Cheat if invest).

(b) In the repeated version, honesty could be sustained by a “grim trigger strategy,” where if your friend ever cheats you, you will never invest with him again. The friend can get an extra $130 = 140 – 10 any one time, but will lose $10 every time thereafter. Then he will not cheat you if 10 > 130r, or r < 10/130 =7,69%. Then 100(1 + r) < 130 is also true, so it is optimal for you to invest in your friend’s business so long as he is following the equilibrium strategy, that is, not cheating first.

(c) If the rate of interest is 10%, the above agreement cannot be sustained. Suppose an alternative agreement is that your friend gets x and you get 140 – x. By cheating you once, he can get an extra 140 – x, but will lose x each period thereafter. He will not cheat you if x > 0.1(140 – x), or 1.1x > 14, or x > 14/1.1. So x = 12,73 will do. Note that this still leaves you a return of 127,27 for your 100, which is adequate when the rate of interest is 10%.

2. Pros: {it’s enough to point two pros, to get 4 points}

   1. at once forgiving – „Don’t be envious”

   2. nice – „Don’t be the first to defect”

   3. provocable – „Reciprocate both cooperation and defection”

   4. clear – „Don’t be too clever”

Cons: It echoes the mistakes back and forth, no way of saying “enough is enough”.