You have to decide whether to invest $100 in a friend’s
enterprise, where in a year’s time the money will increase to $130. You have
agreed that your friend will then repay you $120, keeping $10 for himself. But
instead he may choose to run away with the whole $130. 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 1 r) next year.
(a) Draw the game tree for this situation and show the
rollback equilibrium. 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 $130 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 repayment 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.
(b) 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?
(c) If the 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 invest with your friend and he
repays as agreed?