Question

In this normal form game, the players are a borrower and a bank.

borrower \ bank [] lend at 5% [] not lend

pay back [] 15,5 [] 0,0

default [] 120, -100 [] 0,0

The bank can lend at 5% or not lend, in which case neither is any better or worse off. If the bank lends, the borrower uses the money to get a benefit worth $120, and chooses between paying back $105 (for a net payoff of $15) and defaulting (for a net benefit of $120), in which case the bank must mark a loss of $100.

a. What are the Nash equilibria of this normal form game?

b. Consider the infinitely repeated game. Take the “cooperative action profile” to be (pay back, lend at 5%). Define a clean history and define a dirty history in terms of it.

c. Define the bank’s and the borrower’s grim-trigger strategies, in terms of clean/dirty histories.

d. What are the borrower’s and the bank’s minimal patience ?, for the grim-trigger strategy profile to be a subgame perfect Nash equilibrium? (Identify two deltas.)

e. Consider many borrowers who interact with this bank. Say half of them is sufficiently patient, and half insufficiently patient (relative to the delta in part d). What is the present discounted value of the bank’s profit, assuming its patience is ? = .95?

Answer 1

If the off chance that the bank loans, the borrower utilizes the cash to get an advantage worth $120, and picks between paying back $105(for a net payoff of $15) and defaulting (for a net benefit of $120), in which case the bank must mark a loss of $100.

a)

5. What are the Nash equilibrium of this normal form game

There is one NE in unadulterated methodologies that is (default, not loan). Note that when bank loans, borrower will default and when it doesn't loan, borrower is apathetic. When borrower pay backs, banks will lend and when they default, bank won't lend. This makes only one NE, (default, not lend).

b)

5. Consider the infinitely repeated game. Take the “cooperative action profile” to be (pay back, lend at 5%). Define a clean history and define a dirty history in terms of it.

In this strategy, borrower will always pay and so the payoff each period is 15. With a discount of d, the accumulated discounted profit for borrower is 15 + 15d + 15d^2.......... = 15/1 - d. Similarly, the accumulated discounted profit for bank is 5 + 5d + 5d^2.......... = 5/1 - d

c)

5. Define the bank’s and the borrower’s grim-trigger strategies, in terms of clean/dirty histories.

If borrower cheats, it will earn 120 in 1 period and due to retaliation by bank, he will earn 0 in every period thereafter (punishing strategy is default, don't lend). The accumulated discounted profit for borrower is 120 + 0d + 0d^2..... = 120. From the viewpoint of bank, this profit is 0.

d10. What are the borrower’s and the bank’s minimal patience d, for the grim-trigger strategy profile to be a sub game perfect Nash equilibrium? (Identify two deltas.)

For borrower, this randomization implies

120 = 15/1-d

120 - 120d = 15

120d = 105

d = 0.875

For bank, d is 0.