Banks make money by taking deposits and invest them in business projects. The investments in business projects are typically illiquid meaning that the investment is tied up in the project and cannot be freed up. However, deposits are typically liquid meaning that a depositor can at any point in time demand that the bank return the deposit. Consider a bank that has 100 depositors. Each depositor has put $1 in the bank. The bank has taken 80 of those dollars and invested them in a business project. The remaining $20 are sitting in reserves in the bank. The game is a one period game. At the beginning of the period, each depositor decides whether to ask the bank to return his/her deposit. A depositor who receives his/her deposit back makes a payoff of 0 in the game. If there a fewer withdrawals than available reserves, the bank can honor the withdrawals and continue operations to the end of the game. A depositor who maintains the deposit with the bank receives a payoff of 1 if the bank is still in operation at the end of the period. If there are more withdrawal requests than can be satisfied with the available reserves, a withdrawal request is satisfied with probability equal to reserves divided by withdrawal requests (so, if 40 depositors withdraw, given reserves of $20, each request is honored with probability 20/40 = 1/2). When the bank cannot honor all of its withdrawal requests it declares bankruptcy and any depositor who still has deposit with the bank receives a payoff of −1 (this includes the depositors who attempted to withdraw but were denied).

1. Determine the pure strategy Nash equilibria of this game.

2. Discuss the role of government provided deposit insurance in this game. Deposit insurance is a government guarantee that a depositor can always recover his/her deposit with the bank.