Problem 6: Consider the following game between n friends. Each friend simultaneously chooses

whether to invest 1 dollar or to not invest any money.1 The friends then invest this total amount

in a bank account at an interest rate of 10% which is then divided equally among all n friends.

As a result, if k total dollars are invested, and person i chose to invest, his/her utility is given

by:

(1.1) k/n 1

On the other hand if person i decided not to invest and k total dollars are invested, his/her utility

is given by:

(1.1) k/n

Part a: Suppose that n = 2. Write down the payoff matrix of this game. What are the pure

strategies that are strictly dominated?

Part b: Suppose now that n = 3. Again write down the payoff matrix of this game. What are the

pure strategies that are strictly dominated?

Part c: [Extra Credit] Suppose now that there are ten friends. What are the pure strategies that

are strictly dominated? Justify your answer.

For parts d and e, suppose that the profit that the investment yields changes so that k dollars

yields a surplus of 3k2 dollars. As a result, if player i chose to invest, and k dollars was invested by

all the players, then player i's utility is:

1

n(3k2) 1.

On the other hand, if player i chose not to invest, and k dollars in total was invested by all the

players, then player i's utility is:

1

n(3k2).

Part d: Suppose that n = 2. What are the pure strategies that are strictly dominated? Justify

your answer. Discuss how this answer relates to your answer from part a.

Part e: [Extra Credit] For what values of n is choosing to invest a strictly dominated strategy?