Question

Suppose there is a joint project that generates an amount 10 for each of the individuals A and B; provided the sum total of A's and B's investments in the project is at least 7: Suppose A's investment is denoted by x and B's by y; where x and y can take any values in {0,1,2,3,4,...,8,9,10}. A player always has to incur the cost of the investment(equal to x or y respectively) and gets the benefit if and only if the project goes ahead.Suppose each player is interested in his own net benefit (benefit-cost) and players choose their investment levels simultaneously. The following relate to pure strategies.

(a) Is there a Nash equilibrium in which both players invest 0?

(b) Is there a Nash equilibrium in which one player invests 0 and the other does not?

(c) What is the lowest payoff a player can get in any Nash equilibrium?

Answer 1

Nash Equilibrium means that there shouldn't be any unilateral deviation.

a) There is no Nash Equilibrium when players invest 0.

Net Benefit = 0

Now, consider the following deviation:

Individual A invests Rs 3 and Individual B invests Rs 4 in the project.

Net Benefit of A = 10-3= 7 > 0 ( when player invests 0)

Net Benefit of B= 10-4 =6 >0

Thus, both individuals have an incentive in the unilateral deviation.

b) There is no Nash Equilibrium when one invests 0 and other doesn't.

Net Benefit= 0

Now, consider the following deviation:

If Individual A deviates because the other player does not then the minimum that Individual A has to contribute is 7.

So Net Benefit to Individual A= 3(which is greater than 0) and to Individual B= 10.

So Individual A or Individual B will deviate.

Hence no Nash Equilibrium.

c) It is evident from part (a) that the lowest payoff a player can get in any Nash Equilibrium is 6.