Question

xi (2,2) | (0.10) Consider the infinite repeated version of the game discount factor δ for which each player obtaining an average payoff around 5 is sustained by a SPNE strategy.

Answer 1

Looking at the game matrix and solving by the standard Nash equilibrium rules, we can see that Y strictly dominates X for both the players. Thus SPNE is (Y,Y) and payoff to each player is 1.

If instead we consider an infinitely repeated game with a discount factor $\delta$ and average it to 5,we get:

$5 = 1 + \frac{1}{1+\delta } +\frac{1}{(1+\delta ) ^2}+\frac{1}{(1+\delta ) ^3}+............$

This is because discounted present value of any investment is given as -

$PV = \frac{FV}{(1+r)^n}$ where PV = present value, FV = future value, r = discount rate and n = no. of years.

Considering infinitely repeated game, the right side forms a GP.

Sum of infinite GP = $\frac{a}{1-r}$

where a = 1 and r = $\frac{1}{1+\delta }$ in our case.

Thus Sum = $\frac{1}{1 - \frac{1}{1+\delta }} = \frac{1+\delta }{\delta }$

Thus $5 = \frac{1+\delta }{\delta } \Rightarrow 5\delta =1+\delta$

$\Rightarrow \delta = \frac{1}{4} = 25\%$