Question

Two farmers, Rohit and Harish, graze their animals on a common land. They can choose to use this common resource ‘lightly’ or ‘heavily’ and the resulting strategic interaction may be described as a simultaneous-move game. The payoff matrix is given below: 

The minimum value of the discount rate (where the discount rate is less than one) under infinite repetition of the game where the threat strategy (“Graze lightly if the opponent also grazes lightly, whereas, if the opponent reneges then always graze heavily in all the future periods”), is a Sub-game Perfect Nash Equilibrium (SPNE) and, both the farmers graze their animals lightly is _________ (round off to one decimal place).

Hint:

In repeated games, the minimum discount rate can be found by comparing the present value of cooperating to the present value of defecting.

Answer 1

Correct Answer: 0.6

In a repeated game with discounting, the players will cooperate if the discounted value of future cooperation outweighs the benefit of defection. The strategy described in the question is known as trigger strategy. We can find the minimum discount rate \( \delta \) at which the players will both cooperate.
From the matrix, the payoff of both grazing lightly is \( 40 \) for each. If Rohit deviates to grazing heavily, his payoff will be \( 55 \), but Harish will get \( 20 \). Therefore, for the strategy to be a SPNE, the payoff from cooperating (i.e., grazing lightly) must be greater than or equal to the payoff from defecting (i.e., grazing heavily). This gives the inequality:
\[ 40 \geq \delta \cdot 40 + (1 - \delta) \cdot 55 \] Solving this inequality for \( \delta \), we get the minimum value of \( \delta \) to be \( 0.6 \). Thus, the minimum value of the discount rate is \( 0.6 \).