2 marksGame TheoryGame theory and strategic behaviourRepeated games and cooperationHardabout 4 min

GATE 2021 Question 31: Game theory and strategic behaviour

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:

	Harish: Graze Lightly	Harish: Graze Heavily
Rohit: Graze Lightly	40, 40	20, 55
Rohit: Graze Heavily	55, 20	30, 30

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).

Answer and worked solution

Answer: 0.6

Step 1 - Identify the punishment (Nash reversion) payoff. (Graze Heavily, Graze Heavily) is a Nash equilibrium of the one-shot stage game (each player's best response to the other playing Heavily is also Heavily, since $30>20$ ), giving each farmer a payoff of 30 forever if punishment is triggered.

Step 2 - Value of perpetual cooperation. If both farmers graze lightly forever, each earns 40 every period. The present value is:

$V_{coop} = \frac{40}{1-\delta}$

Step 3 - Value of deviating once. If Rohit deviates (grazes Heavily) while Harish still grazes Lightly this period, Rohit earns the one-shot deviation payoff of 55. From next period onward, Harish triggers permanent punishment (both graze Heavily forever), so Rohit earns 30 in every subsequent period:

$V_{deviate} = 55 + \delta\cdot\frac{30}{1-\delta}$

Step 4 - Set up the sustainability condition. For grim-trigger cooperation to be an SPNE, cooperating must be at least as good as deviating:

$\frac{40}{1-\delta} \ge 55+\delta\cdot\frac{30}{1-\delta}$

Solving this as an equality to find the threshold:

$40 = 55(1-\delta)+30\delta$ $40 = 55-55\delta+30\delta$ $40 = 55-25\delta$ $25\delta = 15 \quad\Rightarrow\quad \delta = 0.6$

Therefore the minimum discount rate sustaining cooperation is 0.6.

The trap: 0.27

Students sometimes forget that the one-time deviation payoff (55) is earned immediately and undiscounted, while the punishment phase begins only from the NEXT period — mixing up which terms get discounted by $\delta$ and which don't is the most common source of error in grim-trigger threshold calculations.

Game theory and strategic behaviour has come up 61 times in 6 exams since 2008.

What this question tests

This question is on Repeated games and cooperation, part of Game theory and strategic behaviour in Game Theory.

How repeated play can sustain cooperation that one-shot play cannot.

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