2 marksGame TheoryGame theory and strategic behaviourPure-strategy Nash equilibriumHardabout 3 min

GATE 2021 Question 27: Game theory and strategic behaviour

Consider an intersection of roads without any traffic light. Two cars A and B approach an intersection and they want to proceed as indicated by respective arrows in the diagram (Car A travelling one direction, Car B travelling a crossing direction toward the same intersection point). If both proceed without stopping and there is an accident, then A would have a payoff of −100 and B would have a payoff of −500 (since B is responsible for the accident). If one stops, and the other proceeds then the payoff is: −5 and 10, respectively (stopper gets −5, proceeder gets 10). If both of them stop, then it takes a little longer to reach their respective destinations, they have a payoff of −5 each. Find the Pure Strategy Nash Equilibrium (PSNE) of the players (Car drivers).

Top-down view of a four-way road intersection with no traffic light, drawn with dash-dot centre lines. Car A (green) approaches from the left, and a curved arrow shows it turning right at the centre, heading down the lower road; Car B (blue) approaches from the right, and a straight arrow shows it going straight ahead to the left, across Car A's turning path.

Select all that apply.

Answer and worked solution

Answer: (B) and (C)

Payoff matrix (Car A payoff, Car B payoff):

	B: Stop	B: Proceed
A: Stop	(−5, −5)	(−5, 10)
A: Proceed	(10, −5)	(−100, −500)

Checking each strategy profile for whether either player wants to unilaterally deviate:

(Stop, Stop): If A deviates to Proceed (while B stays at Stop), A's payoff rises from −5 to 10 — A wants to deviate. Not an equilibrium.

(Stop, Proceed): If A deviates to Proceed, A's payoff falls from −5 to −100 — A does NOT want to deviate. If B deviates to Stop, B's payoff falls from 10 to −5 — B does NOT want to deviate either. This IS an equilibrium.

(Proceed, Stop): By the same logic (symmetric to the case above), neither player wants to unilaterally deviate. This IS an equilibrium.

(Proceed, Proceed): If A deviates to Stop, A's payoff rises from −100 to −5 — A wants to deviate. Not an equilibrium.

This is a classic anti-coordination ('Chicken') game: it has exactly two pure-strategy Nash equilibria, where one driver proceeds and the other stops (in either arrangement) — both (B) and (C) are valid PSNE. This matches the MSQ (multi-select) format of the question, which allows for more than one correct option.

The trap: (A) (Car A, Car B) = (Stop, Stop)

Students sometimes assume '(Stop, Stop)' must be the equilibrium since it seems like the 'safest' or most cooperative-looking outcome, without formally checking that either driver has an incentive to unilaterally deviate to Proceed once they know the other is stopping (since Proceeding while the other Stops yields a strictly better payoff, 10 vs −5).

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

What this question tests

This question is on Pure-strategy Nash equilibrium, part of Game theory and strategic behaviour in Game Theory.

Outcomes from which no player wants to deviate, and classic games that illustrate them.

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Question from the GATE 2021 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.