Which of the following statements is true concerning a Nash equilibrium?

Question

BlackTom AI · Accepted Answer

A Nash equilibrium is a strategy profile in which no player can increase their payoff by unilaterally changing their own strategy.

1) Relevant knowledge points and concepts
- Nash equilibrium: a profile (one strategy for each player) such that given the other players’ strategies, no single player can do better by deviating alone.
- Dominant strategy: a strategy that yields a higher payoff regardless of what others do; Nash equilibrium does not require dominant strategies to exist.
- Verification of a Nash equilibrium is done by checking unilateral deviations for each player.

2) Step-by-step reasoning
- Start with the formal check: pick a candidate strategy profile s*. For each player i, compare i’s payoff at s* with the payoff if i changes to any alternative strategy while others remain at s*. If none of those unilateral changes yield a higher payoff, s* is a Nash equilibrium.
- This yields the two standard equivalent verbal definitions: “no player has an incentive to change their strategy given the strategies chosen by other players” and “each player is doing the best they can given the strategies chosen by others.”

3) Example with intermediate steps (payoff check)
Consider a simple 2-player game with payoff matrix:
- (C,C) = (3,3)
- (D,D) = (1,1)
- (C,D) = (0,5)
- (D,C) = (5,0)
Check (D,D): if Player 1 is playing D and Player 2 plays D, Player 1’s payoff is 1. If Player 1 unilaterally switches to C while Player 2 stays D, Player 1’s payoff becomes 0 (worse). Symmetrically for Player 2. Thus neither player can unilaterally improve: (D,D) is a Nash equilibrium.

4) Why the correct answer is correct
Option “It is an outcome in which no player has an incentive to change their strategy given the strategies chosen by other players” directly matches the formal definition: it focuses on unilateral incentives and the lack of profitable deviation, which is the essence of Nash equilibrium.

5) Why the other options are incorrect or less appropriate
- Option 1 (requires all players have a dominant strategy): incorrect because Nash equilibria can exist without any dominant strategies; dominant strategies are sufficient but not necessary.
- Option 2 (each player is doing the best they can given others): this is an equivalent restatement of the Nash equilibrium definition, so it is correct in substance. In a single-answer multiple-choice context, option 3 is the textbook phrasing selected here.
- Option 4 (all options correct): incorrect because option 1 is not generally true.

6) Conclusion
The precise and authoritative description is option 3: “It is an outcome in which no player has an incentive to change their strategy given the strategies chosen by other players,” which matches the provided answer.

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