Question 4.82 Choose the correct answerRead the following statements about dominant strategies and select the correct one.

Question

BlackTom AI · Accepted Answer

A dominant strategy is one that gives a player a higher payoff no matter what the other players do; a Nash equilibrium is a profile of strategies where each player's choice is a best response to the others. These are distinct concepts: existence (or non‑existence) of dominant strategies does not determine how many Nash equilibria a game can have.

Step‑by‑step reasoning:
1. Relevant knowledge points:
   - Dominant strategy: best for a player regardless of opponents’ actions.
   - Best response: a strategy that maximizes a player’s payoff given the opponents’ strategies.
   - Nash equilibrium: every player is playing a best response to the others.
2. If no player has a dominant strategy, players’ best responses depend on others’ choices, but consistent mutual best responses can still exist.
3. Example (Battle of the Sexes / simple coordination 2×2):
   - Strategies: Opera (O) or Football (F) for each player.
   - Payoffs: (2,1) at (O,O); (1,2) at (F,F); (0,0) at mismatches.
   - Best responses: If Player 2 chooses O, Player 1 best responds with O (2>0). If Player 2 chooses F, Player 1 best responds with F (1>0). Player 1 has no single dominant strategy because the best response depends on Player 2. Same for Player 2.
   - Yet (O,O) and (F,F) are both Nash equilibria because in each of these profiles neither player can improve by unilaterally deviating.
4. Therefore, absence of dominant strategies can coincide with multiple Nash equilibria.

Why the correct answer is correct:
- Statement b captures this: even when no player has a dominant strategy, there can be multiple Nash equilibria (as the example shows).

Why other options are incorrect:
- a: Incorrect — there can be more than one Nash equilibrium without dominant strategies.
- c: Incorrect — many games have no dominant strategy for at least one player.
- d: Incorrect — Nash equilibria can exist without any dominant strategies.

Conclusion: The correct choice is b.

Question 4.82 Choose the correct answerRead the following statements about dominant strategies and select the correct one.单项选择题

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