Final answer:
In simultaneous-move games, it is true that each player can have at most one dominant strategy and if each player has a dominant strategy, the game will have at least one Nash equilibrium. It is generally false that players have the same dominant strategy just because they have the same strategy combinations, and it's not true that each player always has at least one dominant strategy. The statement A and Bis true in this case.
Step-by-step explanation:
In simultaneous-move games, certain statements about dominant strategies hold true. Addressing the options provided, we can assert the following:
- A. Each player can have at most one dominant strategy. This is true because a dominant strategy is one that results in the highest payoff for a player no matter what the other players do.
- B. If each player has a dominant strategy the game will have at least one Nash equilibrium. This statement is also true. A Nash equilibrium is a situation where each player is making the best decision possible, taking into account the decisions of the other players, and no player has anything to gain by changing only their own strategy.
- C. If each player has the same strategy combinations they will have the same dominant strategy. This is typically false because the dominant strategy is context-dependent and can vary based on the payoffs associated with each strategy.
- D. Each player always has at least one dominant strategy. This is not true; some games do not have a dominant strategy for every player.
The classic example given is that of the prisoner's dilemma, where both prisoners would opt to confess because confessing is the dominant strategy for both. This leads them to a worse overall outcome (10 years of jail time combined) than if they cooperated (only 4 years of jail time combined).