Question

In a two-person Nash equilibrium, each player is making a best response to what? In a dominant strategy equilibrium, each player is making a best response to what?

Short answer

In a Nash equilibrium, each player is responding to the other player's strategy; in a dominant strategy equilibrium, to all possible strategies.

Step-by-step solution

Understand Nash Equilibrium

In a Nash equilibrium, each player's strategy is a best response to the strategies chosen by the other player(s). This means that no player has an incentive to change their strategy, given the strategy of the other player(s). Thus, in a two-person Nash equilibrium, each player is making a best response to the strategy of the other player.

Identify Dominant Strategy Equilibrium

A dominant strategy equilibrium occurs when each player has a strategy that is best regardless of what the other player does. In this equilibrium, each player is making a best response to all possible strategies of the other player because their strategy is the best in all scenarios. Therefore, in a dominant strategy equilibrium, each player is making a best response to any possible strategy that the other player might choose.

Key concepts

Nash Equilibrium

Nash Equilibrium is a central concept in game theory that refers to a situation where players, in a strategic interaction, choose strategies that constitute the best response to one another. To understand this better, imagine a two-player game where each player's payoff depends not only on their choice but also on the choice of the other player. In a Nash Equilibrium, neither player can benefit by unilaterally changing their strategy while the other player's strategy remains the same. This implies that each player's choice is optimal, given the choice of the other player. Here are a few key points about Nash Equilibrium:

• It captures the idea of mutual best responses, meaning each player is doing the best they can, given the other player’s strategy.
• No player has anything to gain by changing only their own strategy.
• It may not always result in the optimal outcome for all players involved. In some cases, the equilibrium may lead to suboptimal payoffs for both players.

To summarize, a Nash Equilibrium is the steady state of a strategic interaction where each player’s strategy is optimal given the other’s strategy.

Dominant Strategy Equilibrium

A Dominant Strategy Equilibrium is an even stronger concept than Nash Equilibrium. Here, a player has a dominant strategy if it is the best strategy regardless of what the other player does. This is a more robust decision criterion because the chosen strategy works best in any scenario presented by the opponent's choice. In such games, each player's optimal choice does not depend on expectations about the other's choice. This leads to an outcome where each player adopts their dominant strategy, hence the name dominant strategy equilibrium. Key characteristics of a dominant strategy equilibrium include:

• Independence from the opponent’s strategies, as each player's strategy is universally superior.
• It guarantees the best possible outcome for a player regardless of the opponent's actions.
• If a dominant strategy exists for both players, the equilibrium is straightforward to predict.

It's important to note that not every game has a dominant strategy for each player, but when it does, it greatly simplifies the decision-making process.

Best Response

The best response concept is a crucial element in both Nash and dominant strategy equilibria. A player's best response is the strategy or set of strategies that yields the highest payoff, given the strategies chosen by others in the game. In the context of Nash Equilibrium: - Each player's strategy is a best response to the strategies of the others. In the context of Dominant Strategy Equilibrium: - A player's strategy is the best response under any circumstance, as it dominates all others. Considerations about best response strategies include:

• Flexibility, as players adjust their strategies to maximize payoffs based on the opponent's choices.
• Strategic anticipation, where players predict opponents' strategies and choose their strategies accordingly.
• Adaptive learning in repeated interactions, as players refine their understanding of what constitutes a best response over time.

Understanding best responses helps players in strategic games to focus on strategies that maximize their own outcomes, given the choices of others.