Question

Suppose that in some two-player game, \(s_{1}\) is a rationalizable strategy for player 1. If, in addition, you know that \(s_{1}\) is a best response to \(s_{2}\), can you conclude that \(s_{2}\) is a rationalizable strategy for player 2? Explain.

Short answer

No, knowing \(s_1\) is a best response doesn't ensure \(s_2\) is rationalizable. Further analysis is required.

Step-by-step solution

Understanding Rationalizable Strategies

A rationalizable strategy is one that survives repeated deletion of strictly dominated strategies. Each player's strategy is based on their beliefs about their opponent's strategy, and only strategies that are best responses to some beliefs are rationalizable.

Analyzing Best Response

Since we know that \(s_1\) is a best response to \(s_2\), this means that \(s_1\) is the optimal strategy for player 1, given that player 2 plays \(s_2\). However, best responses only indicate optimal play for a specific belief about the opponent's strategy, not the rationalizability of that opponent's strategy.

Determining \(s_2\) Rationalizability

To check if \(s_2\) is rationalizable for player 2, we should determine whether \(s_2\) is definable as a best response to some belief about the strategies of player 1. Rationalizability of \(s_2\) requires more information beyond \(s_1\) being a best response.

Conclusion Using Given Information

Given only that \(s_1\) is a best response to \(s_2\), we cannot conclude that \(s_2\) is rationalizable. Rationalizability of \(s_2\) requires further analysis about player 1's strategies and beliefs. \(s_2\) could potentially be a non-surviving strategy under some beliefs.

Key concepts

Rationalizable Strategies

In game theory, understanding rationalizable strategies is key to analyzing how players make decisions consciously based on their expectations. These strategies are determined by evaluating what a player believes their opponent might do. All strategies are considered, and only those that consistently make sense as reactions to some plausible beliefs survive this rationalization process.

For a strategy to be rationalizable, it must stand the test of stability against the repeated elimination of strictly worse strategies. This involves:

• An iterative process where a player disregards strategies that are never beneficial, no matter what they believe the opponent might play.
• Checking if there is at least one belief about the opponent's actions that validates this strategy as the best response for achieving the player's objectives.

Rationalization thus ensures both players consistently choose strategies that make sense of the situation, given what they believe the other might do.

Best Response

A best response in two-player games is the action that produces the most favorable outcome for a player, assuming the other player's strategy is held constant. This concept is crucial because it sheds light on what each player should do to maximize their payoff, given their belief about the opponent's move.

Let's outline the characteristics of a best response:

• Conditional Optimality: It is the optimal decision when a player assumes the opponent is sticking with their strategy.
• Not Universally Optimal: It does not suggest it's the best action in a general sense but rather optimal under specific conditions.
• Dependence on Beliefs: It emphasizes that the player's action depends heavily on what they think the other might do.

While a best response is crucial for strategic planning, it doesn't necessarily mean that the other strategy (the one being responded to) is rationalizable. This necessitates further investigation beyond just knowing an opponent's current strategy.

Two-Player Game Analysis

Analyzing a two-player game involves understanding the interaction between the players' strategies, potential outcomes, and corresponding payoffs. This framework sets the stage for examining strategic decisions critically. Let's delve into key aspects:

• Interdependence: Each player's decision impacts the other's payoffs, making strategic thinking essential.
• Game Representation: A common way to represent such games is through matrices or payoff tables, which illustrate outcomes contingent on players' choices.
• Simultaneous Consideration: Players must weigh not only their preferences but also how their opponent is evaluating the scenario.

In the original exercise's context, these components help determine if a strategy like \( s_2 \) is rationalizable. It's essential to conduct thorough game analysis to expose strategies' viability by exploring the dynamics between players' beliefs and resultant actions.