Question

Is the following statement true or false? If it is true, explain why. If it is false, provide a game that illustrates that it is false. "If a Nash equilibrium is not strict, then it is not efficient."

Short answer

False; a non-strict Nash Equilibrium can be efficient, as shown in the provided example.

Step-by-step solution

Understand Nash Equilibrium

A Nash Equilibrium is a situation in a game where no player can benefit by unilaterally changing their strategy, given the other players' strategies remain unchanged. An equilibrium is 'strict' if each player's strategy is the unique best response to the other players' strategies.

Define Efficiency in Game Theory

In game theory, an outcome is considered 'efficient' if there is no other outcome that makes at least one player better off without making any other player worse off. This is often referred to as Pareto Efficiency.

Identify the Statement's Implication

The statement claims that if a Nash Equilibrium is not strict, it cannot be efficient. Hence, we need to establish whether a non-strict Nash Equilibrium can still be Pareto efficient.

Example of a Non-Strict Nash Equilibrium

Consider the game with the following payoff matrix:\[\begin{array}{c|c|c} & ext{Strategy A} & ext{Strategy B} \\hline ext{Strategy A} & (2,2) & (0,0) \\hline ext{Strategy B} & (0,0) & (2,2) \\end{array}\]Both (Strategy A, Strategy A) and (Strategy B, Strategy B) are Nash Equilibria. However, they are not strict because a player can switch strategies without becoming worse off.

Check Efficiency of the Example

In the given example, both Nash Equilibria (Strategy A, Strategy A) and (Strategy B, Strategy B) provide payoffs of (2,2), which are Pareto efficient. There are no alternative outcomes that improve one player's payoff without reducing the other's. Thus, these equilibria are efficient.

Conclusion of the Statement

The provided example demonstrates that a non-strict Nash Equilibrium can be efficient, since changing strategies does not improve the joint outcomes. Hence, the original statement is false.

Key concepts

Nash equilibrium

In game theory, a Nash equilibrium is a fundamental concept denoting a situation where no participant can gain by solely changing their own strategy. This assumes that the other players' strategies remain constant. Typically, a Nash equilibrium occurs when each player's chosen strategy is optimal, given the strategies of all other players. Imagine a simple game like rock-paper-scissors. Here, if each player chooses randomly, they reach a state of Nash equilibrium since deviating from randomness wouldn't benefit any player unless others also deviate.

• Each player plays optimally, assuming others do not change.
• Nash equilibrium doesn't always result in the best joint outcome.
• It can sometimes lead to suboptimal outcomes for the group.

Understanding Nash equilibrium helps to predict stable outcomes of strategic interactions, even if those outcomes are not the absolute best for all involved. Both strict and non-strict equilibria exist, with 'strict' meaning there's a unique best response strategy for each player.

Pareto efficiency

Pareto efficiency, often referenced as Pareto optimality, is a key concept in economics and game theory where a situation is reached when no individual can be made better off without making someone else worse off. In simpler terms, it's about resource allocation where one cannot improve without someone else losing. This concept is critical when analyzing the results of strategic games.

• Pareto efficiency doesn't imply fairness or equality.
• An efficient outcome might not favor all participants equally.
• There can be multiple Pareto efficient outcomes in a game.

A common misuse is assuming that Pareto efficiency is synonymous with societal welfare maximization, which is not always the case. In game theory, we strive to find strategies leading to these outcomes but understand that efficiency speaks only to the allocation aspect, not necessarily about the individual satisfaction of players.

strict equilibrium

A strict equilibrium refers to a scenario in game theory where each player's strategy is the uniquely best response to the others. This is a subset of Nash equilibrium where no player has anything to gain by changing only their own strategy, but with an additional condition that the selected strategy is the only one that yields the best results.

• Strict equilibria are stable.
• Players have an incentive to stick to their path.
• These equilibria reduce strategy deviation.

For an example, think of a game where businesses decide on pricing strategies. If a particular price point yields the highest profit when others keep their prices constant, and any deviation leads to a loss, this pricing strategy forms a strict equilibrium. While beneficial in terms of predictability and stability, strict equilibria are not always present in every game and are generally less common than non-strict equilibria.