Equilibrium selection

What Is Equilibrium Selection?

Equilibrium selection is a core concept within game theory, a branch of economics and mathematics that studies strategic interaction among rational agents. It addresses the challenge that arises when a game, or a situation involving strategic decision-makers, has multiple possible stable outcomes, known as equilibria. When multiple Nash equilibria exist, equilibrium selection provides criteria and theoretical frameworks to predict which specific outcome players are most likely to choose or coordinate on. This problem is fundamental because the existence of several stable points can lead to ambiguity about expected behaviors and results.

History and Origin

The concept of equilibrium selection gained significant prominence through the work of economists John C. Harsanyi and Reinhard Selten. Their contributions, alongside John F. Nash Jr.'s foundational work on the Nash equilibrium, were recognized with the Nobel Memorial Prize in Economic Sciences in 1994 for their pioneering analysis of equilibria in the theory of non-cooperative games.³⁶,³⁵

Harsanyi extensively contributed to the study of game theory, particularly in the analysis of games with incomplete information, known as Bayesian games, and also made important contributions to equilibrium selection.,³⁴ Selten was instrumental in refining the Nash equilibrium concept for analyzing dynamic strategic interactions.³³ The collaboration between Harsanyi and Selten resulted in their seminal work, "A General Theory of Equilibrium Selection in Games," which aimed to provide a method for choosing a unique equilibrium among multiple possibilities.³² Their efforts highlighted that while the Nash equilibrium identifies stable states where no player has an incentive to unilaterally change their strategy given others' choices, it often does not specify which of these multiple equilibria will be realized when players are guided by rationality.

Key Takeaways

• Equilibrium selection is a problem in game theory that arises when multiple stable outcomes (equilibria) exist in a strategic interaction.
• Its purpose is to identify which specific equilibrium is most likely to be chosen or coordinated upon by rational players.
• Key criteria for equilibrium selection include Pareto dominance, which favors outcomes where no player can be made better off without making another worse off, and risk dominance, which favors outcomes that are less risky.
• The concept is crucial for predicting outcomes in economic models, business strategies, and public policy decisions where strategic decision-making by multiple parties is involved.
• Limitations of equilibrium selection theories often stem from assumptions of perfect rationality and complete information, which may not hold in real-world scenarios.

Interpreting Equilibrium Selection

Interpreting equilibrium selection involves analyzing various criteria and factors that can influence players' choices among multiple equilibria. Two prominent concepts used in equilibrium selection are Pareto dominance and risk dominance.

• Pareto Dominance (Payoff Dominance): An equilibrium is considered Pareto dominant if it yields a higher payoff for at least one player without making any other player worse off, compared to other equilibria. In simple terms, it's the "best" outcome for the group. Players might prefer a Pareto-dominant equilibrium because it offers collectively superior results.³¹,
• Risk Dominance: An equilibrium is risk dominant if it is less risky for the players, meaning it has a larger "basin of attraction." This implies that players are more likely to converge on this equilibrium even if there's uncertainty about the other players' actions. In situations where players prioritize minimizing potential losses or are uncertain about others' choices, they might opt for the risk-dominant equilibrium.³⁰,²⁹

Sometimes, these criteria can conflict, meaning an equilibrium might be Pareto dominant but not risk dominant, or vice versa.²⁸,²⁷ This highlights the complexity of equilibrium selection, as different psychological or situational factors might lead players to prioritize different aspects. Understanding these interpretations helps in predicting market behavior and strategic outcomes by considering the underlying motivations and information structures of the players involved.

Hypothetical Example

Consider two technology companies, Alpha Corp and Beta Inc., deciding on a new standard for wireless charging. They face a coordination game where the success of their product depends on adopting the same standard as their competitor.

Scenario:

• Standard A: A newer, more efficient technology, but requires a larger initial investment.
• Standard B: An older, less efficient technology, but cheaper to implement.

The payoffs (in millions of dollars) are as follows:

Here, (10, 10) means Alpha gets $10M, and Beta gets $10M.

Analysis:

1. Multiple Equilibria: There are two pure-strategy Nash equilibria:

   • Both choose Standard A (10, 10)
   • Both choose Standard B (5, 5)
     In both cases, neither company can unilaterally switch standards and improve its payoff. For example, if Alpha chooses Standard A, Beta's best response is Standard A (10 vs. 0). If Alpha chooses Standard B, Beta's best response is Standard B (5 vs. 0). The same logic applies to Beta.

2. Equilibrium Selection Problem: The challenge is that both (Standard A, Standard A) and (Standard B, Standard B) are stable, but they yield different payoffs. Which one will be "selected"?

3. Applying Selection Criteria:

   • Pareto Dominance: The (Standard A, Standard A) equilibrium with payoffs (10, 10) Pareto dominates (Standard B, Standard B) with payoffs (5, 5). Both companies are strictly better off with Standard A.
   • Risk Dominance: To determine risk dominance, one would analyze the consequences of deviating from an expected choice. If players are highly risk-averse, they might lean towards the equilibrium that minimizes potential losses from miscoordination, even if it offers lower maximum gains. In this simplified example, if a company fears the other will choose B, then choosing B might be seen as less risky than choosing A and potentially ending up with a (0,0) payoff if the other chooses B. However, the higher payoff of (10,10) makes Standard A the payoff-dominant choice.

Given the substantial difference in potential gains, the companies would likely engage in pre-game communication or external factors might influence their incentives to coordinate on Standard A.

Practical Applications

Equilibrium selection finds extensive practical applications across various financial and economic domains, particularly where strategic interactions and potential coordination failures are prevalent.

• Financial Markets: In financial markets, equilibrium selection helps explain how investors and traders coalesce around certain pricing conventions or investment strategies, even when multiple stable outcomes are theoretically possible. For example, in situations of market sentiment or herd behavior, collective actions can lead to self-fulfilling prophecies, selecting one equilibrium over others.
• Monetary and Fiscal Policy: Governments and central banks often face coordination problems when setting public policy. For instance, in international macroeconomic policy coordination, countries must decide on policies that consider the actions of other nations. Game theory, including equilibrium selection, is applied to model how central banks and governments might coordinate their monetary and fiscal policies to achieve desired economic outcomes, such as controlling inflation or stimulating growth.²⁶,²⁵,²⁴,²³ A paper from the Federal Reserve discussed how policymakers might "coordinate on a good Nash equilibrium" when multiple solutions exist, highlighting the importance of information sharing and the choice of policy instruments.²² The stability of the interbank market, for example, can be viewed through the lens of coordination games, where banks' individual decisions to lend or borrow influence the overall market equilibrium.²¹,²⁰,¹⁹
• Oligopoly and Market Behavior: In economic models of oligopolies (markets with a few dominant firms), equilibrium selection is crucial for predicting pricing strategies, production levels, and advertising decisions. Firms must anticipate competitors' actions, and if multiple Nash equilibria exist (e.g., both high prices or both low prices), understanding which equilibrium is likely to be chosen is vital for strategic analysis.¹⁸,,¹⁷
• Auctions and Bidding: In auctions, bidders' strategies can lead to multiple equilibria. Equilibrium selection helps determine the likely outcome, especially in complex bidding scenarios like first-price sealed-bid auctions.¹⁶

Limitations and Criticisms

While equilibrium selection provides valuable frameworks for predicting outcomes in strategic interactions, it is not without its limitations and criticisms.

One primary criticism centers on the assumptions of perfect rationality and complete information. Many game theory models assume that players are perfectly rational agents who have full knowledge of the game's structure, payoffs, and other players' preferences.¹⁵,¹⁴ In reality, individuals often operate with limited or imperfect information, are influenced by emotions, or exhibit bounded rationality. These real-world complexities can lead to deviations from theoretically predicted equilibria.¹³ The presence of "incredible threats" in dynamic games, where a player's optimal strategy might involve a threat that would be irrational to carry out if the situation arose, is another challenge that simple Nash equilibrium analysis sometimes fails to address, leading to the development of refinements like subgame perfection.¹²

Furthermore, even with various selection criteria like Pareto dominance and risk dominance, ambiguity can persist. These criteria may sometimes select different equilibria, leading to conflicting predictions about the game's outcome.¹¹,¹⁰ In some cases, no unique equilibrium can be identified, leaving the selection problem unresolved.⁹,⁸ The applicability of these criteria can also be limited in games with a large number of players or highly complex strategy spaces.⁷

Critics from fields like behavioral economics argue that traditional game theory often overlooks the role of learning, adaptation, and evolutionary processes in shaping strategic behavior over time.⁶,⁵ For instance, players might converge on an equilibrium through repeated interactions and learning rather than a single, perfectly rational calculation. Some philosophical critiques of similar equilibrium concepts, such as "reflective equilibrium," also point to the potential for indeterminacy, where different individuals or even the same person at different times might arrive at different equilibria.⁴ This highlights that even sophisticated equilibrium selection methods may not fully capture the dynamic and often irrational elements of human risk management and decision-making.

Equilibrium Selection vs. Nash Equilibrium

The terms "equilibrium selection" and "Nash equilibrium" are closely related within game theory, but they refer to distinct concepts.

A Nash equilibrium is a fundamental solution concept in non-cooperative games where each player's chosen strategy is optimal, given the strategies of all other players. In a Nash equilibrium, no player can improve their individual outcome by unilaterally changing their strategy, assuming the other players keep theirs unchanged. It represents a stable state where there is no incentive for any single player to deviate., A game can have one, many, or no Nash equilibria.

Equilibrium selection, on the other hand, specifically addresses the challenge that arises when a game has multiple Nash equilibria. While the Nash equilibrium concept identifies these stable points, it does not provide a mechanism for choosing which one will actually be played or observed in practice. The "equilibrium selection problem" is precisely this ambiguity. Therefore, equilibrium selection is a refinement or an extension of the Nash equilibrium concept. It involves applying additional criteria—such as Pareto dominance, risk dominance, or other behavioral assumptions—to narrow down the set of possible Nash equilibria to a single, most plausible outcome. In essence, the Nash equilibrium defines what stable outcomes exist, while equilibrium selection attempts to explain why one of those stable outcomes might be chosen over others in a real-world strategic setting. This distinction is vital for accurate strategic analysis and prediction.

FAQs

What is the core problem that equilibrium selection tries to solve?

The core problem equilibrium selection tries to solve is the ambiguity that arises when a strategic interaction, or game, has multiple stable outcomes (Nash equilibria). Without further criteria, it's unclear which of these stable outcomes players will converge on. Equilibrium selection provides methods to predict the most likely outcome.

##³# Why is equilibrium selection important in finance and economics?
Equilibrium selection is crucial in finance and economics because it helps predict actual market behavior and economic outcomes in situations where multiple stable states are possible. This is particularly relevant in areas like oligopoly competition, asset pricing, and central bank public policy decisions, where understanding which equilibrium will prevail allows for better forecasting and strategic planning.,

#²#¹# How do players typically decide which equilibrium to select in a game?
Players might decide which equilibrium to select based on various factors, including:

• Payoff Dominance: Choosing the equilibrium that yields the highest payoffs for all players.
• Risk Dominance: Selecting the equilibrium that minimizes potential losses or is perceived as less risky.
• Focal Points: Relying on shared cultural understanding, historical precedent, or salient features of the game to coordinate their decision-making.
• Communication: Explicit or implicit communication between players can facilitate coordination on a specific equilibrium, especially in international relations or business negotiations.