Multiple Nash Equilibrium Quiz

A game has multiple Nash equilibria when more than one strategy profile satisfies the mutual best-response condition simultaneously. Each equilibrium is self-enforcing on its own: at each one, no player benefits from deviating unilaterally. Multiple equilibria are common in coordination games, battle of the sexes games, and games with symmetric structures, where more than one stable outcome can emerge from rational self-interested behavior.

When a game has multiple Nash equilibria, the standard Nash concept alone cannot predict which equilibrium rational players will reach. If Player 1 expects Left and Player 2 expects Right, both end up miscoordinated and earn zero. Focal points, prior communication, social conventions, or repeated interaction are needed to coordinate on one equilibrium. This equilibrium selection problem is a major limitation of Nash equilibrium as a predictive tool in coordination settings.

It is entirely possible for a game to have multiple Nash equilibria of different types at the same time. For example, some coordination games have two pure strategy Nash equilibria and one mixed strategy Nash equilibrium. Each satisfies the mutual best-response condition independently. The existence of pure strategy equilibria does not preclude the simultaneous existence of a mixed strategy equilibrium in the same game, making multiple equilibrium types a common feature of intermediate game theory analysis.

The Battle of the Sexes has two pure strategy Nash equilibria: both at Opera and both at Football. At each, both players are coordinated and neither would benefit from deviating unilaterally since miscoordination produces a worse outcome for both. A third mixed strategy Nash equilibrium also exists where each player randomizes with specific probabilities that leave the other indifferent. All three satisfy the Nash equilibrium condition, making this a classic example of multiple equilibria.

Multiple Nash equilibria are not equally likely in practice. Some equilibria are more salient due to focal points, such as natural prominence or cultural convention, while others may be risk-dominant or Pareto superior. Schelling's focal point theory, risk dominance refinements, and Pareto dominance criteria all help select among multiple equilibria. Rational players can also coordinate through communication, pre-play negotiation, or social norms, making some equilibria far more likely to be reached than others.

A Pareto-dominant Nash equilibrium makes every player at least as well off as any alternative Nash equilibrium, with at least one player strictly better off. When such an equilibrium exists, it provides a natural focal point for equilibrium selection since rational players who can communicate would prefer it. However, Pareto dominance does not guarantee coordination: if players are uncertain about each other's expectations, they may still fail to select the Pareto-dominant equilibrium.

Games with multiple Nash equilibria present a genuine equilibrium selection challenge since the basic Nash concept does not predict which equilibrium rational players will reach. Coordination games commonly exhibit multiple equilibria. Refinements like Pareto dominance and risk dominance help narrow predictions. The claim that all equilibria produce identical payoffs is false: different Nash equilibria in the same game often yield different payoffs, which is precisely why equilibrium selection matters strategically and analytically.

At (A,A), neither firm earns more by switching to B since (B vs A) yields only 1. At (B,B), neither firm earns more by switching to A since (A vs B) also yields only 1. Both strategy profiles satisfy the no-unilateral-deviation condition, making them both Nash equilibria. This network effects coordination game has two pure strategy Nash equilibria, with (A,A) being Pareto dominant since it yields higher payoffs for both firms than (B,B).

Thomas Schelling introduced the concept of focal points to explain how players coordinate in games with multiple equilibria. A focal point is a solution that stands out due to its salience, natural prominence, symmetry, or cultural convention. Even without communication, players who share common knowledge about what seems natural or obvious can independently converge on the same focal point equilibrium. Focal points explain real-world coordination successes in situations with multiple stable outcomes.

Risk dominance captures the idea that when players are uncertain about which equilibrium rivals will play, they gravitate toward the strategy that performs best on average against uncertainty. A risk-dominant equilibrium is the one each player would choose if they believed the rival was equally likely to play any available strategy. It represents the equilibrium that is most resilient to strategic uncertainty, which is distinct from the Pareto-dominant equilibrium that maximizes payoffs under coordination.

Schelling's focal point concept explains real-world coordination in exactly this type of game. Without communication, players rely on shared expectations about what is natural, prominent, or culturally obvious. A famous example is two strangers both choosing Grand Central Station as a meeting point in New York without prior arrangement. Rational players converge on the focal equilibrium because each expects the other to do the same, making the expectation self-fulfilling and producing successful coordination.

Because Nash equilibrium alone cannot predict which of multiple equilibria rational players will reach, game theorists developed various refinements. Trembling hand perfect equilibrium eliminates equilibria that are not stable against small mistakes. Subgame perfect equilibrium rules out non-credible threats in sequential games. These and other refinements impose additional conditions beyond the basic Nash criterion to select among equilibria and produce sharper, more credible predictions about strategic behavior.

When two Nash equilibria exist and one makes all players at least as well off as the other, the Pareto-dominant equilibrium provides a strong selection criterion. At (X,X), both players earn 7, which is strictly greater than the 4 each earns at (Y,Y). Since (X,X) is Pareto superior to (Y,Y) with neither player worse off, Pareto dominance unambiguously favors (X,X) as the more natural prediction. Rational players who can communicate would unanimously prefer to coordinate on (X,X).

The equilibrium selection problem arises because Nash equilibrium alone does not identify which of multiple stable outcomes will be realized. Focal points provide a behavioral mechanism for coordination. Pareto dominance and risk dominance are analytical refinements that prioritize certain equilibria on welfare and robustness grounds respectively. Nash's existence theorem is relevant to whether equilibria exist at all, not to which specific equilibrium among multiple ones will be selected.

In oligopoly pricing games, multiple Nash equilibria can exist. If both firms are charging high prices and neither would earn more by cutting prices given the rival's current price, high mutual pricing is a Nash equilibrium. Even without communication, this equilibrium can be self-sustaining through rational best-response behavior. The observed coordination reflects one of potentially several equilibria in the pricing game, which illustrates how Nash equilibrium theory explains stable pricing patterns in concentrated markets.