Multiple Equilibria and Equilibrium Selection

Why Equilibrium Selection Matters

A game can have more than one Nash equilibrium for the same players, payoffs, and information. In that case, equilibrium analysis identifies a set of outcomes consistent with rational behavior, rather than a unique prediction. An empirical model must address a further question: which equilibrium is played in each observed instance of the game?

This distinction matters because observed choices reflect both the incentives in the game and the process that determines which equilibrium is realized. If the selection process is not modeled, variation in outcomes may be attributed to payoff parameters when it instead reflects different equilibria being played. Conversely, imposing one equilibrium by convention can make predictions look more precise than the game itself warrants.

Definition: For a game with payoff parameters $\theta$, let $\mathcal{E}(\theta)$ denote its set of equilibria. Equilibrium multiplicity occurs when $\mathcal{E}(\theta)$ contains more than one equilibrium. An equilibrium selection rule specifies how an equilibrium is chosen from $\mathcal{E}(\theta)$, possibly as a function of observed or unobserved variables.

An equilibrium is a profile of strategies in which each player’s strategy is a best response to the other players’ strategies. In a game with multiple equilibria, each equilibrium satisfies this condition; Nash equilibrium alone does not say which one will be played. The earlier material on normal-form games and Nash equilibrium provides the best-response framework used below.

A Coordination Game with Multiple Equilibria

Consider two players who each choose action A or B. Both prefer to match the other player, but matching on A gives a higher payoff than matching on B.

	Player 2: A	Player 2: B
Player 1: A	(2, 2)	(0, 0)
Player 1: B	(0, 0)	(1, 1)

If Player 2 chooses A, Player 1 prefers A because $2>0$. If Player 2 chooses B, Player 1 prefers B because $1>0$. The same reasoning applies to Player 2. Thus (A,A) and (B,B) are both pure-strategy Nash equilibria. The equilibrium (A,A) gives both players higher payoffs, but its payoff advantage does not by itself ensure it is selected.

Worked Example: Finding Pure and Mixed Equilibria

Use the payoff table above to identify the pure equilibria and calculate the symmetric mixed-strategy equilibrium. Let $q$ be the probability that Player 2 chooses A. If Player 1 chooses A, the expected payoff is $2q$. If Player 1 chooses B, the expected payoff is $1-q$. In a mixed equilibrium, Player 1 must be indifferent between these actions:

2q=1-q \quad\Longrightarrow\quad 3q=1 \quad\Longrightarrow\quad q=\frac{1}{3}.

By symmetry, Player 1 also chooses A with probability $1/3$. Each player is indifferent at this probability, and each opponent action is assigned positive probability, so the symmetric mixed-strategy profile is an equilibrium. The game therefore has three equilibria: the two pure equilibria and this mixed equilibrium.

The mixed equilibrium is not the same thing as selection between pure equilibria. In the mixed equilibrium, each player randomizes within a single equilibrium strategy profile. Under a rule that selects between the two pure equilibria, players instead coordinate on (A,A) in some instances and (B,B) in others. These mechanisms imply different distributions of observed action profiles.

This example also illustrates a source of multiplicity: strategic complementarity in the choice to match. A player’s preferred action depends on the other player’s action, and each pure profile makes the other player’s action a best response. More generally, multiple equilibria can arise when agents’ actions reinforce one another, although the precise conditions depend on the game.

Selection Rules in Empirical Models

An empirical model needs a rule, or an explicit set of possibilities, for moving from the equilibrium set to observed outcomes. Common approaches differ in what they assume about coordination, randomness, and unobserved heterogeneity.

Approach	What it specifies	Implication
Deterministic selection	A particular equilibrium is chosen for each observed game, often by a fixed convention or a stated criterion.	Conditional on the game and rule, the equilibrium outcome is fixed.
Stochastic selection	Each equilibrium receives a selection probability, conditional on the game or observed variables.	Repeated instances of the same observed game can produce different equilibria.
Selection through latent variables	Unobserved factors affect which equilibrium is reached or selected.	Observed outcomes can vary with latent factors, even when observed payoffs are unchanged.

A deterministic convention may be useful for making a model’s predictions definite, but it is an additional assumption, not a consequence of equilibrium. A refinement or a criterion such as payoff dominance may narrow the relevant equilibrium set under specified conditions. It should not be described as an empirically verified selection mechanism unless evidence supports that interpretation. In the coordination game, choosing (A,A) because it gives both players higher payoffs is a possible modeling rule; Nash equilibrium alone does not require it.

A stochastic selection rule assigns probabilities to equilibria. Those probabilities must be nonnegative and sum to one over the equilibria considered. They may depend on observed covariates, such as market characteristics, or on unobserved variables. In applications, the rule must also be consistent with the game: it cannot assign positive probability to a profile that is not an equilibrium under the model’s assumptions.

Mixture representation: Let $E$ index the equilibria included in the model, and let $\pi_e(x)=P(E=e\mid X=x)$ be the probability of selecting equilibrium $e$ given observed covariates $X=x$. Then $\pi_e(x)\geq 0$, $\sum_e\pi_e(x)=1$, and the probability of an observed outcome $Y=y$ is $$ P(Y=y\mid X=x)=\sum_e P(Y=y\mid E=e,X=x)\pi_e(x). $$ This expression allows for outcome variation within an equilibrium as well as variation across selected equilibria.

The formula follows by partitioning the event $Y=y$ according to which equilibrium was selected. The equilibrium events are mutually exclusive and exhaustive under the selection model, so the law of total probability gives $P(Y=y\mid X=x)=\sum_e P(Y=y,E=e\mid X=x)$. For each term, the product rule gives $P(Y=y,E=e\mid X=x)=P(Y=y\mid E=e,X=x)P(E=e\mid X=x)$. Substituting the definition of $\pi_e(x)$ yields the mixture representation.

In a model where each equilibrium implies one deterministic outcome, $P(Y=y\mid E=e,X=x)$ is either zero or one. If outcomes can also vary because of mixed strategies or other modeled randomness, that conditional probability need not be zero or one. The representation therefore separates uncertainty about selection across equilibria from uncertainty within an equilibrium.

Worked Examples of Selection

Worked Example: Comparing Deterministic and Stochastic Rules

Suppose 100 hypothetical markets have the same observed payoff structure as the coordination game. In every market, the model includes the pure equilibria (A,A) and (B,B). The observed data contain (A,A) in 68 markets and (B,B) in 32 markets.

Under a deterministic rule that always selects the payoff-dominant equilibrium (A,A), the model predicts 100 occurrences of (A,A). It does not match the observed split unless the model also introduces additional outcome variation or heterogeneity.

Under a stochastic rule, suppose each market independently selects (A,A) with probability $\pi$, and (B,B) with probability $1-\pi$. The likelihood of observing 68 outcomes of (A,A) and 32 of (B,B), ignoring the common combinatorial factor, is $\pi^{68}(1-\pi)^{32}$. The log-likelihood is $68\log(\pi)+32\log(1-\pi)$. Differentiating and setting the result to zero gives

\frac{68}{\pi}-\frac{32}{1-\pi}=0 \quad\Longrightarrow\quad 68(1-\pi)=32\pi \quad\Longrightarrow\quad \hat{\pi}=0.68.

Thus, within this simple stochastic-selection model, the fitted probability of selecting (A,A) is 0.68. This estimate describes the stipulated selection rule; it does not establish that selection is genuinely random or that this rule is uniquely supported. Other assumptions about unobserved heterogeneity or the process of coordination could also account for variation across markets.

Worked Example: Unobserved Types and Outcome Differences

Suppose two groups of hypothetical markets, $X=0$ and $X=1$, face identical observed payoff matrices, each with the two pure coordination equilibria. The outcome $Y$ equals 1 when (A,A) is selected and 0 when (B,B) is selected. An unobserved market type $U$ affects selection. Suppose the selection probabilities are $P(Y=1\mid X=0)=0.8$ and $P(Y=1\mid X=1)=0.3$, because the distribution of $U$ differs across the two groups.

The mean outcome in group $X=0$ is $0.8(1)+0.2(0)=0.8$. In group $X=1$, it is $0.3(1)+0.7(0)=0.3$. The difference in group means is therefore $0.3-0.8=-0.5$. If there are 50 markets in each group, the pooled mean is $(50(0.8)+50(0.3))/100=0.55$.

The observed difference does not, by itself, show that the observed group characteristic changed the payoffs or strategic incentives: by assumption, the observed games are identical. The difference can arise because the unobserved types associated with the groups lead to different equilibrium selection probabilities. If an analyst attributes the entire difference to a payoff effect, the model may misrepresent how the outcomes were generated.

What Observed Outcomes Can and Cannot Reveal

When the selected equilibrium is unobserved, an empirical model often observes actions or outcomes but not the selection event itself. The mixture representation clarifies the interpretation problem: observed outcome probabilities depend on both the outcome distribution within each equilibrium and the selection probabilities $\pi_e(x)$. Changes in observed outcomes can therefore reflect changes in payoffs, changes in selection probabilities, or both.

The distinction is especially important when selection depends on unobserved variables that are related to observed covariates. If those variables are omitted, estimated relationships between covariates and outcomes can combine incentive effects with shifts in the composition of selected equilibria. A model that fixes one equilibrium suppresses this source of variation by assumption. A model that allows selection probabilities to vary introduces additional quantities that must be supported by restrictions or data.

The mixture formula is a way to organize these possibilities, not a guarantee that the selection probabilities are identified from observed outcomes. If several equilibria generate overlapping outcomes, different combinations of within-equilibrium outcome probabilities and selection weights may imply the same observed distribution. The interpretation of estimated effects consequently depends on the maintained assumptions about the game and the selection process.

Common Mistakes

Treating equilibrium multiplicity as random behavior by definition. Multiple equilibria mean there is more than one equilibrium solution. Random selection is a separate modeling assumption.
Assuming the payoff-dominant equilibrium must be played. A higher payoff in one equilibrium does not imply that players coordinate on it. A selection criterion must be stated and justified.
Confusing mixed strategies with selection across equilibria. A mixed equilibrium involves randomization within a strategy profile. A selection rule can instead assign probabilities to distinct equilibrium profiles across instances.
Interpreting outcome differences as payoff effects alone. If equilibrium selection varies with unobserved factors, observed differences can reflect selection as well as changes in incentives.
Using equilibrium conditions to claim a unique prediction. If several equilibria satisfy those conditions, the model has not made a unique outcome prediction until a selection rule or additional restriction is supplied.

Key takeaway: Equilibrium analysis determines which outcomes are consistent with strategic behavior. Equilibrium selection determines how those outcomes are distributed in the data. Empirical interpretations must keep these two parts of the model distinct.

Check Your Understanding

Answer each question using the definitions and examples above.

In the coordination game, why are both (A,A) and (B,B) Nash equilibria even though (A,A) gives higher payoffs?
How does a stochastic selection rule differ from a mixed-strategy equilibrium?
If three equilibria receive selection probabilities of 0.2 and 0.5 for the first two, what must the third probability be?
State one reason observed outcome differences across groups might arise even when the observed payoff matrices are identical.
What does a model assume when it always predicts one equilibrium, and why is that assumption not implied by Nash equilibrium alone?