Closed-form CE-ML Estimator

• Closed-form CE-ML is a statistical estimator that inversely learns agent payoffs in 2x2 games using observed action frequencies under correlated equilibrium assumptions.
• It exploits the tractable structure of the CE polytope to derive closed-form solutions for both the equilibrium distribution and payoff parameters via payoff ratios α and β.
• Empirical evaluations reveal that CE-ML delivers superior accuracy and computational efficiency compared to ICE and LBR-ML in scenarios such as coordination and traffic games.

A Closed-form Correlated Equilibrium Maximum-Likelihood Estimator (CE-ML) provides a statistically efficient and interpretable method for inverse learning of agent payoffs in $2\times2$ games under the assumption that observed joint action frequencies are generated according to a Correlated Equilibrium (CE). The CE-ML estimator exploits the tractable combinatorial structure of the CE polytope in strict coordination games and yields parameters that are directly consistent with empirical frequencies, leveraging a closed-form solution for both the equilibrium distribution and underlying payoff parameters. This approach is specialized for scenarios where agent strategies coordinate through CE, and offers explicit trade-offs between interpretability, computational efficiency, and fidelity to observed behavior (Salazar et al., 15 Jan 2026).

1. Inverse Learning in $2\times2$ Games and Correlated Equilibrium

Consider a two-player game with each player $i \in \{1,2\}$ having actions $A_1 = \{a_1^1,a_1^2\}$, $A_2 = \{a_2^1,a_2^2\}$, leading to four possible joint action profiles $A = \{a(1),a(2),a(3),a(4)\}$, where $a(1) = (a_1^1,a_2^1)$, $a(2) = (a_1^1,a_2^2)$, $a(3) = (a_1^2,a_2^1)$, $a(4) = (a_1^2,a_2^2)$. Each player’s utility for a profile $2\times2$0 is specified via a feature mapping and linear parameterization: $2\times2$1, with $2\times2$2. Observing $2\times2$3 i.i.d. samples $2\times2$4 drawn from an unknown equilibrium strategy $2\times2$5, inverse game-theoretic learning asks for parameters $2\times2$6 such that $2\times2$7 as a CE matches empirical action frequencies.

A joint distribution $2\times2$8 over $2\times2$9 is a correlated equilibrium for payoffs $i \in \{1,2\}$0 if, for each player $i \in \{1,2\}$1, any deviation $i \in \{1,2\}$2 satisfies

$i \in \{1,2\}$3

2. Structure of the $i \in \{1,2\}$4 CE Polytope

Under the no strictly dominated strategies assumption, the CE polytope of a $i \in \{1,2\}$5 coordination game has exactly five extreme points (vertices) $i \in \{1,2\}$6 [Calvo-Armengol 2003]. Each CE is a mixture over these vertices, parameterized as $i \in \{1,2\}$7 with $i \in \{1,2\}$8. Of particular importance is $i \in \{1,2\}$9, the unique interior CE, for which the equilibrium probabilities are determined by payoff ratios:

• Define

$A_1 = \{a_1^1,a_1^2\}$0

• The interior CE probabilities are:

$A_1 = \{a_1^1,a_1^2\}$1

The other four vertices are degenerate CEs, placing probability mass on one or two pure profiles.

3. Maximum-Likelihood Estimation: Closed-Form Solution

Given action counts $A_1 = \{a_1^1,a_1^2\}$2 and empirical frequencies $A_1 = \{a_1^1,a_1^2\}$3, the log-likelihood under CE parameterization is

$A_1 = \{a_1^1,a_1^2\}$4

Due to non-concavity of the log-likelihood in $A_1 = \{a_1^1,a_1^2\}$5, the optimum always lies at a CE vertex. The estimator proceeds by:

• Computing $A_1 = \{a_1^1,a_1^2\}$6 for $A_1 = \{a_1^1,a_1^2\}$7 (log-likelihood under each vertex).
• Selecting $A_1 = \{a_1^1,a_1^2\}$8 and setting $A_1 = \{a_1^1,a_1^2\}$9.
• Optimizing $A_2 = \{a_2^1,a_2^2\}$0 for the selected vertex.

When $A_2 = \{a_2^1,a_2^2\}$1 (interior CE), partial derivatives with respect to $A_2 = \{a_2^1,a_2^2\}$2 and stationarity yield closed-form MLEs:

$A_2 = \{a_2^1,a_2^2\}$3

Substituting definitions, this translates to linear constraints on $A_2 = \{a_2^1,a_2^2\}$4:

$A_2 = \{a_2^1,a_2^2\}$5

With normalization (e.g., fixing $A_2 = \{a_2^1,a_2^2\}$6 or specifying a component), this yields $A_2 = \{a_2^1,a_2^2\}$7 explicitly. If another vertex is selected, the constraints collapse to linear equalities reflecting the induced payoff ordering for pure or edge CEs.

4. Assumptions, Regularity, and Computational Properties

CE-ML relies on:

• Absence of strictly dominated strategies, so $A_2 = \{a_2^1,a_2^2\}$8 remain finite and the polytope has five vertices.
• Sufficient data support: $A_2 = \{a_2^1,a_2^2\}$9, $A = \{a(1),a(2),a(3),a(4)\}$0 to ensure $A = \{a(1),a(2),a(3),a(4)\}$1 are defined.
• Unique maximal vertex for the likelihood (ties resolved arbitrarily).

Computation is efficient: single-pass statistics and closed-form solutions yield total complexity $A = \{a(1),a(2),a(3),a(4)\}$2. In degenerate data scenarios (e.g., samples occupy just two action cells), if denominators in the MLE vanish, any $A = \{a(1),a(2),a(3),a(4)\}$3 consistent with the observed payoff orderings suffices.

Numerical stability is ensured by selecting vertices that allocate zero mass to unobserved profiles when $A = \{a(1),a(2),a(3),a(4)\}$4 for some $A = \{a(1),a(2),a(3),a(4)\}$5.

5. Illustrative Application: Chicken-Dare Game

In the "chicken-dare" case [Bestick et al. 2013], 1000 simulated samples yield frequencies $A = \{a(1),a(2),a(3),a(4)\}$6, $A = \{a(1),a(2),a(3),a(4)\}$7, $A = \{a(1),a(2),a(3),a(4)\}$8, $A = \{a(1),a(2),a(3),a(4)\}$9. Applying the closed-form:

$a(1) = (a_1^1,a_2^1)$0

The interior vertex likelihood dominates, so payoff ratio constraints directly recover $a(1) = (a_1^1,a_2^1)$1 and $a(1) = (a_1^1,a_2^1)$2 up to scale, matching the true payoffs within numerical tolerance. This outcome demonstrates the estimator's ability to precisely reconstruct underlying game parameters when agent behavior is CE-conforming.

6. Empirical Evaluation and Performance

CE-ML was evaluated alongside Inverse Correlated Equilibrium (ICE) and Logit Best Response ML (LBR-ML) estimators on synthetic and SUMO traffic interaction data, with four primary experiments:

• E1 (Chicken via CE): As $a(1) = (a_1^1,a_2^1)$3 increases, CE-ML MAE/RMSE improves ($a(1) = (a_1^1,a_2^1)$4), besting ICE and LBR-ML.
• E2 (Traffic, maximum-entropy CE): CE-ML achieves MAE $a(1) = (a_1^1,a_2^1)$5, ICE $a(1) = (a_1^1,a_2^1)$6, LBR-ML $a(1) = (a_1^1,a_2^1)$7, with CE-ML also giving the best KL and prediction accuracy.
• E3 (Traffic with signaling device): CE-ML identifies the correct mixture vertex ($a(1) = (a_1^1,a_2^1)$8) and 86.4% decision accuracy (ICE: 41.8%).
• E4 (No coordination): CE-ML fails (non-CE data), but LBR-ML with fitted $a(1) = (a_1^1,a_2^1)$9 attains 72.6% accuracy, equaling the best fixed rationality baseline.

7. Comparison to Logit Best Response ML and Practical Recommendations

Criterion	CE-ML	LBR-ML
Interpretability	Explicit payoff-ratio (α,β); mixture vertex interpretable	Includes rationality λ; models stochastic adaptation
Computational Cost	$a(2) = (a_1^1,a_2^2)$0, closed-form solution	Requires $a(2) = (a_1^1,a_2^2)$1 linear system, nonconvex optimization, $a(2) = (a_1^1,a_2^2)$2
Behavioral Assumptions	One-shot correlation device, perfect regret consistency	Repeated logit best responses, bounded rationality; robust to non-CE behavior

CE-ML achieves fast, closed-form inverse learning for small $a(2) = (a_1^1,a_2^2)$3 games when agent behavior plausibly arises from a CE, such as in coordinated, signaled, or regulated environments. In settings without a central correlating device—such as unregulated traffic or when stochastic adaptation is prominent—LBR-ML better captures bounded rationality and noisy, non-equilibrium patterns, albeit with higher computational overhead and additional parameters (Salazar et al., 15 Jan 2026).