Testing for Monotone Equilibrium Strategies in Games of Incomplete Information

This paper proposes a unified statistical framework for testing the monotonicity of Bayesian Nash equilibrium strategies in games of incomplete information by reformulating the problem as a test of moment inequalities, a method validated through simulations and applied to detect cartels in procurement auctions.

The Gist

Imagine you are a detective trying to solve a mystery, but the suspects (the players in a game) are hiding their true motives (their "types," like how much they really want something or how much it costs them to make it). You can only see what they do (their actions, like the bids they place in an auction), not what they think.

This paper is about building a new, powerful lie detector test for these hidden motives.

The Big Problem: The "Secret Handshake"

In many games—like auctions, contests for prizes, or companies competing to sell goods—economists assume players act logically: the more you want something, the more aggressively you act.

• If you really love a painting, you bid high.
• If you are desperate to win a contract, you bid low.
• If you are a strong competitor, you try harder.

This is called a Monotone Strategy. It's like a "secret handshake" of the economic world. If this rule holds, we can trust our economic models. But if players are cheating, colluding, or acting irrationally, this rule breaks.

The Catch: We can't see the players' minds. We only see their hands (their bids). For a long time, economists didn't have a good way to check if the "secret handshake" was actually happening without guessing what was in the players' heads.

The Solution: The "Reverse Engineer" Trick

The authors (Hsu, Li, Liu, and Takahashi) came up with a clever way to check the handshake without needing to see the minds.

Think of it like this: Imagine you are watching a magician pull a rabbit out of a hat. You can't see the rabbit inside the hat (the private type), but you can see the hat and the rabbit (the action).

The authors realized that if the magician is following the rules (monotone strategy), the relationship between the hat and the rabbit has a very specific shape. They call this the "Quasi-Inverse."

• The Old Way: Try to guess what's in the hat based on the rabbit. (Hard and error-prone).
• The New Way: Look at the rabbit and the hat together. If the rabbit gets bigger, does the hat get bigger in a smooth, predictable way? If the relationship gets "wobbly" or goes backward, the magician is cheating (or the players are colluding).

They proved mathematically that checking if the strategy is monotone is exactly the same as checking if this "Quasi-Inverse" relationship is smooth and steady.

How the Test Works: The "Smoothness" Check

Once they turned the problem into checking this relationship, they didn't need complex, messy math. They turned it into a simple "Smoothness Test."

Imagine you are drawing a line on a graph.

• The Rule: The line must always go up (or stay flat). It can never dip down.
• The Test: The authors created a statistical tool (a "Cramér–von Mises" statistic) that looks at the line and asks: "Is this line dipping anywhere?"

They broke the line down into thousands of tiny segments and checked each one. If even a tiny segment dips down, the test says, "Alert! The monotone strategy is broken!"

They use a computer simulation (called "bootstrapping") to figure out what a "normal" wobble looks like versus a "suspicious" wobble.

Why This Matters: Catching the Cheaters

Why do we care if a line goes up or down? Because in the real world, broken monotonicity often means collusion.

• The Analogy: Imagine a group of bidders at an auction. If they are all competing fairly, the person who wants the item the most will bid the highest. The bids will be perfectly sorted.
• The Collusion: If the bidders are a "cartel" (a secret gang), they might agree to let one person win with a low bid, or they might bid weirdly to confuse the auctioneer. This messes up the "sorted" order. The line on the graph starts to wiggle and dip.

Real-World Detective Work

The authors tested their new lie detector on asphalt paving auctions (road construction projects) in Sweden, Finland, and California.

1. California (The Control): This market is known to be competitive. The test looked at the bids and said, "Everything looks smooth. The line goes up perfectly. No cheating detected." (The test passed).
2. Sweden & Finland (The Suspects): These markets had known history of cartels.
   • In Sweden, the test found a big dip in the line. It screamed, "Something is wrong here!" This matched the history of known cartels.
   • In Finland, the evidence was weaker, but still present.

The Takeaway

This paper gives economists a new, easy-to-use tool to check if players in a game are behaving rationally or if they are up to no good.

• Before: We had to guess if players were colluding based on complicated theories.
• Now: We can look at the data, run this "smoothness test," and get a clear "Yes/No" answer on whether the players are following the rules of fair competition.

It's like giving the police a new fingerprint scanner that works even when the suspect is wearing gloves. We can now detect the "fingerprint" of collusion just by looking at the actions, without ever needing to see the secret thoughts behind them.

Technical Summary

1. Problem Statement

In games of incomplete information (Bayesian games), a fundamental prediction is that equilibrium strategies are monotone in private types (e.g., higher valuations lead to higher bids, lower costs lead to higher output). This monotonicity is crucial for:

• Identification: It allows econometricians to invert observed actions to recover latent private types (structural estimation).
• Validity: If monotonicity fails, the structural model and its counterfactuals are invalid.
• Collusion Detection: In auctions, deviations from monotone competitive bidding can signal cartel behavior.

The Gap: While identification and estimation of Bayesian games are well-developed, there is a lack of rigorous statistical tools to test the monotonicity of equilibrium strategies. Existing methods are limited (e.g., restricted to first-price auctions or relying on complex nonparametric concavity tests like the Least Concave Majorant). Furthermore, existing tools struggle with game heterogeneity (observed and unobserved).

2. Methodology

The authors develop a unified framework to test for monotonicity by transforming the problem from unobserved types to observed actions.

A. Theoretical Foundation: Rationalization and Quasi-Inverse

The paper establishes a Rationalization Result under the Symmetric Independent Private Type (SIPT) paradigm:

1. Equivalence: A differentiable Bayesian Nash Equilibrium (BNE) strategy is monotone in private types if and only if the quasi-inverse equilibrium strategy is monotone in observed actions.
2. Definition of Quasi-Inverse ($\xi$): The quasi-inverse is defined as the ratio of the derivatives of two reduced-form objects identified from data:
   $$\xi(b) = \frac{T'(b)}{P'(b)}$$
   Where:
   • $P(b)$: Expected allocation rule (probability of winning/obtaining the good).
   • $T(b)$: Expected payment rule (expected cost or transfer).
   • $b$: Observed action (bid, effort, output).
3. Identification: Since $P(b)$ and $T(b)$ are functions of observed actions and known institutional rules, $\xi(b)$ is nonparametrically identified from the distribution of observed actions.

B. Test Construction: Moment Inequalities

Instead of testing the derivative of $\xi(b)$ directly (which requires density estimation), the authors reformulate the null hypothesis ($H_0: \xi(b)$ is weakly increasing) into a countable set of moment inequalities involving only unconditional expectations.

• Transformation: Using a lemma adapted from Hsu and Shen (2021), the monotonicity condition is transformed into:
  $$\nu(b_1, b_2, q) = M(b_2, q)W(b_1, q) - M(b_1, q)W(b_2, q) \leq 0$$
  Where $M$ and $W$ are integrals of the reduced-form functions weighted by a known function $h(b)$.
• Advantage: This avoids the "curse of dimensionality" associated with estimating densities and allows the use of simple sample analogs for estimation.

C. Statistical Inference

• Test Statistic: A Cramér–von Mises (CvM) type statistic is constructed based on the estimated moment inequalities.
• Critical Values: The authors use a nonparametric bootstrap combined with the Generalized Moment Selection (GMS) method (Andrews and Shi, 2013). This approach:
  • Controls the asymptotic size of the test.
  • Ensures consistency against fixed alternatives.
  • Avoids the "least favorable configuration" problem, improving power.
• Heterogeneity:
  • Observed Heterogeneity: The test is extended to control for covariates (e.g., auction size, prize value) nonparametrically or via semiparametric homogenization (scaling actions by a function of covariates).
  • Unobserved Heterogeneity: The framework provides valid testable implications even when unobserved game characteristics exist, by integrating them out of the moment conditions.

3. Key Contributions

1. Unified Framework: The paper moves beyond first-price auctions to cover a broad class of Bayesian games with continuous actions, including Tullock contests, public good provision, and Cournot competition.
2. Methodological Innovation: It replaces the Least Concave Majorant (LCM) approach (used in Liu and Vuong, 2021) with a moment inequality approach. This is more flexible, easier to implement, and naturally accommodates covariates and heterogeneity.
3. Handling Heterogeneity: It provides the first formal testing procedures for monotonicity that explicitly handle both observed (via covariates) and unobserved (via integration) game heterogeneity, a critical issue in empirical structural modeling.
4. Empirical Application: It provides the first formal test of monotonicity in real-world data, applied to asphalt paving procurement auctions to detect cartels.

4. Results

Simulation Studies

• Size Control: Monte Carlo simulations demonstrate that the test correctly controls the size (Type I error) at the nominal level (e.g., 10%) when the null hypothesis holds.
• Power: The test exhibits strong power against alternatives where monotonicity is violated. Power increases with sample size and the magnitude of the violation.
• Robustness: The results are robust to different choices of tuning parameters (e.g., the number of moment inequalities).

Empirical Application: Asphalt Paving Auctions

The authors applied the test to datasets from Sweden, Finland, and California (Aaltio et al., 2025):

• California (Competitive Market): The test failed to reject the null hypothesis of monotonicity. This is consistent with the market being competitive and free of collusion.
• Sweden & Finland (Cartel Markets):
  • Sweden: The test rejected monotonicity in several specifications (particularly for auctions with 5 bidders), providing strong evidence of non-competitive behavior.
  • Finland: Results showed weak evidence against monotonicity.
• Interpretation: The rejection of monotonicity in Sweden aligns with historical evidence of cartel activity. The authors note that the Swedish cartel was more successful at mimicking competitive behavior in some aspects, but the monotonicity test successfully detected the underlying structural deviation that other methods might miss.

5. Significance

• Validation of Structural Models: This test serves as a crucial "sanity check" for structural econometric models. If monotonicity is rejected, researchers cannot rely on standard inversion methods to identify private types.
• Antitrust and Policy: The method offers a powerful new tool for competition authorities to detect collusion in procurement auctions and other strategic environments without needing to observe private costs or valuations directly.
• Theoretical Extension: By extending the rationalization logic from auctions to general allocation-transfer games, the paper bridges the gap between theoretical game theory (existence of monotone equilibria) and empirical testing.

In summary, the paper provides a rigorous, flexible, and implementable solution to a long-standing problem in the econometrics of games: how to statistically verify that agents are behaving according to a monotone equilibrium strategy in the presence of unobserved types and game heterogeneity.