[Paper Review] Games of Incomplete Information Played By Statisticians

Annie Liang|arXiv (Cornell University)|Oct 15, 2019

Sports Analytics and Performance4 citations

TL;DR

This paper proposes a confidence metric for strategic predictions in incomplete information games where players are statistical learners with common knowledge of a set of learning rules but may interpret data differently. As data grows large, the analyst's confidence in a prediction converges to certainty if the prediction is strictly rationalizable in the limit, providing a quantitative framework for robustness that generates novel comparative statics, such as increased speculative trade with high-dimensional data.

ABSTRACT

Players are statistical learners who learn about payoffs from data. They may interpret the same data differently, but have common knowledge of a class of learning procedures. I propose a metric for the analyst's "confidence" in a strategic prediction, based on the probability that the prediction is consistent with the realized data. The main results characterize the analyst's confidence in a given prediction as the quantity of data grows large, and provide bounds for small datasets. The approach generates new predictions, e.g. that speculative trade is more likely given high-dimensional data, and that coordination is less likely given noisy data.

Motivation & Objective

• To address the limitation of common prior assumptions in incomplete information games, which fail to capture persistent belief disagreement in real-world settings.
• To model players as statistical learners who interpret data differently but share common knowledge of plausible learning rules.
• To develop a quantitative confidence metric for strategic predictions that reflects uncertainty due to heterogeneous belief formation.
• To characterize the asymptotic behavior of this confidence metric as data grows large, and to bound its behavior for finite datasets.
• To generate new, testable comparative statics—such as increased speculative trade with high-dimensional data and reduced coordination with noisy data—under this framework.

Proposed method

• Define a belief restriction where each player assigns probability 1 to all others holding plausible beliefs, forming a common knowledge structure over a set of learning rules.
• Construct a confidence set for a strategic prediction as the interval between the minimum and maximum probability that the prediction holds across all plausible beliefs derived from the learning rules.
• Use a metric dP to measure the distance between belief distributions induced by different learning rules, ensuring uniform convergence under a key assumption.
• Apply rate bounds to quantify how quickly confidence sets converge to their asymptotic limits, depending on data quantity and learning rule convergence speed.
• Extend the framework to approximate common p-belief and to equilibrium-based predictions, maintaining convergence under mild conditions.
• Use the limiting behavior of learning rules to derive conditions under which confidence sets converge to {1} (certainty) or {0} (refutation) as data grows.

Experimental results

Research questions

• RQ1 How does the analyst's confidence in a strategic prediction evolve as the amount of data available to players increases?
• RQ2 Under what conditions does the confidence set for a prediction converge to certainty (i.e., {1}) or refutation (i.e., {0}) in the limit of large data?
• RQ3 To what extent can the confidence set for a prediction differ from its asymptotic limit when only finite data is available?
• RQ4 How do the structure of the learning rules and the dimensionality of the data affect the analyst's confidence in predictions like speculative trade or coordination?
• RQ5 Can the framework generate new, empirically testable comparative statics that are not possible under the common prior assumption?

Key findings

• If the set of learning rules satisfies a uniform convergence property, then the analyst’s confidence set converges to {1} for any action that is strictly rationalizable in the limit, and to {0} for any action that is not rationalizable in the limit.
• When learning rules do not satisfy uniform convergence, confidence sets may fail to converge continuously to {1} even with arbitrarily large data, due to unbounded convergence rates across rules.
• For finite datasets, the paper provides bounds on the deviation of the confidence set from its asymptotic limit, depending on data quantity, learning rule convergence speed, and the strictness of the prediction at the limit.
• The framework predicts that speculative trade is more likely when agents learn from high-dimensional data, a result not derivable under the common prior assumption.
• Coordination is less likely when agents observe noisy data, again a novel comparative static that emerges from the model’s structure.
• The confidence metric can be extended to equilibrium predictions, where convergence to {1} or {0} holds under analogous conditions for strict Bayesian Nash equilibria.