Statistical Games (2402.15892v1)

Abstract: This work contains the mathematical exploration of a few prototypical games in which central concepts from statistics and probability theory naturally emerge. The first two kinds of games are termed Fisher and Bayesian games, which are connected to Frequentist and Bayesian statistics, respectively. Later, a more general type of game is introduced, termed Statistical game, in which a further parameter, the players' relative risk aversion, can be set. In this work, we show that Fisher and Bayesian games can be viewed as limiting cases of Statistical games. Therefore, Statistical games can be viewed as a unified framework, incorporating both Frequentist and Bayesian statistics. Furthermore, a philosophical framework is (re-)presented -- often referred to as minimax regret criterion -- as a general approach to decision making. The main motivation for this work was to embed Bayesian statistics into a broader decision-making framework, where, based on collected data, actions with consequences have to be made, which can be translated to utilities (or rewards/losses) of the decision-maker. The work starts with the simplest possible toy model, related to hypothesis testing and statistical inference. This choice has two main benefits: i.) it allows us to determine (conjecture) the behaviour of the equilibrium strategies in various limiting cases ii.) this way, we can introduce Statistical games without requiring additional stochastic parameters. The work contains game theoretical methods related to two-player, non-cooperative games to determine and prove equilibrium strategies of Fisher, Bayesian and Statistical games. It also relies on analytical tools for derivations concerning various limiting cases.

• The paper introduces a game-theoretic framework that unifies Bayesian and Frequentist inference by mapping statistical decisions to Nash equilibria.
• It develops and analyzes Fisher and Bayesian games using hypergeometric distributions and isoelastic utility functions.
• The study demonstrates practical applications in AI and decision science by extending hypothesis testing to strategic error analysis.

An Overview of Statistical Games

The paper "Statistical Games: A Playful Approach to Statistics" by József Konczer presents a novel paradigm in statistical inference and decision science through the lens of game theory. It explores the intersection of statistics with a strategic, game-theoretic framework, offering a fresh perspective on classical statistical problems. This approach posits a model where two-player non-cooperative games serve as analogues for statistical processes, allowing the integration of Frequentist and Bayesian concepts in equilibrium strategies. This essay explores the primary contributions and implications of this work, emphasizing the mathematical rigor and potential future applications in AI.

Main Contributions

The paper introduces several types of two-player games, most notably Fisher and Bayesian games, which serve as foundational blocks in this theoretical framework. These games are not only mathematical constructs but also illustrative models for understanding statistical inference. In such setups, one player, representing a statistician, makes decisions under uncertainty by sampling data, while the other player, symbolizing nature or an adversary, controls the scenario. The central innovation lies in mapping statistical decisions to strategic equilibria.

One key result is the establishment of symmetrical Nash equilibria for these games, derived from hypergeometric distributions reflecting sampling processes. Such equilibria, backed by formal proofs, substantiate the paper's argument for viewing statistical inference through game-theoretic lenses. An intriguing aspect is the introduction of a generalized statistical game, unifying Bayesian and Fisher games by considering the players' relative risk aversion. This unification is further enriched by a connection to isoelastic utility functions, which naturally arise in repeated multiplicative betting situations.

The paper also extends the classical methodology of hypothesis testing to game theory, offering an interpretation where Type I and Type II errors relate directly to strategic decisions within these games. The adoption of generalized entropies, including Rényi and Tsallis types, also illustrates the potential to interpolate between classical statistical paradigms, aligning with expected utility theory principles.

Strong Numerical Results

The work provides detailed analyses of specific cases, such as the smallest nontrivial statistical games and their equilibria. Through mathematical modeling and numerical simulations, the paper elucidates how certain parameter choices lead to optimal strategies, like the golden ratio appearing as a solution in a simple Bayesian game. These findings not only affirm the theoretical consistency of the approach but also highlight intriguing mathematical phenomena arising from strategic interactions under statistical contexts.

Theoretical and Practical Implications

The implications of this research extend significantly into both theoretical and practical domains. Theoretically, it challenges conventional interpretations of probability and statistics by grounding them in strategic decision-making processes. This viewpoint suggests potential refinements in how uncertainty and decision contexts are modeled across various scientific disciplines. Practically, the game-theoretic models could enrich AI systems, particularly in areas involving uncertain and adversarial environments, such as automated decision-making and reinforcement learning.

Furthermore, by suggesting that probability concepts might not stem solely from frequencies or beliefs but emerge from equilibria in statistical games, the paper prompts a reconsideration of foundational aspects in probability theory. Such an interpretation could inspire novel approaches in statistical education, offering students a dynamic and interactive understanding of uncertainty.

Future Directions

The paper concludes by outlining several avenues for further exploration, including expanding the framework to handle more complex decision problems and integrating it with AI methodologies. There is also a suggestion to explore the implications for operations research, economics, and evolutionary biology, areas inherently dealing with strategies under uncertainty.

In summary, József Konczer's "Statistical Games" presents a compelling case for the integration of game theory and statistics, offering a robust framework that bridges conceptual gaps between disparate statistical methodologies. Its mathematical rigor, coupled with practical implications, provides a fertile ground for future interdisciplinary research, potentially transforming traditional views on statistical inference and decision science.