[Paper Review] Neural Networks, Game Theory and Time Series Generation

Richard Metzler|arXiv (Cornell University)|Dec 19, 2002

Neural Networks and Applications73 references3 citations

TL;DR

This dissertation investigates the interplay between neural networks, game theory, and time series generation, introducing antipredictable time series that foil standard prediction algorithms. It analyzes simple and continuous perceptrons, decision tables, and minority game variants, demonstrating how neural agents can learn optimal strategies in zero-sum games and how complex, chaotic dynamics emerge in learning systems.

ABSTRACT

This dissertation highlights connections between the fields of neural networks, game theory and time series generation. The concept of antipredictability is explained, and the properties of time series that are antipredictable for several prototypical prediction algorithms (neural networks, Boolean funtions etc.) are studied. The Minority Game provides a framework in which antipredictability arises naturally. Several variations of the MG are introduced and compared, including extensions to more than two choices, and the properties of the generated time series are analysed. A learning algorithm is presented by which a neural network can find a good mixed strategy in zero-sum matrix games. In a certain limit, this algorithm is a stochastic variation of the "fictitious play" learning algorithm.

Motivation & Objective

• To understand the limits of time series prediction by constructing antipredictable sequences that foil specific algorithms.
• To analyze the dynamical and statistical properties of time series generated by neural networks, particularly perceptrons with Hebbian learning.
• To explore the emergence of complex behavior—such as chaos and intermittent dynamics—in continuous perceptrons and decision tables.
• To investigate how neural networks and learning rules can converge toward Nash equilibria in two-player zero-sum games.
• To generalize the Minority Game to multi-choice settings and study the role of stochastic and neural strategies in collective decision-making.

Proposed method

• Uses simple and continuous perceptrons with Hebbian learning rules to generate binary and continuous time series with controlled autocorrelation functions.
• Applies mean-field theory and Lyapunov exponent calculations to analyze chaotic dynamics in continuous perceptrons.
• Employs De Bruijn graphs and graph-theoretic methods to analyze cycle structures in decision tables and antipersistent time series.
• Introduces stochastic strategies in the Minority Game and solves them analytically for small and large populations.
• Adapts Hebbian learning rules for matrix games, showing convergence toward Nash equilibrium strategies in zero-sum games.
• Uses Monte Carlo simulations and analytical approximations to validate results in high-dimensional and complex systems.

Experimental results

Research questions

• RQ1 What are the structural and dynamical properties of time series that are inherently unpredictable by specific prediction algorithms?
• RQ2 How do the weight dynamics and output statistics of perceptrons relate to the autocorrelation structure of generated time series?
• RQ3 What role do cycle length and attractor dimension play in the predictability and chaotic behavior of neural time series?
• RQ4 Can stochastic and neural strategies in the Minority Game achieve optimal performance, and how do they compare to random or evolutionary strategies?
• RQ5 To what extent can Hebbian learning in neural networks approximate Nash equilibria in two-player zero-sum games?

Key findings

• Antipredictable time series exist for any prediction algorithm, with properties such as suppressed correlations and complex cycle structures that foil standard predictors.
• The Confused Bit Generator (CBG) based on Hebbian learning in a simple perceptron can generate binary sequences with tailored autocorrelation functions.
• Continuous perceptrons exhibit high-dimensional chaotic dynamics with positive Lyapunov exponents and intermittent behavior, especially for large system sizes and high gain.
• Decision tables with dynamic antipersistence generate time series with predictable cycle lengths and distributions, analyzable via De Bruijn graphs.
• Stochastic strategies in the Minority Game achieve analytical solutions and show robustness to noise, with performance comparable to deterministic strategies.
• Hebbian-based learning rules in two-player games can approximate Nash equilibria, particularly in small payoff matrices, and generalize to biased pattern learning with modified rules.