Cycles, determinism and persistence in agent-based games and financial time-series: part I

The Minority Game (MG), the Majority Game (MAJG) and the Dollar Game ($G) are important and closely related versions of market-entry games designed to model different features of real-world financial markets. In a variant of these games, agents measure the performance of their available strategies over a fixed-length rolling window of prior time-steps. These are the Time Horizon MG/MAJG/$Gs (THMG, THMAJG, TH$G). Their probabilistic dynamics may be completely characterized in Markov-chain formulation. Games of both the standard and TH variants generate time-series that may be understood as arising from a stochastically perturbed determinism because a coin toss is used to break ties. The average over the binomially distributed coin tosses yields the underlying determinism. In order to quantify the degree of this determinism and of higher-order perturbations, we decompose the sign of the time-series they generate (analogous to a market price time-series) into a superposition of weighted Hamiltonian cycles on graphs—exactly in the TH variants and approximately in the standard versions. The cycle decomposition also provides a ‘dissection’ of the internal dynamics of the games and a quantitative measure of the degree of determinism. We discuss how the outperformance of strategies relative to agents in the THMG—the ‘illusion of control’—and the reverse in the THMAJG and TH$G, i.e. genuine control, may be understood on a cycle-by-cycle basis. The decomposition offers a new metric for comparing different game dynamics with real-world financial time-series and a method for generating predictors. We apply the cycle predictor to a real-world market, with significantly positive returns for the latter. Part I provides an overview of the paper and its methodologies with an appendix for the mathematical details of the Markov analysis of the THMG, THMAJG and TH$G. Part I also describes the cycle predictor and applies it to real-world financial series. Part II performs further analyses of the cycle decomposition method as applied to the time-series generated by agent-based models to gain insight into the ‘illusion of control’ that certain of these games demonstrate, i.e. the fact that the strategies outperform the agents that deploy them. Part II also illustrates both numerical and analytic methods for extracting cycles from a given time-series and applies the method to a number of different real-world data sets, in conjunction with an analysis of persistence.