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

The Minority Game (MG), the Majority Game (MAJG) and the Dollar Game ($G) are important and closely-related versions of market-entry games designed to model different features of real-world financial markets. In a variant of these games, agents measure the performance of their available strategies over a fixed-length rolling window of prior time-steps. These are the so-called Time Horizon MG/MAJG/$G (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 as well a new metric for comparing different game dynamics to real-world financial time-series and a method for generating predictors. We apply the cycle predictor a real-world market, with significantly positive returns for the latter.