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The present article constitutes part II of a series of two reports in which we study the decomposition of synthetic and real financial time-series into a superposition of weighted Hamiltonian cycles on graphs. Part II further analyses the cycle-decomposition method introduced in part I for the Minority Game (MG), the Majority Game (MAJG) and the Dollar Game ($G), in order 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. We also illustrate both numerical and analytical methods for extracting cycles from a given time-series and apply the method to a number of different real-world data sets, in conjunction with an analysis of persistence. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the Minority, Majority and Dollar Games (MG, MAJG, $G) agents compete for rewards, acting in accord with the previously best-performing of their strategies. Different aspects/kinds of real-world markets are modelled by these games. In the MG, agents compete for scarce resources; in the MAJG agents imitate the group to exploit a trend; in the $G agents attempt to predict and benefit both from trends and changes in the direction of a market. It has been previously shown that in the MG for a reasonable number of preliminary time steps preceding equilibrium (Time Horizon MG, THMG), agents’ attempt to optimize their gains by active strategy selection is “illusory”: the hypothetical gains of their strategies is greater on average than agents’ actual average gains. Furthermore, if a small proportion of agents deliberately choose and act in accord with their seemingly worst performing strategy, these outperform all other agents on average, and even attain mean positive gain, otherwise rare for agents in the MG. This latter phenomenon raises the question as to how well the optimization procedure works in the THMAJG and TH$G. We demonstrate that the illusion of control is absent in THMAJG and TH$G. This provides further clarification of the kinds of situations subject to genuine control, and those not, in set-ups a priori defined to emphasize the importance of optimization. [ABSTRACT FROM AUTHOR]

Human beings like to believe they are in control of their destiny. This ubiquitous trait seems to increase motivation and persistence, and is probably evolutionarily adaptive [J.D. Taylor, S.E. Brown, Psych. Bull. 103, 193 (1988); A. Bandura, Self-efficacy: the exercise of control (WH Freeman, New York, 1997)]. But how good really is our ability to control? How successful is our track record in these areas? There is little understanding of when and under what circumstances we may over-estimate [E. Langer, J. Pers. Soc. Psych. 7, 185 (1975)] or even lose our ability to control and optimize outcomes, especially when they are the result of aggregations of individual optimization processes. Here, we demonstrate analytically using the theory of Markov Chains and by numerical simulations in two classes of games, the Time-Horizon Minority Game [M.L. Hart, P. Jefferies, N.F. Johnson, Phys. A 311, 275 (2002)] and the Parrondo Game [J.M.R. Parrondo, G.P. Harmer, D. Abbott, Phys. Rev. Lett. 85, 5226 (2000); J.M.R. Parrondo, How to cheat a bad mathematician (ISI, Italy, 1996)], that agents who optimize their strategy based on past information may actually perform worse than non-optimizing agents. In other words, low-entropy (more informative) strategies under-perform high-entropy (or random) strategies. This provides a precise definition of the “illusion of control” in certain set-ups a priori defined to emphasize the importance of optimization. [ABSTRACT FROM AUTHOR]

A correction to the article ""Illusion of Control" in Time-Horizon Minority and Parrondo Games," that was published in the 2007 issue is presented.