”Illusion of control” in Time-Horizon Minority and Parrondo Games

Abstract

Please replace the paragraph immediately before equation (7) with: "Actions are drawn from a reduced strategy space (RSS) of dimension r = 2 m+1 . Each action is associated with a strategy k and a history µ(t). There are 2 m histories. Together, all actions by history and strategy (converted for convenience from {O,1}) can be represented in table form as a dim(RSS)×2 m binary matrix (i.e., a µt k ∈ {−1, 1}). In this case, the table reads:"

Illusory versus genuine control in agent-based games

“…In other words, for a range of m (given N, S), agent performance is better than what strategy performance would be in a game with no agents optimizing. The phenomenon discussed in [2] is that when optimizing, and averaged over all actual agents and their component strategies in a given realization, and then averaged over many such initial quenched disorder states, agents in the TH variant of the MG nonetheless underperform the mean of their own measured strategy performance and do so in all where a subset of agents "freeze" their choice of strategy: One virtual strategy permanently higher value than any other. These frozen agents in general do outperform the mean of all strategies in a given realization as well as the mean of their own S original strategies: They perform precisely as well as their best.…”

“…In Ref. [2], we consider generalizations of the MG in which more clever agents use this insight to develop strategies based on their awareness of prior-level effects: when agents use a boundless recursion scheme to learn and optimize their strategy, the game converges to an equilibrium with fully symmetric mixed strategies where agents randomize their choice at each time step with unbiased coin tosses! The crowding mechanism also predicts that the smaller the parameter 2 m /N, the larger the illusion-of-control effect.…”

“…c-agents "anti-optimize"-at each time-step they deploy that strategy with the fewest virtual points accumulated over τ, rather than the strategy with the most points as do s-agents. In [2], we studied this phenomenon in the "crowded regime" ( , ) where crowd and anti-crowd formation is likely and the minority mechanism causes the larger-sized crowds to lose. In this regime, c-agents that choose their worst performing strategy consistently outperform s-agents that choose their best.…”

“…In fact, the opposite is true. (In the MG proper the situation is more complex and is discussed at length in [2]. Here we note only that two conditions must hold for the statement to be false: (1) m ≥ m c ; (2) the system must be allowed to reach equilibrium.…”

“…Our main result with respect to the THMG may be stated concisely from the pe ility theory and has been detailed extensively in [2]: Throughout the space of parameters (N, m, S, eq τ τ ), the mean payoff of agents' strategies (as calculated by each agent averaged over all strategies and agents in a realization) not only surpasses the mean payoff of supposedlyoptimizing agents (averaged over all given agents), but the respective cumulative distribution functions (CDF) of payoffs show a first-order stochastic dominance of strategies over agents. Thus, were the option available to them, agents would behave in a risk-averse fashion (concave utility function) by switching randomly between strategies rather than optimizing.…”