  In the Minority, Majority and Dollar Games (MG, MAJG, $G), synthetic agents
compete for rewards, at each time-step acting in accord with the previously
best-performing of their limited sets of strategies. Different components
and/or aspects of real-world financial markets are modelled by these games. In
the MG, agents compete for scarce resources; in the MAJG gents imitate the
group in the hope of exploiting a trend; in the $G agents attempt to
successfully predict and benefit from trends as well as 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 calculated hypothetical gains of their individual
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 outper-form 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 MAJG and $G. We demonstrate that the
illusion of control is absent in MAJG and $G. In other words, low-entropy (more
informative) strategies under-perform high-entropy (or random) strategies in
the MG but outperform them in the MAJG and $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.
