"Illusion of control" in Minority and Parrondo Games
The efficient punishment theory is a theory of mind, which attempts to give a rational answer to the paradoxal evidence that h
“Illusion of Control” in Minority and Parrondo Games 
J.B. Satinover1,a and D. Sornette2,b
1Laboratoire de Physique de la Matière Condensée, CNRS UMR6622  and Université des 
Sciences, Parc Valrose, 06108 Nice Cedex 2, France 
2Department of Management, Technology and Economics, ETH Zurich, CH-8032 Zurich, 
Switzerland 
Abstract: 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 [1,2]. 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 [3] 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 Minority game [4,5] and the 
Parrondo Games [6,7], that agents who optimize their strategy based on past information 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 set-ups a priori defined to emphasize the importance of optimization. 
PACS. 89.75.-k Complex systems – 89.65Gh Economics; econophysics, financial markets, 
business and management – 02.50.Le Decision theory and game theory  
1 Introduction 
The success of science and technology, with the development of ever more sophisticated 
computerized integrated sensors in the biological, environmental and social sciences, illustrate 
the quest for control as a universal endeavor. The exercise of governmental authority, the 
managing of the economy, the regulation of financial markets, the management of 
corporations, and the attempt to master natural resources, control natural forces and influence 
environmental factors all arise from this quest. Langer’s phrase, “illusion of control” [3] 
describes the fact that individuals appear hard-wired to over-attribute success to skill, and to 
underestimate the role of chance, when both are in fact present. Whether control is genuine or 
merely perceived is a prevalent question in psychological theories. The sequel presents two 
rigorously controlled mathematical set-ups demonstrating generic circumstances in which 
optimizing agents perform worse than non-optimizing or random agents. 
2 Minority Games 
We first study Minority games (MGs), which constitute a sub-class of market-entry 
games. MGs exemplify situations in which the “rational expectations” mechanism of standard 
economic theory fails. This mechanism in effect asks, “what expectational model would lead 
to collective actions that would on average validate the model, assuming everyone adopted 
it,”?[8]. In minority games, a large number of interacting decision-making agents, each 
aiming for personal gain in an artificial universe with scarce resources, try to anticipate the 
actions of others on the basis of incomplete information. Those who subsequently find 
themselves in the minority group gain. Therefore, expectations that are held in common 
negate themselves, leading to anti-persistent behavior both for the aggregate behavior and for 
                                                 
a jsatinov@princeton.edu
b dsornette@ethz.ch
mailto:jsatinov@princeton.edu
mailto:dsornette@ethz.ch
individuals. Minority games have been much studied as repeated games with expectation 
indeterminacy, multiple equilibria and inductive optimization behavior. 
Consider the Time-Horizon MG (THMG), where N players have to choose one out of 
two alternatives at each time step based on information represented as a binary time series 
A(t). Those who happen to be in the minority win. Each agent is endowed with S strategies. 
Each strategy gives a prediction for the next outcome A(t) based on the history of the last m 
realizations A(t-1), …, A(t-m) (m is called the memory size of the agents). Each agent holds 
the same number S of (in general different) strategies among the total number 22
strategies. The S strategies of each agent are chosen at random but once and for all at the 
beginning of the game. At each time t, in the absence of better information, in order to decide 
between the two alternatives for A(t), each agent uses her most successful strategy in terms of 
payoff accumulated in a rolling window of finite length τ up to the last information available 
at the present time t. This is the key optimization step. If her best strategy predicts A(t)=+1 
(resp. -1), she will take the action ai(t) = - 1 (resp. +1). The aggregate behavior A(t) = Σi=1N  
ai(t) is then added to the information set available for the next iteration at time t+1. The 
corresponding instantaneous payoff of agent i is given by – sign[ai(t)  A(t)] (and similarly for 
each strategy for which it is added to the τ-1 previous payoffs). As the name of the game 
indicates, if a strategy is in the minority (ai(t)  A(t) < 0), it is rewarded. In other words, agents 
in MG try to be anti-imitative. The richness and complexity of minority games stem from the 
fact that agents strive to be different. Previous investigations have shown the existence of a 
phase transition between an inefficient regime and an uncorrelated phase as the parameter 
2m/N is increased, with the size of the fluctuations of A(t) (as measured by its normalized 
variance σ2/N) falling below the random coin-toss limit for large m’s when agents always use 
their highest scoring strategy [5]. The phenomenon discussed here is not directly related to 
these effects as it is observed in all regimes. 
Our main result may be stated concisely from the perspective of utility theory: 
Throughout the space of parameters (N, m, S, τ), the mean payoff of strategies not only 
surpasses the mean payoff of supposedly-optimizing 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. This result generalizes when comparing optimizing agents with S>1 strategies 
with agents having only one strategy (or equivalently m identical strategies), when the single 
strategies are actually implemented (which takes into account the impact of the strategy 
itself). The same result is also found when comparing optimizing agents with agents flipping 
randomly between their S strategies. Agents are supposed to enhance their performance by 
choosing adaptively between their available strategies. In fact, the opposite is true: The 
optimization method is found to be strictly a method for worsening performance! 
This “illusion of control” effect is observed for all N, m, S and τ . Indeed, extensive 
numerical simulations show that all the phenomena we derive for the THMG with the 
simplest parameter values are found in the MG with arbitrary parameters. We use the Markov 
chain formalism for the THMG [8,9] to obtain the following theoretical prediction for the 
average gains ΔWAgent and ΔWStrategy, respectively of optimizing agents and of their strategies 
[10]:  
Agent DNW A μΔ = ⋅  (1) 
( )12 ˆStrategy NW μ κ μΔ = ⋅s ⋅  (2) 
where the brackets denote a time average. μ is a ( )m τ+ -bit “path history” (sequence of 1-bit 
states); μ
 is the normalized steady-state probability vector for the history-dependent 
( ) (m mτ τ+ × +  transition matrix , where a given element T̂
1,t t
Tμ μ − represents the transition 
probability that 1tμ −  will be followed by tμ ; DA  is a 
(2 m )τ+ -element vector listing the 
particular sum of decided values of A(t) associated with each path-history; ˆμs  is the table of 
points accumulated by each strategy for each path-history; κ  is a (2 m )τ+ -element vector listing 
the total number of times each strategy is represented in the collection of N agents.  As shown 
in the supplementary material,  may be derived from T̂ DA , ˆμs  and , the number of 
undecided agents associated with each path history.  Thus agents’ mean gain is determined by 
the non-stochastic contribution to A(t) weighted by the probability of the possible path 
histories. This is because the stochastic contribution for each path history is binomially 
distributed about the determined contribution. Strategies’ mean gain is determined by the 
change in points associated with each strategy over each path-history weighted by the 
probability of that path.   
We find excellent agreement between the numerical simulations and the analytical 
predictions (1) and (2) for the THMG. For instance, for m=2, S=2, τ=1 and N=31,  
‚∆W Ú=-0.22 for both analytic and numerical methods (Agent payoff per time step averaged over 
time and over the optimizing agents) compared with ‚∆W Ú=-0.06 also for both analytic 
and numerical methods (corresponding payoff for individual strategies). In this numerical 
example, the average payoff of individual strategies is larger than for optimizing agents by 
0.16 units per time step.  The numerical values of the predictions (1) and (2) are obtained by 
implementing each agent individually as a coded object.
Strategy
Taking into account the impact of strategies modifies these results quantitatively but not 
qualitatively. The mean per-agent per-step payoff ‚∆WNon-OptÚ accrued by non-optimizing 
agents (because they have only one fixed strategy, or equivalently their S strategies are 
identical) is larger than the payoff ‚∆WAgentÚ of optimizing agents. In general, this 
comparative advantage decreases with their proportion. For example, with m=2, S=2, τ=1 and 
N=31, and 2500 random initializations and n optimizing agents, ‚∆WNon-OptÚ-‚∆WAgentÚ= 
(2.380, 2.270, 2.289, 2.275, 2.145, 2.060, 2.039, 1.994, 1.836, 1.964)  for n=1, 2, …, 10. 
More generally, the following ordering holds: payoff of individual strategies > payoff of non-
optimizing agents > payoff of optimizing agents. The first inequality is due to the fact that not 
all individual strategies are implemented and the theoretical payoff of the non-implemented 
strategies does not take into account what their impact would have been (had they been 
implemented). Implementation of a strategy tends to decrease its performance (this is similar 
to the market impact of trading strategies in financial markets associated with slippage and 
market friction). Non-optimizing agents by definition always implement their strategies. 
However, the higher payoff of non-optimizing compared with optimizing agents shows that 
the illusion-of-control effect is not due to the impact mechanism, but is a genuine observable 
effect.  
310−×
The amplitude of the illusion-of-control effect highlights the distinction between 
optimizing agents with S maximally distinct strategies (in the sense of Hamming distance) 
and non-optimizing agents with S identical strategies. It is helpful to generalize this 
dichotomy by characterizing the degree of similarity between the S strategies of a given agent 
using the Hamming distance dH between strategies (the Hamming distance between two 
strings of equal length is the number of positions for which the corresponding symbols are 
different). Non-optimizing agents with S identical strategies corresponds to dH=0. In contrast, 
optimizing agents with S maximally distinct strategies have large dH’s. Since agents with 
dH=0 out-perform agents with large dH, it is natural to ask whether the ranking of dH could be 
predictive of the ordering of agents’ payoffs. A non-zero Hamming dH implies that there are at 
least two strategies among the S strategies of the agent which are different. But, if dH is small, 
the small difference between the S strategies makes the optimization only faintly relevant and 
one can expect to observe a payoff similar to that of non-optimizing agents, therefore larger 
than for optimizing agents with large dH’s. This intuition is indeed confirmed by our 
calculations: the average payoff per time step is a decreasing function of dH, as originally 
discussed in [11,12].  
 The illusion-of-control effect suggests that the initial set-up of MG in terms of S fixed 
strategies per agent is evolutionarily unstable. It is thus important to ask what happens when 
agents are allowed to replace strategies over time based on performance. A number of authors 
have investigated this issue, adding a variety of longer-term learning mechanisms on top of 
the short-term adaptation that constitutes the basic MG [12-18]. Inter alia, Ref.[13] 
demonstrates that if agents are allowed to replace strategies over time based on performance, 
they do so by ridding themselves of those composed of the more widely Hamming-distant 
tuples. Agents that start out composed of identical strategies do not change at all; those 
composed of strategies close in Hamming space change little. Similarly, the authors of [12] 
explicitly fixed agents with tuples of identical strategies and found they performed best. 
Another important finding in [12] is that the best performance attainable is equivalent to that 
obtained by agents choosing their strategies at random. Note that learning only confers a 
relative advantage. In general, agents that learn out-perform agents that don’t. This is 
certainly true for this privileged subset of agents among standard ones. But the performance 
of learning agents approaches a maximum most closely attained by agents where the 
hamming distance between strategies is 0. These agents neither adapt (optimize) nor learn. 
One might say that when learning is introduced, the system learns to rid itself of the illusory 
optimization method that has been hampering it.  
There are exceptions, of course. Carefully designed privileges and certain kinds of 
learning can yield superior results for a subset of agents, and occasionally for all agents. But 
the routine outcome is that both optimization and straightforward learning cannot improve on 
simple chance. The fact that the optimization method employed in the MG yields the opposite 
of the intended consequence, and that learning eliminates the method, leads to an important 
question. We pose it carefully so as to avoid introducing either privileged agents or learning: 
Is the illusion-of-control so powerful in this instance that inverting the optimization rule could 
yield equally unanticipated and opposite results? The answer is yes: If the fundamental 
optimization rule of the MG is symmetrically inverted for a limited subset of agents who 
choose their worst-performing strategy instead of their best, those agents systematically 
outperform both their strategies and other agents. They also can attain positive gain.  Thus, 
the intuitively self-evident control over outcome proffered by the MG “optimization” strategy 
is most strikingly shown to be an illusion. Even learning and evolutionary strategies generally 
at best rid the system of any optimization method altogether. They do not attain the kind of 
results obtained simply by allowing some agents to reverse the method altogether. We discuss 
elsewhere the phenomena that arise as the proportion of agents choosing their best performing 
strategy and of agents choosing their worst performing strategy are varied for different 
parameters of the MG. We emphasize here only the fact that extensive numerical studies 
confirm that the phenomenon here indicated persist over a very wide range of parameters for 
both the MG and the THMG. Hence, having a portfolio of S strategies to choose from is 
counter-productive: (diversification + optimization) performs on average worse than a single 
fixed strategy. 
Intuitively, the illusion-of-control effect in MG results from the fact that a strategy that 
has performed well in the past becomes crowded out in the future due the minority 
mechanism: performing well in the recent past, there is a larger probability for a strategy to be 
chosen by an increasing number of agents, which inevitably leads to its demise. This 
argument in fact also applies also to all the strategies which belong to the same reduced set; 
their number is 22 2
m , equal to the ratio of the cardinality of the set of all strategies to the 
cardinality of the set of reduced strategies. Thus, the crowding mechanism operates from the 
fact that a significant number of agents have at least one strategy in the same reduced subset 
among the 2m reduced strategy subsets. Optimizing agents tend on average to adapt to the past 
but not the present. They choose an action a(t) which is on average out-of-phase with the 
collective action A(t). In contrast, non-optimizing agents average over all the regimes for 
which their strategy may be good and bad, and do not face the crowding-out effect. The 
crowding-out effect also explains simply why anti-optimizing agents over-perform: choosing 
their worst strategy ensures that it will be the least used by other agents in the next time step, 
which implies that they will be in the minority. The crowding mechanism also predicts that 
the smaller the parameter 2m/N, the larger the illusion-of-control effect. Indeed, for large 
values of 2m/N, it becomes more and more probable that agents have their strategies in 
different reduced strategy classes, so that a strategy which is best for an agent tells nothing 
about the strategies used by the other agents, and the crowding out mechanism does not 
operate. Thus, regions of successful optimization, if they occur at all, are more likely at higher 
values of  2m/N . (See supplementary material for further details.) 
This leads to the conclusion that there is a profound clash between optimization on the 
one hand and minority payoff on the other hand: an agent who optimizes identifies her best 
strategy, but in so doing by her “introspection”, she somehow knows the fate of the other 
agents, that it is probable that the other agents are also going to choose similar strategies, … 
which leads to their underperformance since most of them will then be in the majority. It 
follows then that an optimizing agent playing a standard minority game should optimize at a 
second order of recursion in order to win: Her best strategy allows her to identify the class of 
best strategies of others, which she thus must avoid absolutely to be in the minority and to win 
(given that other players are just optimizing at the first order as in the standard MG). 
Generalization to ever more complex optimizing set-ups, in which agents are aware of prior-
level effects up to some finite recursive level, can in principle be iterated ad infinitum.  
Actually, the game theory literature on first-entry games shows that the resulting 
equilibria depend on how agents learn [19]: with reinforcement learning, pure equilibria 
involve considerable coordination on asymmetric outcomes where some agents enter and 
some stay out; learning with stochastic fictitious plays leads to symmetric equilibria in which 
agents randomize over the entry decisions. There may even exist asymmetric mixed 
equilibria, where some agents adopt pure strategies while others play mixed strategies. We 
consider the situation where agents use a boundless recursion scheme to learn and optimize 
their strategy so that the equilibrium corresponds to the fully symmetric mixed strategies 
where agents randomize their choice at each time step with unbiased coin tosses. Consider a 
MG game with N agents total, NR of which employ such a fully random symmetric choice. 
The remaining NS = N-NR   “special” agents (with NR >> NS) will all be one of three possible 
types: agents with S fixed strategies that choose their best (respectively worst) performing 
strategy to make the decision at the next step (referred to above as anti-optimizing) and agents 
with a single fixed strategy. Our simulations confirm that these three types of agents indeed 
under-perform on average the optimal fully symmetric purely random mixed strategies of the 
NR agents (see Fig. 4 of the Supplementary materials). Here, pure random strategies are 
obtained as optimal, given the fully rational fully informed nature of the competing agents. 
The particular results are sensitive to which strategies are available to the special agents and 
to their proportion. Their underperformance in general requires averaging over all possible 
strategies and S-tuples of strategies. (In the supplementary material we show sample 
numerical results for NS = 1). 
3 Parrondo Games  
We now turn to the illusion-of-control effect in the Parrondo games (PG). The basic Parrondo 
effect (PE) was first identified as a game-theoretic equivalent to the directional drift of 
Brownian particles in a time-varying “ratchet”-shaped potential [20,21], wherein two or more 
individually fair (or losing) games yield a net winning outcome if alternated periodically or 
randomly. Consider N > 1 s-state Markov games , iG { }1,2, ,i ∈ … N , and their N 
s s× transition matrices, . For every ( )ˆ iM ( )ˆ iM , denote its vector of s winning probabilities 
conditional on each of the s states as ( ) ( ) ( ) ( ){ }1 2, ,i i i sip p p=p …  and its steady-state equilibrium 
distribution vector as ( ) ( ) ( ) ( ){ }1 2, , ,i i i sπ π πΠ = i… . For each game, the steady-state probability of 
winning is therefore ( ) ( ) ( )i iwinP = ⋅Πp
. Consider also a sequence of randomly alternating with 
individual time-averaged proportion of play
0,1 ,  1
∈ =∑ . The transition matrix for the 
combined sequence of games is the convex linear combination with 
conditional winning probability vector 
1 2( , , , ) ( )
γ γ γ γ
≡ ∑M M…
( , ,..., )
γ γ γ γ
= ∑p p and steady-state probability 
vector ( 1 2, , , n )γ γ γΠ …  (which is a complex nonlinear mixture of the   
Π 's ). The steady-state 
probability of winning for the combined game is therefore 
 ( ) ( ) (1 2 1 2 1 2
, , , , , , , , ,N N
)Nγ γ γ γ γ γ γ γ γ= ⋅ Πp… … …  (3) 
A PE occurs whenever (and in general it is the case that)  
 ( ) ( 1 2
, , ,
i win win
P P )γ γ γγ
≠∑ …       i.e.       ( ) ( ) ( ) (1 2 1 2, , , , , ,
γ γ γ γ γ γγ
⋅ Π ≠ ⋅ Π∑ p p … )N…  (4) 
hence the PE, or “paradox”, when the left hand sides of (4) are less than zero and the right-
hand sides greater. 
Many variants have been studied, including capital-dependent multi-player PG (MPPG) 
[22,23]: At (every) time-step t, a constant-size subset of all participants is randomly re-
selected actually to play. All participants keep individual track of their own capital but do not 
alternate games independently based on it. Instead this data is used to select which game the 
participants must use at t. The chosen game is the one which, given the individual values of 
the capital at and the known matrices of the two games and their linear convex 
combination, has the most positive expected aggregate gain in capital, summed over all 
participants. This rule may be thought of as a static optimization procedure—static in the 
sense that the “optimal” choice appears to be known in advance. It appears exactly 
quantifiable because of access to each player’s individual history. If the game is chosen at 
random, the change in wealth averaged over all participants is significantly positive. But when 
the “optimization” rule is employed, the gain becomes a loss significantly greater than that of 
either game alone. The intended “optimization” scheme actually reverses the positive 
(collective) PE. The reversal arises in this way: the “optimization” rule causes the system to 
spend much more time playing one of the games, and individually, any one game is losing.  
Here, we present a more natural illustration of the illusion-of-control: while the MG is 
intrinsically collective, PGs are not. Neither the capital- nor the history-dependent variations 
require a collective setting for the PE to appear as shown from (4). Thus, the effect is most 
clearly demonstrated in a single-player implementation with two games under the most 
natural kind of optimization rule: at time t, the player plays whichever game has accumulated 
the most points (wealth) over a sliding window of τ prior time-steps from to 1t − t τ− . Under 
this rule, a “current reversal” (reversal of a positive PE) appears. By construction, the 
individual games ( )1M̂ and ( )2M̂ played individually are both losing; random alternation 
between them is winning (the PE effect (4)), but unexpectedly, choosing the previously best-
performing game yields losses slightly less than either ( )1M̂ and ( )2M̂  individually: the PE is 
almost entirely eliminated. Furthermore, if instead the previously worst performing game is 
chosen, the player does better than either game and even much better than the PE from 
random game choice. Under the choose-best optimization rule, two matrices and ( )1M̂ ( )2M̂  do 
not form a linear convex sum. Instead, the combined game is represented by an 
( ) (s s )τ τ+ × +  transition matrix ( )1,2Q̂  with conditional winning probabilities 
( ){ }1 1 2 2 1 212 1 1j j j j j jq p p p p p pα β β α β β⎡ ⎤ ⎡ ⎤= + − + − +⎣ ⎦ ⎣ ⎦j with 1,2, , 2j s= …  and indices 
( ) [ ] [ ] [ ](121,4 1,   1,2 1j Mod j j j Mod jα β= − + = − − )+ . (Under the choose-worst rule 
( ){ }1 1 2 2 1 212 1 1j j j j j jq p p p p p pα β β α β β⎡ ⎤ ⎡= − + + + −⎣ ⎦ ⎣ j ⎤⎦ ). Using matrices with the same values 
as studied in [24,25], the one-player two-game history-dependent PE is as follows: ( )1M̂ and 
( )2M̂ have respective winning probabilities  and . Alternated at 
random in equal proportion , . If the previously winning 
game is selected, , while if the previously losing one is, . The 
mechanism for this illusion-of-control effect characterized by the reversing of the PE under 
optimization is not the same as for the MG, as there is no collective effect and thus no-
crowding out of strategies or games. As seen from 4), the PE results from a distortion of the 
steady-state equilibrium distributions 
1 0.494winP =
2 0.495winP =
( )1 2 0.5γ γ= = 1 2
0.5, 0.5 0.501winP
γ γ= = =
(1,2) 0.496bestwinP =
(1,2) 0.507bestwinP =
Π  and   
Π  into a vector   
Π  (for the n=2 
version) which is more co-linear to the conditional winning probability vector   
p  than in 
the case of each individual game (this is just a geometric restatement of the fact that the 
combined game is winning). One can say that each game alternatively acts at random so as to 
better align these two vectors on average under the action of the other game. Choosing the 
previously best performing game amounts to removing this combined effect, while choosing 
the previously worst performing game tends to intensify this effect. 
We have identified two classes of mechanisms operating in the Minority games and in the 
Parrondo games in which optimizing agents obtain suboptimal outcomes compared with non-
optimizing agents. These examples suggest a general definition: the “illusion of control” 
effect occurs when low-entropy strategies (i.e. which use more information) under-perform 
random strategies (with maximal entropy). The illusion of control effect is related to bounded 
rationality as well as limited information [26] since, as we have shown, unbounded rational 
agents learn to converge to the symmetric mixed fully random strategies. It is only in the 
presence of bound rationality that agents can stick with optimization scheme on a subset of 
strategies. Our robust message is that, under bounded rationality, the simple (large-entropy) 
strategies are often to be preferred over more complex elaborated (low-entropy) strategies. 
This is a message that should appeal to managers and practitioners, who are well-aware in 
their everyday practice that simple solutions are preferable to complex ones, in the presence 
hat it can be 
ly to help formulate better strategies and tools for management. 
, Gilles Daniel and especially Yannick Malevergne for 
elpful feedback on the manuscript. 
the ubiquitous uncertainty. 
More examples should be easy to find. For instance, control algorithms, which employ 
optimal parameter estimation based on past observations, have been shown to generate broad 
power law distributions of fluctuations and of their corresponding corrections in the control 
process, suggesting that, in certain situations [27], uncertainty and risk may be amplified by 
optimal control. In the same spirit, more quality control in code development often decreases 
the overall quality which itself spurs more quality control leading to a vicious circle [28]. In 
finance,  there are many studies suggesting that most fund managers perform worse than 
random [29] and strong evidence that over-trading leads to anomalously large financial 
volatility [30]. Let us also mention the interesting experiments in which optimizing humans 
are found to perform worse than rats [301]. We conjecture that the illusion-of-control effect 
should be widespread in many strategic and optimization games and perhaps in many real life 
situations. Our contribution is to put this question at a quantitative level so t
studied rigorous
nowledgements 
We are grateful to Riley Crane
Appendix A. Analytic Methods and Simulations 
A1 The Minority Game: Choosing the Best Strategy 
In the simplest version of the Minority Game (MG) with N agents, every agent has S = 2 
strategies and m = 2. In the Time Horizon Minority Game (THMG), the point (or score) table 
associated with strategies is not maintained from the beginning of the game and is not ever 
growing. It is a rolling window of finite length τ (in the simplest case 1τ = ). The standard 
MG reaches an equilibrium state after a finite number of steps stt . At this point, the dynamics 
and the behavior of individual agents for a given initial quenched disorder in the MG are 
indistinguishable from an otherwise identical THMG with sttτ ≥ . Extensive numerical 
simulations show that all the phenomenon we discuss in the THMG with the simplest 
parameters are found in the MG with arbitrary parameters and in the THMG with τ of 
generally arbitrary length and parameters. In particular, the message of our communication 
holds true: agents under-perform strategies.  
The fundamental result of the MG is generally cast in terms of system volatility: 2 Nσ . All 
variations of agent and strategy reward functions depend on the negative sign of the majority 
vote. Therefore both agent and strategy “wealth” (points, whether “real” or hypothetical) are 
inverse or negative functions of the volatility: The lower the value of   2 Nσ , the greater the 
mean “wealth” of the “system”, i.e., of agents. However, this mean value is scarcely ever 
compared to the comparable value for the raw strategies of which agents are composed. Yet 
agents are supposed to enhance their performance by choosing adaptively between their 
available strategies. In fact, the opposite is true: The optimization method is strictly a method 
for worsening performance.  
To emphasize the relation of the MG to market-games and the illusion of optimization, we 
transform the fundamental result of the MG from statements on the properties of  2 Nσ  to 
change in wealth, i.e., W tΔ Δ  for agents and  W tΔ Δ  for strategies. We again use the simplest 
possible formulation—if an agent’s actual (or strategy’s hypothetical ) vote places it in the 
minority, it scores  points, otherwise 1+ 1− . Formally: At every discrete time-step t, each 
agent independently re-selects one of its S strategies. It “votes” as the selected strategy 
dictates by taking one of two “actions,” designated by a binary value: 
 ( ) { }1,0 ,   ,ia t i t∈ ∀  (3) 
The state of the system as a whole at time t is a mapping of the sum of all the agents’ 
actions to the integer set { }12N N− , where  is the number of 1 votes and . 
This mapping is defined as : 
1N 0 1N N N= −
 ( ) ( ) 1
0A t a t N N
N= − = −∑  (4) 
If ( ) 2NA t > , then the minority of agents will have chosen 0 at time t ( ); if 0N N< 1
( ) 2NA t < , then the minority of agents will have chosen 1 at time t ( ). The minority 
choice is the “winning” decision for t . This is then mapped back to 
1N N< 0
{ }0,1 : 
 ( ) ( ) ( ) { } { }Sgn   1, 1 0,1sys sysD t A t D t= − ∴ ∈ − + →⎡ ⎤⎣ ⎦  (5) 
For the MG, binary strings of length m form histories ( )tμ , with . For the 
THMG, binary strings of length 
( )dimm μ= ⎡⎣ t ⎤⎦
m τ+  form paths (or “path histories”) [8,9], with 
( )dim tm τ μ+ = , where we define ( )tμ  as a history in the standard MG and tμ as a path in 
the THMG. Then as demonstrated in [8,9], any THMG has a Markov chain formulation. For 
{ } { }, , 2,2,31m S N = , the typical initial quenched disorder in the strategies attributed to each 
of the N agents is represented by the tensor and its symmetrized equivalent Ω̂
(12ˆ ˆ ˆ= +Ψ Ω ΩT ) . Positions along all S edges of  represent an ordered listing of all available 
strategies.  The numerical values in  indicate the number of times a specific strategy-
tuple has been selected. (E.g., for two strategies per agent, S=2, 
ijΩ … Ω̂
2,5Ω =3 means that there are 3 
agents with strategy 2 and strategy 5.) Without loss of generality, we may express  in 
upper-triangular form since the order of strategies in a agent has no meaning. The example 
is a typical such tensor  for S = 2, N = 31.  Ω̂
  (6) 
1 2 0 0 1 1 0 0
0 0 0 0 3 3 1 1
0 0 2 0 1 0 0 0
0 0 0 1 1 0 0 1ˆ
0 0 0 0 1 0 2 1
0 0 0 0 0 2 2 1
0 0 0 0 0 0 2 1
0 0 0 0 0 0 0 0
⎜ ⎟⎜ ⎟
Actions are drawn from a reduced strategy space (RSS) [4,32] of dimension 2m. Each is 
associated with a strategy k and a path tμ . Together they can be represented in table form as a 
( ) ( )dim RSS dim tμ×  binary matrix with elements converted for convenience from 
{ } { }0,1 1, 1→ − + , i.e., { }1,tka
μ ∈ − + , 1 . For m=2, τ=1 ( )tdimτ μ =3, there are 2m + =
possible histories and r=22 reduced strategies (and 2r strategies in total). In this case, the table 
of dimensio  ( ) ( )dim RSS dim tμ× s:  coding for all possible reduced strategies and paths read
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
− − − −
− − + +
− + − +
− + + −
+ − − +
+ − + −
+ + − −
+ + + +
⎟  (7) 
The change in wealth (point gain or loss) associated with each of the 2r =8 strategies for 
the 8 paths (= allowed transitions between the 4 histories) at any time t is then: 
 ( ) ( ) ( ) ( ) ( ){ }, 1 ˆ 2 1 ,2t t tS a Mod tμ μ μδ μ− = × −⎡ ⎤⎣ ⎦
1−  (8) 
[ ],Mod x y is “x modulo y”; and ( )tμ ( )1tμ − label each of the 4 histories { }00,01,10,11  
hence take on one of values { }1,2,3,4 . Equation (8) picks out from (7) the correct change in 
wealth over a single step since the strategies are ordered in symmetrical sequence. 
The change in points associated with each strategy for each of the allowed transitions 
between paths tμ of the last τ  time steps used to score the strategies is: 
 ( ) ( )
t t i t i
μ μ μδ
− − −
= ∑  (9) 
For example, for m = 2 and t = 1, the strategy scores are kept for only a single time-step. 
There is no summation so (9) in matrix form reduces to the score: 
 ( ) ( ), 1t t ts Sμ μ μδ −=  (10) 
or, listing the results for all 8 path histories:  
 ˆˆμ δ=s S  (11) 
ˆδS  is an 8μ8 matrix that can be read as a lookup table. It denotes the change in points 
accumulated over t = 1 time steps for each of the 8 strategies over each of the 8 path-
histories. 
Instead of computing ( )A t , we compute ( )tA μ . Then for each of the 2 possible 8
m τ+ = tμ , 
( )tA μ  is composed of a subset of wholly determined agent votes and a subset of 
undetermined agents whose votes must be determined by a coin toss: 
 ( ) ( ) ( )t D t UA A A tμ μ= + μ  (12) 
Some agents are undetermined at time t because their strategies have the same score and the 
tie has to be broken with a coin toss. ( )U tA μ  is a random variable characterized the binomial 
distribution. Its actual value varies with the number of undetermined agents. This number can 
be explicated (2): 
[ ]( ) ( ) [ ]( ) ( ) ( )
1 1Mod 1,4 1 Mod 1,4 1
Mod 1,2 1
ˆˆ ˆ1
t t m
a a s sδ μ δ μμ μ
− + − +
⎡ ⎤− +⎣ ⎦
⎧ ⎫⎛ ⎞⎡ ⎤− ⊗ ⊗⎨ ⎬⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠⎩ ⎭
 (13) 
“ δ⊗ ” is a generalized outer product, with the product being the Kronecker delta. UN  
constitutes a vector of such values.  The summed value of all undetermined decisions for a 
given tμ is distributed binomially. Similarly (2): 
( ){ }
1 Mod 1,2 1
Sgn s s aμ μ
= ⎡ ⎤− +⎣ ⎦
⎛ ⎡ ⎤⎡ ⎤− •⎜ ⎟⎣ ⎦⎣ ⎦⎝ ⎠
⎞  (14) 
An example of how (13) and (14) can be deduced is given later in the context of the 
original definition of alternate types of agents. Details may also be found in Ref.[9]. We 
define DA  as a vector of the determined contributions to ( )A t for each path tμ . In expression 
(14) tμ numbers paths from 1 to 8 and is therefore here an index. tsμ is the “ tμ
th” vector of net 
point gains or losses for each strategy when at t the system has traversed the path tμ ( i.e., it is 
the  “ tμ
th” element of the matrix ˆˆμ δ=s S in (11)). is a generalized outer product of two 
vectors with subtraction as the product. The two vectors in this instance are the same, i.e., 
sμ . 
“ ” is Hadamard (element-by-element) multiplication and “ ” the standard inner product. 
The index r refers to strategies in the RSS. Summation over r transforms the base-ten code for 
tμ   into { }1, 2,3, 4,1, 2,3, 4 . Selection of the proper number is indicated by the subscript 
expression on the entire right-hand side of (13). This expression yields an index number, i.e., 
selection takes place 1 + Modulo 4 with respect to the value of ( )1tμ − . 
To obtain the transition matrix for the system as a whole, we require the 2 2m mτ τ+ +×  
adjacency matrix that filters out disallowed transitions. Its elements are 
1,t tμ μ
  (15) 
1 0 0 0 1 0 0 0
1 0 0 0 1 0 0 0
0 1 0 0 0 1 0 0
0 1 0 0 0 1 0 0ˆ
0 0 1 0 0 0 1 0
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1
0 0 0 1 0 0 0 1
⎜ ⎟⎜ ⎟
Equations (13), (14) and (15) yield the history-dependent ( ) ( )m mτ τ+ × +  matrix with 
elements 
1,t t
Tμ μ − , representing the 16 allowed probabilities of transitions between the two sets 
of  8 path-histories tμ and 1tμ − : 
 ( ) ( )
( ) ( )( ) { }( )
1 1, ,
Sgn 2 2 Mod , 2 1
t t t t
U tU t U t
D t U t t
A x N
μ μ μ μ
δ μ μ μ
= Γ ×
⎧ ⎫⎪ ⎪⎛ ⎞ ⎡ ⎤× + − +⎨ ⎬⎜ ⎟ ⎣ ⎦⎝ ⎠⎪ ⎪⎩ ⎭
 (16) 
The expression 
( ) ( )1
U tU t
μμ ⎛ ⎞
 in  (16) represents the binomial distribution of 
undetermined outcomes under a fair coin-toss with mean = ( )D tA μ . Given a specific ,  Ω̂
 ( ) ( )  t D tA A tμ μ μ= ∀  (17) 
We now tabulate the number of times each strategy is represented in , regardless of 
coupling (i.e., of which strategies are associated in forming agent S-tuples):  
  (18) ( ) ( ) ( ) ({
1 2 2
ˆ ˆ2 , ,
n n n
τκ σ σ
≡ + = =∑ ∑Ω Ω Ψ …T )}σ
where kσ  is the k
th strategy in the RSS,  and are the kˆ ˆ,k kΩ Ω
th element (vector) in each 
tensor and ( kn )σ  represents the number of times kσ is present across all strategy tuples. 
Therefore 
 ( )1Agent DNW Abs A μΔ = − ⋅  (19) 
 ( )12 ˆStrategy NW μ κ μΔ = ⋅s ⋅  (20) 
with μ  the normalized steady-state probability vector for . Expression T̂ (19) states that the 
mean per-step change in wealth for agents equals –1 times the probability-weighted sum of 
the (absolute value of the) determined vote imbalance associated with a given history. 
Expression (20) states that the mean per-step change in wealth for individual strategies equals 
the probability-weighted sum of the representation of each strategy (in a given ) times the 
sum over the per-step wealth change associated with every history. The –1 in 
(19) reflects the 
minority rule. I.e., the awarding of points is the negative of the direction of the vote 
imbalance. No minus sign is required in (20) as it is already accounted for in (7). 
Figure A1 shows the cumulative mean change in wealth for strategies versus agents 
over time, given (15). 
As first studied in [11,12], and discussed in the body of the manuscript, agent performance 
is inversely proportional to the Hamming distance between strategies within agents. With the 
variation expected of a single example, our sample  given by Ω̂ (6) reproduces this relation as 
shown in Figure A2. Thus agent performance is distributed within  in orderly if complex 
fashion. The mean over many corresponds to a “flat” .  
Ω̂ Ω̂
A2 The Minority Game: Choosing the Worst Strategy 
First, we re-cast the initial quenched disorder on the set of strategies attributed to the N 
agents in a given game realization as a two-component tensor: { }ˆ ˆ ˆ,= + -Ω Ω Ω . represents 
standard (S) agents that adapt as before; represents “counteradaptive” (C) agents that 
instead select their worst-performing strategies. In our example 
(6) then, suppose we select at 
random 3 agents to use the C rule, one each at 1,2 2,6,Ω Ω and 7,8Ω : 
{ }ˆ ˆ ˆ,
1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0
0 0 0 0 3 2 1 1 0 0 0 0 0 1 0 0
0 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 2 1 0 0 0 0 0 0 0 0
0 0 0 0 0 2 2 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
⎧⎛ ⎞ ⎛
⎪⎜ ⎟ ⎜
⎪⎜ ⎟ ⎜
⎪⎜ ⎟ ⎜
⎪⎜ ⎟ ⎜
⎪⎜ ⎟ ⎜
⎨⎜ ⎟ ⎜
⎜ ⎟ ⎜
⎜ ⎟ ⎜
⎜ ⎟ ⎜
⎜ ⎟ ⎜⎜ ⎟ ⎜
⎝ ⎠ ⎝⎩
+ -Ω Ω Ω
 (21) 
For any number of C agents in thus redefined, the analytic expression for  need only 
be modified by decomposing 
Ω̂ T̂
( )D tA μ  accordingly. The new term in ( )D tA μ  makes evident 
the symmetry of the C rule with respect to the S rule, and the lack of privilege of C agents. 
Thus: 
( ) ( ){ }
1 Mod 1,2 1
ˆ ˆ ˆ1 1
t t t t
Sgn s s Sgn s s aμ μ μ μ
= ⎡ ⎤− +⎣ ⎦
⎛ ⎞⎡ ⎤⎡ ⎤ ⎡ ⎤− + + •⎜ ⎟⎣ ⎦ ⎣ ⎦⎣ ⎦⎝ ⎠
∑ Ψ Ψ 
 (22) 
with 
 ( ) ( )1 12 2ˆ ˆ ˆ ˆ ˆ ˆ;   + −= + = ++ +Ψ Ω Ω Ψ Ω ΩT − − T  (23) 
The number of undetermined agent votes remains unchanged. In (13),  need only be 
replaced with (
)ˆ ˆ++ -Ω Ω : 
[ ]( ) ( ) [ ]( ) ( ) ( ) ( )1 1Mod 1,4 1 Mod 1,4 1 Mod 1,2 1
ˆ ˆˆ ˆ1
t t m
a a s sδ μ δ μμ μ
− + − +
⎡ ⎤− +⎣ ⎦
⎧ ⎫⎛ ⎞⎡ ⎤− ⊗ ⊗ +⎨ ⎬⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠⎩ ⎭
+ -Ω ΩT T
(24) 
Results for numerical simulation and analytic calculation are in close agreement even for a 
single short simulation, as illustrated in  
Table A1. 
The 3 C agents of 31 now perform so well that they significantly raise the overall 
performance of the system as detailed in Figure A3. They not only outperform both their own 
strategies and the other S agents on average, they generate net positive gain. The hypothetical 
outperformance of unused relative to used strategies in the MG was first observed in (5). But 
the explicit generation of positive results, by agents simply deploying their unused strategies 
(without privileging), has not been tested. (In the case of S = 2, “unused” are by definition the 
“worst-performing”.)  
We discuss in the manuscript and elsewhere the phenomena that arise as the proportion of 
S and C agents are varied for different parameters of the MG. We emphasize here only the 
fact that extensive numerical studies confirm that the phenomenon here illustrated persist over 
a very wide range of parameters for both the MG and the THMG. 
A3 The Minority Game: Random Agents 
We provide in Figure A4 some numerical results for a MG game with N agents total, NR 
of which employ such a fully random symmetric choice. The remaining NS = N-NR   “special” 
agents (with NR >> NS) will all be one of two possible types: (i) agents with S fixed strategies 
that choose their worst performing strategy to make the decision at the next step (referred to 
above as anti-optimizing); (ii) agents with a single fixed strategy. We use the simplest 
example, that of  NS = 1 (with 1τ = ), to illustrate the fact that in the MG, agents 
allowed/restricted to a fully symmetric random choice outperform agents that attempt to 
optimize. (Note that the outperformance and absolute positive returns associated with a small 
proportion of anti-optimizing agents, requires the remaining agents to optimize, as described 
above. Here the small proportion of or optimizing and anti-optimizing agents compete with 
fully random agents.)  
Figures and Tables 
Figure A1: Mean Strategy versus Agent Cumulative Change in Wealth 
in the THMG. { } { }, , 2,2,31m S N = ; 100 time steps 
Figure A2: Agent wealth as a function of Hamming distance between 
strategy pairs in agents for the example simulation.  
Figure A3: Average wealth variation per time step for different agents. 
In red are shown the wealth variations of the three among the 31 agents 
which use counteradaptive (“C”, choose worst) strategy selection. The 
usual underperformance of agents compared to individual strategies 
when using standard selection rule (“S”, choose best) is shown in the 
blue dots. 
-0.350
-0.300
-0.250
-0.200
-0.150
-0.100
2 3 4 5
random, S=2 strategic, S=2
random, S=3 strategic, S=3
-0.350
-0.300
-0.250
-0.200
-0.150
-0.100
2 3 4 5
random, S=2 strategic, S=2
random, S=3 strategic, S=3
-0.350
-0.300
-0.250
-0.200
-0.150
-0.100
2 3 4 5
random, S=2 strategic, S=2
random, S=3 strategic, S=3
Figure A4: Performance (mean change in wealth per step) of a single optimizing agent versus all other agents 
making a symmetric random choice in a MG-like game. From left to right n=11, 21, 31. S=2,3  m= 2,3,4,5 and 
t=1. Random agents always outperform optimizing agents. Similar results obtain for other values of n, m, S and 
t.Within statistical fluctuations typical for the number of runs/random selection of strategies comprising the 
optimizing agent (100 runs), results for anti-optimizing agents are identical. 
Table A1: Numerical/Analytic Results of THMG with and without 3 C Agents 28 S Agents 
 ‚∆WAgentÚ ‚∆WStrategyÚ 
With –0.14/–0.14 –0.05/–0.05 
Without –0.26/–0.26 –0.05/–0.05 
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	“Illusion of Control” in Minority and Parrondo Games
