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vs.
5.1 Majority
Minority Game
The model extensively used in literature is based on the grand canonical minority
game [11,10,8,9,7]. However, it was not obvious for us if the minority mechanism
is better than the majority one. In fact we found that, in the case of prediction,
both mechanisms are equivalent.
The algorithm of the majority game is very similar to that of the standard
minority game. The only difference is a formula (2) which for majority game
reads
Φ
α
n
(
t
)=
a
α
n
(
t
)
g
[
A
(
t
)]
.
(6)
We consider two time series: the endogenous and exogenous. Considering first
the game with endogenous time series we find a number of differences between
the minority and majority game. In the minority game no one of strategies is
permanently profitable provided that the game is large enough [16]. Hence, the
number of winners and losers changes in time. On the contrary, in the majority
game the number of winners and losers is stable and, on average,
N
(1
1
2
S
)
agents are in majority. The reasoning behind is similar to that presented in
Refs. [14,15,16] and utilizes our observation that the first large oscillation creates
a comprehensive difference in utilities of strategies.
Intriguingly, both games are equivalent, regardless of the payoff, when series
of decisions is exogenous. Originally, in the standard minority game, strategies
predicting sign opposite to
y
(
t
) are rewarded. Assuming that patterns in the
exogenous signal exist, the individuals prefer more often strategies predicting
sgn(
y
(
t
)) incorrectly i.e. most of them fail with prediction. The predictor aggre-
gates decisions of individuals and acts in opposition to the majority, and, predicts
correctly. Contrary to the minority game, in the majority game, strategies that
correctly forecast sgn(
y
(
t
)) are rewarded and most of agents follow strategies
more frequently recognizing patterns. Subsequently, the predictor acts according
to action suggesting the majority and also correctly recognizes patterns. Given
this, the majority and minority game should provide the same quality of predic-
tion. This is confirmed by numerical simulations presented in Fig. 2 (right). As
it is seen, there are no qualitative differences between them. Small distortions
are due to random choice of strategies at the beginning of all simulations.
−
m
N
S
5.2 Tuning of
,
and
Parameters
Considering three parameters:
m
,
N
and
S
,atleast
m
requires different opti-
mization techniques than two others. The optimization of
m
is strictly related to
analysis of time series properties, especially the analysis of the range of depen-
dencies between values of
Y
. Therefore it depends on the researcher's knowledge
about the object. There are different methods of finding this range.
If the process is explicitly known,
m
is equal to the order of this process, e.g.
m
= 3 for AR(3). The predictor with
m
lower than the order cannot work effec-
tively because strategies would incorrectly recognize patterns. Larger values of
m
would introduce additional and unnecessary noise that would degrade the predic-
tion. The latter effect is further illustrated and explained in the section 6.2.
