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2 The Formal Definition of the Minority Game
At each time step
t
,the
n
th agent out of
N
(
n
=1
,...,N
) takes an action
a
α
n
(
t
)
according to some strategy
α
n
(
t
). The action
a
α
n
(
t
) takes either of two values:
−
1 or +1. An aggregated demand is defined as
A
(
t
)=
n
=1
a
α
n
(
t
), where
α
n
refers to the action according to the best strategy, as defined in eq. (1) below.
Agents do not know each other's actions but
A
(
t
) is known to all agents. The
minority action
a
∗
(
t
) is determined from
A
(
t
):
a
∗
(
t
)=
sgn
A
(
t
). Each agent's
memory is limited to
m
most recent winning, i.e. minority, decisions. Each agent
has the same number
S
−
2 of devices, called strategies, used to predict the next
minority action
a
∗
(
t
+1).The
s
th strategy of the
n
th agent,
α
n
(
s
=1
,...,S
),
is a function mapping the sequence
μ
of the last
m
winning decisions to this
agent's action
a
α
n
. Since there is
P
=2
m
possible realizations of
μ
, there is 2
P
possible strategies. At the beginning of the game each agent randomly draws
S
strategies, according to a given distribution function
ρ
(
n
):
n
≥
→
Δ
n
,where
Δ
n
is a set consisting of
S
strategies for the
n
th agent.
Each strategy
α
n
, belonging to any of sets
Δ
n
, is given a real-valued function
U
α
n
which quantifies the utility of the strategy: the more preferable strategy,
the higher utility it has. Strategies with higher utilities are more likely chosen
by agents. There are various choice policies. In the popular
greedy policy
each
agent selects the strategy of the highest utility
α
n
(
t
) = arg max
s
:
α
n
∈Δ
n
U
α
n
(
t
)
.
(1)
If there are two or more strategies with the highest utility then one of them is
chosen randomly. Each strategy
α
n
is given the
payoff
depending on its action
a
α
n
Φ
α
n
(
t
)=
−
a
α
n
(
t
)
g
[
A
(
t
)]
,
(2)
where
g
is an odd
payoff function
, e.g. the steplike
g
(
x
)=sgn(
x
) [5], proportional
g
(
x
)=
x
or scaled proportional
g
(
x
)=
x/N
. The learning process corresponds
to updating the utility for each strategy
U
α
n
(
t
+1)=
U
α
n
(
t
)+
Φ
α
n
(
t
)
,
(3)
such that every agent knows how good its strategies are.
The presented definition is related to genuine MG [4]. If game is used as
the predictor then the feedback effect is destroyed and
μ
is derived from using
exogenous time series.
3 Relation to Other Models
As it is known from other works [4,2] the standard MG exhibits an intriguing
phenomenological feature: a non-monotonic variation of the volatility when the
control parameter is varied. There are two mechanisms potentially responsible for
