Information Technology Reference
In-Depth Information
where
m
is the memory of agents. A
history
H
,e.g.,
[
−
1
,
1
,
1
,...
]
, is a sequence of
−
T
=(0
,
1
,
2
,...
)
.
The winning decision of MG (resp. MJ) is determined by the minority (resp. majority)
group of
1
and
1
representing a
winning decision
h
(
t
)
for each time step
t
∈
∈
R
i,a
is given a
score
U
i,a
(
t
)
so that the
best strategy can make a winning decision. For the last
m
winning decisions, denoted
by
μ
=
h
m
(
t
−
1
or
1
. Each strategy
R
i,a
(
μ
)
∈
R
i,a
determines
−
1)
⊆
H
, agent
i
's strategy
R
i,a
(
μ
)
−
1
or
1
by (1).
Among them, each agent
i
selects his highest scored strategy
R
i
(
μ
)
∈
R
i,a
and makes
a decision
a
i
(
t
)=
R
i
(
μ
)
at time
t
∈
T
. The highest scored strategy is represented by
R
i
(
μ
) = arg max
a∈{
1
,...,s}
U
i,a
(
t
)
,
(2)
which
is
randomly
selected
if
there
are
many
ones.
An
aggregate
value
A
(
t
)=
i
=1
a
i
(
t
)
is called an excess demand. If
A
(
t
)
>
0
, agents with
a
i
(
t
)=
1
win, and otherwise, agents with
a
i
(
t
)=1
win in MG, and vice versa in MJ. Hence the
payoffs
g
MG
i
−
and
g
MJ
i
of agent
i
are represented by
g
MG
i
(
t
+1)=
−a
i
(
t
)
A
(
t
)
and
(3)
g
MJ
i
(
t
+1)=
a
i
(
t
)
A
(
t
)
,
respectively
.
(4)
The winning decision
h
(
t
)=
−
1
or
1
is added to the end of the history
H
,i.e.,
h
m
+1
(
t
)=[
h
m
(
t
1)
,h
(
t
)]
, and then it will be reflected in the next step. After the
winning decision has been turned out, every score is updated by
−
U
i,a
(
t
+1)=
U
i,a
(
t
)
⊕
R
i,a
(
μ
)
·
sgn
(
A
(
t
))
,
(5)
where
⊕
means subtraction for MG (addition for MJ) and
sgn
(
x
)=1(
x
≥
0)
,
=
−
1(
x<
0)
. In other words, the scores of winning strategies are increased by 1, while
those of losing strategies are decreased by 1. We simply say that an agent increases
selling (resp. buying) strategies if the scores of selling (resp. buying) strategies are in-
creased by 1. Likewise the decrement of scores. Notice that the score is an accumulated
value from an initial state in the original MG. In contrast, we define it as a value from
the last
H
p
steps according to [13]. That is, we use
U
i,a
(
t
+1)=
U
i,a
(
t
)
⊕
R
i,a
(
μ
)
·
sgn
(
A
(
t
))
−
U
i,a
(
t
−
H
p
)
.
(6)
The constant
H
p
is not relevant to
m
, but is only used for selecting the highest score.
Analogous to a financial market, the decision
a
i
(
t
)=1
(respectively,
1
)represents
buying (respectively, selling) an asset. Usually, the price of an asset is defined as
−
exp
A
(
t
)
N
p
(
t
+1)=
p
(
t
)
·
.
(7)
2.2
Asset Value Game
The difference between MG and our asset value game is the payoff function. Let
v
i
(
t
)
be agent
i
's mean asset value at time
t
,and
u
i
(
t
)
the number of units of his asset. The
payoff function in AG is defined as
g
A
i
(
t
+1)=
−
a
i
(
t
)
F
i
(
t
)
,
(8)
