attending the bar will exceed N
s
. There is no communication among
patrons and they make decisions using only information on past
attendance and different predictors (e.g., attendance today is the
same as yesterday, or is some average of past attendance).
The minority game is a simple binary choice problem in which
players have to choose between two sides, and those on the minority
side win. Similarly to the El Farol’s bar problem, in the minority
game there is no communication among players and only a given set
of forecasting strategies defines player decisions. The minority game
is an interesting stylized model that may have some financial implica-
tions [2]. But we shall focus further on the models derived specifically
for describing financial markets.
In the known literature, early work on the agent-based modeling of
financial markets can be traced back to 1980 [4]. In this paper, Beja and
Goldman considered two major trading strategies, value investing and
trend following. In particular, they showed that system equilibrium
may become unstable when the number of trend followers grows.
Since then, many agent-based models of financial markets have
been developed (see, e.g., reviews [1, 5], the recent collection [6] and
references therein). We divide these models into two major groups. In
the first group, agents make decisions based on their own predictions
of future prices and adapt their beliefs using different predictor func-
tions of past returns. The principal feature of this group is that price is
derived from the supply-demand equilibrium [7–10].
2
Therefore, we
call this group the adaptive equilibrium models. In the other group, the
assumption of the equilibrium price is not employed. Instead, price is
assumed to be a dynamic variable determined via its empirical relation
to the excess demand (see, e.g., [11, 12]). We call this group the non-
equilibrium price models. In the following two sections, we discuss two
instructive examples for both groups of models, respectively. Finally,
Section 12.4 describes a non-equilibrium price model that is derived
exclusively in terms of observable variables [13].
12.2 ADAPTIVE EQUILIBRIUM MODELS
In this group of models [7–10], agents can invest either in the risk-
free asset (bond) or in the risky asset (e.g., a stock market index). The
risk-free asset is assumed to have an infinite supply and a constant
130 Agent-Based Modeling of Financial Markets
interest rate. Agents attempt to maximize their wealth by using some
risk aversion criterion. Predictions of future return are adapted using
past returns. The solution to the wealth maximization problem yields
the investor demand for the risky asset. This demand in turn deter-
mines the asset price in equilibrium. Let us formalize these assump-
tions using the notations from [10]. The return on the risky asset at
time t is defined as
r
t
¼ (p
t
p
t1
þ y
t
)=p
t1
(12:2:1)
where p
t
and y
t
are (ex-dividend) price and dividend of one share of
the risky asset, respectively. Wealth dynamics of agent i is given by
W
i,tþ1
¼ R(1 p
i,t
)W
i,t
þ p
i, t
W
i,t
(1 þ r
tþ1
)
¼ W
i,t
[R þ p
i,t
(r
tþ1
r)] (12:2:2)
where r is the interest rate of the risk-free asset, R ¼ 1 þ r, and p
i,t
is
the proportion of wealth of agent i invested in the risky asset at time t.
Every agent is assumed to be a taker of the risky asset at price that is
established in the demand-supply equilibrium. Let us denote E
i, t
and
V
i,t
the ‘‘beliefs’’ of trader i at time t about the conditional expect-
ation of wealth and the conditional variance of wealth, respectively. It
follows from (12.2.2) that
E
i,t
[W
i, tþ1
] ¼ W
i,t
[R þ p
i,t
(E
i,t
[r
tþ1
] r)], (12:2:3)
V
i,t
[W
i, tþ1
] ¼ p
2
i
,t
W
2
i
,t
V
i,t
[r
tþ1
] (12:2:4)
Also, every agent i believes that return of the risky asset is normally
distributed with mean E
i,t
[r
tþ1
] and variance V
i,t
[r
tþ1
]. Agents choose
the proportion p
i, t
of their wealth to invest in the risky asset, which
maximizes the utility function U
max
p
i,t
{E
i, t
[U(W
i, tþ1
)]} (12:2:5)
The utility function chosen in [9, 10] is
U(W
i,t
) ¼ log (W
i, t
) (12:2:6)
Then demand p
i,t
that satisfies (12.2.5) equals
p
i,t
¼
E
i,t
[r
tþ1
] r
V
i, t
[r
tþ1
]
(12:2:7)
Agent-Based Modeling of Financial Markets 131

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