154 L. Grilli, M.A. Russo, and A. Sfrecola
more details. Empirical studies also show cluster formations and other anomalies in
financial time series [3, 9].
In order to model the inductive behaviour of financial agents, one of the most
famous examples is the Minority Game (MG) model. The MG is a simple model of
interacting agents initially derived from Arthur’s famous El Farol’s Bar problem [1]. A
popular bar with a limited seating capacity organises a Jazz-music night on Thursdays
and a fixed number of potential customers (players) has to decide whether to go or
not to go to the bar. If the bar is too crowded (say more than a fixed capacity level)
then no customer will have a good time and so they should prefer to stay at home.
Therefore every week players have to choose one out of two possible actions: to stay
at home or to go to the bar. The players who are in the minority win the game.
Since the introduction of the MG model, there have been, to date, 200 papers on
this subject (there is an overview of literature on MG at the Econophysics website).
The MG problem is very simple, nevertheless it shows fascinating properties and
several applications. The underlying idea is competition for limited resources and
it can be applied to different fields such as stock markets (see [5–7] for a complete
list of references). In particular the MG can be used to model a very simple market
system where many heterogeneous agents interact through a price system they all
contribute to determine. In this market each trader has to take a binary decision every
time (say buy/sell) and the profit is made only by the players in the minority group.
For instance, if the price increases it means that the minority of traders are selling and
they get profit from it. This is a simple market where there is a fixed number of players
and only one asset; they have to take a binary decision (buy/sell) in each time step
t. When all players have announced their strategies the prices are madeaccording to
the basic rule that if the minority decides to sell, then the price grows (the sellers get
profit); if the minority decide to buy, then price falls (the buyers get profit).
In this model cooperation is not allowed; players cannot communicate and so they
all get information from the global minority. In order to make decisions, players use
the global history of the past minorities or, in most cases, a limited number of past
minorities that can be considered the time window used by the player. In our case
the global history is given by the time series of price fluctuations. Let us consider
the set of playersi={ 1 ,...,N}whereN∈N(odd and fixed). Indicate withtthe
time step when each player makes a decision. In the market there is one asset and the
possible decision in each time step is buy or sell; as a consequence the playeriat
timetchoosesσit∈{+ 1 ,− 1 }(buy/sell).
In each time stept,letptbe the price of the asset at timet; the minority (the
winning strategy) is determined by
St=−sign log(
pt
pt− 1)
.
Consequently, the time series of price fluctuations is replaced by a time series
consisting of two possible values:+1and−1 (the minority decisions).
In [4] it is shown that often similar results can be obtained by replacing the real
history with an artificial one.