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it will gain a majority soon. That is, the group of buyers cannot keep a majority, a
contradiction. Thus, it is difficult to simulate the bubble phenomenon by MG.
Related Work.
Much work has been done for the purpose of adapting MG to a real fi-
nancial market. For example, first, several authors investigated the majority game (MJ),
consisting of
trend-followers
. Marsili [14] and Martino et al. [15] investigated a mixed
majority-minority game by varying the fraction of trend-followers. Tedeschi et al. [17]
considered agents who change themselves from contrarians to trend-followers, and vice
versa, according to the price movements. Second, another way is to incorporate more
realistic mechanism. A grand canonical minority game (GCMG) [5,10,11] is consid-
ered as one of the most successful models of a financial market. In the GCMG, a set of
agents consist of two groups, called producers and speculators, and the speculators are
allowed not to trade in addition to buy and sell. Third, it is also useful to improve the
payoff function. Andersen and Sornette[1] proposed a different market payoff, called
$-game, in which the timing of strategy evaluation is taken into consideration. Ferreira
and Marsili[9] compared the behavior of the $-game with that of the MG/MJ. The diffi-
culty of the $-game is to evaluate its payoff function because we have to know one step
future result. Kiniwa et al. [12] proposed an improved $-game, in which the timing of
evaluation is delayed until the future result is turned out. Fourth, there are some other
kinds of improvement. Liu et al. [13] proposed a modified MG, where agents accumu-
late scores for their strategies from the recent several steps. Recent work by Challet [4]
proposed a more sophisticated model using asynchronous holding positions which are
driven by some patterns. Finally, two topics [6,8] comprehensively described the history
of minority games, mathematical analysis, and their variations. Beyond the framework
of MG, efforts to reproduce the real market dynamics are continued [16,18].
Motivation.
The purpose of this paper is also to improve MG by the thirdly mentioned
above. Though the framework of MG and its variants seem to be reasonable, we have a
basic question — “Do people always make decisions by using their strategies depending
on the recent history ?” Some people may just take actions by considering losses and
gains. For example, if one has a company's stock which has rapidly risen (resp. fallen),
he will sell (resp. not sell) it soon without using his strategy as illustrated in Figure 1.
Such a situation gives us the idea of an acquisition cost, or a mean asset value. In the
conventional games, like the original MG, an agent forgets the past events and makes a
decision by observing only the price up/down within the memory size
1
. In our game,
however, each agent evaluates the strategies by whether or not the current price exceeds
his mean asset value. Since the mean asset value contains all the past events in a sense,
he can increase his net profit by reducing the mean asset value. We call the game an
asset value game
, denoted by AG.
However, there is still an unsolved problem in AG that stems from the framework
of MG: the payoff function does not give an action, but just adds points to desirable
strategies. Thus, if the adopted strategy is not desirable, the agent has to wait until the
1
Recently, several studies [2,3] in this direction have been made from the viewpoint of evolu-
tionary learning.
