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take into account the possibility of coalition in such a situation [18]. Indeed,
DIY-L has at least three players and usually three or more strategies.
The games with mixed strategy equilibrium are considered dicult to reach
or, if luckily achieved, not stable [5,10,16]. In the corresponding game, MSE is
almost impossible to calculate and depends on both the number of players and
that of strategies. Besides, players in guessing games usually make their decisions
with only the game results, not their past choices.
Ostling et al. have implemented laboratory experiments of SL and found
that there are mainly four kinds of behavioral rules employed (
random
,
stick
,
lucky
,and
strategic
)[12]
1
. Their results may have the following three mean-
ings: First, individuals are bounded rational and heterogeneous [9]. Second, some
subjects try to make use of the information on past winning integers but oth-
ers not, namely there is mixed evidence for whether individuals take care of
foregone payoffs of others [8,17]. Third, some subjects use plural behavioral
rules.
This study simplifies the findings of Ostling et al. [12] and analyzes how
players using simple adaptive learning model or quasi belief-based learning model
behave and learn in a multi-player and multi-strategy game and consequently
the whole game behavior by agent-based computational economics approach
2
.In
particular, we are going to see whether the combination of two learning models
affects the game results or which learning model is more adaptive in each setup.
The rest of this paper is organized as follows: The next section explains the
basic framework of DIY-L. Section 3 presents experimental design and compu-
tational results. Finally, Section 4 gives concluding remarks.
2GameD gn
There are
N
players each of who chooses one positive integer from 1 to
M
(
>
1).
All of them know this setup. The player who submits the smallest integer that
is not chosen by anyone else is a winner. The winner receives a positive payoff,
usually normalized 1, and the losers do zero. If there is no uniquely chosen
integer, all players become losers.
Here we consider DIY-L with
N
3. In case of bi-matrix game,
there are three equilibria, (1) both players choose 1 and (2) one player chooses
1 and the other does 2. But, since one never makes one's opponent a win-
ner so long as (s)he keeps on choosing 1 [12], this kind of game is not worth
investigating.
≥
3and
M
≥
1
For Swedish Lottery (SL), Lowest Unique Bid Auction and Highest Unique Bid
Auction, see the references therein.
2
From the viewpoints of individual learning in agent-based computational economics
literature, it seems that multi-player and multi-strategy games have not been inten-
sively studied as Shoham et al. have pointed out [14]. Indeed, although Vu et al.
have computationally analyzed such games [15], the number of studies is still small.
