Previous computer simulations of the Minority Game (MG) have shown that the average agent number in the winning group (i.e., the minority group) had a maximal value such that the global gain was also maximal when an optimal amount of information was available to all agents . This property was further examined and its connection to financial markets has also been discussed . Here we report the results of an unprecedented real MG played by university staff members who clicked one of two identical buttons (A and B) on a computer screen while clocking in or out of work. We recorded the number of people who clicked button A for 1288 games, beginning on April 21, 2008 and ending on October 31, 2010, and calculated the variance among the people who clicked A as a function of time. We find that variance per person decreases to a minimum and rises to a value close to 1/4 which is the expected value when agents click buttons randomly. Our results are consistent with previous simulation results for the theoretical MG and suggest that our agents had employed more information for their strategies as their experience playing the game grew. We also carried out another experiment in which we forced 101 fish to enter one of the two symmetric chambers (A and B). We repeated the fish experiment 500 times and found that the variance of the number of fish that entered chamber A also decreased to a minimum and then increased to a saturated value, suggesting that fish have memory and can employ more strategies when facing the same situation again and again 

Fei Fang Chung

Ruey-Tarng Liu

Sy-Sang Liaw

English

Nature Precedings

Fish play Minority Game as humans do 
 
Ruey-Tarng Liu, Fei Fang Chung, and Sy-Sang Liaw 
 
Department of Physics, National Chung Hsing University 
 
250 Guo-Kuang Road, 402 Taichung, Taiwan 
 
 
Previous computer simulations of the Minority Game (MG)1  have shown that 
the average agent number in the winning group (i.e., the minority group) had a 
maximal value such that the global gain was also maximal when an optimal 
amount of information was available to all agents 2 . This property was further 
examined 83  and its connection to financial markets has also been discussed 9 . 
Here we report the results of an unprecedented real MG played by university 
staff members who clicked one of two identical buttons (A and B) on a computer 
screen while clocking in or out of work. We recorded the number of people who 
clicked button A for 1288 games, beginning on April 21, 2008 and ending on 
October 31, 2010, and calculated the variance among the people who clicked A as 
a function of time. We find that variance per person decreases to a minimum and 
rises to a value close to 1/4 which is the expected value when agents click buttons 
randomly. Our results are consistent with previous simulation results 2  for the 
theoretical MG and suggest that our agents had employed more information for 
their strategies as their experience playing the game grew. We also carried out 
another experiment in which we forced 101 fish to enter one of the two 
symmetric chambers (A and B). We repeated the fish experiment 500 times and 
found that the variance of the number of fish that entered chamber A also 
decreased to a minimum and then increased to a saturated value, suggesting that 
fish have memory 11,10  and can employ more strategies when facing the same 
situation again and again12 . 
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The Minority Game (MG) is a simple evolutionary game designed for studying, 
among other things, how the actions of selfish players can be coordinated by an 
invisible hand to cooperate for global benefit. In the game, N players have to 
independently choose one of two sides (A and B) and those on the minority side win. 
In a theoretical analysis, one must assume that agents have a fixed number of 
strategies and a fixed level of memory for the winning history. Under these 
assumptions, many interesting properties have been discovered13 . One interesting 
property that has attracted much attention is that the global gain, defined as the 
portion of agents who have won the game, is maximal when the agents employ an 
optimal amount of information 2 . The global gain is related to the variance of the 
number of players choosing a particular side (e.g., side A) in a simple way such that 
variance is negatively correlated to the extent of global gain. Therefore the value of 
variance per player )(A  decreases to a minimum as the amount of information 
available to all agents increases, and then increases to a value close to 1/4, indicating 
that when there is too much information all player choices are effectively random 
(Figure 1). 
Obviously, human players of a real MG do not have a fixed memory and most 
likely will not use binary tables for strategies as in a theoretical MG model, and 
understanding of which essential features of the theoretical MG model would remain 
in a real game could have practical applications. Two reports have been written on 
real MGs played by humans 15,14  but the numbers of players in these two experiments 
(15 and 5, respectively) are so small that their conclusions are not statistically 
significant. 
Beginning in April 2008, with the help of the Office of Personnel and the 
Computer and Information Network Center of National Chung Hsing University 
(NCHU), we set up a time clock system for university staff to play the MG while 
clocking in and out of work. All university staff ( 600~200N ) were required to 
click one of two buttons (A or B) shown on their office work stations to clock in 
before 08:30 and clock out after 16:45. Each button click was recorded automatically 
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with instances denoted as n(A) and n(B), to determine the winning button. The 
general concept of the MG was explained to all staff, and they were instructed by 
memo how to use the system. Staff members were incentivized to play the MG 
seriously by giving away gift certificates redeemable at local convenience stores to 
the top three winners every month. For the first 20 games, we purposely set button A 
on the left-hand side of the screen and button B on the right-hand side. Button B had 
won all these games, suggesting that more people tended to click the button on the 
left-hand side without paying attention to the game. However, the number of winners 
oscillated and approached N/2, indicating that more and more people were playing the 
game seriously. Starting from April 21, 2008, we switched the positions of buttons A 
and B randomly at every game. By October 31, 2010 a total of 1288 games had been 
recorded. In the first phase, consisting of the first 458 games, the winning buttons of 
the previous four games were provided on the screen for reference. In the second 
phase, consisting of the final 880 games, the winning buttons of previous four games 
were still provided but these results had been modified without the knowledge of the 
players in favor of button B by adding a value 0.09N to n(A)16 . 
We calculated the variance per player )(A  and studied its variation as a 
function of time. Note the number of agents iN  at time it  is not a constant in our 
games (Figure 2) since the number of university staff members working on any given 
day was subject to fluctuation. Thus each n(A) has to be multiplied by a factor 
iNN /  before calculating its variance, where  N  is the average value of 
iN  in the time period considered. (See Methods for details.) To find the time 
evolution of )(A , we divided the total 1288 games into 18 time periods (with each 
period marked by a horizontal thick bar in Figure 2) so that each period contains no 
less than 50 games and iN  does not vary too dramatically within most periods. In 
phase 1, )(A  drops to a minimum and rises back to near 1/4. The evolution of the 
variance, as compared with the theoretical variance as a function of information in the 
standard MG (Figure 1), shows the agents employed more information in playing the 
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game as they gained experience with it. According to the simulation results shown in 
Figure 1, )(A will saturate to 1/4 when even more information is available to the 
agents. For our games, since )(A  has passed the minimum and reached 1/4, we did 
not expect it to vary significantly if more games were played the same way. We 
therefore reactivated the game in phase 2, starting from game #459, by adding 9% of 
iN  as virtual players who always chose button A. The manipulated results were 
provided as a reference to the agents, who were not aware of the addition of the 
virtual players. Apparently, the strategies learned from phase 1 did not work in phase 
2 and )(A  began to increase (Figure 2). It took another 350 games before the 
agents adapted to the effect of the virtual players, at which time )(A  decreased 
again, this time to a lower minimum than seen in phase 1. A smaller variance means, 
on average, more agents in each game and so a larger global gain. The increase in 
global gain by adding persistent virtual players was consistent with what had been 
previously predicted in the theoretical MG model16 . After reaching a minimum, 
)(A  increased to near 1/4 and we concluded our human experiment. 
  We also carried out a separate fish experiment, the experimental setup of which is 
shown in Figure 3. The Mosquito fish (Gambusia affinis) is a common fish about 3 
cm long and is ubiquitous in Taiwan’s ponds and rivers. Using a slow-moving mesh, 
we forced N = 101 Mosquito fish in a tank into two symmetric chambers A and B, lit 
from overhead by a lamp to reveal their entrances. The whole system was covered by 
a large piece of black cloth to isolate the fish from environmental cues. Once all the 
fish had entered a chamber, the number of fish in each chamber was quickly counted 
and recorded. We then reduced the water volume in the losing chamber so that the 
50-plus fish inside were forced into a crowded cluster for a few seconds. Presumably, 
fish do not like being forced to be in direct contact with one another and would 
learn 1210  to enter the other chamber when forced to choose next time. We repeated 
this fish MG 500 times (a few fish died during the repeated experiments and were 
replaced by new fish). Variance per fish )(A  as a function of time is given in the 
inset of Figure 2. Qualitatively, the evolution of )(A  is similar to that in the human 
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experiment, and reaches a minimum even faster than in the human case. However, 
)(A  is surprisingly large in the fish experiment, ranging from 0.55 to 1.25. Upon 
closer investigation, we found that the Mosquito fish have a tendency to form groups 
when escaping17 . Assuming that a group averages 3 fish when entering the chambers, 
the variance per group is then given by one third of the value presented in Figure 2, 
thus its value begins with 0.4, drops to a minimum of 0.19, and then saturates to 1/4, a 
pattern very similar to the human case. 
 
Methods 
In the human game, we divided the 1288 games into 18 samples such that the 
number of games L in each sample is no less than 50 while minimizing the variance of 
iN  in each sample. To calculate the variance of the number of agents choosing 
button A, we modified the value n(A). While pretending that all games in a given 
sample were played by  N  agents, we multiplying it by the factor iNN / , 
where  N  is the mean of iN  in the sample 

 L
i
iNL
N
1
1 . Since the 
standard deviation of a sample is proportional to the square root of the sample size, 
we compared variance per agent between different samples by multiplying n(A) by 
another factor  N/1 . Thus the actual value used in calculating variance per 
agent )(A  of the number of agents choosing button A was iNNn /)( A . 
We have checked for possible errors due to the inconstancy of iN  using 
random samples of length L. Take 400 N  and ii rNN  , where ir  is a 
random integer between -100 and 100. We used iNNn /)( A  to calculate the 
variance per agent )(A  and found that the random fluctuation of ir  has little effect 
on )(A . The average value of )(A  is close to 1/4 as expected. The standard 
deviation18  of )(A  is inversely proportional to L . For 50L , the standard 
deviation is close to 0.05.  
 
 
 
 
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Acknowledgments: This work is supported by grants from the National Science Council under the 
grant number NSC97-2112-M005-001, and the National Centre for Theoretical Sciences of Taiwan. 
 
 
Correspondence and requests for materials should be addressed to S.-S. Liaw (liaw@phys.nchu.edu.tw) 
 
 
Supplementary Information 
1. Human MG records (file: HumanMG.txt) 
2. Fish MG records (file: FishMG.txt) 
 
 
 
 
 
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Figure 1 
Theoretical variance per agent )(A  as a function of information )/2log( NM  in the 
standard MG, where M stands for the memory length in the theoretical model. The dashed line 
indicates the variance value 1/4 for the case of random choices. 
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
V
ar
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nc
e 
pe
r a
ge
nt
0 1 2
Information  
 
Figure 2 
Evolution of the variance per agent )(A  for human MG and fish MG (inset). In the fish experiment, 
N =101. In the human MG, the number iN  of players (dot) was not constant. In both phases 
(diamond for phase 1, circle for phase 2), the variance per agent was qualitatively similar: decreasing to 
a minimum and then increasing to near the value 1/4. Variances per agent were calculated for periods 
(indicated by solid bars) each containing more than 50 games. The dashed line indicates the variance 
value 1/4 for the case of random choices. Mosquito fish aggregate into groups, each containing on 
average 3 fish so that its )(A  is three times greater than the results of the human MG. 
 
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Figure 3 
Fish MG experimental setup, in which 101 Mosquito fish in tank C were forced by a moving mesh to 
enter either chamber A or B which were lighted by a lamp above. Tank C was covered by a piece of 
black cloth to isolate the fish from environmental cues.  
 
 
C 
A B 
motor 
mesh 
A 
C 
B 
Mosquito fish 
C 
A B 
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Fish play Minority Game as humans do

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Fish play Minority Game as humans do

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