Twisted Strategy Bolsters Minority Cooperator Populations
Akihiro Takahara
1
and Tomoko Sakiyama
2a
1
Information Systems Science, Graduate School of Science and Engineering, Soka University, Tokyo, Japan
2
Department of Information Systems Science, Faculty of Science and Engineering, Soka University, Tokyo, Japan
Keywords: Spatial Prisoner’s Dilemma, Populations, Minority.
Abstract: Defectors tend to survive in the spatial prisoner's dilemma. Thus, many studies have sought to keep the
cooperator alive. Here, we aimed to enhance the survival of the cooperator by considering the memory length
in the spatial prisoner's dilemma. In the proposed model, all players are assigned a memory length. Based on
this memory length, players updated their strategies to those that were harder to choose in the past only when
the score of each neighbor with the same strategy was high. This above strategy update rule therefore
alleviates a disadvantageous situation for the player. In this paper, we focused on two cases where the
cooperators were initially in the minority and observed their evolution over time. The results showed that the
model eventually strives to maintain the cooperator population even when it was initially low.
1 INTRODUCTION
Cooperative behaviors are characteristic of several
animals including humans (Smith and Price, 1973).
Game theory presents the evolution of cooperation
among defective players (Nowak and May 1992,
Marko et al. 2022). In classical game theory, players
have two different strategies: the cooperative strategy
or the defector strategy. Defectors earn higher payoffs
against the opponent if the opponent is cooperative.
However, defectors earn a low payoff against the
defector opponent (Doebeli and Hauert, 2004, Hauert
and Doebeli, 2005). On the other hand, cooperators
share payoffs with each other if they mutually interact
with each other. Using the payoff matrix, classical
game theory has revealed that cooperators cannot
survive under some conditions (Doebeli and Hauert,
2004). To this end, many models have been
developed for the sake of the evolution of cooperative
players (Qin et al. 2018, Sakiyama and Arizono,
2019, Sakiyama, 2021).
Recently, we developed a spatial prisoner’s
dilemma (SPD) model called the twisted PD (TPD)
model, where players considered the past occurrence
of each strategy for themselves and sometimes
ignored the classical strategy update rule (Takahara
and Sakiyama, 2023). At that time, players adopted
an unlikely strategy. As a result, the TPD model
a
https://orcid.org/0000-0002-2687-7228
outperformed the classical SPD. In fact, studies have
revealed that introducing memory to players in the
system facilitates cooperation (Danku et al. 2019,
Deng et al. 2017, Javarone, 2016).
In this paper, we analyzed the flexibility of the
TPD model by considering a situation where the
cooperator population was a minority in the initial
spatial distribution. In other words, most of the
population was a defector. Under these conditions, a
cooperative population developed in the TPD model
over time.
2 METHODS
2.1 Simulation Environments
A 100 × 100 square lattice was formed. Players were
placed in all cells and initially assigned a cooperator
(C) or defector (D) strategy. There were two types of
initial distributions of strategies: one where the value
of initial density of defector r was set to 0.5, 0.9, 0.95,
or 0.99 while a random uniform distribution was used
for the players, and one where cooperators were
placed on the center cell and its neighboring four cells
in a fixed distribution, while the remaining players
were defectors. We therefore assessed the
Takahara, A. and Sakiyama, T.
Twisted Strategy Bolsters Minority Cooperator Populations.
DOI: 10.5220/0012262000003636
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 16th International Conference on Agents and Artificial Intelligence (ICAART 2024) - Volume 1, pages 175-178
ISBN: 978-989-758-680-4; ISSN: 2184-433X
Proceedings Copyright © 2024 by SCITEPRESS Science and Technology Publications, Lda.
175
performance of the model where cooperators were
initially in the minority.
Payoff was set to T = b, R = 1, and S = P = 0 based
on the payoff matrix shown in Table 1, where T > R
> P > S. The parameter b that determines T was set to
1 < b < 2 (Nowak & May, 1992). A player with
strategy D received T if the neighboring player was
assigned strategy C. A player with strategy C received
S if the neighboring player was assigned strategy D.
If both strategies were D, the player earned P.
However, if both strategies were C, the player
received R. We used the Neumann neighborhood and
periodic boundary conditions. Individual players
interacted with players above, below, and to the left,
and right of them. Each trial was included 1000 time
steps.
Table 1: Payoff matrix.
nei
g
hbo
r
C D
Player C
𝑅1 𝑆0
D
𝑇𝑏 𝑃0
2.2 Model Description of SPD
The iteration was initiated after a strategy was
assigned to each player, who compared
neighborhoods, and strategies based on the payoff
matrix and calculated a score. After completing this
task, they compared score with their neighbors and
memorized the strategy of the neighbor with the
highest score. The strategy of each player was then
synchronously updated to the learned strategy.
However, the strategy was not updated if there were
multiple nearby players with the same highest score
but different strategies.
2.3 Model Description of the TPD
Model
Here, we describe the twisted Prisoner’s Dilemma
model (TPD model) (Takahara & Sakiyama, 2023),
where every player is assigned a length of memory of
value p that was constant between trials. After the
score was calculated, each player reflected on his or
her previous strategy. The length of the past
considered is from t (current) to t p, and the number
of experienced cooperative strategies was recorded in
the parameter Count_c.
If neighboring players had the same strategy while
the player had a lower score, the player updated their
strategy using one of the two following probabilities:
The player will update its strategy to C with the
following probability:
1- (Count_c)/p
The player will update its strategy to D with the
following probability:
(Count_c)/p
If the above conditions were not satisfied, the rule
of the SPD model was applied for the strategy update.
The strategy of each player synchronously
updated. In this model, the strategy update rule to use
the values of p that was different from the SPD model
rules was not executed until t was greater than p. The
proposed model was based on the following concept:
the player changes their behavior when their score is
lower than that of neighbors who have the same
strategy.
3 RESULTS
3.1 Defector Density
First, the r was set to 0.5, 0.9, 0.95, and 0.99, whereas
p was fixed at 10. The defector density over 1000 time
steps was calculated by averaging 10 trials. The
results are shown in Figure 1. We found that an
initially large defector populations did not affect the
evolution of cooperators, though cooperators did not
survive if r was set to 0.99. This is perhaps because
not enough cooperators are placed, and they cannot
interact with each other.
Figure 1: Defector density for various values of r (0.5, 0.9,
0.95, 0.99).
Next, we switched the initial distribution of
players to the second condition, where each
cooperator was placed on the center cell and its
ICAART 2024 - 16th International Conference on Agents and Artificial Intelligence
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neighboring four cells. The remaining players were
defectors. Results were compared with r = 0.99 as
shown in Figure 2, where the defector density was
lower than r = 0.99 and remained around 0.80 for any
value of b. Although the initial density was much
higher than r = 0.99 in the density with a fixed initial
distribution, the defector did not increase as much as
r = 0.99.
Figure 2: Defector density of the two initial distributions.
3.2 Spatial Distribution
Next, we compared the spatial distribution of the
fixed initial distribution between the TPD model and
the SPD model. Here, we set the parameter b = 1.9.
As shown in Figure 3, the distribution of
cooperators in the TPD model was spread out from
the center at t = 10 but sparse at t = 1000. However,
the distribution remained constant over time in the
SPD model.
In both models, the score of the player at the
center was 4 at t = 1, and the neighbors of the centered
player adopted strategy C. Their strategy did not
change because the player in the neighborhood with
the highest score is the one in the center. In the TPD
model, this process is repeated until t = 9 according
to the SPD model rule where p = 10. Therefore, the
distribution of strategies did not change until t = 9.
However, neighbors of the centered player
considered previous strategies and followed an
unusual update rule because their own strategy earned
a score lower than that of the centered player, even
though their strategies were the same at t = 10. As a
result, they adopted strategy D according to the rules
of the TPD model.
Even though players outside of the region of
interest described above adopted strategy D, their
scores were lower than the player whose strategy was
D and who neighbors the player in C, so those players
were likely to change their strategy to C. As a result,
a diamond-like shape formed, and repeated many
times; a constant number of players with strategy C
survived at t = 1000. However, in the SPD model, the
strategy distribution maintained its shape and did not
deviate from the initial distribution even at t = 1000.
Therefore, the cross-like shape in the spatial
distribution of the TPD model during early stages
contributed to cooperator survival.
Figure 3: Spatial distribution of fixed initial density in two
models.
4 CONCLUSIONS
In this paper, we evaluated the TPD model in two
cases where cooperators were in the minority of the
population. In the first case, players of each strategy
were randomly distributed according to the defector
density parameter r. As we considered cooperative
populations as a minority group, the parameter r had
high values. We found that cooperators could evolve
despite their low initial density. A fixed distribution
was used in the second case, where only five players
in the center of the system adopted cooperative
strategies while others were initially defectors.
However, the number of cooperators increased over
time. Interestingly, the initial number of cooperators
in the second condition was lower than that of the first
condition with r = 0.99, and the final cooperator
population in the former was higher than the latter,
suggesting that the initial placement of cooperators
influences outcomes.
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177
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