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