Tuning of the Update Timing Will Stop the Defector Invasion in the

Spatial Game Theory

Akihiro Takahara and Tomoko Sakiyama

Department of Information Systems Science, Faculty of Science and Engineering, Soka University, Tokyo, Japan

Keywords: Prisoner’s Dilemma, Cooperator, Defector, Interval.

Abstract: Since the defectors tend to survive in the Spatial prisoner’s dilemma, many studies have proposed models for

the purpose of survivals of cooperators. But many of those models are not realistic. Therefore, in this study,

we proposed a model that considered the player’s decision-making time based on previous research. In the

proposed model, the defectors decrease the probability of the strategy update while the cooperators increase

the probability of the strategy update. In this paper, we investigate the defector density and the spatial

distribution by setting two different system sizes: 50×50 and 200×200. Since the results were very similar to

each system size, we found that the proposal model was not affected by the system size. Furthermore, ever if

the payoff parameter regarding a defector vs. a cooperator increased, the defector density did not increase

rapidly, which was against the conventional model. We compared the spatial distribution of the proposal

model with the conventional model and found that cooperators were widely maintained in the proposal model

while they were partially maintained in the conventional model. Thus, the proposed model that introduces

decision-making times of players is a realistic model and contributes to the survival of cooperators.

1 INTRODUCTION

Humans demonstrates cooperative behaviours that

facilitate the prosperity of human populations

(Maynard Smith and Price, 1973). Cooperators can

build a win-win relationship within a group of people

having similar characteristics. Among selfish people

however, cooperators may be exploited. Nonetheless,

selfish behaviors are also one of the characteristics of

humans. The game theory presents how

selfish/cooperative behaviours are evolved in

population systems (Nowak and May, 1992, Doebeli

and Hauert, 2005, 2022, Marko et al. 2022). In the

basic game theory, players select one strategy from

two strategies. Defectors can win alone within a

group of cooperators. However, defectors get almost

nothing when facing with people having the same

strategy. On the other hand, cooperators can be

mutually benefitable while they are deprived of

profits when facing people having the against

strategy. In the classical spatial game theory,

cooperators appeared to be extinct in some situations

(Hauert and Doebeli, 2005, Szabó and Toké, 1998).

To solve this issue, a lot of studies have focused on

https://orcid.org/0000-0002-2687-7228

the evolution of cooperators. Some of them have

considered the importance of external disturbances

while others have considered the evolution of link

weights and so on (Szolnoki and Perc, 2014, Qin et

al. 2018, Shen et al. 2018).

In this paper, we develop a spatial prisoner’s

dilemma (PD) model by considering a relationship

between players’ decision time and the timing of the

strategy-update. Since some of previous models do

not consider the actual actions found in humans, we

refer to a study reporting that human subjects

switched their cooperative behaviours with selfish

behaviours according to the thinking time (Capraro,

2017). In our model, individual players have a fixed

time interval for the strategy update. The

conventional model only considers the rules of battle

and strategy update in Prisoner's Dilemma. They

sometimes modulate that timing based on their

strategy. As a result, we found that the population

system in our model confirmed the survival of

cooperators. Interestingly, we could not find any

advantages of cooperators’ survival if players held a

given time interval and did not coordinate it. Thus,

the coordination of the update timing can be related

Takahara, A. and Sakiyama, T.

Tuning of the Update Timing Will Stop the Defector Invasion in the Spatial Game Theory.

DOI: 10.5220/0011716200003485

In Proceedings of the 8th International Conference on Complexity, Future Information Systems and Risk (COMPLEXIS 2023), pages 13-16

ISBN: 978-989-758-644-6; ISSN: 2184-5034

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

to the propagation of cooperators/defectors in the

populations.

2 METHODS

2.1 Simulation Environments

We introduced players on two-dimensional square

lattice having 200×200 or 50×50 system size. Each

player was set on individual cells. One of two

strategies (defector (D) or cooperator (C) was

assigned to individual players. Payoff can be R=1,

T=b, S=P=0 satisfying T>R>P>S. Parameter 𝑏 that

determines 𝑇 can be set to 1<𝑏<2 (Nowak and

May, 1992). In that paper, S=P is applied, as P is a

positive number and does not lose generality when P

is less than 1.

As the boundary condition, we adopted the

periodic boundary condition. Von-Neumann

neighbourhood was used for players’ interaction.

Each simulation experiment lasted for 500-time steps.

2.2 SPD Model

Here, we defined the spatial prisoner’s dilemma

(SPD) model as the control model. In the SPD model,

two sub-models exist in each time iteration. In the

first sub-model, individual players earn scores by

interacting with the neighbouring players. More

strictly, each player plays against all neighbouring

players who located above, below, right, and left of

the player and earns scores based on the payoff

matrix.

In the second sub-models, each player updates its

strategy by comparing its earned scores with the

neighbouring players’ scores. Players replace their

strategy with the new one’s strategy that has the

highest score among neighbouring. The strategy

update will stop for the player if more than two

players among itself and neighbouring players have

the same highest scores but have different strategies

with each other.

The classical SPD has a problem that defectors

increase as the parameter 𝑏 increases after the system

reaches a steady state.

2.3 Proposed Model

Here, we proposed two different models. One is the

Interval PD (IPD) model. The other one is the

Update-Modification PD model. Both of two models

are based on the cognitive experiment using the

human subjects where subjects presented various

thinking time, which depended on whether they

behaved like a defector or not (Capraro, 2017).

In the report, researchers found that people were

more likely to choose the selfish action when having

much thinking time. On the other hands, people were

more likely to choose the cooperative action when

having short time.

2.3.1 IPD Model

In the IPD model, individual players have a unique

variable named as interval. The variable interval is

the time interval between two consecutive strategy

updates. Each player is randomly given a different

random natural value of interval, and the value is

fixed at the end of each trial. The minimum value is

1, the maximum value is 𝑝, and 𝑝 can be set as a

parameter. Thus, players have a chance to update

their strategy every interval time steps.

2.3.2 UMPD Model

We modified the IPD model and named the modified

model as the Update-Modification Prisoner’s

Dilemma (UMPD) model.

The featuring point of this model is that we

introduced a probability into the strategy update

event. Therefore, players in this model do not always

have a chance to update their strategy every interval

time steps. The probability is implemented using the

variable interval.

If the strategy of player is D (defector) at the

strategy-update timing, the strategy can be updated

with the following probability.

1 / interval (1

)

On the other hand, if own strategy of player is C

(cooperator) at the strategy-updated strategy timing,

the strategy can be updated with the following

probability.

1 – (1 / interval) (2

)

These two rules are based on the fact that the

human subjects present long/short thinking time

when they behave like a defector/cooperator.

Therefore, players in this model tend to maintain the

current strategy using the variable interval. At the

same time however, the probability is dependent on

the current strategy of players.

COMPLEXIS 2023 - 8th International Conference on Complexity, Future Information Systems and Risk

3 RESULTS

The initial distribution probability 𝑟 for defectors was

fixed at 0.5. It means that density of defector and

cooperator were equal. We set 1000-time steps as one

trial. Parameter 𝑏 was set to 1<𝑏<2. The

maximum interval 𝑝 was fixed at 3. It means that each

player had an interval randomly from three interval

values: 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙  1,2 and 3. The value of interval

was assigned to individual players at start of each trial.

We set two different system sizes. One was 50×50.

The other one was 200×200. We examine the effect

of changing system size of the proposal model.

3.1 Invasion of Defectors

In this subsection, we checked the relationship between

parameter 𝑏 and the defector density. Here, the

average density of defectors was calculated using the

defector density at the end of trial (𝑡  1000) over 10

trials.

First, we show results using 50×50 system.

According to Figure 1, which shows the relationship

between parameter 𝑏 and the average defector density

for three models, model having the largest defector

density is the UMPD model in 1<𝑏<1.5. But as

the parameter increased, the UMPD model performed

better than other two models since the defector density

did not become large in the UMPD model while the

defector density increased sharply in the other models.

Next, we show results using 200×200 system size.

Figure 2 demonstrates the relationship between

parameter 𝑏 and the defector density. Consequently,

the UMPD model did not perform well in the 1<𝑏<

1.5. However, as the parameter increased, the UMPD

model again performed better than other models.

We found that there was no difference regarding

the defector density and its relationship with the

parameter 𝑏 between 50×50 system size and 200×200

system size. Therefore, the UMPD model is a flexible

model regardless of the system size.

3.2 Spatial Distribution

We also checked the spatial distribution. Here, we

present the defector/cooperator distributions using the

UMPD model and the SPD model at t = 1, 10, 20 and

30. Parameter 𝑏 was fixed to 1.9 for the UMPD model.

First, Figure 3 shows the spatial distribution of the

SPD model. Cooperators partially survive while

defector occupies most of space at t = 10. Then, each

cooperator seems to spread vertically and

horizontally around the certain cooperators. After that,

some cooperators additionally spread vertically and

horizontally as a time goes on. However, cooperators

do not appear in other places. Consequently,

cooperators can be maintained partially.

Next, Figure 4 shows the spatial distribution of the

UMPD model. Cooperators widely survive at t = 10,

which is contrary to the case of SPD model. After that,

there is no significant change in the spatial distribution.

Therefore, cooperators can be survived easier in the

UMPD model than the SPD model.

Figure 1: The defector density in the three models (the

system size: 50×50). 𝑝3.

Figure 2: The defector density in the three models (the

system size: 200×200). 𝑝3.

Figure 3: Spatial distribution of the SPD model. Blue: D

(defector), Yellow: C (cooperators).

Tuning of the Update Timing Will Stop the Defector Invasion in the Spatial Game Theory

Figure 4: Spatial distribution of the UMPD model. Blue: D

(defector), Yellow: C (cooperators).

4 CONCLUSIONS

In this paper, we proposed the UMPD model for the

purpose of maintaining the cooperator. In the

proposed model, decision-making time for strategy

update is probabilistically introduced into the SPD

model. Consequently, the UMPD model is easier to

maintain the cooperator than the conventional model.

In the SPD model, the defector density tends to

increase as the parameter 𝑏 increases. On the other

hands, in the UMPD model, the defector density is

unlikely to be increased as the parameter 𝑏 increases.

Therefore, it is considered that the model is less

affected by the parameter than conventional model.

We were also able to get similar results even after the

system size replacement.

According to the previous research by V.Caprero

(2017), the longer human subjects have thinking time,

the easier human subjects behave selfishly. Similarly,

human subjects behave cooperatively if they have

short thinking time. Similar effects were introduced

on our model. Therefore, we introduced realistic

assumptions in the model. As a result, players having

the cooperative strategy tend to be maintained if they

have relatively short thinking time. This is considered

that the cooperator density stabilized by introducing

the probability of strategy update that (1 / interval)

and (1 - (1 / interval)) into the conventional model.

Since some players do not update their strategies

when interval = 1, we added a rule to change the value

of interval after a strategy update was made. However,

since the result was the same as before the change, it

is considered to be less affected by the value of

interval = 1. Detailed results will appear in another

paper (Takahara and Sakiyama, 2023).

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