The Optional Prisoner’s Dilemma in a Spatial Environment: Coevolving

Game Strategy and Link Weights

Marcos Cardinot, Colm O’Riordan and Josephine Grifﬁth

Information Technology, National University of Ireland Galway, Galway, Ireland

Keywords:

Coevolution, Optional Prisoner’s Dilemma Game, Spatial Environment, Evolutionary Game Theory.

Abstract:

In this paper, the Optional Prisoner’s Dilemma game in a spatial environment, with coevolutionary rules for

both the strategy and network links between agents, is studied. Using a Monte Carlo simulation approach, a

number of experiments are performed to identify favourable conﬁgurations of the environment for the emer-

gence of cooperation in adverse scenarios. Results show that abstainers play a key role in the protection of

cooperators against exploitation from defectors. Scenarios of cyclic competition and of full dominance of

cooperation are also observed. This work provides insights towards gaining an in-depth understanding of the

emergence of cooperative behaviour in real-world systems.

1 INTRODUCTION

Evolutionary game theory in spatial environments

has attracted much interest from researchers who

seek to understand cooperative behaviour among ra-

tional individuals in complex environments. Many

models have considered the scenarios where partici-

pant’s interactions are constrained by particular graph

topologies, such as lattices (Szab

o and Hauert, 2002;

Nowak and May, 1992), small-world graphs (Chen

and Wang, 2008; Fu et al., 2007), scale-free graphs

(Szolnoki and Perc, 2016; Xia et al., 2015) and, bi-

partite graphs (G

omez-Garde

nes et al., 2011). It has

been shown that the spatial organisation of strategies

on these topologies affects the evolution of coopera-

tion (Cardinot et al., 2016).

The Prisoner’s Dilemma (PD) game remains one

of the most studied games in evolutionary game the-

ory as it provides a simple and powerful framework to

illustrate the conﬂicts in the formation of cooperation.

In addition, some extensions of the PD game, such

as the Optional Prisoner’s Dilemma game, have been

studied in an effort to investigate how levels of coop-

eration can be increased. In the Optional PD game,

participants are afforded a third option — that of ab-

staining and not playing and thus obtaining the loner’s

payoff (L). Incorporating this concept of abstinence

leads to a three-strategy game where participants can

choose to cooperate, defect or abstain from a game

interaction.

The vast majority of the spatial models in previ-

ous work have used static and unweighted networks.

However, in many social scenarios that we wish to

model, such as social networks and real biological

networks, the number of individuals, their connec-

tions and environment are often dynamic. Thus, re-

cent studies have also investigated the effects of evo-

lutionary games played on dynamically weighted net-

works (Huang et al., 2015; Wang et al., 2014; Cao

et al., 2011; Szolnoki and Perc, 2009; Zimmermann

et al., 2004) where it has been shown that the coevo-

lution of both networks and game strategies can play

a key role in resolving social dilemmas in a more re-

alistic scenario.

In this paper we adopt a coevolutionary spatial

model in which both the game strategies and the link

weights between agents evolve over time. The in-

teraction between agents is described by an Optional

Prisoner’s Dilemma game. Previous research on spa-

tial games has shown that when the temptation to de-

fect is high, defection is the dominant strategy in most

cases. We believe that the combination of both op-

tional games and coevolutionary rules can help in the

emergence of cooperation in a wider range of scenar-

ios.

The aims of the work are, given an Optional Pris-

oner’s Dilemma game in a spatial environment, where

links between agents can be evolved, to understand

the effect of varying the:

• value of the link weight amplitude (the ratio ∆/δ).

• value of the loner’s payoff (L).

Cardinot, M., O’Riordan, C. and Grifﬁth, J.

The Optional Prisoner’s Dilemma in a Spatial Environment: Coevolving Game Strategy and Link Weights.

DOI: 10.5220/0006053900860093

In Proceedings of the 8th International Joint Conference on Computational Intelligence (IJCCI 2016) - Volume 1: ECTA, pages 86-93

ISBN: 978-989-758-201-1

• value of the temptation to defect (T ).

By investigating the effect of these parameters, we

aim to explore the impact of the link update rules and

to investigate the evolution of cooperation when ab-

stainers are present in the population.

Although some work has considered coevolv-

ing link weights when considering the Prisoner’s

Dilemma in a spatial environment, to our knowledge

the investigation of an Optional Prisoner’s Dilemma

game on a spatial environment, where both the strate-

gies and link weights are evolved, has not been stud-

ied to date.

The results show that cooperation emerges even

in extremely adverse scenarios where the temptation

of defection is almost at its maximum. It can be ob-

served that the presence of the abstainers are funda-

mental in protecting cooperators from invasion. In

general, it is shown that, when the coevolutionary

rules are used, cooperators do much better, being also

able to dominate the whole population in many cases.

Moreover, for some settings, we also observe inter-

esting phenomena of cyclic competition between the

three strategies, in which abstainers invade defectors,

defectors invade cooperators and cooperators invade

abstainers.

The paper outline is as follows: Section 2 presents

a brief overview of the previous work in both spa-

tial evolutionary game theory with dynamic networks

and in the Optional Prisoner’s Dilemma game. Sec-

tion 3 gives an overview of the methodology em-

ployed, outlining the Optional Prisoner’s Dilemma

payoff matrix, the coevolutionary model used (Monte

Carlo simulation), the strategy and link weight update

rules, and the parameter values that are varied in order

to explore the effect of coevolving both strategies and

link weights. Section 4 features the results. Finally,

Section 5 summarizes the main conclusions and out-

lines future work.

2 RELATED WORK

The use of coevolutionary rules constitute a new trend

in evolutionary game theory. These rules were ﬁrst in-

troduced by Zimmermann et al. (2001), who proposed

a model in which agents can adapt their neighbour-

hood during a dynamical evolution of game strategy

and graph topology. Their model uses computer sim-

ulations to implement two rules: ﬁrstly, agents play-

ing the Prisoner’s Dilemma game update their strat-

egy (cooperate or defect) by imitating the strategy of

an agent in their neighbourhood with a higher pay-

off; and secondly, the network is updated by allowing

defectors to break their connection with other defec-

tors and replace the connection with a connection to

a new neighbour selected randomly from the whole

network. Results show that such an adaptation of the

network is responsible for an increase in cooperation.

In fact, as stated by Perc and Szolnoki (2010),

the spatial coevolutionary game is a natural upgrade

of the traditional spatial evolutionary game initially

proposed by Nowak and May (1992), who consid-

ered static and unweighted networks in which each

individual can interact only with its immediate neigh-

bours. In general, it has been shown that coevolving

the spatial structure can promote the emergence of co-

operation in many scenarios (Wang et al., 2014; Cao

et al., 2011), but the understanding of cooperative be-

haviour is still one of the central issues in evolutionary

game theory.

Szolnoki and Perc (2009) proposed a study of the

impact of coevolutionary rules on the spatial version

of three different games, i.e., the Prisoner’s Dilemma,

the Snow Drift and the Stag Hunt game. They intro-

duce the concept of a teaching activity, which quanti-

ﬁes the ability of each agent to enforce its strategy on

the opponent. It means that agents with higher teach-

ing activity are more likely to reproduce than those

with a low teaching activity. Differing from previ-

ous research (Zimmermann et al., 2004, 2001), they

also consider coevolution affecting either only the de-

fectors or only the cooperators. They discuss that, in

both cases and irrespective of the applied game, their

coevolutionary model is much more beneﬁcial to the

cooperators than that of the traditional model.

Huang et al. (2015) present a new model for the

coevolution of game strategy and link weight. They

consider a population of 100 × 100 agents arranged

on a regular lattice network which is evolved through

a Monte Carlo simulation. An agent’s interaction is

described by the classical Prisoner’s Dilemma with a

normalized payoff matrix. A new parameter, ∆/δ, is

deﬁned as the link weight amplitude and is calculated

as the ratio of ∆/δ. They found that some values of

∆/δ can provide the best environment for the evolu-

tion of cooperation. They also found that their coevo-

lutionary model can promote cooperation efﬁciently

even when the temptation of defection is high.

In addition to investigations of the classical Pris-

oner’s Dilemma on spatial environments, some exten-

sions of this game have also been explored as a means

to favour the emergence of cooperative behaviour.

For instance, the Optional Prisoner’s Dilemma game,

which introduces the concept of abstinence, has been

studied since Batali and Kitcher (1995). In their

work, they proposed the opt-out or “loner’s” strategy

in which agents could choose to abstain from play-

ing the game, as a third option, in order to avoid co-

The Optional Prisoner’s Dilemma in a Spatial Environment: Coevolving Game Strategy and Link Weights

operating with known defectors. There have been a

number of recent studies exploring this type of game

(Xia et al., 2015; Ghang and Nowak, 2015; Olejarz

et al., 2015; Jeong et al., 2014; Hauert et al., 2008).

Cardinot et al. (2016) discuss that, with the introduc-

tion of abstainers, it is possible to observe new phe-

nomena and a larger range of scenarios where coop-

erators can be robust to invasion by defectors and can

dominate.

However, the inclusion of optional games with co-

evolutionary rules has not been studied yet. There-

fore, our work aims to combine both of these trends

in evolutionary game theory in order to identify

favourable conﬁgurations for the emergence of coop-

eration in adverse scenarios, where, for example, the

temptation to defect is very high.

3 METHODOLOGY

The goal of the experiments outlined in this section

is to investigate the environmental settings when co-

evolution of both strategy and link weights of the

Optional Prisoner’s Dilemma on a weighted network

takes place.

Firstly, the Optional Prisoner’s Dilemma (PD)

game will be described; secondly, the spatial environ-

ment is described; thirdly, the coevolutionary rules for

both the strategy and link weights are described and

ﬁnally, the experimental set-up is outlined.

In the classical version of the Prisoner’s Dilemma,

two agents can choose either cooperation or defec-

tion. Hence, there are four payoffs associated with

each pairwise interaction between the two agents.

In consonance with common practice (Huang et al.,

2015; Nowak and May, 1992), payoffs are character-

ized by the reward for mutual cooperation (R = 1),

punishment for mutual defection (P = 0), sucker’s

payoff (S = 0) and temptation to defect (T = b, where

1 < b < 2). Note that this parametrization refers to the

weak version of the Prisoner’s Dilemma game, where

P can be equal to S without destroying the nature of

the dilemma. In this way, T > R > P ≥ S maintains

the dilemma.

The extended version of the PD game presented

in this paper includes the concept of abstinence, in

which agents can not only cooperate (C) or defect (D)

but can also choose to abstain (A) from a game inter-

action, obtaining the loner’s payoff (L = l) which is

awarded to both players if one or both abstain. As de-

ﬁned in other studies (Cardinot et al., 2016; Szab

o and

Hauert, 2002), abstainers receive a payoff greater than

P and less than R (i.e., P < L < R). Thus, considering

the normalized payoff matrix adopted, 0 < l < 1. The

payoff matrix and the associated values are illustrated

in Tables 1 and 2.

Table 1: The Optional Prisoner’s Dilemma game matrix.

C D A

Table 2: Payoff values.

Payoff Value

Temptation to defect (T) ]1,2[

Reward for mutual cooperation (R) 1

Punishment for mutual defection (P) 0

Sucker’s payoff (S) 0

Loner’s payoff (L) ]0,1[

In these experiments, the following parameters are

used: a 100×100 (N = 100

) regular lattice grid with

periodic boundary conditions is created and fully pop-

ulated with agents, which can play with their eight

immediate neighbours (Moore neighbourhood). We

adopt an unbiased environment in which initially each

agent is designated as a cooperator (C), defector (D)

or abstainer (A) with equal probability. Also, each

edge linking agents has the same weight w = 1, which

will adaptively change in accordance with the interac-

tion.

Monte Carlo methods are used to perform the Op-

tional Prisoner’s Dilemma game. In one Monte Carlo

(MC) step, each player is selected once on average.

This means that one MC step comprises N inner steps

where the following calculations and updates occur:

• Select an agent (x) at random from the population.

• Calculate the utility u

of each interaction of x

with its eight neighbours (each neighbour repre-

sented as agent y) as follows:

= w

, (1)

where w

is the edge weight between agents x

and y, and P

corresponds to the payoff obtained

by agent x on playing the game with agent y.

• Calculate U

the accumulated utility of x, that is:

∑

y∈Ω

, (2)

where Ω

denotes the set of neighbours of the

agent x.

• In order to update the link weights, w

between

agents, compare the values of u

and the average

ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications

accumulated utility (i.e.,

= U

/8) as follows:











+ ∆ if u

− ∆ if u

otherwise

, (3)

where ∆ is a constant such that 0 ≤ ∆/δ ≤ 1.

• In line with previous research (Huang et al., 2015;

Wang et al., 2014), w

is adjusted to be within the

range of 1 − δ to 1 + δ, where δ (0 < δ ≤ 1) de-

ﬁnes the weight heterogeneity. Note that when ∆

or δ are equal to 0, the link weight keeps constant

(w = 1), which results in the traditional scenario

where only the strategies evolve.

• In order to update the strategy of x, the accumu-

lated utility U

is recalculated (based on the new

link weights) and compared with the accumulated

utility of one randomly selected neighbour (U

If U

> U

, agent x will copy the strategy of agent

y with a probability proportional to the utility dif-

ference (Equation 4), otherwise, agent x will keep

its strategy for the next step.

p(s

= s

) =

−U

8(T − P)

, (4)

where T is the temptation to defect and P is the

punishment for mutual defection. This equation

has been considered previously by Huang et al.

(2015).

Simulations are run for 10

MC steps and the frac-

tion of cooperation is determined by calculating the

average of the ﬁnal 1000 MC steps. To alleviate the

effect of randomness in the approach, the ﬁnal results

are obtained by averaging 10 independent runs.

The following scenarios are investigated:

• Exploring the effect of the link update rules

by varying the values of ∆/δ. Speciﬁcally,

the value of the link weight amplitude ∆/δ

is varied for a range of ﬁxed values of the

loner’s payoff (l = [0.0, 1.0]), temptation to de-

fect (b = [1.0, 2.0]) and the weight heterogeneity

(δ = (0.0, 1.0]).

• Investigating the evolution of cooperation when

abstainers are present in the population.

• Considering snapshots of the evolution of the pop-

ulation over time at Monte Carlo steps of 0, 45,

1113, and 10

• Investigating the relationship between ∆/δ, b and

l. Speciﬁcally, the values of b, l and ∆/δ are var-

ied for a ﬁxed value of δ (δ = 0.8).

It is noteworthy that a wider range of values for

l, b and δ were considered in our simulations, but

for the sake of simplicity, we report only the values

of l = {0.0, 0.6}, b = {1.18, 1.34, 1.74, 1.90} and

δ = {0.2, 0.4, 0.8}, which are representative of the

outcomes at other values also.

4 RESULTS

In this section, we present some of the relevant ex-

perimental results of the simulations of the Optional

Prisoner’s Dilemma game on the weighted network.

4.1 Varying the Values of ∆/δ

Figure 1 shows the impact of the coevolutionary

model on the emergence of cooperation when the link

weight amplitude ∆/δ varies for a range of ﬁxed val-

ues of the loner’s payoff (l), temptation to defect (b)

and weight heterogeneity (δ). In this experiment, we

observe that when l = 0.0, the outcomes of the coevo-

lutionary model for the Optional Prisoner’s Dilemma

game are very similar to those in the classical Pris-

oner’s Dilemma game (Huang et al., 2015). This re-

sult can be explained by the normalized payoff matrix

adopted in this work (Table 1). Clearly, when l = 0.0,

there is no advantage in abstaining from playing the

game, thus agents choose the option to cooperate or

defect.

In cases where the temptation to defect is less than

or equal to 1.34 (b ≤ 1.34), it can be observed that

the level of cooperation does not seem to be affected

by the increment of the loner’s payoff, except when

the advantage in abstaining is very high, i.e., l ≥ 0.8.

However, these results highlight that the presence of

the abstainers may protect cooperators from invasion.

Moreover, the difference between the traditional case

(∆/δ = 0.0) for l = {0.0, 0.6} and all other values of

∆/δ is strong evidence that our coevolutionary model

is very advantageous to the promotion of coopera-

tive behaviour. Namely, when l = 0.6, in the tra-

ditional case with a static and unweighted network

(∆/δ = 0.0), the cooperators have no chance of sur-

viving; in this scenario, when the temptation to defect

b is low, abstainers always dominate, otherwise, when

b is high, defection is always the dominant strategy.

However, when the coevolutionary rules are used, co-

operators do much better, being also able to dominate

the whole population in many cases.

4.2 Presence of Abstainers

In addition to the fact that the levels of cooperation are

usually improved in the coevolutionary model, as the

The Optional Prisoner’s Dilemma in a Spatial Environment: Coevolving Game Strategy and Link Weights

Figure 1: Relationship between cooperation and link weight amplitude ∆/δ when the loner’s payoff (l) is equal to 0.0 (left)

and 0.6 (right).

value of the loner’s payoff increases we also observe

newer phenomena.

In the classical Prisoner’s Dilemma in this type of

environment, when the defector’s payoff is very high

(i.e., greater than 1.75) defectors spread quickly and

dominate the environment. However, Figure 1 also

shows that, for some values of l, it is possible to reach

high levels of cooperation even when the temptation

of defection b is almost at its peak.

Therefore, abstainers seem to help the popula-

tion to increase their fraction of cooperation in many

cases, but mainly in the case where the link weight

amplitude ∆/δ is higher than 0.7. This is usually a bad

scenario in the classical Prisoner’s Dilemma game.

4.3 Snapshots at Different Monte Carlo

Steps

In order to further explain the results witnessed in the

previous experiments, we investigate how the popu-

lation evolves over time. Figure 2 features the time

course of cooperation for three different values of

∆/δ = {0.0, 0.2, 1.0}, which are some of the criti-

cal points when b = 1.9, l = 0.6 and δ = 0.8. Based

on these results, in Figure 3 we show snapshots for the

Monte Carlo steps 0, 45, 1113 and 10

for the three

scenarios shown in Figure 2.

We see from Figure 2 that for the traditional case

(i.e., ∆/δ = 0.0), abstainers spread quickly and reach

a stable state in which single defectors are completely

isolated by abstainers. In this way, as the payoffs ob-

tained by a defector and an abstainer are the same,

neither will ever change their strategy. In fact, even

if a single cooperator survives up to this stage, for

the same aforementioned reason, its strategy will not

change either.

When ∆/δ = 0.2, it is possible to observe some

sort of equilibrium between the three strategies. They

reach a state of cyclic competition in which abstain-

ers invade defectors, defectors invade cooperators and

cooperators invade abstainers.

This behaviour, of balancing the three possible

outcomes, is very common in nature where species

with different reproductive strategies remain in equi-

librium in the environment. For instance, the same

scenario was observed as being responsible for pre-

serving biodiversity in the neighbourhoods of the Es-

cherichia coli, which is a bacteria commonly found in

the lower intestine of warm-blooded organisms. Ac-

cording to Fisher (2008), studies were performed with

three natural populations mixed together, in which

one population produces a natural antibiotic but is im-

mune to its effects; a second population is sensitive to

the antibiotic but can grow faster than the third popu-

lation; and the third population is resistant to the an-

tibiotic.

Because of this balance, they observed that each

population ends up establishing its own territory in the

environment, as the ﬁrst population could kill off any

other bacteria sensitive to the antibiotic, the second

population could use their faster growth rate to dis-

place the bacteria which are resistant to the antibiotic,

and the third population could use their immunity to

displace the ﬁrst population.

Another interesting behaviour is noticed for

∆/δ = 1.0. In this scenario, defectors are dominated

by abstainers, allowing a few clusters of cooperators

to survive. As a result of the absence of defectors,

cooperators invade abstainers and dominate the envi-

ronment.

4.4 Investigating the Relationship

between ∆/δ, b and l

To investigate the outcomes in other scenarios, we ex-

plore a wider range of settings by varying the values

of the temptation to defect (b), the loner’s payoff (l)

ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications

Figure 2: Progress of the fraction of cooperation ρ

during a Monte Carlo simulation for b = 1.9, l = 0.6 and δ = 0.8.

Figure 3: Snapshots of the distribution of the strategy in the Monte Carlo steps 0, 45, 1113 and 10

(from left to right) for

∆/δ equal to 0.0, 0.2 and 1.0 (from top to bottom). In this Figure, cooperators, defectors and abstainers are represented by

the colours blue, red and green respectively. All results are obtained for b = 1.9, l = 0.6 and δ = 0.8.

and the link weight amplitude (∆/δ) for a ﬁxed value

of weight heterogeneity (δ = 0.8).

As shown in Figure 4, cooperation is the dominant

strategy in the majority of cases. Note that in the tra-

The Optional Prisoner’s Dilemma in a Spatial Environment: Coevolving Game Strategy and Link Weights

Figure 4: Ternary diagrams of different values of b, l and ∆/δ for δ = 0.8.

ditional case, with an unweighted and static network,

i.e., ∆/δ = 0.0, abstainers dominate in all scenarios il-

lustrated in this ternary diagram. In addition, it is also

possible to observe that certain combinations of l, b

and ∆/δ guarantee higher levels of cooperation. In

these scenarios, cooperators are protected by abstain-

ers against exploitation from defectors. In most cases,

for populations with the loner’s payoff, l = [0.4, 0.8],

cooperation is promoted the most.

Although the combinations shown in Figure 4 for

higher values of b (b > 1.8) are just a small subset

of an inﬁnite number of possible values, it is clearly

shown that a reasonable fraction of cooperators can

survive even in an extremely adverse situation where

the advantage of defecting is very high. Indeed, our

results show that some combinations of high values

of l and ∆/δ such as for ∆/δ = 1.0 and l = 0.6, can

further improve the levels of cooperation, allowing for

the full dominance of cooperation.

In summary, we see that the use of a coevolution-

ary model in the Optional Prisoner’s Dilemma game

allows for the emergence of cooperation.

5 CONCLUSIONS AND FUTURE

WORK

In this paper, we studied the impact of abstinence in

the Prisoner’s Dilemma game using a coevolutionary

spatial model in which both game strategies and link

weights between agents evolve over time. We consid-

ered a population of agents who were initially organ-

ised on a lattice grid where agents can only play with

their eight immediate neighbours. Using a Monte

Carlo simulation approach, a number of experiments

were performed to observe the emergence of coopera-

tion, defection and abstinence in this environment. At

each Monte Carlo time step, an agent’s strategy (co-

operate, defect, abstain) and the link weight between

agents can be updated.

The payoff received by an agent after playing with

another agent is a product of the strategy played and

the weight of the link between agents. We explored

the effect of the link update rules by varying the val-

ues of the link weight amplitude ∆/δ, the loner’s pay-

off l, and the temptation to defect b. The aims were

to understand the relationship between these parame-

ters and also investigate the evolution of cooperation

when abstainers are present in the population.

Results showed that, in adverse scenarios where

b is very high (i.e. b ≥ 1.9), some combinations of

high values of l and ∆/δ, such as for ∆/δ = 1.0 and

l = 0.6, can further improve the levels of cooperation,

even resulting in the full dominance of cooperation.

When ∆/δ = 0.2, b = 1.9 and δ = 0.8, it was pos-

sible to observe a balance among the three strategies,

indicating that, for some parameter settings, the Op-

tional Prisoner’s Dilemma game is intransitive. In

other words, such scenarios produce a loop of dom-

inance in which abstainer agents beat defector agents,

defector agents beat cooperator agents and cooperator

agents beat abstainer agents.

In summary, the difference between the outcomes

of ∆/δ = 0.0 (i.e., a static environment with un-

weighted links) and ∆/δ = (0.0, 1.0] clearly showed

that the coevolutionary model is very advantageous

to the promotion of cooperative behaviour. In

most cases, with populations with the loner’s pay-

off, l = [0.4, 0.8], cooperation is promoted the most.

Moreover, results also showed that cooperators are

protected by abstainers against exploitation from de-

fectors.

Although recent research has considered coevolv-

ing game strategy and link weights (Section 2), to our

knowledge the investigation of such a coevolutionary

model with optional games has not been studied to

date. We conclude that the combination of both of

these trends in evolutionary game theory may shed

additional light on gaining an in-depth understanding

of the emergence of cooperative behaviour in real-

world scenarios.

Future work will consider the exploration of dif-

ferent topologies and the inﬂuence of a wider range

of scenarios, where, for example, agents could rewire

their links, which, in turn, adds another level of com-

ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications

plexity to the model. Future work will also involve

applying our studies and results to model realistic sce-

narios, such as social networks and real biological

networks.

ACKNOWLEDGEMENTS

This work was supported by the National Council for

Scientiﬁc and Technological Development (CNPq-

Brazil).

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