Evolutionary dynamics in repeated optional games Fang Chen1 Lei Zhou2and Long Wang1 1Center for Systems and Control College of Engineering Peking University Beijing 100871 China

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Evolutionary dynamics in repeated optional games

Fang Chen1†, Lei Zhou2†and Long Wang1∗

1Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China

2School of Automation, Beijing Institute of Technology, Beijing 100081, China

†These authors contributed equally to this work

∗Corresponding author. E-mail: longwang@pku.edu.cn

Abstract

Direct reciprocity facilitates the evolution of cooperation when individuals interact repeatedly.

Most previous studies on direct reciprocity implicitly assume compulsory interactions. Yet, inter-

actions are often voluntary in human societies. Here, we consider repeated optional games, where

individuals can freely opt out of each interaction and rejoin later. We ﬁnd that voluntary par-

ticipation greatly promotes cooperation in repeated interactions, even in harsh situations where

repeated compulsory games and one-shot optional games yield low cooperation rates. Moreover,

we theoretically characterize all Nash equilibria that support cooperation among reactive strate-

gies, and identify three novel classes of strategies that are error-robust, readily become equilibria,

and dominate in the evolutionary dynamics. The success of these strategies hinges on the eﬀect

of opt-out: it not only avoids trapping in mutual defection but also poses additional threats to

intentional defectors. Our work highlights that voluntary participation is a simple and eﬀective

mechanism to enhance cooperation in repeated interactions.

Introduction

Humans routinely face social dilemmas where mutual cooperation is most beneﬁcial for the group

yet each group member proﬁts more by defecting [1]. Classical metaphors to describe such social

dilemmas include the prisoner’s dilemma and the public goods game (PGG) [2, 3, 4]. Without

additional mechanisms, natural selection generally favors defection in such games, which contrasts

with the reality that cooperation is ubiquitous [5, 6]. This raises a fundamental question about

how cooperation evolves [7, 8]. Based on repeated interactions, one mechanism that is shown to

support the evolution of cooperation is direct reciprocity [9, 7], under which individuals cooperate

conditionally on past interactions. Mathematically, the logic of direct reciprocity can be conveniently

described by the framework of repeated games. Indeed, employing this framework, previous work

has addressed important questions such as which strategies support cooperation and under what

conditions, cooperation evolves [10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

arXiv:2210.08969v2 [physics.soc-ph] 16 Nov 2023

A tacit assumption in most previous studies on direct reciprocity is that interactions are compul-

sory, namely, each individual should participate in every interaction. In reality, participation is often

voluntary and individuals have the freedom to opt out [20, 21, 22, 23]. In this case, the underlying

strategic interactions are better captured by repeated optional games, where individuals are allowed to

abstain from any interaction (and also to resume participation) (see Fig. 1). When individuals choose

to opt out, they become self-suﬃcient and obtain a payoﬀ that is independent of others. This payoﬀ

is often set to be greater than the one received under the social trap of mutual defection and less than

the social optimum with everyone cooperating [23, 24, 25], encouraging individuals to opt out when

mutual defection occurs and to re-establish cooperation if individuals abstain. In repeated optional

games, opt-out can serve as an additional response against co-players’ defection, which guarantees a

safe income and avoids the risk of mutual retaliation. Based on these, opting out conditionally on

past behaviors may become new leverage to force cooperation in repeated optional games.

Nonetheless, existing studies on repeated optional games fail to provide a comprehensive under-

standing of the role that opt-out plays in the evolution of cooperation due to (i) a presupposition of

a small and incomplete set of available strategies [22] and (ii) no focus on cooperation [26, 27]. So

far, it is yet to be known which strategies facilitate the evolution of cooperation in repeated optional

games if all possible strategies of a given complexity are considered and under what conditions, these

strategies dominate. More importantly, it still remains unclear how individuals could strategically

opt out to promote cooperation. Although an interesting ﬁnding in one-shot (non-repeated) optional

PGGs shows that unconditional opt-out (i.e., always opt out) can rescue cooperation if the incentive

for cooperation is high [23, 28, 29], unconditional strategies are easily invaded and cooperation in

one-shot optional games is not stable.

Here, we systematically investigate the eﬀect of voluntary participation on the evolution of co-

operation in repeated games. For a comprehensive analysis, we conduct an exhaustive search for

optimal strategies in the space of reactive strategies. Through evolutionary simulations, we show that

voluntary participation leads to almost full cooperation even in situations where repeated compulsory

games and one-shot optional games yield low propensities for cooperation. Resorting to equilibrium

analysis, we mathematically characterize all Nash equilibria that support cooperation, and identify

three novel classes of strategies that are robust to implementation errors, readily become equilibria,

and dominate in the evolutionary dynamics. In the meanwhile, we ﬁnd that these strategies and their

behaviorally close variants account for the evolutionary advantage under voluntary participation and

are thus key to the promotion of cooperation in repeated optional games. For the success of these

strategies, their eﬀective leverage of opting out against defection is crucial: it oﬀers a safe income

that cannot be exploited, provides a way out of mutual defection, and poses additional threats to

intentional defectors. In addition, when considering the eﬀect of opt-out payoﬀ on cooperation, our

results demonstrate that a small incentive for opt-out is enough to achieve almost full cooperation.

Besides, all our ﬁndings are veriﬁed to be robust to changes in model parameters and to other model

extensions (e.g., failures of opting out). Our work thus highlights that voluntary participation is a

simple and eﬀective mechanism to enhance cooperation in repeated interactions.

a b

Repeated optional PGGs

t t+1 t+kRound

Opting out

(O)

Rejoining

(C / D)

... ...

Defection

Cooperation

Compulsory PGGs

Public goods

Available

actions DefectionCooperation Opt-out

Optional PGGs

Public goods

Fig. 1. In repeated optional public goods games (PGGs), individuals can additionally choose to

opt out of the interaction and rejoin later. a, In a compulsory PGG, each individual has to decide either

to cooperate (marked as green) by contributing an amount, c, to the public goods or to defect (marked as

yellow) by contributing nothing. The total contributions are then multiplied by a factor, r, and equally divided

among all participants, irrespective of whether they cooperate or defect. b, In an optional PGG, individuals

have the additional option to opt out (marked as purple) and gain a payoﬀ σ(0 < σ < (r−1)c) that does not

depend on others’ actions. If there are xindividuals cooperating, ydefecting, and the number of participants

x+y > 1, an individual who cooperates gets xrc/(x+y)−cand an individual who defects gets xrc/(x+y).

If only one individual participates in the game (i.e., x+y= 1), the interaction is canceled and all individuals

get σ.c, In repeated optional PGGs, individuals interact for many rounds of optional PGGs. In each round,

individuals decide to cooperate, defect or opt out depending on the outcome of the previous round. Compared

with repeated compulsory PGGs where individuals are required to participate in every interaction, repeated

optional PGGs allow individuals to opt out of any interaction and to rejoin later.

Results

Repeated optional games. In the following, we introduce the framework of repeated optional

games. Here, we focus on repeated optional PGGs (see illustrations in Fig. 1 and see repeated

optional prisoner’s dilemma games in Section 4 of the Supplementary Information). In such games,

there are n≥2 individuals and they repeatedly play many rounds of optional PGGs. In every round,

each individual can choose one of the three actions, to participate in the game and cooperate (C)

by contributing an amount c > 0 to the public goods, to participate in the game and defect (D) by

contributing nothing, and to opt out (O) and obtain a ﬁxed payoﬀ σ. The total contributions in

the public goods are then multiplied by a multiplication factor r(1 < r < n) and evenly distributed

to all the participants, irrespective of whether they cooperate or defect. If there are xindividuals

cooperating, ydefecting, and at least two individuals participating in the game (i.e., 2 ≤x+y≤n),

the payoﬀ for an individual who cooperates, defects, and opts out is PC(x, y) = xrc/(x+y)−c,

PD(x, y) = xrc/(x+y), and PO(x, y) = σ, respectively. If less than two individuals choose to

participate in the game (i.e., x+y < 2), the interaction is canceled and everyone obtains the payoﬀ σ.

Here, we assume that 0 < σ < (n−1)c, meaning that full cooperation is better oﬀ than full opt-out,

and full opt-out is better oﬀ than full defection [22, 23, 30].

We consider repeated optional PGGs that last for inﬁnitely many rounds in the limit of no dis-

counting (see discounted games in the Supplementary Information). In such games, individuals may

take the whole game history into account to make a decision, and the resulting strategy can be arbi-

trarily complex. To make evolutionary analysis feasible, we focus on reactive strategies where current

actions depend on the number of each action in the previous round [31]. Let (x, y) denote the game

state in the previous round, where xand yare respectively the numbers of individuals who cooperate

and defect. Let G={(x, y)|0≤x, y ≤nand 0 ≤x+y≤n}denote the set of all possible game

states. A reactive strategy can be represented as

p= (pC

x,y, pD

x,y)(x,y)∈G,(1)

where pA

x,y ∈[0,1] is the probability to implement action A(A∈ {C, D}) in the current round and

x,y +pD

x,y ≤1 for all (x, y)∈ G. The probability to opt out is thus pO

x,y = 1 −pC

x,y −pD

x,y. A strategy

is pure if all entries pA

x,y belong to the set {0,1}; otherwise, it is stochastic. We also consider the

eﬀect of trembling hands (i.e., implementation errors), where individuals may mistakenly implement

another random (not intended) action with a small probability ε/2 where ε∈[0,1]. For instance, if a

pure strategy pprescribes cooperation after full cooperation, individuals adopting this strategy may

mistakenly defect or opt out of the game with probability ε/2, and correctly cooperate with the rest

probability 1 −ε. Therefore, when errors are present (ε > 0), the eﬀective strategy for a pure strategy

becomes stochastic and each entry pA

x,y ∈ {ε/2,1−ε}.

When individuals adopt strategies p1,p2,...,pnto play repeated optional PGGs, the game dy-

namics can be modeled as a Markov chain. The state space of the Markov chain is the set of all action

proﬁles. When the eﬀect of trembling hands is considered (ε > 0), the Markov chain is ergodic and

there exists a stationary distribution. In repeated optional PGGs, the payoﬀ for each strategy is the

average gain in this stationary distribution (see Supplementary Information for details).

Evolutionary dynamics. On a longer time scale, we assume that individuals change their strate-

gies. Here, we consider the pairwise comparison process [32, 33, 34, 35, 36, 37], where individuals

imitate successful peers, in a well-mixed population of Nindividuals. At each time step of such

a process, a group of nindividuals is randomly selected from the population. They play repeated

optional PGGs and each obtains the expected payoﬀ π. After that, a random individual lis selected

to update its strategy. It either adopts a random strategy with probability µ(random exploration

or mutation) or implements imitation with the rest probability 1 −µ. If individual limitates, it

randomly chooses a role model k(k̸=l), and adopts its strategy with a probability that depends

on the payoﬀ diﬀerence between individual kand l, i.e., πk−πl. The larger the payoﬀ diﬀerence

is, the more likely kis imitated (see Supplementary Information for details). When the mutation is

present (µ > 0), the resulting evolutionary dynamics are ergodic and it is possible to transit between

any possible strategy conﬁgurations of the population. In this work, we mainly focus on the case of

rare mutations (µ→0) [38], where the evolutionary dynamics spend most of the time in homogenous

populations with everyone adopting the same strategy.

Evolutionary advantage under voluntary participation. To explore the evolution of coopera-

tion in repeated optional PGGs, we run simulations and analyze a “melting pot” of reactive strategies

(in total, 3(n+1)(n+2)/2strategies). We ﬁnd that voluntary participation greatly enhances the coop-

eration rates in repeated optional PGGs (see the blue line in Fig. 2a). In contrast, under the same

conditions, repeated compulsory PGGs (see the red line in Fig. 2a) and one-shot optional PGGs

(see the orange line in Fig. 2a) only yield low propensities for cooperation. This indicates that the

combination of voluntary participation and conditional responses is conducive to cooperation. Here,

opt-out acts as a catalyst for the evolution of cooperation: it not only provides a natural way for

individuals to escape from the social trap of mutual defection but also serves as a stepping stone to

boost cooperation (see Fig. 2b).

In addition, to understand how individuals react in each game state (x, y)∈ G, we calculate the

average tendency for individuals to cooperate, defect, and opt out, i.e., the average strategy (see

Fig. 2c). Our results show that the average strategy exhibits clear and interesting characteristics: (i)

it supports (persistent) cooperation by prescribing cooperation after full cooperation and avoiding

persistent defection (i.e., not to defect after full defection) and opt-out (i.e., not to opt out after full

opt-out); (ii) it actively leverages opt-out to punish defection, such as when someone in a once fully

cooperative group starts to defect (i.e., game state (2,1)) or when full defection occurs.

Equilibrium analysis for repeated optional PGGs. Based on the characteristics reﬂected by

the average strategy, we turn to strategies that support cooperation and try to identify key strategies

therein that promote the evolution of cooperation in repeated optional PGGs. Due to the presence of

implementation errors, such strategies are expected to be error-robust, which means mutual coopera-

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EvolutionarydynamicsinrepeatedoptionalgamesFangChen1†,LeiZhou2†andLongWang1∗1CenterforSystemsandControl,CollegeofEngineering,PekingUniversity,Beijing100871,China2SchoolofAutomation,BeijingInstituteofTechnology,Beijing100081,China†Theseauthorscontributedequallytothiswork∗Correspondingauthor.E-mail:lo...

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Evolutionary dynamics in repeated optional games Fang Chen1 Lei Zhou2and Long Wang1 1Center for Systems and Control College of Engineering Peking University Beijing 100871 China

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