DP2023-11

The Concomitance of Prosociality
and Social Networking Agency*
Danyang JIA
Ivan ROMIC
Lei SHIQi SU
Chen LIU
Jinzhuo LIU
Petter HOLME
Xuelong LI
Z h e n WA N G
Revised June 24, 2024

* The Discussion Papers are a series of research papers in their draft form, circulated to encourage
discussion and comment. Citation and use of such a paper should take account of its provisional character.
In some cases, a written consent of the author may be required.

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The concomitance of prosociality and social networking agency
Danyang Jia1,2† , Ivan Romić2,3† , Lei Shi4 , Qi Su5,6,7 , Chen Liu8 , Jinzhuo
Liu9 , Petter Holme3,10 , Xuelong Li and Zhen Wang1,2*
1 School of Cybersecurity, Northwestern Polytechnical University, Xi’an, China.
2 School of Artificial Intelligence, OPtics and ElectroNics (iOPEN), Northwestern

Polytechnical University, Xi’an, China.
3 Center for Computational Social Science, Kobe University, Kobe, Japan.
4 School of Statistics and Mathematics, Yunnan University of Finance and Economics,

Kunming, China.
5 Department of Automation, Shanghai Jiao Tong University, Shanghai, China.
6 Key Laboratory of System Control and Information Processing, Ministry of Education ofChina, Shanghai, China.
7 Shanghai Engineering Research Center of Intelligent Control and Management,

Shanghai, China.
8 School of Ecology and Environmental Sciences, Northwestern Polytechnical University,

Xi’an, China.
9 School of Software, Yunnan University, Kunming, China.
10 Department of Computer Science, Aalto University, Espoo, Finland.
11 Institute of Artificial Intelligence (TeleAI), China Telecom Corp Ltd, Beijing, China.

*Corresponding author(s). E-mail(s): w-zhen@nwpu.edu.cn;† These authors contributed equally to this work.
Abstract
The awareness of individuals regarding their social network surroundings and their capacity touse social connections to their advantage are well-established human characteristics. Economic
games, incorporated with network science, are frequently used to examine social behaviour. Traditionally, such game models and experiments artificially limit players’ abilities to take varied
actions toward distinct social neighbours (i.e., to operate their social networks). We designed
an experimental paradigm that alters the degree of social network agency to interact withindividual neighbours, and applied it to the prisoner’s dilemma (N = 735), trust game (N= 735), and ultimatum game (N = 735) to investigate cooperation, trust, and fairness. The
freedom to interact led to more prosocial behaviour across all three economic games andresulted in higher wealth and lower inequality compared to controls without such freedom.
These findings suggest that human behaviour is more prosocial than current science indicates.
Keywords: behavioural science, networks, cooperation, prosociality

1

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Social networks coordinate society. They embody
an evolved cognizance of social surroundings
and the capacity to use these networks advantageously [1, 2]. Intrinsically linked to social
networks is another fundamental societal feature—cooperation. The maintenance of social networks is a cooperative act, and conversely, withoutsocial networks, cooperation would not exist at thescale and complexity observed even in simple societies [3]. Consequently, any explanation of thesefoundational cultural elements must encompass
both. However, the majority of studies investigating the origins of cooperation overlook thecapacity of humans to actively manage their social
networks—their social networking agency [4–6].
Similarly, studies examining how individuals use
their social networks seldom explore the evolutionary roots predating the demographic transition [3]. The burgeoning field of studying cooperation in populations structured by social networks
can be categorized into four areas. First, a lineof research originating from theoretical population
biology [6], primarily aimed at reconciling the costof prosociality with its emergence in prehistoricsocieties, occasionally uses social network structure as a contextual backdrop for the decisions ofrational agents [7–9]. Second, there is a sustainedinterest in how network structure and evolutionary game dynamics can collectively generate complex patterns [10]. Third, anthropologists have alongstanding tradition of documenting networksof gift-giving, food-sharing, and other elementsof cooperative community building [11]. A fourtharea combines networks and cooperative decisions
in studies of optimal network formation in economic game theory [12]. Despite the diversity ofthese efforts, a fundamental question persists: How
does the ability to actively use one’s social network
influence cooperation?
Using one’s social network necessitates the
freedom to behave differently towards various individuals in one’s social environment. Indeed, in thegame-theoretic study of cooperation, any restriction on this freedom would stem from social
conventions that form part of the dilemma underinvestigation. Thus, in our study, we bridge thegap in the literature by modulating between situations that retain this basic freedom and theartificially constrained setup of players interacting
uniformly with their entire neighbourhood, which
has become the conventional approach.

We present the results of 15 experimental
setups and reveal that the ability to act differently towards individual neighbours universallyenhances cooperation, trust, and fairness, leading
to higher payoffs and reduced inequality. Additionally, we identified three distinct behavioural
phenotypes for players in each game and demonstrated how the distribution of these phenotypes
shifts as the ability to exploit their social network
increases in the population. Finally, we developed numerical models for each type of game toprovide a deeper understanding of our observed
experimental results.

Laboratory experiments
We employed repeated spatial versions of three
economic games—the prisoner’s dilemma, thetrust game, and the ultimatum game—to examine cooperation, trust, and fairness. These games
were chosen due to their extensive use in prosocial
behaviour research [13, 14]. The classical prisoner’s dilemma involves two players who simultaneously decide between cooperation and defection.
If one player defects while the other cooperates,
the defector gains more than if both had cooperated. Consequently, defection becomes a rational,
self-regarding solution, leading to a Nash equilibrium where both players defect and earn less thanif they had cooperated [15].
Unlike the prisoner’s dilemma, the standard
trust and ultimatum games proceed sequentiallyover two stages [14]. In the trust game, a trustor
can share an endowment with a trustee. The
entrusted amount is typically tripled, and thetrustee then decides how much to return. Selfinterest dictates that the trustor should send nothing and the trustee should return nothing [16].
The ultimatum game involves a proposer
dividing an endowment and a responder accepting
or rejecting the division. If accepted, the division proceeds; if rejected, neither player receives
anything. A self-regarding proposer offers theminimum, and a self-regarding responder accepts
it [17].
Contrary to theoretical predictions, experiments reveal more prosocial human behaviour.
For instance, in one-shot games, players cooperate
approximately half of the time in the prisoner’s
dilemma, trustors send about 50% of their endowment on average in the trust game, and trustees

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return slightly less. In the ultimatum game, proposers typically offer between 40% and 50% of theendowment, which responders accept while regularly rejecting smaller offers [18, 19]. In repeatedgames, players generally exhibit higher levels ofcooperation, trust, and fairness compared to oneshot games, as repeated interactions allow forstrategy development based on past experiencesand expectations of future encounters [20, 21].
The discrepancy between theoretical predictions and experimental outcomes in games on
networks unfolds in a slightly different manner.
Here, theory suggests that homogeneous networks
should promote prosocial behaviour relative towell-mixed populations and heterogeneous networks relative to homogeneous ones [22, 23]. However, numerous laboratory experiments have failed
to find a significant difference between cooperationlevels in well-mixed populations, homogeneous
networks, and heterogeneous networks [24], except
under very specific conditions [25].
To investigate whether the ability to act differently towards individual neighbours enhances
prosocial behaviour, we designed games on a

Fig. 1 Schematics of our experiment
for the three economic games. (A) In
the prisoner’s dilemma, players choose
between cooperation and defection, withpayoffs dependent on their own actions
and those of their neighbours. The trust
and ultimatum games are two-stage
games. In the trust game’s first stage, thetrustor decides the number of tokens tosend to the trustee, who then receives
triple the amount and determines how
many to return to the trustor. The firststage of the ultimatum game involves the
proposer deciding on the distribution of agiven number of tokens. In the second
stage, the responder opts to accept or
reject the proposal. If accepted, bothplayers receive the proposed amounts; if
rejected, both receive nothing. (B)Differences between constrained and freeplayers in the prisoner’s dilemma, and thefirst stage of the trust and ultimatum
games. Constrained players make decisions
at the neighbourhood level, while freeplayers make individualized decisions foreach neighbour. For the prisoner’s
dilemma, this manifests as the freedom tochoose actions for each neighbour
separately, and in the trust and ultimatum
games as the freedom to entrust or propose
varied amounts to different neighbours.

lattice network with two player types, constrained and free. We tailored the basic setup foreach game’s specifics (Fig. 1). In the prisoner’s
dilemma, constrained players were restricted tocooperating or defecting with all neighbours
simultaneously, as in standard models of games on
networks. Free players, conversely, could choose
actions for individual neighbours (Fig. 1A).
The trust game comprised two stages. First,
all players acted as trustors with neighbours, thenas trustees. Constrained trustors could entrust thesame token amount to all neighbours, while freetrustors could vary their contributions (Fig. 1B).
In the second stage, there was no differentiation
between constrained and free trustees, and thereturned amounts were distributed equally amongneighbours.
Similarly, the ultimatum game proceeded in
two stages. In the proposer stage, constrained
proposers could offer an identical token amountto all neighbours, whereas free proposers had
the flexibility to propose individualized amounts(Fig. 1B). In the responder stage, both constrained and free players had the option to accept

Springer Nature 2021 LATEX templateor reject proposals from each neighbour independently.
We employed a control group and four treatment groups for each game. The control group wascomposed solely of constrained players, representing a conventional spatial game theory setup. In
contrast, the four treatment groups saw a progressive reduction in these constraints by incrementally increasing the fraction of free players within
the total population to low (e.g. 25%), medium(e.g. 50%), high (e.g. 75%), and finally, a full (e.g.
100%) free-player population. For comprehensive
details, refer to the Methods section.

Cooperation, trust, and fairness
Allowing players to interact differently with theirindividual neighbours enhances prosociality, leading universally to increased cooperation, trust,and fairness (Fig. 2). This is evident in the prisoner’s dilemma as a steady rise in the frequency
of cooperation with an increasing fraction of freeplayers in the population. Compared to the control group, cooperation increases progressively inmixed populations with varying densities of freeplayers, peaking in the population of free players(Fig. 2A). The trust game results display a comparable pattern, with the lowest entrusted amount in
the population of trustors. This is succeeded by asteady increase in mixed populations, culminating
in a population composed entirely of free trustors(Fig. 2B). As the trust game is a two-stage game,we also noted a similar increasing pattern in theamount returned by trustees in the second stage(Extended Data Fig. 1A).
In the ultimatum game, the proposed amounts
in the population of constrained proposers arelower than those in the population of free proposers (Fig. 2C). However, mixed populations
exhibit less sensitivity to the increasing fractionof free players. The proposed amounts are slightly
lower in treatments with a low and medium fraction of free players compared to the populationof constrained proposers, but slightly higher inthe population with a high fraction of free players. A similar pattern is observed for the acceptedamounts in the second stage (Extended Data
Fig. 1B).
We then examined how players use their social
networking agency over time (Extended Data

Fig. 2). In populations of free players, the firstround cooperation, entrusted amount, and proposed amount increase considerably relative topopulations of constrained players. In mixed populations, these variables generally increase witha higher fraction of free players, but not consistently. Across the rounds, the trend is stable and often slightly negative, rarely significant(Extended Data Table 1 and 2).
These findings indicate that free players
express their prosocial tendencies from the firstround, rather than relying solely on learning andoptimisation through repeated interactions.
An examination of the impact of higher prosociality on wealth reveals that in all three economic
games, wealth generally increases, more so among
free players compared to their constrained counterparts (Extended Data Fig. 3A-C). Inequality
is typically lower in populations of constrained
and free players, with the latter having lower levels. However, in mixed populations, we observe
higher inequality at lower fractions of free players,which then decreases with an increased proportion of free players. This pattern is consistentin both the prisoner’s dilemma and trust game
(Extended Data Fig. 3D, E). In the ultimatum
game, inequality rises with more free players but
falls entirely in the free population (Extended
Data Fig. 3F). A closer look at the different stagesof the games reveals additional differences between
the trust and ultimatum games. In the former,
wealth increases for both trustors and trustees as
the number of free players grows (Extended Data
Fig. 3G), while in the latter, proposers’ wealth initially rises and then falls, but responders’ wealthconsistently increases (Extended Data Fig. 3H).
Inequality trends are also distinct. In the trust
game, it increases for trustors with more freeplayers and varies for trustees (Extended Data
Fig. 3I), whereas, in the ultimatum game, it
steadily rises for both proposers and responders(Extended Data Fig. 3J).
These findings imply that the low inequality
observed in control groups is a result of the restrictive environment, where players cannot optimize
their strategies and, thus, often resort to antisocial behaviour. Mixed populations experience anincrease in inequality as free players utilize theirsocial networking agency. However, in entirely free
populations, the playing field is level, leading toSpringer Nature 2021 LATEX template
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1Fig. 2 Social networking agency universally enhances cooperation, trust, and fairness. Displayed are theresults across the control group and four treatment groups in (A) the prisoner’s dilemma, (B) the trust game, and (C) theultimatum game. Each game comprized a control group with a population of constrained players and three treatmentswith mixed populations, including low, medium, and high fractions of free players, as well as a treatment with apopulation entirely composed of free players. An increase in the fraction of free players in the population leads to elevatedlevels of cooperation, trust, and fairness, as indicated by the frequency of cooperation, the entrusted amount, and the
proposed amount, respectively. The second stage of the trust and ultimatum games exhibits a similar pattern (Extended
Data Fig. 1). Statistical significance was evaluated using the Bonferroni correction.

both the highest wealth and the lowest levels of
inequality.

Behavioural phenotypes
Next, we derived behavioural phenotypes for eachgame using explanatory data mining (Fig. 3).
Experimental studies often classify behaviour intoa few stable clusters [26, 27], a pattern also evident
in our data. In the prisoner’s dilemma, players are
categorized as prosocial, conditional, or antisocial
based on their overall cooperation frequency andthe probability of cooperating in response to theprevious round’s circumstances (Fig. 3A). A population of constrained players primarily consistsof antisocial players who defect in most interactions. As the fraction of free players increases,
there is a significant rise in the number of conditional cooperators, who typically employ the“tit-for-tat” strategy—defecting in response todefection and cooperating in response to cooperation—and prosocial players, who generally cooperate. In the free-player population, conditional
cooperators are the majority, followed by prosocial players. This suggests that the high prevalenceof antisocial players in the control group likely

results from the constrained environment rather
than inherent antisocial tendencies.
In the trust game, we identified behavioural
phenotypes—prosocial, neutral, and antisocial—
distinguished by the average overall trust theyextend and the average amounts in response to
the previous round’s circumstances. Similar to theprisoner’s dilemma, the constrained trustor population primarily consists of antisocial players.
However, increasing the proportion of free players results in a gradual rise in the number ofprosocial trustors, eventually making them the
majority, while the numbers of neutral and antisocial trustors decrease (Fig. 3B). Extracting the
behavioural types for trustees revealed an increasing prevalence of the neutral phenotype as the
proportion of free players rises, with the numberof prosocial trustees also increasing and antisocial trustees decreasing (Extended Data Fig. 4).
We also tested whether players’ behaviour is consistent across different stages of the game—for
instance, whether a prosocial trustor is also a
prosocial trustee. Our findings confirm this, as
prosocial trustors return the highest amounts
when acting as trustees, followed by neutral andthen antisocial trustors (Extended Data Fig. 5).

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The defined clustering factors could notdetect conditional behaviour in the trust game,as free trustors may respond to lower trustee
returns by redistributing their endowment among
other neighbours, thus maintaining their average
entrusted amounts. To confirm the existence ofconditional behaviour, we examined how trustors
react to trustee returns and found that trustors
tend to increase their entrusted amounts in
response to receiving larger returns from trustees(Extended Data Fig. 6).
Behavioural phenotypes in the ultimatum
game, characterized by the average proposed
amounts overall and in response to the previous
round’s circumstances, reflect those observed in
the trust game (Fig. 3C). However, the distinction
between antisocial and neutral players is less pronounced here than in the trust game. This can be
attributed to the second-stage players in the ultimatum game controlling the rejection punishment
mechanism, thereby maintaining minimal offers at
relatively higher levels. This difference is also evident in the distribution of behavioural phenotypes
across the control and treatment groups. In the

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Fig. 3 Prosocial behaviour
emerges more readily inincreasingly free populations. The
behavioural phenotypes displayed arederived from explanatory data mining
based on six empirical behaviouralvariables. In all three economic games,
Clustering Factor A represents theoverall result (cooperation frequency,
entrusted amount, and proposed
amount, respectively), while clusteringfactors B-F depend on the circumstancesof the previous round (for more details,
see Methods). A) In the prisoner’s
dilemma, we identify prosocial,conditional, and antisocial behavioural
phenotypes. The fraction of prosocialand conditional players increases withthe fraction of free players. B) and C)
In the trust and ultimatum games, aneutral phenotype emerges in place ofthe conditional phenotype. In the trust
game, a higher fraction of free players
results in prosocial trustors becoming
the majority. In the ultimatum game,prosocial proposers are slightly more
represented than the other two
phenotypes, but only in the final
treatment. For additional results see
Extended Data Fig. 4 to 7.

Antisocial

0.75
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constrained population, neutral proposers are predominant, followed by antisocial proposers. Conversely, prosocial proposers are most prevalent in
the free population, although the difference fromthe other two phenotypes is not significant. The
increase in prosocial proposers primarily occurs
at the expense of neutral proposers, while thenumber of antisocial proposers remains relatively
stable between constrained and free populations.
The behavioural types of responders mirror thoseof proposers (Extended Data Fig. 4). Antisocial
responders dominate in the constrained population, while prosocial responders are slightly moreprevalent in the free population. Similar to thetrust game, we found that proposers’ behaviour
remained consistent when they assumed the roleof responders (Extended Data Fig. 5).
To assess conditional behaviour in the ultimatum game, we examined proposers’ reactions
to the acceptance and rejection of their previous
round’s proposals. We found a positive relationship with both outcomes (Extended Data Fig. 7).
However, the narrow range of data points for proposals reacting to acceptance suggests that once
proposers identify acceptable levels, they tend
not to deviate significantly. Conversely, the wider

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Fig. 4 Computational and theoretical analyses elucidate the observed experimental results. (A) For theprisoner’s dilemma, we present experimental, computational, and theoretical results for both the frequency of cooperationand the average payoff. (B) For the trust game, we display experimental, computational, and theoretical results for theentrusted amount, the returned amount, and the average payoff. (C) For the ultimatum game, we present experimental,
computational, and theoretical results for the proposed amount, rejected amount, and average payoff.

range of data points in response to rejection indicates that proposers often increase their offers
following a rejection, while some may offer less,possibly shifting their focus to other, presumably
more agreeable, neighbours.
In summary, these results suggest that social
networking agency promotes the proliferation of
conditional behaviour in economic games. Furthermore, the similarity between the average
entrusted and proposed amounts of antisocial andneutral players in these games to those observedin classical two-player experiments suggests that
constrained players perceive their neighbourhood
as a single entity, potentially affecting their understanding of fairness and trust. In contrast, freeplayers regard each neighbour as an individual,
facilitating a more prosocial distribution of theirendowment.

Numerical analysis
To gain a deeper understanding of the mechanisms
driving our results, we developed a model basedon the simplest assumptions that align simulated
behaviour with that of human participants: firstorder conditional strategies, heterogeneity, andrandom mutation.
The general model, which is applicable to all
three economic games, comprises an interaction
structure and the evolution of strategies over time.
Specifically, agents were positioned on a lattice
network and randomly assigned as either constrained or free players. At each time step, they
updated their strategies based on probabilities
extrapolated from experimental data. The generalmodel was further refined to accommodate thespecifics of each game (see Methods and Supplementary Information for additional details).
In the prisoner’s dilemma, computational
and theoretical analyses centred on probabilities
derived from players’ behaviours in the first andprevious rounds, successfully approximating theexperimental results (Fig. 4A).
For the trust game, a more complex model
was necessary. It required estimating both theamounts and probabilities for both stages of theSpringer Nature 2021 LATEX template

game: the entrusted and returned amounts in theinitial round, and the probabilities of trustors
entrusting certain amounts based on their neighbours’ behaviour in the previous round (Fig. 4B).
The ultimatum game necessitated the mostintricate modelling. It involved estimating variables based on initial round proposal amounts
and acceptance thresholds, as well as the probabilities of proposing certain amounts and acceptance thresholds, contingent upon neighbours’
behaviours in the previous round (Fig. 4C).
In summary, our model provides an algorithmic blueprint for understanding human behaviourin the prisoner’s dilemma, trust game, and ultimatum game, while revealing the distinct levels ofcomplexity among these economic games.

findings lay a foundation for further theoreticaland experimental exploration of trust and ultimatum games on networks—topics almost absent in
the current literature.
If social networks—crucial in coordinatingsocial life, including acts of prosociality—also
instil a more prosocial mindset in people more
directly, one might speculate that cooperation andsocial networking ability emerged concurrently.
This aligns with the interdependence hypothesis of anthropology [35], which posits that whencommunities of prehistoric humans began facing competition from other communities, manyof the defining characteristics of social organisingappeared—fundamentally a “group-mindedness”
characterized by a “collective intentionality” [35].

Discussion

References

The study demonstrates that the freedom tointeract differently with individual neighbours
enhances cooperation, trust, and fairness, compared to the unrealistic constraint prevalent in
the literature—that agents must behave identically towards their entire social neighbourhood.
We observe higher payoffs and lower inequality
than in the constrained reference experiments.
These findings suggest that much of the literature
on social dilemmas on social networks underestimates the prosociality of the actual situations
their studies aim to model [23]. Reality is morecooperative than science has thus far suggested.
By overcoming the constraints of traditional
network-based experiments, this study paves theway for many interesting future directions. An
evident next step towards increasing realism isto remove the anonymity of the experiments(which we maintained to isolate our topical mechanism) to enable human social sensing [28, 29].
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A. Tarancón, J.A. Cuesta, A. Sánchez,
Y. Moreno, Heterogeneous networks do notpromote cooperation when humans play aprisoner’s dilemma.
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Humans display a ‘cooperative phenotype’
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A. Sánchez, Humans display a reduced setof consistent behavioral phenotypes in dyadic
games. Sci. Adv. 2(8), e1600,451 (2016)[28] C.F. Camerer, E. Fehr, in Foundations ofhuman sociality, ed. by J. Henrich, R. Boyd,
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Springer Nature 2021 LATEX template[30] D. Balliet, J. Wu, C.K.W. De Dreu, Ingroupfavoritism in cooperation: a meta-analysis.
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11

Methods
Experiment
Structure and setup. The experimental design
embedded three economic games – the prisoner’s
dilemma, the trust game, and the ultimatum game– into a network structure. Volunteers were placedon a 7 × 7 two-dimensional lattice and randomly
assigned as constrained or f ree player types. In
each round, constrained players could only perform identical actions towards all four neighbours,
while free players could vary their actions.
Each game included a control group of solely
constrained players. Additionally, four treatmentgroups were established, each with varying fractions of free players: 25%, 50%, 75%, and 100%.
Over 50 rounds, players interacted with theirfour neighbours, with player types and neighbours
remaining constant.
In the prisoner’s dilemma, constrained players had to uniformly decide to either cooperate or
defect with all neighbours. Conversely, free players could tailor their strategies for each neighbour(Fig. 1).
The trust and ultimatum games unfolded in
two stages. The first stage involved the classification into constrained and free players, with nodifferentiation in the second stage. This design
aimed to assess whether the presence of free players would influence prosocial behaviour even in a
scenario with minimal deviations from the control
group setup.
In the trust game’s first stage, all participants,
acting as trustors, decided the amount of tokens to
entrust to their neighbours. In the second stage, allplayers, now trustees, decided how many tokens toreturn (Fig. 1A). Constrained trustors could only
entrust an identical token amount to all neighbours, while free trustors could vary the amounts
to each neighbour (Fig. 1B). Both types of trustees
could only select a single amount to be returned,
which was then equally divided among all fourneighbours.
In the ultimatum game’s first stage, all participants, acting as proposers, decided on a token division for their neighbours. In the second stage, now
responders, they chose to either accept or reject
the proposed token distribution. If a proposal was
rejected, both the proposer and the responder
received no payoff (Fig.1A). Constrained proposers had to offer the same token amount to

all neighbours, while free proposers could propose
varied amounts (Fig. 1B). Both types of responders could individually accept or reject proposals.
The trust and ultimatum games were preregistered (https://aspredicted.org/KFF YQK). The
prisoner’s dilemma game was not included in thepreregistration as some data had already beencollected by the time the preregistration was
submitted.
Player recruitment. Across all three games, we
conducted 15 sessions, each with 49 volunteers
and three replications per treatment, resulting ina total of 45 sessions. We recruited 2205 undergraduate students from seven universities in thenorthern, northwestern, and southwestern regionsof China. The participants were 54.3% female and45.7% male, with an average age of 19.7 years. A
more detailed demographic breakdown is available
in the Supplementary Information.
Each volunteer participated in a single session.
To minimize the impact of known interactions,
we ensured that the participants were from different classes and remained anonymous throughout
the experiment. Upon arrival at the computerlab, volunteers were randomly assigned to individual computer cubicles. The screens displayed
the instructions for the experimental procedures(see Supplementary Information). The formal
experiment began only after volunteers confirmed
their understanding of the instructions througha questionnaire. Participants played 50 consecutive rounds of the game, accumulating tokens. To
avoid end-of-treatment effects, the exact numberof rounds was not disclosed to the participants.
Gameplay. In each round of the prisoner’s
dilemma game, volunteers had 30 seconds to makea decision using a personalized gameplay interface.
The trust and ultimatum games, being two-stage
games, required volunteers to play two roles ineach round. In the trust game, volunteers initially
acted as trustors, with 30 seconds to decide thenumber of tokens to entrust to their neighbours.
Subsequently, they assumed the role of trustees,
with another 30 seconds to determine the number of tokens to return. An additional 30 seconds
was allocated for reviewing the results. In the ultimatum game, volunteers first acted as proposers,with 30 seconds to decide on a token division proposal for their neighbours. They then switchedSpringer Nature 2021 LATEX template

roles to responders, with another 30 seconds toaccept or reject the proposal. The final 30 seconds
were allocated for reviewing the results. Examples
of interfaces for all three economic games can befound in the Supplementary Information.
Payoffs. We used the o-Tree platform to create
the gameplay interface [36]. Each session lasted
approximately 2 hours on average. In the prisoner’s dilemma game, participants were endowedwith an initial 50 tokens. Conversely, in the trust
and ultimatum games, participants received a perround allocation of five tokens. To incentivize
participation, the total tokens accrued over all
rounds were converted into real monetary rewards
at a rate of RMB 1.0 per token. A show-up feeof RMB 20 was also awarded to each participant.
Participants with a negative token balance were
only eligible to receive the show-up fee. Uponcompletion of the experiments, participants wererequired to sign to verify their earnings.
In the prisoner’s dilemma game, the average
earnings amounted to RMB 443, with a range fromRMB 20 to RMB 1024.
In the trust game, the average earnings stoodat RMB 321, with a range from RMB 20 to
RMB 676. For trustors, the calculation of payoffs
included only the amounts returned by trustees, ameasure implemented to prevent potential endowment hoarding throughout the repeated game.
In the ultimatum game, the average earnings
were RMB 249, ranging from RMB 91 to RMB
350. Given that the proposer had to divide theirendowment among four neighbours, each of whom
could individually accept or reject the proposal,
payoffs were computed by dividing the retained
endowment by the number of neighbours who
accepted the proposal and adding it to the payoffs
from the accepted proposals.
Ethical approval declarations. The ethicalboards of all involved universities approved theexperiments.
Specific methods used for data analysis andfigure generation. Fig. 3 presents the results ofclustering analysis on the experimental results inthe prisoner’s dilemma, trust game, and ultimatum game. Data mining was based on six empiricalbehavioural variables (denoted as A - F in Fig. 3).
In all three economic games, clustering factor A

represents the average overall result, while clustering factors B - F are conditional on circumstancesin the previous round.
In prisoner’s dilemma, B to F represent cooperation probabilities and depend on the cooperative behaviour of focal player and their neighbours
in the previous round:A - cooperation frequency;
B - probability of playing C after focal player
played C in the previous round;C - probability of playing C after focal player
played D in the previous round;
D - Probability of playing C after up to one
neighbour played C;
E - probability of playing C after exactly twoneighbours played C;
F - probability of playing C after at least threeneighbours played C.
In the trust game, B to F represent the average amounts sent by the trustor and depend on
the behaviours of the trustors and trustees in theprevious round:A - entrusted amount;B - entrusted amount after the trustor sent fouror more in the previous round;C - entrusted amount after the trustor sent lessthan four in the previous round;D - entrusted amount when up to one neighbouring trustee returned an amount equal to or
greater than the trustor’s previous send;E - entrusted amount when exactly two neighbouring trustees returned an amount equal toor greater than the trustor’s previous send;F - entrusted amount when at least three neighbouring trustees returned an amount equal toor greater than the trustor’s previous send.
In the ultimatum game, B to F represent the
average amounts proposed by the proposer anddepend on the amounts proposed by proposers
and those accepted by responders in the previous
round:
A - proposed amount;B - proposed amount after the proposer sent fouror more in the previous round;C - proposed amount after the proposer sent lessthan four in the previous round;D - proposed amount when up to two neighbouring responders accept the proposal in theprevious round;Springer Nature 2021 LATEX template
13
E - proposed amount when exactly three neighbouring responders accept the proposal in theprevious round;F - proposed amount when exactly four neighbouring responders accept the proposal in theprevious round.
Model
Structure and setup. By adhering to a setof simple assumptions—first-order conditional
strategies, heterogeneity, and randomness—wereplicate the experiment using computational andmathematical models. This approach aims to facilitate a more comprehensive understanding of themechanisms that underpin the outcomes observedin our experiment.
All three economic games have the same structure: they are set on a homogeneous network withperiodic boundaries, comprising G nodes with adegree of k. Each node on the network is occupied by an individual agent, resulting in a totalof G agents who are labelled as G = {1, · · · , G}.
The connections between nodes are represented by{wij }i,j∈G , where wij = 1 (resp. wij = 0) signifies
a connection (resp. no connection) between nodes
i and j. A fraction F of players are randomly designated as free, while the remaining players areclassified as constrained. We denote the player
type in node i as vi , where vi = 1 (resp. vi = 0)
indicates a free agent (resp. constrained agent).
Prisoner’s dilemma. The prisoner’s dilemma
is a single-stage game, structured as previously
described. Agents decide whether to cooperate
or defect and receive payoffs based on a payoff
matrix.
Let sij denote player i’s action towards j,
where sij = 1 means cooperation and sij = 0
defection. For each free agent, the strategy is
described by (pf ; pfC , pfD ), and, for each constrained agent, the strategy is (pc ; pcCkC , pcDkC ).
The notations are explained below. The system
updates are as follows:
In time step t = 0, player i with vi = 1,
towards a neighbour j, chooses cooperation withprobability pf (i.e. sij = 1) and defection (i.e.
sij = 0) otherwise. Player i with vi = 0, cooperates with probability pc —that is, sij = 1 for allj ∈ Ni (the set of i’s neighbours)—and defects
otherwise.

At each time step, all players update theiractions across all interactions. We begin with theaction update of a free agent i towards j.
(i) For sji = 1, agent i takes action sij = 1 (resp.
sij = 0) with probability pfC (resp. 1 − pfC ) in
the next round;(ii) For sji = 0, agent i takes action sij = 1 (resp.
sij = 0) with probability pfD (resp. 1 − pfD ) in
the next round;
Next, we consider the action updating of a constrained agent i towards all j ∈ Ni (the set of i’sneighbours).
(i) For sij = 1, if agent i has kC cooperative neighbours, agent i takes action sij = 1(resp. sij = 0) with probability pcCkC (resp.
1 − pcCkC ) in the next round;(ii) For sij = 0, if agent i has kC cooperative neighbours, agent i takes action sij = 1(resp. sij = 0)with probability pcDkC (resp.
1 − pcDkC ) in the next round;
Supplementary Information B.1 contains thenumerical simulation and theoretical analysis.
Trust game. The trust game unfolds in twostages, involving agents positioned within a predefined structure. In the first stage, agents, acting
as trustors, allocate a certain number of tokens totheir neighbours. This allocation is then tripled. In
the second stage, agents, now acting as trustees,
decide how many of the received tokens to returnto their neighbours.
For simplicity, we divide the entrusted amountinto several intervals and study the agents’entrusted amount in different intervals, denoted
by S = {S1 , S2 , · · · , Ss }. Let Aij (t) denote agent
i’s entrusted amount to neighbour j at time t
and Ri (t) denote agent i’s returned amount toeach neighbour. Note that constrained agent i’s
entrusted amounts are identical for all neighbours,
whereas the free agent’s entrusted amounts varyfor each neighbour. We make the following two
assumptions:(i) The entrusted amount of constrained agent
i to j at time step t, (i.e. Aij (t)) depends
on the entrusted amount to him by hisneighbours
in the previous time step (i.e.
P(1/k) ℓ∈G wℓi Aℓi (t−1)), while the entrusted
amount of a free agent i to j at time step t(i.e. Aij ) depends on the entrusted amount toSpringer Nature 2021 LATEX template

him by his neighbour j in the previous time
step (i.e. Aji (t − 1));(ii) The returned amount of an agent i (i.e. Ri (t))
depends on the entrusted P
amount to i in
the current round (i.e. (1/k) ℓ∈G wℓi Aℓi (t)).
Let λc (resp. λf ) denote the ratio of the
returned amount to the entrusted amountof a constrained agent (resp.Pa free agent).
That is, Ri (t) = (3λc /k) ℓ∈G wℓi Aℓi (t)for a constrained
agent and Ri (t) =P(3λf /k) ℓ∈G wℓi Aℓi (t) for a free agent.
Each agent’s strategy refers to his current
entrusted amount in response to the neighbours’
entrusted amount in the previous round, and his
current returned amount in response to the neighbours’ entrusted amount in the current round.
We use (x, y) to describe an entrusted amountin response to neighbours, where x represents
the entrusted amount from neighbours in theprevious round, and y represents the agent’s
entrusted amount in the current round. All possible entrusted amounts in response to neighbours are {(x, y)}x,y∈{1,2,··· ,s} . Let pcx,y (resp. pfx,y )denote the probability of a constrained agent(resp. free agent) taking entrusted amount Sy
in response to neighbours’ entrusted amount Sx .
Moreover, let pcx and pfx denote the entrusted
amount at the first round.
Supplementary Information B.2 contains thenumerical simulation and theoretical analysis.
Ultimatum game. The ultimatum game isplayed in two stages within the previously
described structure. In the first stage, agents act
as proposers and propose how to split an amountof tokens between themselves and neighbours. In
the second stage, agents act as responders anddecide whether to accept or reject the proposal. Ifthey accept the proposal, proposers and responders receive payoffs as proposed, otherwise theyreceive nothing.
For simplicity, we divide the proposed amountinto several intervals and study the agents’ proposed amount in different intervals, denoted by
S = {S1 , S2 , · · · , Ss }. In the responder stage, theset of optional responses is also S.
Let Pij (t) denote agent i’s proposed amount to
j at time step t, and Ri (t), the threshold for agent
i to accept the proposal. Note that constrained
agents make the same proposal for all neighbours,
while free agents make different proposals against

different neighbours. Every constrained agent isassumed to have an identical threshold of acceptance for all neighbours, while a free agent has
four possibly different thresholds of acceptanceagainst four neighbours. We make the following
assumptions:(i) Proposed amount and acceptance thresholdof constrained agent i to j at time step t(i.e. Pij (t) and Ri (t)) depends on their proposed amount in the previous time step andthe perceived acceptance threshold of neighbours (i.e. Pij (t − 1) and R̃ic (t − 1)), including
neighbours’ proposed amount
P and i’s acceptance threshold (i.e. (1/k) ℓ∈G wℓi Pℓi (t − 1)
and Ri (t − 1)). The perceived acceptancethreshold of constrained agent i towards theirneighbours is
R̃ic (t − 1)
=Pij (t − 1)!
1X
wℓi Θ(Piℓ (t − 1) − Rℓ (t − 1)) ∆,
+ 1−
k
ℓ∈G

(1)
where Θ(x) is Heaviside Function with
Θ(x) = 1 for x > 0 and Θ(x) = 0 otherwise, and ∆ is the increment to adjust
one’s perception to the neighbours’ acceptance threshold. An increasing number of i’sneighbours accepting i’s proposed amountmeans that i’s current proposed amount ismore likely to be higher than neighbours’
acceptance threshold;(ii) Proposed amount and acceptance thresholdof free agent i to j depends on their proposed
amount in the previous time step to j and theperceived acceptance threshold of neighbour
j (i.e. Pij (t − 1) and R̃if (t − 1)), including
neighbour j’s proposed amount and i’s acceptance threshold (i.e. Pji (t − 1) and Ri (t − 1)).
The perceived acceptance threshold of a free
agent i towards their neighbours isR̃if (t − 1)
=Pij (t − 1) + (1 − Θ(Pij (t − 1) − Rj (t − 1))) ∆.
(2)
Each agent’s strategy refers to his current
proposed amount and acceptance threshold inSpringer Nature 2021 LATEX template15
response to his proposed amount and perceived
neighbours’ acceptance threshold, neighbours’
proposed amount, and his acceptance threshold in
the previous round. We use (x1 , x2 , y1 , y2 ; z1 , z2 )to describe proposed amount and acceptancethreshold response, where x1 , x2 , y1 , y2 , z1 , z2
respectively represents the agent’s proposed
amount and acceptance threshold in the previous round, the neighbours’ proposed amountand perceived acceptance threshold in theprevious round, and the agent’s proposed
amount and acceptance threshold in the current round. The set of all possible responses is{(x1 , x2 , y1 , y2 ;z1 , z2 )}x1 ,x2 ,y1 ,y2 ,z1 ,z2 ∈{1,2,··· ,s} .
Let pcx1 ,x2 ,y1 ,y2 ; z1 ,z2 (resp. pfx1 ,x2 ,y1 ,y2 ; z1 ,z2 )denote the probability to take proposed amountSz1 and acceptance threshold Sz2 for a constrained agent (resp. a free agent) in the current
round in response to his proposed amount Sx1 andacceptance threshold Sx2 , neighbours’ proposed
amount Sy1 , and perceived acceptance thresholdSy2 in the previous round. Let pcx1 ,x2 (resp. pfx1 ,x2 )denote the probability that a constrained agent(resp. a free agent) uses proposed amount Sx1
and acceptance threshold Sx2 in the first round.
Let Ωi (t) (Ω̄i (t)) denote the set of agents whose
proposed amount is accepted (rejected) by i, givenbyΩi (t) = ℓ wℓi Θ(Pℓi (t) − Ri (t)) ≥ 0 ,Ω̄i (t) = ℓ wℓi Θ(Pℓi (t) − Ri (t)) < 0 .

(3)

In the statistical analysis of experimental data,
a constrained agent i’s acceptance threshold isassumed to beR̂ic (t − 1) =
 min P + max Pℓiℓi ℓ∈Ωi (t−1)ℓ∈Ω̄i (t−1)




2
max
P
+
∆
ℓi


ℓ∈Ω̄i (t−1)
 min Pℓi − ∆ℓ∈Ωi (t−1)|Ωi (t − 1)| > 0,|Ω̄i (t − 1)| > 0;|Ωi (t − 1)| = 0;|Ω̄i (t − 1)| = 0.
(4)

In the interaction with ℓ, Rif is the averaged
value rejected by a free agent i, and the free agent
i’s acceptance threshold is assumed to be(R̂if (t − 1) =Pℓi + ∆|Ωi (t − 1)| = 0;

Rif + ∆|Ω̄i (t − 1)| = 0.

(5)

For the given proposed amount setting
S = {S1 , S2 , · · · , Ss }, the free agent’s strategy
pfx1 ,x2 , pfx1 ,x2 ,y1 ,y2 ; z1 ,z2
and the constrained
agent’s strategy pcx1 ,x2 , pcx1 ,x2 ,y1 ,y2 ; z1 ,z2 can beinferred from the experimental data.
Supplementary Information B.3 contain the
numerical simulation and theoretical analysis.

References[36] Daniel L Chen, Martin Schonger, and Chris
Wickens. otree—an open-source platform
for laboratory, online, and field experiments. Journal of Behavioral and Experimental Finance, 9:88–97, 2016.
Data availability. The Open Science Framework
page for this project (link to be provided) includes
all data from laboratory experiments.
Code availability. The codes for numerical analysis are available on the Open Science Framework(link to be provided).
Author contributions. D. J., I.R., and Z.W.
conceived the project. D.J. and I.R. designed thelaboratory experiments. D.J. and Z.W. organized
the experiments and collected data, supported by
L.S., C.L., J. L., and X.L. D.J. and I.R. analysed data with inputs from and Z.W. and P.H.
D.J., I.R., and P.H. prepared the figures. Q.S. and
D.J. developed the numerical model. I.R. wrote
the manuscript with inputs from D.J. and Z.W.,
and revisions from P.H.
Acknowledgements. Z.W. was supported by the
National Science Fund for Distinguished Young
Scholars, Grant No. 62025602, and the Tencent
Foundation, XPLORER PRIZE 2021; P.H. was
supported by JSPS KAKENHI Grant No. JP
21H04595.

Springer Nature 2021 LATEX template
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Article Title
A Trust game

B

Ultimatum game5
4

Accepted amount

Returned amount

15

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3
2
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Extended Data Fig. 1 The second stage of the trust and ultimatum games follows a similar pattern as thefirst stage. Despite the absence of distinction between constrained and free players in the second stage, an increase in thefraction of free players typically results in a rise in (A) the returned amount in the trust game and (B) the acceptedamount in the ultimatum game. Statistical significance was assessed using the Bonferroni correction.

Springer Nature 2021 LATEX template
Article Title
A Prisoner's Dilemma

B

17

Trust game

0.50
0.25
0.00

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Extended Data Fig. 2 Cooperation, trust, and fairness surge when players are free to act differently withneighbours, often within the first round of the game. The figure depicts the temporal evolution of six variables
across three games, each dataset representing the average of three replicates. The average and the 95% confidence bandsof a time series model fitted to the data are also displayed (Extended Table 1 and 2). (A) In the prisoner’s dilemma, thecooperation frequency shows a downtrend in the control group, a slight downtrend in mixed population treatments, and aslight uptrend in the free-player population treatment. (B) In the trust game, the entrusted amount exhibits a slight
downtrend in the control and treatments with low and high densities of free players but a slight uptrend in the treatment
with a medium density of free players and in the free-player population treatment. The returned amount shows a slight
downtrend across both the control and all treatments. (C) In the ultimatum game, the proposed, accepted, and rejectedamounts show a downtrend in control and across all treatments.

Springer Nature 2021 LATEX template
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Article Title
Prisoner's DilemmaA

Constrained

Free

Combination

Combination

200

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Average wealth

400

300

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Extended Data Fig. 3 Social networking agency increases wealth and decreases inequality. The average
wealth and the Gini coefficient are displayed, categorized by player type and game stage. (A-C) In the prisoner’s dilemma
and trust game, wealth based on player type increases steadily as treatments permit more free players. In the ultimatum
game, wealth differences are subtle. However, free players are wealthier across all mixed treatment groups, reflecting theirability to optimize strategies for each neighbour. (D-F) In the prisoner’s dilemma and trust game, inequality based on
player type initially spikes in the second treatment, where free players are in the minority, and then gradually decreases tolevels lower than in the initial treatment. In the ultimatum game, the situation is reversed. Inequality increases with therising fraction of free players but drops when the population is entirely free, though with smaller variations in overall
inequality levels. (J-K) Wealth disparities are more pronounced between stages of the game, as players in both games
accrue most of their wealth in the second stage. While in the trust game, wealth consistently increases in both stages as thefraction of free players rises, in the ultimatum game, wealth in the proposer stage slightly decreases with a higher fractionof free players. (L-J) In the trust game, inequality based on game stage increases with a higher fraction of free players,
reflecting the fact that more players in the population can optimize their strategies, leading to varying levels of success. In
the ultimatum game, inequality increases in both stages, although overall inequality levels remain relatively similar.

Springer Nature 2021 LATEX template
Article Title

A

Trust game

B

A

F

Ultimatum game
A

B

2
E

4

6

F

B

2
8
10

C

4

E

6

D
1

Antisocial

0.75
0.5
Neutral
0.25
0

C

D

Prosocial
0 0.25 0.5 0.75 1
Fraction of free players

Fraction of players

Fraction of players

119Antisocial

0.75
0.5
0.25
0

NeutralProsocial
0 0.25 0.5 0.75 1
Fraction of free players

Extended Data Fig. 4 Behavioural phenotypes in the second stage of the trust and ultimatum games
mirror those in the first stage. Using the six clustering factors from Fig. 3, we identify prosocial, neutral, andantisocial types in both stages of the games. A) In the second stage of the trust game, unlike the first stage where
prosocial players predominate as the fraction of free players increases, neutral players become more prominent. The
clustering factors represent average values in the current round for (A) the returned amount; (B) the returned amountafter the trustee received four or more in the previous round; (C) the returned amount after the trustee received less thanfour in the previous round; and (D - F) the returned amount after up to one, exactly two, or at least three neighbouring
trustors sent an amount equal to or greater than in the previous round; B) In the second stage of the ultimatum game,patterns resemble the first stage, with a relatively constant number of neutral players, but an increasing number ofprosocial players and a decreasing number of antisocial players. The clustering factors represent average values in thecurrent round for (A) the accepted amount; (B) the accepted amount after receiving a total offer of four or more in theprevious round; (C) the accepted amount after receiving a total offer of less than four in the previous round; and (D - F)
the accepted amount when up to two, exactly three, or exactly four neighbouring proposers propose an amount equal to or
greater than in the previous round.

Springer Nature 2021 LATEX template20

Article Title

A

Trust game

B

A

F

B

Ultimatum game
A
F

B

2

2
4

E

6
D

C

4

E

6

C

D

Extended Data Fig. 5 Player behaviour remains consistent across different stages of the trust andultimatum games. Applying the same clustering factors as in Extended Data Fig. 3 to the behavioural phenotypes
derived from the first stage of games reveals a consistent pattern of behaviour in the second stage. Specifically, if a player
adopts a prosocial, neutral, or antisocial role in the first stage of the trust and ultimatum game, they are likely tomaintain the same behaviour in the second stage.

Springer Nature 2021 LATEX template
Article Title

Average entrusted amountin the current round

Fraction free: 0

Fraction free: 0.25

Fraction free: 0.5

Fraction free: 0.75

21Fraction free: 1

4
3
2
1
0
0

1

2

3

4

0

1

2

3

4
0
1
2
3
4
0
1
Average returned amount in the previous round

2

3

4

0

1

2

3

4

Extended Data Fig. 6 Trustors tend to increase the amounts they entrust in response to larger returns
received in the previous round. The presence of free players allows for more diverse strategies, observable in the
spread of data points. This diversity is evident as some players opt to entrust less despite receiving more fromneighbouring trustees in prior rounds. The regression slopes and their 95% CI are 0.39 with [0.30, 0.48], 0.58 with
[0.53, 0.63], 0.49 with [0.44, 0.54], 0.35 with [0.30, 0.39], and 0.33 with [0.28, 0.39], respectively. The regression intercepts
and their 95% CI are 0.25 with [0.23, 0.28], 0.31 with [0.28, 0.34], 0.43 with [0.38, 0.47], 0.58 with [0.52, 0.63], and 0.63 with
[0.56, 0.71], respectively. The adjusted R2 are 0.11(F = 72.98, P < 10−15 ), 0.48(F = 551.7, P < 10−15 ),0.37(F = 342.2, P < 10−15 ), 0.30(F = 255.5, P < 10−15 ), and 0.19(F = 140.1, P < 10−15 ), respectively.

Springer Nature 2021 LATEX template

Average proposed amountin the current round22

Article Title
Fraction free: 0

5

Fraction free: 0.5

Fraction free: 0.75

Fraction free: 1

4
3
2
1
0
0

Average proposed amountin the current round

Fraction free: 0.25

1

2

3

4

5

0

Fraction free: 0

5

1

2

3

4
5 0
1
2
3
4
5 0
1
Average accepted amount in the previous round

Fraction free: 0.25

Fraction free: 0.5

2

3

4

5

0

1

Fraction free: 0.75

2

3

4

5

4

5

Fraction free: 1

4
3
2
1
0
0

1

2

3

4

5

0

1

2

3

4
5 0
1
2
3
4
5 0
1
Average rejected amount in the previous round

2

3

4

5

0

1

2

3

Extended Data Fig. 7 Proposers tend to increase their offers in response to acceptance and rejection in
the previous round. This pattern is apparent from the narrow data range in the low fraction of free players’ treatment,suggesting proposers adhere to successful offer amounts. In treatments with a majority of free players, some proposers opt
to offer less despite previous acceptance, indicating a willingness to test established relationships and adjust responder
expectations. However, when offers are rejected, proposers tend to increase their subsequent offers, suggesting they learn
from rejection and attempt to make offers more appealing. The steeper slope with higher densities of free players mightsuggest that in a freer population, proposers encounter a variety of responder thresholds for acceptance, prompting themto adjust their offers upward to find a successful compromise. In the case of acceptance, the regression slopes and their 95%
CI are 0.85 with [0.81, 0.88], 0.99 with [0.98, 1.01], 1.00 with [0.99, 1.01], 0.98 with [0.96, 0.99], and 0.99 with [0.98, 1.01],
respectively. The regression intercepts and their 95% CI are 0.10 with [0.07, 0.13], −0.02 with [−0.03, −0.01], −0.03 with[−0.04, −0.02], −0.04 with [−0.06, −0.02], and −0.06 with [−0.09, −0.04], respectively. The adjusted R2 are 0.80 (F = 2320,
P < 10−15 ), 0.97 (F = 2.11 × 104 , P < 10−15 ), 0.98 (F = 2.74 × 104 , P < 10−15 ), 0.95 (F = 1.14 × 104 , P < 10−15 ), and0.96 (F = 1.45 × 104 , P < 10−15 ), respectively. In the case of rejection, the regression slopes and their 95% CI are 0.09
with [0.03, 0.15], 0.27 with [0.19, 0.35], 0.58 with [0.51, 0.66], 0.47 with [0.36, 0.58], and 0.90 with [0.80, 1.00], respectively.
The regression intercepts and their 95% CI are 0.71 with [0.68, 0.75], 0.59 with [0.55, 0.63], 0.42 with [0.38, 0.47], 0.47 with
[0.42, 0.53], and 0.30 with [0.24, 0.37], respectively. The adjusted R2 are 0.02 (F = 10.16, P < 0.01), 0.09 (F = 46.86,
P < 10−10 ), 0.33 (F = 239, P < 10−15 ), 0.13 (F = 69.18, P < 10−15 ), and 0.39 (F = 311, P < 10−15 ), respectively.

Springer Nature 2021 LATEX template
Article Title

Prisoner’s dilemma

Trust game

Cooperation
Summary

Coefficients

23

Entrusted

Returned

Estimate t-statistic p-value Estimate t-statistic p-value Estimate t-statistic p-value(model)Constant
Trend
0% free

Stationarity

0.123

3.631

< 10−3

0.435

2.411

0.020

0.315

3.561

< 10−3

α2 −0.002−2.9380.005−0.002−1.4990.141−0.003−2.2330.031−3−4.271 < 10

−0.281−2.7730.008 −0.289−0.545−0.283−2.1930.034

α1

α3 −0.583

Auto-correlative α4 −0.064

0.589

2
Radj
= 0.297

Constant
Trend
25% freeStationarity

α1

0.002

0.678

2.126

0.039

0.735

2.293

0.027

α2 −0.0004−1.2060.234−0.003−1.781

0.082−0.007−2.0700.044−3.691 < 10

−0.242−2.217

0.032−0.276−2.5050.016−0.187−0.098−0.6610.512−0.185−1.5010.140

α3 −0.352

−3

0.852

2
Radj
= 0.0852
Radj= 0.150

2.695

0.010

1.328

3.131

0.003

2.080

3.657

< 10−3

α2 −0.00004 −0.147

0.884

0.003

1.949

0.058

−0.006−2.0960.042

α3 −0.363−2.6930.010−0.437−3.1300.003 −0.640

Auto-correlative α4 −0.067−0.4610.647−0.029−0.232

0.818

Constant
Trend
Stationarity

α1

0.150

Constant−3.714 < 10−3−0.358

0.722

2
Radj
= 0.289

0.006

0.116

0.908

0.209

0.710

0.481

0.597

1.444

0.156

α2 −0.001−1.1030.276−0.002−2.4890.017−0.007−2.869

0.006

α3

0.018

0.161

0.873

−0.042−0.5370.594−0.109−1.2280.226

Auto-correlative α4 −0.264−1.5130.137−0.216−1.4090.166−0.145−1.201

0.236

Trend
Stationarity

α1−0.0502
Radj= 0.1642= 0.144
Radj

2
Radj
= 0.0962= 0.054
Radj

Constant

100% free

0.357

3.297

2
Radj
= 0.20275% free−0.9302
Radj= 0.325

0.125

Auto-correlative α4 −0.025

50% free−0.0992
Radj= 0.244−4.336 < 10−4

α1

2
Radj
= 0.158

0.198

1.656

0.105

1.360

2.354

0.023

1.098

1.846

0.072

Trend

α2 0.0004

0.926

0.359

0.002

1.275

0.209

−0.004−1.876

0.067

Stationarity

α3 −0.254−1.5820.121−0.340−2.311

0.026−0.209−1.7570.086

Auto-correlative α4 −0.225−1.256

0.216

0.052

0.373

0.711

−0.074−0.608

0.547

2
Radj
= 0.1322
Radj= 0.0772
Radj= 0.094

Extended Data Table 1 Time series model for prisoner’s dilemma and trust game. We fitted equation
Ct − Ct−1 = α1 + α2 t + α3 Ct−1 + α4 (Ct−1 − Ct−2 ) + ϵt to the data displayed in Extended Data Fig. 2A and 2B, where
αi , i = 1, 2, 3, 4 are the regression coefficients, and ϵt is a normally distributed error term with a zero mean and an
unknown variance. The regression coefficients have the following interpretation: α1 is a constant term, α2 ̸= 0 indicatestrend, α3 < 0 indicates stationarity, and α4 ̸= 0 indicates auto-correlation. The results show that while the small negative
trend is predominant, it becomes slightly positive in entirely free populations of players for the cooperation frequency andentrusted amount. The trends are, however, rarely significant. There is no significant auto-correlation, but stationarity
stops being significant in treatments with high and entirely free populations of players.

Springer Nature 2021 LATEX template24

Article Title

Ultimatum game
Proposed
Summary

Coefficients

Accepted

Rejected

Estimate t-statistic p-value Estimate t-statistic p-value Estimate t-statistic p-value(model)Constant

0% free

α1

1.178

5.318

< 10−5

0.957

3.307

0.002

0.262

3.214

0.002

Trend

α2 −0.003−5.766 < 10−6 −0.001−0.8180.418−0.004−3.0070.004

Stationarity

α3 −0.354−5.193 < 10−5 −0.338−3.2100.002−0.544−3.368

0.002

−1.67

−0.1580.876−0.047−0.3420.734

Auto-correlative α4 −0.195

0.102−0.0202
Radj= 0.413

Constant
Trend
25% free2
Radj= 0.167

1.821

3.314

0.002

2.581

4.695

< 10−4

0.181

3.687

< 10−3

α2 −0.004

−3.20

0.003

−0.002−3.2840.002−0.003−3.162

0.003

−4

α1

α3 −0.586−3.3290.002−0.917−4.672 < 10

−0.624−4.152 < 10−3

Auto-correlative α4 −0.177−1.318

0.194

0.032

0.217

0.049

0.341

Stationarity

2
Radj= 0.350

Constant
Trend
50% freeStationarity

α1

2.535

α2 −0.009
α3 −0.784

2
Radj= 0.432

α12Radj= 0.287

2.349

3.126

0.003

0.107

3.026

0.004

−0.006−3.3860.002−0.001−2.3600.023−3−3.655 < 10

−0.784−3.1190.003−0.478−3.690 < 10−3

−1.158−0.118−0.6660.509−0.208−1.982−3.643 < 10

0.253

2
Radj
= 0.330

0.054

2
Radj
= 0.300< 10−5

0.712

2.102

0.041

1.051

2.434

0.019

0.144

5.268

Trend

α2 −0.002−1.6170.113−0.003−2.1550.037−0.002−4.360 < 10−4

Stationarity

α3 −0.217−2.1940.034−0.329−2.4600.018−0.791−6.213 < 10−6

Auto-correlative α4 −0.275−2.1670.036−0.198−1.3570.182−0.043−0.3952
Radj= 0.174

Constant
Trend
100% free

0.734

−3

2
= 0.370
Radj

Constant

0.830< 10−33.626

Auto-correlative α4 −0.175

75% free2Radj= 0.2482
Radj= 0.189

0.695

2
Radj
= 0.477

1.193

1.969

0.055

2.911

4.035

< 10−3

0.100

2.491

0.017

α2 −0.002−2.5830.013−0.0004 −0.418

0.678−0.002−1.9430.058

α1

α3 −0.330−1.9340.060−0.850−3.984 < 10−0.702−3.591 < 10−3

Auto-correlative α4 −0.057−0.2800.781−0.022−0.136−0.279−1.981

Stationarity

2
Radj= 0.100−3

2
Radj= 0.353

0.893

0.054

2
Radj
= 0.494

Extended Data Table 2 Time series model for ultimatum game. We fitted equation
Ct − Ct−1 = α1 + α2 t + α3 Ct−1 + α4 (Ct−1 − Ct−2 ) + ϵt to the data displayed in Extended Data Fig. 2C, where αi ,i = 1, 2, 3, 4 are the regression coefficients, and ϵt is a normally distributed error term with a zero mean and an unknown
variance. The regression coefficients have the following interpretation: α1 is a constant term, α2 ̸= 0 indicates trend,
α3 < 0 indicates stationarity, and α4 ̸= 0 indicates auto-correlation. In all cases, the trend is slightly negative, but theresults are less frequently significant in treatments with high and entirely free populations of players. There is noauto-correlation, but stationarity stops being significant at high and entirely free populations of proposers.

Springer Nature 2021 LATEX template
Article Title

Appendix A

1

Supplementary Information

Experimental Setup. The experiment incorporated three classic games into a network: the Prisoner’s
Dilemma Game (PDG), the Trust Game (TG), and the Ultimatum Game (UG). Players, positioned on a
7 × 7 two-dimensional lattice, were randomly designated as either constrained or free. Constrained players
had to adopt the same action towards all four neighbours, whereas free players could vary their actions.
The proportion of these player types in the network represented the group’s decision-making freedom. To
augment this freedom, we devised five treatments for each game, each with a different ratio of free players:
entirely constrained, 25% free, 50% free, 75% free, and entirely free. In all games, players interacted with
their four neighbours over 50 rounds, with their type and neighbours remaining constant. In the PDG,
constrained players selected a single strategy for all neighbours, while free players could differentiate theirstrategies. The TG and UG each consisted of two stages, with the decision-making flexibility differing
between constrained and free players only in the first stage. In the TG, each player acted as both trustor
and trustee, determining the amount to send and return to neighbours. In the second stage, tokens sentby neighbours were tripled. In the UG, each player served as both proposer and responder, deciding how
to distribute and respond to five tokens among their neighbours. Responders could react independently
to each neighbour.
Player Recruitment and Implementation. The experimental framework spanned 45 sessions, with
the PDG conducted from May 2020 to June 2021, the TG from April to October 2021, and the UG
from November to December 2021. Each session involved 49 subjects, and each treatment was conducted
over three sessions, resulting in 15 sessions per game (PDG, TG, and UG). We enlisted 2205 undergraduate volunteers from seven universities across north (Shanxi Normal University and Tianjin Universityof Technology), northwest (Northwestern Polytechnical University, Xijing University, and Xi’an International University), and southwest (Yunnan University and Yunnan University of Finance and Economics)
China, with a female representation of 54.3% and an average age of 19.7 years (Table. C1). Each volunteer
participated in one session only, with no repeat participation allowed. To mitigate the effects of known
interactions, we ensured that subjects were from different classes and maintained anonymity throughout the experiments. Upon arrival at the computer lab, volunteers were randomly assigned to individualcomputer cubicles. The experiment’s procedures and details were displayed on the computer screen, andformal participation commenced only after volunteers confirmed their understanding of the instructions.
Participants played 50 consecutive rounds of games, accruing tokens. The total number of rounds wasundisclosed to prevent end-of-treatment effects. For the PDG, the initial endowment was 50 tokens. In
each round, players (both constrained and free) had 30 seconds to make a decision using a personalized
gameplay interface (Fig. A6). For the TG and UG, the initial endowment was zero tokens, with eachplayer receiving five tokens at the start of each round. Each round comprised two stages. In the TG, during the first stage, each participant, acting as a trustor, decided how many of the five tokens to send totheir four neighbours. In the second stage, each participant, now a trustee, decided how many tokens toreturn to their neighbours. Players (both constrained and free) had 30 seconds to send and return tokens
using the personalized gameplay interface, followed by a 30-second review period (Fig. A7-A10). In theUG, during the first stage, each participant, as a proposer, decided how to divide five tokens between
themselves and their four neighbours. In the second stage, each participant, now a responder, decided
which proposals to accept or reject. Players (both constrained and free) had 30 seconds to propose andrespond using the personalized gameplay interface, followed by a 30-second review period (Fig. A11-A14).
The gameplay interface was developed on the o-Tree platform for laboratory, online, and field experiments. Each session lasted approximately 2 hours. As an incentive, the final token tally was converted
into a monetary payoff at a rate of RMB 1.0 per token, in addition to a RMB 20 show-up fee. If thetoken total was negative, the participant received only the show-up fee. At the experiment’s conclusion,
volunteers verified their payoffs by signing. The average earnings were RMB 443 (ranging from RMB 20
to RMB 1024) for the PDG, RMB 321 (ranging from RMB 20 to RMB 676) for the TG, and RMB 249(ranging from RMB 91 to RMB 350) for the UG.

Springer Nature 2021 LATEX template2

Article Title

Ethical approval declarations. The ethical boards of all involved universities approved the experiments.
Instructions. The English version of the gameplay instructions displayed to participants before thestart of the game is presented here. In this translated version, we used the terms constrained and f ree
to denote player types. However, to avoid framing effects, we used Chinese terms in the original version,
which can be translated as node and link players (see, e.g., Fig. A6 through Fig. A14).
Prisoner’s dilemma game
Welcome to our game experiment!
Please read the following instructions carefully. If you encounter any issues during the game, raise yourhand, and our expert staff will assist you. This experiment is anonymous; a computer system will assign
each participant a random ID number that cannot be traced back to you. Please avoid communicatingwith other players during the game. The game may last up to 1.5 hours. If you anticipate not staying forthe entire duration, please inform us now.
1. Background and objective: This game experiment aims to examine the decision-making patternsof players when faced with binary choices. Each participant will select either strategy A or strategy B
to interact with four neighbours. Upon posting your decisions, you will receive a payoff based on thestrategies chosen by you and your four neighbours. The payoff matrix is as follows: if both you and youropponent choose strategy A, you will each earn four tokens. If both of you choose strategy B, neitherof you earns any tokens. If you and your opponent choose different strategies, the player who chooses
strategy A loses two tokens, while the player who chooses strategy B gains six tokens.

A
B



A
+4
+6

B

−2
0

2. Gameplay rules: During the game, you will be positioned on a two-dimensional lattice and randomly
assigned a player type: constrained or free. These player types differ in their strategic flexibility: constrained players choose a single strategy for all neighbours, while free players can select different strategies
for each neighbour. The game comprises an undetermined number of rounds. Your total payoff in eachround will be the sum of your payoffs from interactions with your four neighbours.
3. Experiment interface: The gameplay uses a custom computer interface comprising a single screen:• The screen displays your information, the information of your four neighbours, and the results fromthe preceding round. This includes the strategies you and your neighbours employed and the resulting
payoffs. Based on the previous rounds’ strategies and outcomes, you must make decisions for the current
round within a 30-second timeframe. Upon decision-making, clicking the “Next” button initiates thesubsequent round. Failure to press the button within the allotted time results in automatic progression
to the next round.
4. Monetary payout: Upon game completion, your final token accumulation is visible. Our staff will
convert this total into a real monetary payout at a rate of RMB 1 per token. Additionally, a show-up feeof RMB 20 is guaranteed, regardless of your performance.

Springer Nature 2021 LATEX template
Article Title

Constrained

Free

N1

N1

A
N4

B
A

A

You
A

B
N3

A
A
B

N2

N4

A
B

3

B

You

B
B

N2

A

A
N3

Figure A1 Schematic diagram of gameplay for the prisoner’s dilemma game. The constrained player (left) employs thesame strategy with all neighbours, while the free player (right) can use different strategies for each neighbour. In this instance,
the constrained player adopts strategy A for all neighbours, whereas the free player employs strategy A for neighbours N1
and N3 and strategy B for neighbours N2 and N4. According to the payoff matrix, the cumulative payoffs for the constrained
and free players are −2 and eight tokens, respectively.

Springer Nature 2021 LATEX template4

Article Title

Trust game
Welcome to our game experiment!
Please read the following instructions carefully. If you encounter any issues during the game, raise yourhand, and our expert staff will assist you. This experiment is anonymous; a computer system will assign
each participant a random ID number that cannot be traced back to you. Please avoid communicatingwith other players during the game. The game may last up to two hours. If you anticipate not staying
for the entire duration, please inform us now.
1. Background and objective: This game experiment aims to examine player decision-making patterns
when tasked with sending and returning tokens amongst themselves. Each player assumes the roles ofboth trustor (token sender) and trustee (token returner). Initially, you will decide on an amount to send
to your four neighbours. Subsequently, you will receive tokens from your neighbours, which will be tripled,and you must decide how many to return. Your payoff is determined by the number of tokens returnedby your neighbours and the number you sent.
2. Gameplay rules: During the game, you will be positioned on a two-dimensional lattice and randomly
assigned a player type: constrained or free. These types differ in their flexibility in sending tokens toneighbours; constrained players can only send equal amounts to each neighbour, while free players canvary the amounts. Each player starts each round with five tokens. The game comprises an indeterminate
number of rounds, each consisting of two stages. In the first stage, as a trustor, you decide how manyof the five tokens to retain and how many to send to your neighbours. In the second stage, as a trustee,
you receive tokens from your neighbours, which are tripled, and you must decide how many to return.
The return amount is equally divided amongst the four neighbours. Thus, as a trustee, your decisionaffects the entire neighbourhood, not individual neighbours. Your total payoff each round is the sum of
the payoffs from the two stages. In the first stage, as a trustor, your payoff is the tokens received fromneighbours minus the tokens you sent (Fig. A2). In the second stage, as a trustee, your payoff is thetokens received from neighbours minus the tokens you returned (Fig. A3).
Your role:
TrustorConstrained

Free

N1

N1

0.75
N4

1.0
0.75

0.75

You
0.25
N3

0.75

2.0
0.75
0.5

N2

N4

1.0
0.0

0.75

You
0.25

0.5
0.5

N2

0.5

N3

Figure A2 Schematic diagram of gameplay as a trustor. The constrained player (left) can only distribute tokens equally
among their neighbours. In this instance, out of five tokens, the constrained player opts to retain two tokens and distribute
three tokens among their neighbours. The free player (right) also chooses to retain two tokens and distribute three tokens
among their neighbours. However, the free player has the flexibility to decide the number of tokens to send to each individual
neighbour.

3. Experiment interface: The gameplay employs a custom computer interface comprising three screens:Springer Nature 2021 LATEX template
Article Title
Your role:
Trustee5N10.75×3

0.5
1.0×3
0.5

You
0.25×3

N4

0.5
0.5×3

N2

0.5

N3
Figure A3 Schematic diagram of gameplay as a trustee. The trustee receives tokens sent by their neighbours after thetokens are multiplied by a compounding factor of 3. The trustee returns 2 tokens to neighbours, which are then equally
split between the neighbours, with each neighbour receiving 0.5 tokens.

• On the Trustor screen, you will decide how many of your five tokens to retain and how many to
distribute among your four neighbours. As a constrained player, you can enter a value in any one of thefour boxes, and the remaining three will automatically display the same value. As a free player, youmust input values for each box separately. In both scenarios, the central box displays the total numberof tokens you are distributing. If the sum of tokens exceeds five, the system will prompt you to entera valid amount. You have 30 seconds to complete this stage. Once ready, click the “Next” button toproceed.
• On the trustee screen, you will see the tokens sent by your four neighbours. The central box will display
these tokens, multiplied by a factor of three. You must then decide how many tokens to return to yourneighbours. Enter a value in any one of the four boxes, and the remaining three will automaticallydisplay the same value. The central box shows the total number of tokens you are returning. You have
30 seconds to complete this stage. Once ready, click the “Next” button to proceed.
• On the Result screen, you can review key information for 30 seconds. This includes your actions
as a trustor, such as the number of tokens you distributed, the number of tokens returned by yourneighbours, and your payoff. It also includes your actions as a trustee, such as the number of tokens
received from your neighbours, the number of tokens you returned, and your payoff. The screen will
display your balance from the previous and current rounds. After reviewing the results, click the “Next”
button to start the next round. If you do not press the button within the allotted time, the system willautomatically proceed to the next screen.
4. Monetary payout: Upon the game’s conclusion, your final balance will be visible. This balance will
be converted into a tangible monetary payout by our staff at a rate of 1 token to RMB 1.0. Additionally,
you will receive a participation fee of RMB 20, which is independent of your performance during theexperiment.

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Article Title

Ultimatum game
Welcome to our game experiment!
Please read the following instructions carefully. If you encounter any issues during the game, raise yourhand, and our expert staff will assist you. This experiment is anonymous; a computer system will assign
each participant a random ID number that cannot be traced back to you. Please avoid communicatingwith other players during the game. The game may last up to two hours. If you anticipate not staying
for the entire duration, please inform us now.
1. Background and objective: This game experiment aims to examine players’ decision-making patterns when tasked with dividing a sum of tokens between themselves (as proposers) and other players(as responders). Each participant will assume both roles. Initially, you will propose a division of tokens
between yourself and your four neighbours. Subsequently, you will respond to your neighbours’ proposals by either accepting or rejecting them. The payoff you receive will depend on the responses to yourproposal and your reactions to your neighbours’ proposals.
2. Gameplay rules: During the game, you will be positioned on a two-dimensional lattice and randomly
assigned as either a constrained player or a free player. These player types differ in their ability to divide
a sum of tokens among their neighbours. Constrained players can only distribute tokens equally among
the neighbours, while free players have the flexibility to allocate tokens differently for each neighbour.
Each round provides each player with an endowment of five tokens. The game comprises an indeterminate
number of rounds, each consisting of two stages. In the first stage, you act as a proposer and decide
how to divide five tokens between yourself and your four neighbours. In the second stage, you act as a
responder, receiving offers from four neighbours and deciding which proposals to accept or reject.
Your total payoff in each round will be the sum of payoffs from the two stages. In the proposal stage, youwill receive a payoff only if your neighbours accept your proposal (Fig. A4). In the responder stage, youwill receive a payoff only from the proposals you decide to accept (Fig. A5).
Your role:
Proposer

Constrained

Free

N1

N1

0.75
N4

1.0
0.75

0.75

You
0.25
N3

0.75

1.5
0.75
0.5

N2

N4

1.0
0.2

0.75

You
0.25

0.8
0.5

N2

0.5

N3

Figure A4 Schematic diagram of gameplay as a proposer. The constrained player (left) can only distribute tokens equally
among neighbours. In this instance, the player proposes to retain two out of five tokens and distribute the remaining threeequally. The free player (right) also proposes to keep two tokens but has the flexibility to distribute the remaining tokens
among neighbours individually. The tokens retained by the proposer are divided by four (the number of neighbours), andthe proposer receives a payoff only from neighbours who accept the proposal. If a proposal is rejected, both the proposerand neighbour receive no payoff. For instance, if three neighbours accept the offer in our diagram, the proposer receives
2 × 3/4 = 1.5 tokens. If two neighbours accept the offer, the payoff is 2 × 2/4 = 1, and so on.

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Article Title
Your role:
Responder7

1.0
AcceptYou

Accept0.5

N2

Reject

0.25

N4

0.75

AcceptN1

N3
Figure A5 Schematic diagram of gameplay as a responder. The focal responder receives proposals from four neighbours,
N1-4, and decides which to accept or reject. In this example, the responder accepts proposals from neighbours N1, N2, andN4, and rejects the offer from neighbour N3. Consequently, the total payoff received by the focal responder is 0.75+0.5+1 =2.25.

3. Computer interface: The gameplay employs a custom computer interface comprising three screens:• On the proposer screen, you are tasked with distributing five tokens amongst yourself and four neighbours. As a constrained player, you can assign a value to any one of the four boxes, and the remaining
three will automatically display the same value. As a free player, you must individually assign values
to each of the four boxes. In both scenarios, a central box will display the total number of tokens you
propose to distribute amongst your neighbours. If your proposal exceeds five tokens, the system willprompt you to submit a valid proposal. You have 30 seconds to complete this stage. Once ready, click
the “Next” button to proceed.
• On the responder screen, you will view proposals from your four neighbours. For each proposal, youmust decide whether to accept or reject the offer within a 30-second timeframe. Once you have madeyour decisions, click the “Next” button to advance.
• The result screen allows you to review key information for a duration of 30 seconds. This includes youractions as a proposer: your proposal to four neighbours, their responses, the acceptance status of yourproposal, and your payoff. As a responder, you can view the proposals of your four neighbours, theiracceptance status, and your payoff. The screen also displays your endowment balance from the previous
round and the current round. After reviewing the results, click the “Next” button to initiate the next
round. Failure to press the button within the specified time will result in the system automaticallyprogressing to the next screen.
4. Monetary payout: Upon game completion, you can view your final balance. Our staff will convert
this balance into a real monetary payout at a rate of RMB 1 per token. Additionally, you will receive ashow-up fee of RMB 20, regardless of your performance during the experiment.

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Article Title

A 【Round 2】
本页面剩余时间 (Time left on this screen) 0:18
Payoff matrix

A

B

A

+4

-2

B

+6

0
邻居1: 点玩家
Neighbor 1: Constrained player邻居选择 (Neighbor chose): A邻居收益 (Neighbor payoff): 8
你: 点玩家
You: Constrained player邻居4: 边玩家
Neighbor 4: Free player邻居2: 点玩家
Neighbor 2: Constrained player你上一轮选择 (Your previous choice): A你的收益 (Your payoff): 4

邻居选择 (Neighbor chose): A邻居收益 (Neighbor payoff): -2

你当轮的选择 (Your current choice):A
B邻居选择 (Neighbor chose): B
邻居收益 (Neighbor payoff): 24

邻居3: 边玩家
Neighbor 3: Free player邻居选择 (Neighbor chose): B
邻居收益 (Neighbor payoff): 14
下一页(Next)

你的总收益 (Your total payoff): 56+4=60

B 【Round 2】
本页面剩余时间 (Time left on this screen) 0:18
Payoff matrix

A

B

A

+4

-2

B

+6

0
邻居1: 点玩家
Neighbor 1: Constrained player邻居选择 (Neighbor chose): A邻居收益 (Neighbor payoff): 8
你选择 (You chose): B

邻居4: 边玩家
Neighbor 4: Free player邻居选择 (Neighbor chose): A邻居收益 (Neighbor payoff): -2
你选择 (You chose): B

你: 边玩家
You: Free player
你的收益 (Your payoff): 8

邻居2: 点玩家
Neighbor 2: Constrained player邻居选择 (Neighbor chose): B
邻居收益 (Neighbor payoff): 24
你选择 (You chose): A

邻居3: 边玩家
Neighbor 3: Free player邻居选择 (Neighbor chose): B
邻居收益 (Neighbor payoff): 14
你选择 (You chose): A
邻居1: 点玩家
Neighbor 1: Constrained player邻居4: 边玩家
Neighbor 4: Free player你当轮的选择 (Your current choice):AB
你: 边玩家
You: Free player你当轮的选择 (Your current choice):A
B

邻居2: 点玩家
Neighbor 2: Constrained player你当轮的选择 (Your current choice):A
B

邻居3: 边玩家
Neighbor 3: Free player你当轮的选择 (Your current choice):AB你的总收益 (Your total payoff): 62+8=70

Figure A6 Constrained and free players’ interface in the prisoner’s dilemma game.

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Article Title

A阶段1：分享 (Stage 1: Send) 【Round 6】
本页面剩余时间 (Time left on this screen) 0:18
点类型玩家向四个方向的邻居分享的点数需要保持一致；边类型玩家向四个方向的邻居分享的点数可以不同。
(Constrained players should entrust the same amount to each neighbor. Free players can entrust different amounts.)
分享点数应该大于或等于0，小数点后不多于2位，且总共不超过5点。
(The entrusted amount should be no less than zero, have at most 2 decimal places, and not exceed 5 tokens.)
邻居1: 点玩家
Neighbor 1: Constrained playerYour role:
Trustor

分享 (Send)
邻居4: 边玩家
Neighbor 4: Free player
分享 (Send)

0.28

0.28

tokens

你: 点玩家
You: Constrained player你有5单位 (You have 5 tokens).

tokens

共分享 (Send totally): 1.12

邻居2: 点玩家
Neighbor 2: Constrained player
分享 (Send)

0.28

tokens邻居3: 边玩家
Neighbor 3: Free player
分享 (Send)

0.28

tokens清空(Clear)B

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阶段2：返还 (Stage 2: Return) 【Round 6】
本页面剩余时间 (Time left on this screen) 0:18
玩家向四个方向的邻居返还的点数需要保持一致。
(Players should return the same amount to each neighbor. )返还点数应该大于或等于0，小数点后不多于2位，且总共不超过6.03点。
(The returned amount should be no less than zero, have at most 2 decimal places, and not exceed 6.03 tokens.)
邻居1: 点玩家
Neighbor 1: Constrained playerYour role:
Trustee邻居分享 (Neighbor sent): 0.41
你返还 (You return) 1.3

tokens邻居4: 边玩家
Neighbor 4: Free player

你: 点玩家
You: Constrained player邻居2: 点玩家
Neighbor 2: Constrained player

邻居分享 (Neighbor sent): 0.06

翻倍后,你从邻居获得 (After multiplication,
you received from neighbors):

邻居分享 (Neighbor sent): 0.82

你返还 (You return) 1.3

tokens

2.01×3=6.03
共返还 (Return totally): 5.2

你返还 (You return) 1.3

tokens邻居3: 边玩家
Neighbor 3: Free player
邻居分享 (Neighbor sent): 0.72
你返还 (You return) 1.3

tokens清空(Clear)Figure A7 Constrained trustor and trustee’s interface in the trust game.

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Article TitleC结果 (Result) 【Round 6】
本页面剩余时间 (Time left on this screen) 0:18

Your role:
Trustor邻居1: 点玩家
Neighbor 1: Constrained player你分享 (You sent): 0.28
邻居返还 (Neighbor returned): 0.98

邻居4: 边玩家
Neighbor 4: Free player你分享 (You sent): 0.28
邻居返还 (Neighbor returned): 0.38

你: 点玩家
You: Constrained player你分享1.12, 收到3.95, 你的净收益是(You sent 1.12, and received 3.95,
your payoff is ): +2.83

邻居2: 点玩家
Neighbor 2: Constrained player你分享 (You sent): 0.28
邻居返还 (Neighbor returned): 0.89

邻居3: 边玩家
Neighbor 3: Free player你分享 (You sent): 0.28
邻居返还 (Neighbor returned): 1.7

Your role:
Trustee邻居1: 点玩家
Neighbor 1: Constrained player
邻居分享 (Neighbor sent): 0.41
你返还 (You returned): 1.3

邻居4: 边玩家
Neighbor 4: Free player
邻居分享 (Neighbor sent): 0.06
你返还 (You returned): 1.3

你: 点玩家
You: Constrained player你收到6.03，返还5.2，你的净收益是(You received 6.03, and returned 5.2,
your payoff is): +0.83

邻居2: 点玩家
Neighbor 2: Constrained player
邻居分享 (Neighbor sent): 0.82
你返还 (You returned): 1.3

邻居3: 边玩家
Neighbor 3: Free player
邻居分享 (Neighbor sent): 0.72
你返还 (You returned): 1.3

上一轮余额 (Balance in the previous round): 32.56
当前余额 (Balance in the current round): 32.56+2.83+0.83=36.22
Figure A8 Constrained player’s result interface in the trust game.

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Article Title

A阶段1：分享 (Stage 1: Send) 【Round 8】
本页面剩余时间 (Time left on this screen) 0:18
点类型玩家向四个方向的邻居分享的点数需要保持一致；边类型玩家向四个方向的邻居分享的点数可以不同。
(Constrained players should entrust the same amount to each neighbor. Free players can entrust different amounts.)
分享点数应该大于或等于0，小数点后不多于2位，且总共不超过5点。
(The entrusted amount should be no less than zero, have at most 2 decimal places, and not exceed 5 tokens.)
邻居1: 点玩家
Neighbor 1: Constrained playerYour role:
Trustor

分享 (Send)
邻居4: 点玩家
Neighbor 4: Constrained player
分享 (Send)

1.25

0.19

tokens你: 边玩家
You: Free player邻居2: 边玩家
Neighbor 2: Free player你有5单位 (You have 5 tokens).

tokens

共分享 (Send totally): 3.92

分享 (Send)

1.36

tokens邻居3: 边玩家
Neighbor 3: Free player
分享 (Send)

1.12

tokens清空(Clear)B

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阶段2：返还 (Stage 2: Return) 【Round 8】
本页面剩余时间 (Time left on this screen) 0:18
玩家向四个方向的邻居返还的点数需要保持一致。
(Players should return the same amount to each neighbor.)
返还点数应该大于或等于0，小数点后不多于2位，且总共不超过11.46点。
(The returned amount should be no less than zero, have at most 2 decimal places, and not exceed 11.46 tokens.)
邻居1: 点玩家
Neighbor 1: Constrained playerYour role:
Trustee邻居分享 (Neighbor sent): 0.85
你返还 (You return) 1.4

tokens邻居4: 点玩家
Neighbor 4: Constrained player

你: 边玩家
You: Free player邻居2: 边玩家
Neighbor 2: Free player

邻居分享 (Neighbor sent): 0.29

翻倍后,你从邻居获得 (After multiplication,
you received from neighbors):

邻居分享 (Neighbor sent): 1.06

你返还 (You return) 1.4

tokens3.82×3=11.46
共返还 (Return totally): 5.6

你返还 (You return) 1.4

tokens邻居3: 边玩家
Neighbor 3: Free player
邻居分享 (Neighbor sent): 1.62
你返还 (You return) 1.4
清空(Clear)Figure A9 Free trustor and trustee’s interface in the trust game.

tokens下一页
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Article TitleC结果 (Result) 【Round 8】
本页面剩余时间 (Time left on this screen) 0:18

Your role:
Trustor邻居1: 点玩家
Neighbor 1: Constrained player你分享 (You sent): 0.19
邻居返还 (Neighbor returned): 0.23

邻居4: 点玩家
Neighbor 4: Constrained player你分享 (You sent): 1.25
邻居返还 (Neighbor returned): 0.35

你: 边玩家
You: Free player
你分享3.92, 收到5.23, 你的净收益是(You sent 3.92, and received 5.23,
your payoff is ): +1.31

邻居2: 边玩家
Neighbor 2: Free player你分享 (You sent): 1.36
邻居返还 (Neighbor returned): 2.15

邻居3: 边玩家
Neighbor 3: Free player你分享 (You sent): 1.12
邻居返还 (Neighbor returned): 2.5

Your role:
Trustee邻居1: 点玩家
Neighbor 1: Constrained player
邻居分享 (Neighbor sent): 0.85
你返还 (You returned): 1.4

邻居4: 点玩家
Neighbor 4: Constrained player
邻居分享 (Neighbor sent): 0.29
你返还 (You returned): 1.4

你: 边玩家
You: Free player你收到11.46，返还5.6，你的净收益是(You received 11.46, and returned 5.6,
your payoff is): +5.86

邻居2: 边玩家
Neighbor 2: Free player
邻居分享 (Neighbor sent): 1.06
你返还 (You returned): 1.4

邻居3: 边玩家
Neighbor 3: Free player
邻居分享 (Neighbor sent): 1.62
你返还 (You returned): 1.4

上一轮余额 (Balance in the previous round): 39.81
当前余额 (Balance in the current round): 39.81+1.31+5.86=46.98
Figure A10 Free player’s result interface in the trust game.

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Article Title

A阶段1：提议 (Stage 1: Propose) 【Round 2】
本页面剩余时间 (Time left on this screen) 0:18
点类型玩家向四个方向的邻居提议的点数需要保持一致；边类型玩家向四个方向的邻居提议的点数可以不同。
(Constrained players should propose the same amount to each neighbor. Free players can propose different amounts.)提议点数应该大于或等于0，小数点后不多于2位，且总共不超过5点。
(The proposed amount should be no less than zero, have at most 2 decimal places, and not exceed 5 tokens.)
邻居1: 点玩家
Neighbor 1: Constrained playerYour role: Proposer

提议 (Propose)邻居4: 边玩家
Neighbor 4: Free player

0.82

tokens

你: 点玩家
You: Constrained player邻居2: 点玩家
Neighbor 2: Constrained player你能分配5点(You can distribute 5 tokens).
提议 (Propose)

0.82

tokens你给邻居的总提议 (Your total proposal):

提议 (Propose)

0.82

tokens3.28
邻居3: 边玩家
Neighbor 3: Free player
提议 (Propose)

0.82

tokens清空(Clear)B

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阶段2：回应 (Stage 2: Response) 【Round 2】
本页面剩余时间 (Time left on this screen) 0:18
邻居1: 点玩家
Neighbor 1: Constrained playerYour role:
Responder邻居提议 (Neighbor proposed): 0.75
你的回应 (Your response)：
接受(Accept)

邻居4: 边玩家
Neighbor 4: Free player拒绝(Reject)

你: 点玩家
You: Constrained player邻居2: 点玩家
Neighbor 2: Constrained player邻居提议 (Neighbor proposed): 0.95

邻居提议 (Neighbor proposed): 1.05

你的回应 (Your response)：你的回应 (Your response)：

接受(Accept)

接受(Accept)拒绝(Reject)拒绝(Reject)

邻居3: 边玩家
Neighbor 3: Free player邻居提议 (Neighbor proposed): 0.51
你的回应 (Your response)：
接受(Accept)拒绝(Reject)

清空(Clear)Figure A11 Constrained proposer and responder’s interface in the ultimatum game.

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Article TitleC结果 (Result) 【Round 2】
本页面剩余时间 (Time left on this screen) 0:18
邻居1: 点玩家
Neighbor 1: Constrained playerYour role:
Proposer你提议 (You proposed): 0.82
邻居的回应 (Neighbor’s response):
接受(Accept)

邻居4: 边玩家
Neighbor 4: Free player

你: 点玩家
You: Constrained player邻居2: 点玩家
Neighbor 2: Constrained player你提议 (You proposed): 0.82
邻居的回应 (Neighbor’s response):

3个邻居同意你的提议, 你的总收益是(3 neighbors agreed to your proposal,你提议 (You proposed): 0.82
邻居的回应 (Neighbor’s response):

接受(Accept)

your total payoff is): 1.29

拒绝(Reject)

邻居3: 边玩家
Neighbor 3: Free player你提议 (You proposed): 0.82
邻居的回应 (Neighbor’s response):
接受(Accept)

Your role:
Responder邻居1: 点玩家
Neighbor 1: Constrained player邻居提议 (Neighbor proposed): 0.75
你的回应 (Your response): 接受(Accept)
你的收益 (Your payoff): 0.75

邻居4: 边玩家
Neighbor 4: Free player

你: 点玩家
You: Constrained player邻居2: 点玩家
Neighbor 2: Constrained player邻居提议 (Neighbor proposed): 0.95
你的回应 (Your response): 接受(Accept)

你的总收益是 (Your total payoff is):邻居提议 (Neighbor proposed): 1.05
你的回应 (Your response): 接受(Accept)

你的收益 (Your payoff): 0.95

2.75

你的收益 (Your payoff): 1.05

邻居3: 边玩家
Neighbor 3: Free player邻居提议 (Neighbor proposed): 0.51
你的回应 (Your response): 拒绝(Reject)你的收益 (Your payoff): 0

上一轮余额 (Balance in the previous round): 4.52
当前余额 (Balance in the current round): 4.52+1.29+2.75=8.56
Figure A12 Constrained player’s result interface in the ultimatum game.

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Article Title

A阶段1：提议 (Stage 1: Propose) 【Round 5】
本页面剩余时间 (Time left on this screen) 0:18
点类型玩家向四个方向的邻居提议的点数需要保持一致；边类型玩家向四个方向的邻居提议的点数可以不同。
(Constrained players should propose the same amount to each neighbor. Free players can propose different amounts.)提议点数应该大于或等于0，小数点后不多于2位，且总共不超过5点。
(The proposed amount should be no less than zero, have at most 2 decimal places, and not exceed 5 tokens.)
邻居1: 点玩家
Neighbor 1: Constrained playerYour role: Proposer

提议 (Propose)邻居4: 边玩家
Neighbor 4: Free player

tokens0.35

你: 边玩家
You: Free player邻居2: 点玩家
Neighbor 2: Constrained player你能分配5点(You can distribute 5 tokens).
提议 (Propose)

1.21

tokens你给邻居的总提议 (Your total proposal):

提议 (Propose)

1.32

tokens3.84
邻居3: 边玩家
Neighbor 3: Free player
提议 (Propose)

tokens0.96

下一页(Next)

清空(Clear)B

阶段2：回应 (Stage 2: Response) 【Round 5】
本页面剩余时间 (Time left on this screen) 0:18
邻居1: 点玩家
Neighbor 1: Constrained playerYour role:
Responder邻居提议 (Neighbor proposed): 0.3
你的回应 (Your response)：
接受(Accept)

邻居4: 边玩家
Neighbor 4: Free player拒绝(Reject)你: 边玩家
You: Free player邻居2: 点玩家
Neighbor 2: Constrained player邻居提议 (Neighbor proposed): 2.1

邻居提议 (Neighbor proposed): 1.12

你的回应 (Your response)：你的回应 (Your response)：

接受(Accept)

接受(Accept)拒绝(Reject)拒绝(Reject)

邻居3: 边玩家
Neighbor 3: Free player邻居提议 (Neighbor proposed): 0.05
你的回应 (Your response)：
接受(Accept)拒绝(Reject)

清空(Clear)Figure A13 Free proposer and responder’s interface in the ultimatum game.

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15

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Article TitleC结果 (Result) 【Round 5】
本页面剩余时间 (Time left on this screen) 0:18
邻居1: 点玩家
Neighbor 1: Constrained playerYour role:
Proposer你提议 (You proposed): 0.35
邻居的回应 (Neighbor’s response):拒绝(Reject)

邻居4: 边玩家
Neighbor 4: Free player

你: 边玩家
You: Free player邻居2: 点玩家
Neighbor 2: Constrained player你提议 (You proposed): 1.21
邻居的回应 (Neighbor’s response):

3个邻居同意你的提议, 你的总收益是(3 neighbors agreed to your proposal,你提议 (You proposed): 1.32
邻居的回应 (Neighbor’s response):

接受(Accept)

your total payoff is): 0.87

接受(Accept)

邻居3: 边玩家
Neighbor 3: Free player你提议 (You proposed): 0.96
邻居的回应 (Neighbor’s response):
接受(Accept)

Your role:
Responder邻居1: 点玩家
Neighbor 1: Constrained player邻居提议 (Neighbor proposed): 0.3
你的回应 (Your response): 拒绝(Reject)你的收益 (Your payoff): 0

邻居4: 边玩家
Neighbor 4: Free player

你: 边玩家
You: Free player邻居2: 点玩家
Neighbor 2: Constrained player邻居提议 (Neighbor proposed): 2.1
你的回应 (Your response): 接受(Accept)

你的总收益是 (Your total payoff is):邻居提议 (Neighbor proposed): 1.12
你的回应 (Your response): 接受(Accept)

你的收益 (Your payoff): 2.1

3.22

你的收益 (Your payoff): 1.12

邻居3: 边玩家
Neighbor 3: Free player邻居提议 (Neighbor proposed): 0.05
你的回应 (Your response): 拒绝(Reject)你的收益 (Your payoff): 0

上一轮余额 (Balance in the previous round): 16.12
当前余额 (Balance in the current round): 16.12+0.87+3.22=20.21
Figure A14 Free player’s result interface in the ultimatum game.

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Article Title

Appendix B

17

Model analysis

B.1

Prisoner’s dilemma

B.1.1

Numerical simulation

To perform the numerical simulation based on the model delineated in the Methods section, we derived
probabilities for agents’ initial strategies and strategy updating from experimental data (an overview
of variables is in Table C2). We denoted probabilities as (pc ; pcCkC , pcDkC ) for constrained agents and(pf ; pfC , pfD ) for free agents.
Initial strategies. For the agents’ initial strategy, we assumed that the probability that agents choose
to cooperate is related to the behaviour of neighbours. Therefore, based on the experimental data fromthe first round, for free and constrained agents, we calculated their proportion of cooperation, denoted
as pf and pc , respectively. Therefore, at time t = 0, towards a neighbor j cooperated (i.e., sij = 1) withprobability pf and defected (i.e., sij = 0) otherwise. Agent i with vi = 0 cooperated with probability pc(i.e., sij = 1 for all j ∈ Ni ; the set of i’s neighbours) and defected otherwise.
Strategy updating. In the strategy updating process, we assumed that the free agent i’s (i.e., vi = 1)
strategy towards neighbor j at time t + 1 was determined by the neighbor j’s strategy sji at time t. The
constrained agent i’s (i.e., vi = 0) strategy sij towards neighbor j ∈ Ni (the set of i’s neighbors) at timet + 1 was determined by agent i’s strategy and the number of cooperative neighbors kC (kC = 0, 1, 2, 3, 4).
Free agent i updated their strategy towards neighbor j as below.
(i) For sji = 1 at time t, agent i chose strategy sij = 1 (resp. sij = 0) with probability pfC (resp. 1 − pfC )at time t + 1;(ii) For sji = 0 at time t, agent i chose strategy sij = 1 (resp. sij = 0) with probability pfD (resp. 1 − pfD )at time t + 1;
Constrained agent i updated their strategy towards neighbor j ∈ Ni as below.
(i) For sij = 1, if kC neighbors of agent i chose strategy sji = 1 at time t, agent i chose strategy sij = 1(resp. sij = 0) with probability pcCkC (resp. 1 − pcCkC ) at time t + 1;(ii) For sij = 0, if kC neighbors of agent i chose strategy sji = 1 at time t, agent i chose strategy sij = 1(resp. sij = 0) with probability pcDkC (resp. 1 − pcDkC ) at time t + 1;
Notably, in each time step, free agents independently updated their strategy against each neighbour
according to the aforementioned rules. In contrast, constrained agents updated their strategy towards
the entire neighbourhood.
Initialization. The simulation was initialized with constrained and free agents randomly distributedon an L × L square lattice with a periodic boundary and a von Neumann neighbourhood. The fraction offree agents was denoted by F , where a set of F = 0.00 : 0.05 : 1.00 was investigated. Each agent engaged
in a repeated PDG with their neighbours and accumulated a payoff based on the same payoff matrix as
the experiment. For a complete Monte Carlo simulation, we observed the cooperation frequency and theaverage payoff on the lattice of size L = 100 over 104 time steps, of which the last 2×103 has up to a stablestate. All results were averaged over 1000 independent realizations for a fixed set of free agent densities.

B.1.2

Theoretical analysis

Our analysis assumes that one’s action updating depends on one’s joint action profile last round rather
than the comparison of participants’ payoffs. Let ρf (t) (resp. ρc (t)) denote the cooperation frequency of
free (resp. constrained) players at time t. Here we adopt the mean-field approach to approximate thechange in ρf (t) and ρc (t).

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Article Title
The change in the cooperation frequency of free players, i.e., ρf (t), isF ρf (t)pDC +F (1−ρf (t))pDD
dt ρf (t) = 1 − ρf (t) +(1−F)ρc (t)pDC +(1−F )(1−ρc (t))pDDf(t)(1−pCC )+F (1−ρf (t))(1−pCD )− ρf (t) (1−FF)ρρ c (t)(1−p.
cCC )+(1−F )(1−ρ (t))(1−pCD )(B1)

The probability that a constrained player has kC cooperative neighbors is
C(kC ) =kCk!
F ρf (t) + (1 − F )ρc (t)kC !(k − kC )!
k−kC,
1 − F ρf (t) − (1 − F )ρc (t)(B2)

The change in the cooperation frequency of constrained players, i.e., ρc (t), isdt ρc (t) =(1 − ρc (t))

k
X

k
X

C(kC )pcDkC − ρc (t)kC =0C(kC )(1 − pcCkC ).

(B3)

kC =0

The cooperation frequency of the population is
ρC =F ρf + (1 − F )ρc .

(B4)

The accumulated payoff of one player isF ρ2f rw+F ρf (1−ρf )sp+F(1−ρf )ρf tm+F (1−ρf )2 pn

Q =kF +(1−F )ρf ρc rw+(1−F )ρf (1−ρc )spfcfc+(1−F )(1−ρ )ρ tm+(1−F )(1−ρ )(1−ρ )pn)

F ρc ρf rw+F ρc (1−ρf )sp
c
f
+F
(1−ρ
tm+F (1−ρc )(1−ρf )pn + k(1 − F )  +(1−F )ρ)ρ2c rw+(1−F.
)ρc (1−ρc )spccc 2+(1−F )(1−ρ )ρ tm+(1−F )(1−ρ ) pn

(B5)

where rw, sp, tm, pn respectively correspond to the payoff under mutual cooperation, unilateral
cooperation, unilateral defection, and mutual defection.

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B.2

Trust game

B.2.1

Numerical simulation

19

To perform the numerical simulation based on the model outlined in the Methods section of the paper,
we inferred several aspects of the agents’ behaviour from the experimental data (Table C3 presents
the variables overview). For the entrusted amount in the first stage of the game, we took the average
entrusted amount of the whole experiment as a reference and considered two entrusted amount intervals
S = {S1 , S2 }, where S1 ∈ [0, 0.75), and S2 ∈ (0.75, 5.0]. Constrained and free agents updated theirstrategies in the first stage following (pcx ; pcx,y ) and (pfx ; pfx,y ) response rules; in the second stage ofthe game, they followed a fixed return ratio of λc and λf respectively. Therefore, the specific values
corresponding to pcx , pcx,y , pfx , pfx,y , λc , and λf were also inferred from the experimental data.
Initial strategies. For the agents’ entrusted amount in the first round, we inferred from the data
the frequency of the constrained agents’ entrusted amount in intervals S1 and S2 , respectively, denotedas {pcx }x∈{1,2} = {pc1 , pc2 }, and the mean of the constrained agents’ entrusted amount in each interval,
denoted as {Sxc }x∈{1,2} = {S1c , S2c }. Similarly, we inferred the frequency of the free agents’ entrusted
amount in intervals S1 and S2 , respectively, denoted as {pfx }x∈{1,2} = {pf1 , pf2 }, and the mean of the free
agents’ entrusted amount in each interval, denoted as {Sxf }x∈{1,2} = {S1f , S2f }. Therefore, at time t = 0,agent i’s entrusted amount to neighbour j followed:(i) For vi = 1, agent i selected the entrusted amount interval S1 (resp. S2 ) with probability pf1 (resp.
1 − pf1 ) and selected S1f (resp. S2f ) to entrust to neighbour j;(ii) For vi = 0, agent i selected the entrusted amount interval S1 (resp. S2 ) with probability pc1 (resp.
1 − pc1 ) and selected S1c (resp. S2c ) to entrust to the neighbour.
Strategy updating. In the strategy updating process, all agents updated their entrusted and returned
amounts in each time step for all interactions. For the first stage at each time step, we assumed agents’entrusted amount Sy , y ∈ {1, 2} at time t + 1 was in response to the entrusted amount Sx , x ∈ {1, 2}
received from their neighbours at time t. The combination (Sx , Sy ) encompassed four response scenarios.
Therefore, based on the experimental data, for constrained agents (resp. free agents), we inferred pcx,yfc(resp. pfx,y ) and Sx,y(resp. Sx,y). At each time step for the second stage of the game, we assumed theconstrained agents (resp. free agents) returned the tripled entrusted amount to neighbours at a fixed rate
λc (resp. λf ). As stated earlier, we also inferred the average of the returned amount ratio as λc (resp. λf )for constrained agents (resp. free agents) from the experimental data. Agent i updated their entrusted
amount towards neighbour j as below.
(i) For vi = 1 and Sx = S1 at time t, agent i selected the entrusted amount interval S1 (resp. S2 ) withff
(resp. S1,2
) to entrust to neighbour j at time t + 1;
probability pf1,1 (resp. 1 − pf1,1 ) and selected S1,1(ii) For vi = 1 and Sx = S2 at time t, agent i selected the entrusted amount interval S1 (resp. S2 ) withff
probability pf2,1 (resp. 1 − pf2,1 ) and selected S2,1
(resp. S2,2
) to entrust to neighbour j at time t + 1;(iii) For vi = 0 and Sx = S1 at time t, agent i selected the entrusted amount interval S1 (resp. S2 ) withc (resp. S c ) to entrust to neighbours at time t + 1;
probability pc1,1 (resp. 1 − pc1,1 ) and selected S1,1
1,2(iv) For vi = 0 and Sx = S2 at time t, agent i selected the entrusted amount interval S1 (resp. S2 ) withc (resp. S c ) to entrust to neighbours at time t + 1.
probability pc2,1 (resp. 1 − pc2,1 ) and selected S2,1
2,2
Furthermore, agent i updated their returned amount towards neighbours as below.
(i) For vi = 1, agent i returned the tripled entrusted amount to neighbours at a fixed rate λf .
(ii) For vi = 0, agent i returned the tripled entrusted amount to neighbours at a fixed rate λc .

Initialization. The simulation was initialized with constrained and free agents randomly distributedon an L × L square lattice with a periodic boundary and a von Neumann neighbourhood. The fraction offree agents was denoted F , where a set of F = 0.00 : 0.05 : 1.00 was investigated. Each agent participatedSpringer Nature 2021 LATEX template20

Article Title

in a repeated TG against their neighbours, accruing a cumulative payoff based on the same rules as theexperiment. For a comprehensive Monte Carlo simulation, we observed the averages of entrusted amount,returned amount, and payoff on the lattice of size L = 100 over 104 time steps, with the last 2 × 103
considered a stable state. All results were averaged over 1000 independent realizations for a fixed set offree agent fractions.

B.2.2

Theoretical analysis

The state of the system can be described by the frequency of agents using the entrusted amount Sx . Let
ρfx (t) (resp. ρcx (t)) denote the frequency of free agents (resp. constrained agents) entrusting Sx at time
step t. Here, we analyse the change in ρcx (t) and ρfx (t). For constrained and free agents, there may exist
2s types of neighbours; that is, {(ℓ, X)}ℓ∈{1,2,··· ,s},X∈{c,f } , where (ℓ, X) represents an X-neighbour using
the entrusted amount Sℓ .
We begin with a focal (x, c)-agent. The probability of a configuration with kℓc (ℓ, c)-neighbours and kℓf(ℓ, f )-neighbours is
s

kℓf
Y
k!
kc


((1 − F )ρcℓ (t)) ℓ F ρfℓ (t)sc fℓ=1 kℓ !kℓ ! ℓ=1
C kℓc , kℓf = Q

(B6)


Pc 
with ℓ=1 kℓc + kℓf = k. With such a neighbour configuration, the focal agent is expected to receive
entrusted amounts
s 


 1X
kℓc Sℓ + kℓf Sℓ(B7)
A kℓc , kℓf =kℓ=1

on average from a random neighbour. The returned amount from neighbours depends on the entrusted
amounts they receive. Using the mean-field approximation, an individual with a neighbour using Sx is
expected to receive
1
Ax =
k

Sx + (k − 1)(1 − F )

s
X

s
X
ρcℓ (t)Sℓ + (k − 1)F
ρfℓ (t)Sℓ

ℓ=1

!
(B8)

ℓ=1

from a random neighbour. The returned amount is thenRc = 3λc Ax(B9)

Rf = 3λf Ax(B10)

for a random constrained neighbour andfor a random free neighbour. For the focal agent with Sx , the returned amount averaged over all neighbours
is
s 


 3X
kℓc λc + kℓf λf Ax .
(B11)
R kℓc , kℓf =kℓ=1

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We define [ϕ] to be the label of the entrusted amount in set S closest to value ϕ (e.g. [Sℓ ] = ℓ). Thechange in ρcx (t) is given bydt ρcx (t) =XXρcy (t)fcℓ (kℓ +kℓ )=k

y̸=x

P

X

− ρcx (t)
P

=

X



C kℓc , kℓf pc[A kc ,kf ],[R kc ,kf ],x( ℓ ℓ) ( ℓ ℓ)fcℓ (kℓ +kℓ )=kXρcy (t)yX
C kℓc , kℓf
pc[A kc ,kf ],[R kc ,kf ],y( ℓ ℓ) ( ℓ ℓ)fcℓ (kℓ +kℓ )=k

(B12)

y̸=x



C kℓc , kℓf pc[A kc ,kf ],[R kc ,kf ],x − ρcx (t).
( ℓ ℓ) ( ℓ ℓ)P

Analogously, the change in ρfx (t) is given bydt ρfx (t) =

X

" s
X

ρfy (t)(1 − F )ρcz (t)pfz,[Rc ],x +

z=1

y̸=x

s
X
z=1


s
X
X f
− ρfx (t)  (1 − F )ρcz (t)pz,[Rc ],y +

z=1

=

X

ρfy (t)

" s
X

y

#
F ρfz (t)pfz,[Rf ],x

s
X


X f
F ρfz (t)pz,[Rf ],y 

z=1

y̸=x

(1 − F )ρcz (t)pfz,[Rc ],x +

z=1

s
X

(B13)

y̸=x

#
F ρfz (t)pfz,[Rf ],x − ρfx (t).

z=1

Numerical calculation of the above two sets of equations reveals the stationary distribution of ρcx and ρfx .
The entrusted amount and payoff of a constrained and free agent are separately
Ac =k
Af =k
c

W =k
W f =k

s
X
x=1
s
X
x=1
s
X
x=1
s
X

ρcx Sx ,ρfx Sx ,ρcxh− Sx + (3 − 3λc )hρfx − Sx + (3 − 3λf )

x=1

(B14)

s 
X

(1 − F )ρcℓ + F ρfℓiSℓ + 3 ((1 − F )λc + F λf ) Ax ,

ℓ=1
s 
X


i
(1 − F )ρcℓ + F ρfℓ Sℓ + 3 ((1 − F )λc + F λf ) Ax ,

ℓ=1

where
1
Ax =
k

Sx + (k − 1)s 
X

(1 − F )ρcℓ + F ρfℓ



!
Sℓ

.

(B15)

ℓ=1

The average entrusted amount by an agent is
A = (1 − F )Ac + F Af ,(B16)

and the average returned amount by an agent is
R = 3 [(1 − F )λc + F λf ] (1 − F )Ac + F Af .

(B17)

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Article Title

The average payoff obtained by an agent is
W = F W f + (1 − F )W c .

B.3

Ultimatum game

B.3.1

Numerical simulation(B18)

To perform the simulation based on the model outlined in the Methods section of the paper, we inferred
several aspects of the agents’ behaviour from the experimental data (Table C4 presents the variables
overview). Specifically, in the first stage of the game, we focused on the proposed amount and considered three intervals: S = {S1 , S2 , S3 }, where S1 ∈ [0, 0.6), S2 ∈ [0.6, 0.8), and S3 ∈ [0.8, 5.0]. In thesecond stage, we focused on whether the agents were satisfied with the neighbours’ proposals and defined
the acceptance threshold of the constrained and free agents, respectively, as the basis for the agent toaccept or reject the proposed amount from neighbours. We assumed that the agent’s proposal and acceptance in the current round depended on their own proposal and acceptance, including the neighbours’
proposal and the perceived acceptance of neighbours in the previous round. As a player cannot directly
obtain the real acceptance of the neighbour in the actual interaction, we assumed the agent evaluatedneighbours’ acceptance based on the neighbours’ response to proposals and, hence, defined the perceived
acceptance threshold of the neighbours. The constrained agent evaluated the perceived acceptance ofthe entire neighbourhood, while the free agent evaluated the perceived acceptance of each neighbour.
Constrained agents and free agents updated proposed amount and acceptance threshold following response rules (pcx1 ,x2 , pcx1 ,x2 ,y1 ,y2 ; z1 ,z2 ) and (pfx1 ,x2 , pfx1 ,x2 ,y1 ,y2 ; z1 ,z2 ), where x1 , x2 , y1 , y2 , z1 , z2 ∈ {1, 2, 3},
respectively. Therefore, we inferred the exact values corresponding to pcx1 ,x2 , pcx1 ,x2 ,y1 ,y2 ; z1 ,z2 , pfx1 ,x2 , andpfx1 ,x2 ,y1 ,y2 ; z1 ,z2 from the experimental data.
Initial strategies. For constrained agents (resp. free agents), we counted the frequency of each combination of the proposed amount Sx1 and acceptance threshold R̂x2 , denoted as pcx1 ,x2 (resp. pfx1 ,x2 ), themean of proposed amount, denoted as Sxc 1 ,x2 (resp. Sxf1 ,x2 ), and the mean of the acceptance threshold,denoted as R̂xc 1 ,x2 (resp. R̂xf 1 ,x2 ) in each case. Therefore, at time t = 0, it followed that:(i) For vi = 1, agent i selected a combination of the proposed amount intervals and acceptance threshold intervals with probability pfx1 ,x2 , proposed Sxf1 ,x2 towards their neighbour j, and assigned acceptanceR̂xf 1 ,x2 ;(ii) For vi = 0, agent i selected a combination of the proposed amount interval and acceptance threshold interval with probability pcx1 ,x2 , proposed Sxc 1 ,x2 towards their neighbours, and assigned acceptanceR̂xc 1 ,x2 .
Strategy updating. In the strategy updating process, all agents updated their proposed amount andacceptance threshold in each time step for all interactions. We measured from the experimental data
agents’ proposed amount and acceptance threshold response (x1 , x2 , y1 , y2 ; z1 , z2 ), where the combination contained 36 responses. Specifically, for constrained agents (resp. free agents), we counted thefrequency of each response, denoted as pcx1 ,x2 ,y1 ,y2 ; z1 ,z2 (resp. pfx1 ,x2 ,y1 ,y2 ; z1 ,z2 ), the corresponding average of proposed amount Sxc 1 ,x2 ,y1 ,y2 ; z1 ,z2 (resp. Sxf1 ,x2 ,y1 ,y2 ; z1 ,z2 ), and the average of acceptance threshold
Rxc 1 ,x2 ,y1 ,y2 ; z1 ,z2 (resp. Rxf 1 ,x2 ,y1 ,y2 ; z1 ,z2 ). Hence, it followed that:(i) For vi = 1, if agent i’s interval of the proposed amount and acceptance threshold was x1 and x2 , andneighbour j’s interval of the proposed amount and perceived acceptance threshold was y1 and y2 at time t,

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23

then agent i’ selected a response (x1 , x2 , y1 , y2 ; z1 , z2 ) with probability pfx1 ,x2 ,y1 ,y2 ; z1 ,z2 at time t + 1, proposed S f x1 ,x2 ,y1 ,y2 ; z1 ,z2 towards their neighbour j and assigned acceptance threshold R̂f x1 ,x2 ,y1 ,y2 ; z1 ,z2 ;(ii) For vi = 0, if agent i’s interval of proposed amount and acceptance threshold was x1 and x2 ,and the neighbours’ interval of proposed amount and perceived acceptance threshold was y1 and y2 attime t, then agent i selected a response (x1 , x2 , y1 , y2 ; z1 , z2 ) with probability pcx1 ,x2 ,y1 ,y2 ; z1 ,z2 , proposed
S c x1 ,x2 ,y1 ,y2 ; z1 ,z2 towards the neighbours, and assigned acceptance threshold R̂c x1 ,x2 ,y1 ,y2 ; z1 ,z2 .
Initialization. The simulation was initiated with both constrained and free agents randomly distributed on an L × L square lattice with a periodic boundary and a von Neumann neighbourhood. The
fraction of free agents was denoted F , where a set of F = 0.00 : 0.05 : 1.00 was investigated. Each agent
engaged in a repeated UG with their neighbours, accruing cumulative payoffs based on the same rules as
the experiment. For a comprehensive Monte Carlo simulation, we observed the averages of the proposed
amount, acceptance rate, and payoff on the lattice of size L = 100 over 104 time steps, of which the last
2 × 103 were deemed a stable state. All results were averaged over 1000 independent realizations for a
fixed set of free agent fractions.

B.3.2

Theoretical analysis

The state of the system can be described by the frequency of agents using the proposed amount Sx1 andresponse Sx2 , denoted by x = (x1 , x2 ). The set of all possible states is X = {(x1 , x2 )}x1 ,x2 ∈{1,2,··· ,s} . Let
ρf(x1 ,x2 ) (t) (resp. ρc(x1 ,x2 ) (t)) denote the frequency of free agents (constrained agents) using the proposed
amount Sx1 and response Sx2 at time step t. Here, we analyse the change in ρfx1 ,x2 (t) and ρcx1 ,x2 (t). For
both constrained and free agents, there may exist 2s2 types of neighbours; that is, {x, X)}x∈X ,X∈{c,f } ,where (x, X) represents an X-neighbour using the proposed amount Sx1 and response Sx2 .
We begin with a focal (x, c)-agent. The probability of a configuration with kvc constrained neighbours
and kvf free neighbours is

Y
kf
k!
kc


((1 − F )ρcv (t)) v F ρfv (t) v ,fcv∈X kv !kv ! v∈XC kvc , kvf = Q

(B19)


P
with x∈X kxc + kxf = k. For a constrained agent with state (x1 , x2 ) and configuration (kvc ,kvf ), theaverage proposed amount from the neighbours isv̄1 (v|x) =
1 X c
kv Sv1 + kvf Sv1 ,
k

(B20)

v∈X

and the perceived acceptance of neighbours isv̄2 (v|x) = Sx1 +

!

1 X c
f
kv + kv Θ(Sx1 − Sv2 ) ∆.
1−
k
v∈X

(B21)

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We define [ϕ] to be the label of the allocation in set S closest to value ϕ (e.g. [Sℓ ] = ℓ). The change inρcx (t) is then given bydt ρcx (t) =XXρcz (t)fcv∈X kv +kv =k

z̸=x

P

X

− ρcx (t)
P

=

X


C kvc , kvf pcz1 ,z2 ,[v̄1 (v|z)],[v̄2 (v|z)]; x1 ,x2
X c
px1 ,x2 ,[v̄1 (v|x)],[v̄2 (v|x)]; z1 ,z2fcv∈X kv +kv =kXρcz (t)z

C kvc , kvf

P

z̸=x

(B22)


C kvc , kvf pcz1 ,z2 ,[v̄1 (v|z)],[v̄2 (v|z)]; x1 ,x2 − ρcx (t).

fcv∈X kv +kv =kAnalogously, the change in ρfx , given bydt ρfx (t) =XXρfz (t)(1 − F )ρcv (t) + F ρfv (t) pfz1 ,z2 ,v1 ,[z1+(1−Θ(Sz −Sv ))∆]; x1 ,x2
1

z̸=x

v∈X

− ρfx (t)

X

1

v∈X

=

X
z

2

X f
(1 − F )ρcv (t) + F ρfv (t)px1 ,x2 ,v1 ,[x1 +(1−Θ(Sx −Sv ))∆]; z1 ,z2Xρfz (t)

2

(B23)

z̸=x


(1 − F )ρcv (t) + F ρfv (t) pfz1 ,z2 ,v1 ,[z1+(1−Θ(Sz −Sv ))∆]; x1 ,x2 − ρfx (t).
1

2

v∈X

We then have the stationary distribution of ρcx and ρfx . The average proposed amount (P), acceptance(R, which theoretically approximates the rejected amount), and payoff (Q) in an individual interaction
are respectively
P =FX

ρfx Sx1 + (1 − F )

x∈X

R=F
Q=

X

X

ρcx Sx1 ,x∈X

ρfx Sx2 + (1 − F )

X

x∈X

x∈X

5 X
k

F ρfx + (1 − F )ρcx

ρcx Sx2 ,(B24)F ρfy + (1 − F )ρcy Θ(Sx1 − Sy2 ).

x,y∈X

Note that we set the constrained and free agents to have the same proposal-acceptance set S. Analogously,
we can analyse the case where constrained and free players have possibly different proposal and acceptancesets. As adopted in this study, for a constrained player (resp. free player) with state x, their proposed
amount and acceptance are respectively S1c (x) and S2c (x) (resp. S1f (x) and S2f (x)).

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Appendix C

25

Supplementary tables

Table C1 Basic demographic information on volunteers

Game

Prisoner’s dilemma

Trust game

Ultimatum game

Treatment

RoundsPlayers/Session

Sessions

Players

Women (%)

Age (SD)0% free

50

49

3

147

51

19.46 (1.22)

25% free

50

49

3

147

54

19.30 (1.04)

50% free

50

49

3

147

48

19.52 (1.18)

75% free

50

49

3

147

46

19.18 (1.06)

100% free

50

49

3

147

48

19.24 (1.10)

0% free

50

49

3

147

63

20.01 (0.97)

25% free

50

49

3

147

71

19.54 (1.10)

50% free

50

49

3

147

69

19.71 (1.07)

75% free

50

49

3

147

31

19.97 (0.94)

100% free

50

49

3

147

65

19.44 (1.09)

0% free

50

49

3

147

41

20.03 (0.89)

25% free

50

49

3

147

52

19.96 (0.87)

50% free

50

49

3

147

60

19.93 (0.83)

75% free

50

49

3

147

59

20.00 (0.88)

100% free

50

49

3

147

59

20.24 (0.73)

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Article Title

Table C2 Parameters in the prisoner’s dilemma game

1.1 First round strategy probabilities
constrainedpc

The probability that constrained agents choose cooperation

free

pf

The probability that free agents choose cooperation

1.2 Strategy updating probabilities

constrained

free

pcCkC

The probability that constrained agents choose to cooperate in the next round
if constrained agents cooperating and there are kC cooperative neighbors

pcDkC

The probability that constrained agents choose to cooperate in the next round
if constrained agents defecting and there are kC cooperative neighbors

pfC

The probability that free agents choose to cooperate in the next round if theirneighbors are cooperating

pfD

The probability that free agents choose to cooperate in the next round if theirneighbors are defecting

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27

Table C3 Parameters in the trust game

2.1 Entrusted amount in the first stage of the game
2.1.1 Entrusted amount intervals (S1 , S2 )2.1.2 The first round frequency of entrusted amounts in intervals S1 and S2constrained{pcx }x∈{1,2}

The frequency of constrained agents’ entrusted amounts in intervals S1 and S2

free{pfx }x∈{1,2}

The frequency of free agents’ entrusted amounts in intervals S1 and S2

2.1.3 The first round mean of entrusted amount in each interval
constrained{Sxc }x∈{1,2}

The mean of constrained agents’ entrusted amount in each interval

free{Sxf }x∈{1,2}

The mean of free agents’ entrusted amount in each interval

2.1.4 Entrusted amount response{(x, y)}x,y∈{1,2}The entrusted amount Sy agents choose at time t + 1 in response to the entrusted amount Sx received from their neighbours at time t

2.1.5 Strategy updating probabilities (response rules) for the first stage of the game
constrained

pcx,yThe probability that constrained agents choose the entrusted amount withinterval Sy in response to the neighbours’ entrusted amount with interval Sx

free

pfx,y

The probability that free agents choose the entrusted amount with interval Sy
in response to the neighbours’ entrusted amount with interval Sx

2.1.6 Strategy updating mean (response rules) for the first stage of the game
constrainedcSx,yThe mean of constrained agents’ chosen entrusted amount in interval Sy in
response to the neighbours’ entrusted amount with interval Sx

free

f
Sx,y

The mean of free agents’ chosen entrusted amount in interval Sy in responseto the neighbours’ entrusted amount with interval Sx

2.2 Returned amount in the second stage of the game
2.2.1 Fixed return ration for constrained and free agents
constrainedλc

The fixed return ration of constrained agents

free

λf

The fixed return ration of free agents

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Table C4 Parameters in the ultimatum game

3.1 Proposed amount and acceptance threshold
3.1.1 Proposed amount intervals (S1 , S2 , S3 )3.1.2 The first round frequency of each combination of the proposed amount and acceptance thresholdconstrained

pcx1,x2

The probability that constrained agents choose proposed amount Sx1 andacceptance threshold Sx2

free

pfx1,x2

The probability that free agents choose proposed amount Sx1 and acceptancethreshold Sx2

3.1.3 The first round mean of the proposed amountconstrained

Sxc 1 ,x2

The mean of constrained agents’ proposed amount if the constrained agent
chooses proposed amount Sx1 and acceptance threshold Sx2

free

Sxf1 ,x2

The mean of free agents’ proposed amount if the free agent chooses proposed
amount Sx1 and acceptance threshold Sx2

3.1.4 The first round mean of the acceptance thresholdconstrainedR̂xc 1 ,x2

The mean of constrained agents’ acceptance threshold if the constrained agent
chooses proposed amount Sx1 and acceptance threshold Sx2

freeR̂xf 1 ,x2

The mean of free agents acceptance threshold if the free agent chooses proposed
amount Sx1 and acceptance threshold Sx2

3.1.5 Proposed amount and acceptance threshold response{(x1, x2, y1, y2; z1, z2)}
x1, x2, y1, y2, z1, z2 ∈ {1, 2, 3}

Agent’s proposal x1 and acceptance x2 in the previous round, neighbours’
proposal y1 and perceived acceptance y2 in the previous round, and agent’s
proposal z1 and acceptance z2 in the current round3.1.6 Strategy updating frequency of each response
constrained

pcx1,x2,y1,y2; z1,z2

The probability of each response for constrained agents

free

pfx1,x2,y1,y2; z1,z2

The probability of each response for free agents

3.1.7 Strategy updating average of the proposed amount of each response
constrained

S c x1 ,x2 ,y1 ,y2 ; z1 ,z2

The average of the proposed amount of each response for constrained agentsfree

S f x1 ,x2 ,y1 ,y2 ; z1 ,z2

The average of the proposed amount of each response for free agents

3.1.8 Strategy updating average of acceptance threshold of each response
constrainedR̂c x1 ,x2 ,y1 ,y2 ; z1 ,z2

The average of the acceptance threshold of each response for constrained agentsfreeR̂f x1 ,x2 ,y1 ,y2 ; z1 ,z2

The average of the acceptance threshold of each response for free agents