Social norms of fairness with reputation-based role assignment in the dictator game Qing Li1Songtao Li1Yanling Zhang1aXiaojie Chen2band Shuo Yang1

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Social norms of fairness with reputation-based role assignment in the

dictator game

Qing Li,1Songtao Li,1Yanling Zhang,1, a) Xiaojie Chen,2, b) and Shuo Yang1

1)Key Laboratory of Knowledge Automation for Industrial Processes of Ministry of Education,

School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083,

China

2)School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731,

China

(Dated: 25 October 2022)

A vast body of experiments share the view that social norms are major factors for the emergence of fairness in a

population of individuals playing the dictator game (DG). Recently, to explore which social norms are conducive

to sustaining cooperation has obtained considerable concern. However, thus far few studies have investigated how

social norms inﬂuence the evolution of fairness by means of indirect reciprocity. In this study, we propose an indirect

reciprocal model of the DG and consider that an individual can be assigned as the dictator due to its good reputation.

We investigate the ‘leading eight’ norms and all second-order social norms by a two-timescale theoretical analysis. We

show that when role assignment is based on reputation, four of the ‘leading eight’ norms, including stern judging and

simple standing, lead to a high level of fairness, which increases with the selection intensity. Our work also reveals that

not only the correct treatment of making a fair split with good recipients but also distinguishing unjustiﬁed unfair split

from justiﬁed unfair split matters in elevating the level of fairness.

Fair behavior among unrelated individuals has been found

in a quantity of DG experiments. In the DG, the dictator

is assigned to unilaterally divide a given resource between

her/himself and the recipient, who unconditionally accepts

the division. Making a fair split is remarkable because it is

costly for the dictator, but it is beneﬁcial for the recipient.

As such, making a fair split is in contradiction with the

prediction of standard game theory, which suggests that

entirely rational dictators should dispense nothing to re-

cipients. Evolutionary game theory provides a theoretical

framework for explaining the mismatch between experi-

mental observations and game theory. The explanation of

the mismatch has received ample attention in the recent

past. In our paper, we extend the scenario of the random

role assignment in the classic evolutionary DG model and

take into account the reputation-based role assignment,

namely, an individual with good reputation plays the role

of dictator when her/his opponent is bad. Our research

reveals that the reputation-based role assignment can lead

to a high level of fairness for four of the ‘leading eight’

norms, including stern judging and simple standing, and

that the level of fairness increases with the selection inten-

sity. These results can provide some insights for a better

understanding of the emergence of fairness in the realistic

scenario of reputation.

I. INTRODUCTION

A quantity of canonical experiments using economic games

demonstrated that people tend to exhibit prosocial behaviors

a)yanlzhang@ustb.edu.cn

b)xiaojiechen@uestc.edu.cn

even in one-shot anonymous interactions1. Fair behavior is

one of the most important prosocial behaviors in human soci-

ety, and it is an important undertaking for many social dilem-

mas, such as accessibility dilemma of antibiotics2and stale-

mates in climate talks3. However, how to understand the

emergence of fair behavior among unrelated individuals re-

mains a challenge. As one typical paradigm, the DG has been

often used to study the evolution of fair behavior4. In the

game, the recipient must accept whatever the dictator deliv-

ers, and then the choice of the dictator is not inﬂuenced by the

retributive motive of the recipient5.

A large DG experiment6, which was administered across 15

diverse societies, showed that dictators offered, on average,

37% of the total resource to recipients with a range from 26%

to 47%. However, a meta-analysis of DG experiments7found

a mean offer of 28%. Those results deviate signiﬁcantly from

the prediction by game theory that entirely rational dictators

(exclusively driven by the maximization of their own mon-

etary payoffs) ought to dispense nothing to recipients. One

classic explanation for the mismatch between the theoretical

prediction and experimental observations is about social pref-

erence8, under which an individual’s utility function depends

on not only his own monetary payoff, but also the monetary

payoff of the other one involved in the DG, e.g., the inequity

aversion utility9.

However, a series of experimental ﬁndings cannot be ex-

plained by any utility functions that are based solely on mon-

etary outcomes10, and thus the explanation about social pref-

erences is criticized. Despite the fact that the monetary payoff

outcome of giving or keeping is identical when playing the

DG is compulsory and optional, giving is more likable than

keeping in the former and the verse holds in the latter11,12. A

similar result was also found when the choice set of dictators

is extended13. These results mentioned above imply that be-

yond the social preference, fair behavior reﬂects an intrinsic

desire to adhere to social norm14,15, which is deﬁned as ‘com-

arXiv:2210.12930v1 [cs.GT] 24 Oct 2022

monly known standards of behavior that are based on widely

shared views how group members ought to behave in a given

situation’.

Evolutionary game theory, where one strategy with higher

payoff is more likely to spread among the population16–20,

provides a theoretical framework for studying the effect of

social norm on the evolution of fairness. Under this frame-

work, a social norm is usually enforced by a reputation sys-

tem, where the cost of complying with the social norm can be

efﬁciently reduced21. Here, the social norm works as a top-

down mechanism that impacts the bottom-up behaviors indi-

rectly by generating a reputation uplift/downgrade, underlying

indirect reciprocity22. Note that indirect reciprocity has been

found to be a fundamental mechanism for the evolution of co-

operation in the donation game23–25. Then, a natural question

is how fair behavior in the DG is inﬂuenced by indirect reci-

procity.

In this work, we thereby address the emergence and mainte-

nance of fairness by an indirect reciprocal model. We consider

the random role assignment for individuals when they play

the DG with each other. Furthermore, note that in the real

society, a person with good reputation is more likely to vol-

unteer his time to work at an NGO, donate money to charity,

or give money to a homeless person on the street. Thus, be-

sides the random role assignment, we also investigate a way of

reputation-based role assignment for individuals. In addition,

the widely investigated social norms for the evolution of co-

operation use the ﬁrst-order information (only the action), the

second-order information (the recipient’s reputation and the

action), or the third-order information (the two participants’

reputation and the action) to assess the actor’s reputation. In

an exhaustive research of the third-order social norms26, eight

social norms, called as the ‘leading eight’ norms, were found

to maintain cooperation. In this paper, we concentrate on the

‘leading eight’ norms together with all possible second-order

social norms, and study which social norms can promote the

evolution of fairness in our indirect reciprocal model.

II. MODEL

We consider a ﬁnite well-mixed population with population

size Zand each player in the population is assigned with a bi-

nary reputation, good or bad. Any two players are randomly

chosen from the population to engage in a DG, where one is

the dictator and the other one is the recipient. The 50 −50 di-

vision is widely observed in economic environments of the

real world and the laboratory27. Accordingly we consider

a simpliﬁed version of the DG, where the dictator only has

two optional strategies, 50 −50 division (fair split is abbrevi-

ated as F) and 100 −0 division (unfair split is abbreviated as

N). Since the DG includes two roles of dictator and recipi-

ent, two ways of role assignment will be investigated, that is,

the random role assignment and the reputation-based role as-

signment (see the illustrative plot in Fig. 1). In the random

role assignment, the two participants have the equal possibil-

ity of becoming the dictator. While in the reputation-based

role assignment, the individual with good reputation plays the

FIG. 1. The DG under indirect reciprocity. (A) Pairs of players are

randomly chosen from the well-mixed population to play the DG,

in which the dictator can make a 50 −50 division or a 100 −0 di-

vision with a recipient. Then a third player is randomly chosen as

the observer to report the dictator’s reputation or not. (B) Two ways

of role assignment. In the random role assignment, the two partici-

pants have the same possibility of becoming the dictator. Yet in the

reputation-based role assignment, the good plays the role of dictator

when he interacts with the bad; two players with the same reputation

are randomly chosen as the dictator or the recipient.

role of dictator when the opponent is a bad individual; two

players with the same reputation are randomly chosen as the

dictator or the recipient. In addition, each interaction can be

witnessed by a randomly chosen third player. After observing

and assessing the interaction, the observer chooses to report

the outcome (abbreviated as R) or to be silent (abbreviated as

S). Note that reporting means that the observer shares the dic-

tator’s reputation with all other players, and accordingly bears

a personal cost cR.

Here, we denote the strategy of each player in the game

by a three-letter string sGsBsR, which depicts (1) whether the

player makes a fair division with a good recipient (sG=F)

or not (sG=N)if he ﬁnds himself in the role of dictator, (2)

whether the player makes a fair division with a bad recipient

(sB=F)or not (sB=N)if in the role of dictator, and (3)

whether the player reports (sR=R)or not (sR=S)if in the

role of observer. Following the common practice28, an im-

plementation error is also considered, meaning that a dictator

fails to act fairly when he intends to be fair with probability ε.

The observer uses the social norm to assess the dictator’s rep-

utation. In this work, we consider the ‘leading eight’ norms

and all second-order social norms. Assume that a third-order

social norm is represented by two four-dimensional vectors

SG= (FG,FB,NG,NB)and SB= (FG,FB,NG,NB), which de-

note two cases where the previous reputation of the dictator

is good or bad. Regarding each entry of SGand SB, we set

LM=1, which means that the dictator is labelled with a good

credit when he takes action Lagainst a recipient of reputation

M, and analogously LM=0, which means that the correspond-

ing dictator is regarded as bad. Particularly, a second-order

social norm satisﬁes SG=SB.

After individuals play the game and obtain the payoffs, they

will take the strategy imitation by the pairwise comparison

rule29. To be speciﬁc, in each generation two players are ran-

domly chosen to become the focal one and the model one,

respectively. With a small probability µ, a mutation occurs,

meaning that the focal one randomly adopts one of all candi-

date strategies. Otherwise (with probability 1 −µ), the focal

one xhas an opportunity of imitation, and he imitates the strat-

egy of the model one ywith the following probability

p(x→y) = 1/(1+e−β(gy−gx)),

where βdenotes the intensity of selection, gxand gyare

the payoffs of xand y, respectively. The βstands for how

closely the strategy dynamics relies on the outcome of inter-

actions30–32. For weak selection β→0, the update slightly

depends on the payoffs of individuals and randomness mat-

ters in the strategy selection.

III. METHODS

In this study, we assume that the time scale for strategy se-

lection is much slower than the one for reputation update22,33.

Accordingly, we ﬁrst calculate the steady-state distribution of

the reputation system and the expected payoff of each strat-

egy by considering that the strategies of all players are ﬁxed.

We then compute the ﬁxation probability between any pair of

strategies and the steady-state frequency of each strategy by

allowing players to update their strategies over time.

Given that the strategies of all players are ﬁxed, the rep-

utation system can be described by a Markov chain. When a

population consists of mplayers with strategy X=sX

GsX

BsX

Rand

Z−mplayers using strategy Y=sY

GsY

BsY

R, the state of the rep-

utation system is denoted by a two-dimensional vector (i,j),

which means that there are i∈0,1,··· ,m(j∈0,1,··· ,Z−m)

good players among m X-players (among Z−m Y -players).

Given that the reputation state is (i,j)at time t, we can calcu-

late the transition probability P(i,j;i0,j0), which characterizes

how likely the population will be in state (i0,j0)at time t+1

(see Appendix A for the detailed calculation).

Let V= (v(i,j)) be the steady-state distribution of the repu-

tation system with the transition matrix P= (P(i,j;i0,j0)), and

accordingly we have V P =Vand ∑i,jv(i,j) = 1. Here each

entry v(i,j)means the expected frequency of the state (i,j)

which we observe over the course of the reputation dynam-

ics. Note that the expected payoffs of Xand Yin a population

with m X−players and Z−m Y −players, gX(m)and gY(m),

are respectively calculated as

gX(m) = ∑m

i=0∑Z−m

j=0v(i,j)πX(m,i,j),

gY(m) = ∑m

i=0∑Z−m

j=0v(i,j)πY(m,i,j),(1)

where πX(m,i,j)and πY(m,i,j)are the expected payoffs of X

and Ywhen there are igood players among m X-players and

jgood players among Z−m Y -players (see Appendix B for

their expressions).

We now focus on how players change their strategies over

time, which occurs on a much slower time scale than the

reputation update. In the limit of low mutation µ→0,

the strategy dynamics can be described by an embedded

Markov Chain34. The state space includes eight monomor-

phic populations, each of which adopts one of FFR,FFS,

FNR,FNS,NFR,NFS,NNR, and NNS. The correspond-

ing transition matrix is A= (aXY )8×8, where aXY can be

expressed by the ﬁxation probability between any pair of

strategies (see Appendix C for the detailed expression). The

normalised left eigenvector Φof the stochastic matrix A

associated with the eigenvalue 1 satisﬁes Φ=ΦA. Here

Φ= (φFFR,φFFS,φFNR,φFNS,φNF R,φNFS,φNNR,φNNS)is the

steady-state distribution of the strategy dynamics, where φXis

the fraction of time spent in each monomorphic population of

X. Accordingly by assuming that fF(X)is the fairness level of

the monomorphic population of X, we deﬁne the total level of

fairness FFas the weighted average of fF(X)with the weight

φX,

FF=∑

φXfF(X).(2)

In the monomorphic population of X, we assume that

pF(i|X)is the probability for a player to make a fair division

in the role of dictator or to receive a fair division in the role

of recipient when there are igood players. Then fF(X)is the

weighted average of pF(i|X)with the weight v(i|X), which is

the proportion of time spent in a state of igood players in a

monomorphic population of X. Accordingly, fF(X)is given

fF(X) = ∑

v(i|X)pF(i|X).(3)

For unconditionally fair/unfair strategies, fF(X)is indepen-

dent of v(i|X), i.e., fF(X) = pF(i|X)for X=NNS,NNR,

FFS, or FFR. Then, we have fF(X) = 0 for X=NNS or

NNR and fF(X) = 1−εfor X=FFS and FFR because an

Xdictator is always unfair or always fair except the imple-

menting error, irrespective of the opponent’s reputation. In a

monomorphic population of NFS or FNS, individuals cannot

obtain the reputation information of opponents. Then we as-

sume that they act fairly with a ﬁxed probability and let the

probability be only 0.1 for simplicity. Therefore, we have

fF(X) = pF(i|X) = 0.1 for X=NFS and FNS, suggesting

that NFS and FNS have little effect on the total level of fair-

ness. Different from the above six strategies, fF(FNR)and

fF(NFR)depends closely on v(i|X), which is equal to v(i,0)

in a monomorphic population of X. In addition, according to

the deﬁnition of pF(i|X), we have pF(i|X)for X=NFR or

FNR as follows

pF(i|X) = (1−ε)I(sX

G)i

Z+ (1−ε)I(sX

B)Z−i

when role assignment is random,

pF(i|X)=(1−ε)i(i−1)

Z(Z−1)IsX

G+Z2−Z−i2+i

Z(Z−1)IsX

B,

when role assignment is based on reputation,

(4)

where I(x) = 1 for x=Fand I(x) = 0 for x=N.

IV. RESULTS

A. Evolutionary outcomes for four typical social norms

We study the level of fairness by numerically calculat-

ing the stationary distribution of the strategies and the one

of the reputation system in each monomorphic population.

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Socialnormsoffairnesswithreputation-basedroleassignmentinthedictatorgameQingLi,1SongtaoLi,1YanlingZhang,1,a)XiaojieChen,2,b)andShuoYang11)KeyLaboratoryofKnowledgeAutomationforIndustrialProcessesofMinistryofEducation,SchoolofAutomationandElectricalEngineering,UniversityofScienceandTechnologyBeijing,B...

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Social norms of fairness with reputation-based role assignment in the dictator game Qing Li1Songtao Li1Yanling Zhang1aXiaojie Chen2band Shuo Yang1

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