Department of Mathematical Informatics, The University of Tokyo, Bunkyo, Tokyo 113-8656, 7-3-1 Hongo, Japan

Abstract

Background

Indirect reciprocity is a mechanism for cooperation in social dilemma situations. In indirect reciprocity, an individual is motivated to help another to acquire a good reputation and receive help from others afterwards. Another aspect of human cooperation is ingroup favoritism, whereby individuals help members in their own group more often than those in other groups. Ingroup favoritism is a puzzle for the theory of cooperation because it is not easily evolutionarily stable. In the context of indirect reciprocity, ingroup favoritism has been shown to be a consequence of employing a double standard when assigning reputations to ingroup and outgroup members. An example of such a double standard is the situation in which helping an ingroup member is regarded as good, whereas the same action toward an outgroup member is regarded as bad.

Results

We analyze a computational model of indirect reciprocity in which information sharing is conducted groupwise. In our model, individuals play social dilemma games within and across groups, and the information about their reputations is shared within each group. We show that evolutionarily stable ingroup favoritism emerges even if all the players use the same reputation assignment rule regardless of group (i.e., a single standard). Two reputation assignment rules called simple standing and stern judging yield ingroup favoritism; under these rules, cooperation with (defection against) good individuals is regarded as good (bad) and defection against bad individuals is regarded as good. Stern judging induces much stronger ingroup favoritism than does simple standing. Simple standing and stern judging are evolutionarily stable against each other when groups employing different assignment rules compete and the number of groups is sufficiently large. In addition, we analytically show as a limiting case that homogeneous populations of reciprocators that use reputations are unstable when individuals independently infer reputations of individuals, which is consistent with previously reported numerical results.

Conclusions

Our results suggest that ingroup favoritism can be promoted in indirect reciprocity by the groupwise information sharing, in particular under the stern judging assignment rule.

Background

Behavioral nature of humans depends on the economy of reputations, where praise and blame often lead to gain and loss of material benefits

Another facet of human cooperation is that an individual often cooperates with members in the same group and not with others, a phenomenon called ingroup favoritism

If maintaining a good reputation is a concern, why do individuals want to discriminate between ingroup and outgroup fellows? One of the present authors has shown that ingroup favoritism is evolutionarily stable in various situations when only group-level reputations are available in regard to outgroup members

In the context of indirect reciprocity, group structure may play a crucial role in spreading reputations of individuals via rumor and gossip. In general, individuals interact more frequently with ingroup members than with outgroup members

In the present study, we explore a scenario of ingroup favoritism without resorting to rules that apply double standards. In practice, humans may not differentiate between ingroup and outgroup coplayers with regard to their action rules or reputation assignment rules. We analyze a group-structured model of indirect reciprocity, in which an individual’s reputation is shared by each group but not between groups. We study the case in which all the players use the same reputation assignment rule and the case in which players in different groups use different reputation assignment rules. We show that ingroup favoritism can emerge when players simply implement reputation-based decision making and do not favor ingroup members. Because of the assumed groupwise information sharing and some reputation assignment error, ingroup and outgroup members tend to possess good and bad reputations, respectively, without further assumptions. In particular, ingroup favoritism is strong when individuals adopt a reputation assignment rule called stern judging, under which helping bad individuals is regarded as bad.

Methods

Model

We consider an infinitely large population of players divided into

To know a recipient’s reputation, the donor consults the unique information source, called the observer, that is shared by the group to which the donor belongs. Therefore, players in different groups may perceive different reputations (i.e., G or B) of the same player. The observer in each group independently assigns a reputation to the donor and shares it with the other players in the observer’s group. Observers intend the predefined reputation assignment toward a donor’s action but may assign a reputation opposite to the intention. The

Behavior of different observers in different groups (

**Behavior of different observers in different groups (**
**
M
**

Observers assign reputations according to a common reputation assignment rule unless otherwise stated. We principally compare three rules: image scoring (IM), simple standing (ST), and stern judging (JG)

Three reputation assignment rules

**Three reputation assignment rules.** Image scoring (IM), simple standing (ST), and stern judging (JG). The rows represent the donor’s actions (i.e., C and D), the columns represent the recipient’s reputations (G and B), and G and B inside the boxes represent the reputations that observers assign to the donor.

After sufficiently many rounds of the donation game involving reputation updates, the reputation distribution in the eyes of each group-specific observer reaches a unique equilibrium. In the equilibrium, we measure the quantities of interest such as the fractions of G players, the probability of cooperation, and their dependence on groups.

Analysis methods

Equilibria of the reputation dynamics

Table

**Symbol**

**Meaning**

Number of groups

Probability that a donor and recipient in a one-shot game are in the same group

**
r
**∈{G, B}

Reputation vector of a player in the eyes of

_{
k
}(**
r
**)

Probability that a player in group **
r
**

_{−k
}(**
r
**)

Probability that a player outside group **
r
**

Donor’s action to a recipient having reputation

Φ_{
r
}(^{
′
})

Probability that an observer assigns reputation ^{
′
}∈{C, D}

_{in}(

Probability that a player in the eyes of an ingroup observer has reputation

_{out}(

Probability that a player in the eyes of an outgroup observer has reputation

We examine the stability of a homogeneous population of DISC players. Each player bears a reputation vector, **
r
**=(

The summation on the right-hand side of Eq. (1) represents the average over the recipient’s reputation vector **
r
**

**Rule**

**Φ**
_{
G
}**(C, G)**

**Φ**
_{
G
}**(D, G)**

**Φ**
_{
G
}**(C, B)**

**Φ**
_{
G
}**(D, B)**

Φ_{G}(_{B}(_{G}(

IM

1−

1−

ST

1−

1−

1−

JG

1−

1−

We reduce Eq. (1) to mean field dynamics of two reputation distributions. First, we apply summation

where
**(b)**. With probability _{in}(^{
′
}), the recipient belongs to the donor and observer’s group, and has reputation ^{
′
} (Figure
_{out}(^{
′
}), the recipient does not belong to the donor and observer’s group, and has reputation ^{
′
}(Figure

Five possible situations of the reputation update

**Five possible situations of the reputation update.** Observations are made by ingroup observers in **(a)** and **(b)**, and by outgroup observers in **(c)**, **(d)**, and **(e)**.

Second, by applying summation

where

The three terms inside the curly brackets on the right-hand side of Eq. (4) correspond to the three situations shown in Figure
**(d)**, and **(e)**. With probability _{in}(^{
′
})_{out}(^{
′′
}), the recipient belongs to the donor’s group, which differs from the observer’s group, and has reputation ^{
′
} and ^{
′′
}in the eyes of the donor and observer, respectively (Figure
_{out}(^{
′
})_{in}(^{
′′
}), the recipient belongs to the observer’s group, which differs from the donor’s group, and has reputation ^{
′
} and ^{
′′
}in the eyes of the donor and observer, respectively (Figure
_{out}(^{
′
})_{out}(^{
′′
}), the recipient belongs to a group different from the donor’s and observer’s groups, and has reputation ^{
′
} and ^{
′′
} in the eyes of the donor and observer, respectively (Figure

By setting d_{in}(_{out}(**
J
**> 0 and Tr

Stability against invasion by ALLC and ALLD mutants

We check the evolutionary stability of a homogeneous population composed of DISC players against invasion by an infinitesimal fraction of mutants adopting ALLC or ALLD. The payoff to a DISC resident player is given by

and those to ALLC and ALLD mutants are given by

and

respectively. In Eqs. (6) and (7),

and

where

Cooperativeness

DISC donors cooperate exclusively with G recipients. Therefore, in each stable equilibrium, the probability of cooperation, which we call the cooperativeness, toward ingroup and outgroup recipients is given by

Measurement of ingroup bias

To quantify the degree of ingroup bias, we measure the difference between ingroup and outgroup cooperativeness, defined by

When

Results

Table

**Rule**

**Stability condition**

**
ψ
**

**
ρ
**

IM

Unstable

0

ST

1−

Eq. (14)

JG

1−

Eq. (18)

IM

Under IM, the equilibrium fractions of G players in the eyes of ingroup and outgroup observers are both equal to

ST

Under ST, DISC players almost always cooperate with ingroup recipients, i.e.,

The fraction of G players in the eyes of outgroup observers is given by

Therefore, DISC players almost always cooperate with both ingroup and outgroup recipients unless ^{2})). Because donors defect slightly more often against outgroup than ingroup recipients, weak ingroup favoritism occurs (i.e., ^{2})).

Equations (5), (6), and (7) yield the payoff differences given by

and

Therefore, the stability condition (Eq. (8)) reads

ALLC mutants invade DISC players if

JG

Under JG, DISC players have the same cooperativeness toward ingroup recipients as under ST, i.e.,

The fraction of G players in the eyes of outgroup observers is given by

Therefore, DISC players cooperate with outgroup recipients with probability 1/2. In contrast to the case of ST, frequent intergroup interaction considerably reduces cooperation under JG (i.e.,

The payoff differences are given by

and

The stability condition reads

The DISC population is resistant to invasion by ALLC mutants when ^{2}, ALLD mutants invade the population of DISC players. The cooperation is stable down to a small value of

Under both ST and JG, in particular JG, ingroup favoritism emerges. This is because the donors (equivalently, ingroup observers) and outgroup observers generally perceive different reputations of the same player due to the assignment error (see Figures
**(d)**, and 3**(e)**). For example, if a donor defects against a recipient whose reputation is B in the eyes of the donor’s group members, the donor receives a G reputation from the ingroup observer. However, if the same recipient has a G reputation in the eyes of the outgroup observer, the outgroup observer assigns B to the donor under ST and JG. As another example, if a donor cooperates with a recipient whose reputation is G in the eyes of the donor’s group members, the donor receives G from the ingroup observer. However, if the recipient has a B reputation in the eyes of the outgroup observer, the outgroup observer assigns B to the donor under JG. As these examples suggest, different groups may perceive the opposite reputations of the same players in a long run. Players in the same group coordinate the subjective information about a given player’s reputation, whereas those in different groups do not. This discrepancy causes ingroup favoritism.

Numerical results

We compare the theoretical results with numerical results obtained from individual-based simulations in Figure

Equilibria for a population of DISC players under ST and JG

**Equilibria for a population of DISC players under ST and JG. ****(a)** Cooperativeness (**(b)** ingroup bias (

We also examine the error-prone case in which donors fail to help recipients (i.e., select D when the donors intend C) with probability

Equilibria for a population of DISC players under action implementation error

**Equilibria for a population of DISC players under action implementation error. ****(a)** Cooperativeness (**(b)** ingroup bias (

Mixed assignment rules

We have shown that JG leads to strong ingroup favoritism, whereas ST does not. To examine the transition between the two regimes, we consider an assignment rule denoted by MX, which is a mixture of JG and ST. In a one-shot game under MX, observers independently assign reputations by using JG with probability _{G}(C,_{G}(D,_{G}(C,_{G}(D,

The results under MX are shown in Figure
**(f)** indicate the values of

Equilibria and the stability conditions for a population of DISC players under MX

**Equilibria and the stability conditions for a population of DISC players under MX. ****(a)** Cooperativeness (**(b)** ingroup bias (**(a)** and **(b)**, we set (**(c)**–**(f)** Stability conditions. The homogeneous population of DISC players is stable in the shaded parameter regions. We set **(c)**(**(d)**(**(e)**(**(f)**(

Heterogeneous assignment rules

We have assumed that all the groups use a common reputation assignment rule. In this section, we numerically examine a case in which observers in different groups use different reputation assignment rules. We consider a situation in which

Numerically obtained equilibria with **(b)**, respectively. As the number of JG groups (i.e., _{ST} and _{JG} for ST and JG groups, respectively) decreases, and ingroup bias (_{ST} and _{JG} for ST and JG groups, respectively) increases. Figure
**(d)** shows the difference between the payoff to a player in a ST group and that to a player in a JG group (i.e., _{JG}−_{ST}) when _{JG}−_{ST} is positive. Therefore, if observers update their assignment rules according to an evolutionary dynamics (e.g., group competition
_{JG}−_{ST} is positive when _{JG}−_{ST} is negative only when

Equilibria for a population of DISC players under heterogeneous assignment rules

**Equilibria for a population of DISC players under heterogeneous assignment rules. ****(a)**, **(b)** Cooperativeness (_{ST}and _{JG}) and ingroup bias (_{ST} and _{JG}) for groups employing ST and JG. **(c)**, **(d)** Payoff difference between a player in a ST group and that in a JG group (_{JG}−_{ST}). We set **(a)** and **(c)**, **(b)** and **(d)**, and vary the number of JG groups (i.e.,

Discussion

In the present study, we showed that ingroup favoritism emerges in a group-structured model of indirect reciprocity. In our model, players share information about reputations in each group but not across different groups. We assumed that a player’s action purely depends on the coplayer’s reputation; players do not refer to the group identity of the coplayers or use other types of prejudice. We also assumed that observers impartially assess ingroup and outgroup donors. We analyzed the model using a mean-field approximation and numerical simulations. Ingroup favoritism occurs under both simple standing (ST) and stern judging (JG) assignment rules. The cooperativeness is reduced by the frequent intergroup interactions, i.e., small

Different mechanisms govern ingroup favoritism in our model and that observed in psychological experiments

We implemented the group structure by controlling probabilities of ingroup and outgroup interactions (i.e.,

In public reputation models, all the players have access to a common information source that provides the reputation values of the players. Therefore, a donor and observer perceive the same reputation of a recipient such that they do not suffer from the discrepancy of reputations. In public reputation models without group structure of the population, ST and JG realize evolutionarily stable cooperation

In private reputation models, each player individually collects others’ reputations such that a reputation of a player varies between individuals. In contrast to the case of public reputation models, a homogeneous population of discriminators is invaded by unconditional cooperators in private reputation models. A mixture of discriminators and unconditional cooperators is often stable under variants of ST

For intermediate

One of the present authors previously studied a model of ingroup favoritism on the basis of indirect reciprocity

Group competition models of indirect reciprocity were previously studied

Uchida and Sigmund analyzed competition between assignment rules by using replicator dynamics

Conclusion

To explore the possibility of spontaneous ingroup favoritism in indirect reciprocity, we analyzed a social dilemma game in a population with group structure. We showed that the degree of ingroup bias depends on the reputation assignment rule. In particular, considerable ingroup favoritism occurs under the so-called JG assignment rule, whereby observers assign bad reputations to players helping bad players. Ingroup favoritism has been considered to be an evolutionary outcome

Appendices

A Numerical methods in the case of the homogeneous assignment rule

We prepare a population of ^{3}DISC players divided into **
R
**= (

After repeating ^{5}rounds of the donation game, we calculate the fraction of G players in group ^{2} runs of the simulation.

B Numerical methods in the case of the heterogeneous assignment rule

To analyze heterogeneous populations, we assume that observers in groups 1,2,⋯,

The probability that a recipient in group

The ingroup bias of the players in group

The payoff to the players in group

The cooperativeness, ingroup bias, and payoff to the players in groups employing JG and ST are defined by
^{2}runs for each parameter set to generate Figure

Competing interests

The authors have no competing interests to declare.

Author’s contributions

MN and NM designed the model. MN derived the analytical and numerical results. MN and NM wrote the paper. Both authors read and approved the final manuscript.

Acknowledgements

We thank Shoma Tanabe for careful reading of the manuscript. MN acknowledges the support provided through Grants-in-Aid for Scientific Research (No. 10J08999) from JSPS, Japan. NM acknowledges the support provided through Grants-in-Aid for Scientific Research (No. 23681033) and Innovative Areas “Systems Molecular Ethology”(No. 20115009) from MEXT, Japan.