## Abstract

Parochialism occurs when an individual mentally codes the population into in-group and out-group members and cooperates only with in-group members. Conditional cooperation of this kind is distinct from strategies such as tit-for-tat where the decision to cooperate is conditioned on others’ previous actions. Parochialists practice in-group favoritism by cooperating with others, conditional on their spatial proximity or cultural similarity. We consider an evolutionary model of local interaction with three types of strategies: altruists “always cooperate”; egoists “always defect”; and parochialists cooperate only with neighbors within a certain radius on a spatial network. In the model, we provide a new explanation for why parochialism is durable and can stably remain prevalent in human populations. The main driving force is the homophily effect. Interestingly, the homophily effect leads to the prevalence of parochialists but not altruists who benefit more from homophilic association, because altruists are invaded by parochialists. The two groups can coexist only if egoists buffer their direct interaction. Accordingly, the proportion of egoists can be greater than that of altruists in our model, contrary to the result of Eshel et al. (1998). Simulations show, for most parameter values, that the socially optimal cooperation radius (achieving the greatest mean fitness across the entire population) is two and that narrow in-group parochialism is prevalent regardless of frequency of interaction within a society.

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## Notes

- 1.
- 2.
Other important theoretical contributions to modeling the evolution of cooperation used different approaches, modeling social interaction without a spatial network structure. For example, Axelrod (1984) used computer tournaments to show that conditional cooperation, or reciprocal altruism, as exhibited by the tit-for-tat strategy was evolutionarily advantageous. Conditional cooperation in the tit-for-tat strategy conditions on other players’ previous actions, which is conceptually distinct from altruistic and parochial strategies. The altruistic strategy does not condition cooperation on any features or behaviors of other players. And parochialism conditions cooperation on others’ spatial proximity (as a representation of cultural similarity) but not on other players’ past actions. Frank (1988) stressed the role of emotions, arguing that people cooperate by helping others thought to be deserving of help because there are emotional rewards for doing so. Güth and Yaari (1992) and Güth (1995) developed the so-called

*indirect evolutionary approach*in which preferences are treated as endogenous to an evolutionary process and actions are determined by Nash equilibrium. - 3.
In economics, the concept of “social identity” is often based on this concept of “mental coding” of others based on which group they belong to (e.g., ethnicity). For example, Akerlof and Kranton (2000) show that this kind of social identity has significant effects on decisions about contributions to public goods. For more on the economics of identity, see Davis (2010) and Berg et al. (2018).

- 4.
William Graham Sumner’s (1906) book,

*What the Social Classes Owe to Each Other*, introduced the concepts of*in-group*,*out-group*and*ethnocentrism*, defined as a preference for in-group over out-group members. - 5.
Vampire bats share food reciprocally with a relatively small set of unrelated in-group others, although they preferentially share food with kin (Cosmides and Tooby 1992). In a similar pattern, ground squirrels are more likely to call out an alarm warning others when a predator approaches, which is potentially very costly, if there are in-group members nearby.

- 6.
Because each individual has their own subjective criteria for distinguishing in-group from out-group members (according to some well-defined metric measuring distance between one’s own versus another’s traits), these models essentially deal with conditional cooperation based on closeness or similarity, rather than based on groups the membership of which is transitive. Riolo et al. (2001) and Bowles and Gintis (2004b) who use threshold-based similarity to determine the boundary between the in-group and the out-group are concerned with closeness-based cooperation, just as in our paper. In contrast, Tajfel et al. (1971), Axelrod et al. (2004), Bernhard et al. (2006), Hammond and Axelrod (2006), and Choi and Bowles (2007) are concerned with group-based cooperation.

- 7.
This modeling idea is similar to the concept of network reciprocity (e.g., Nowak 2006; Ohtsuki et al. 2006) in the sense that interactions on a network structure favor cooperation. However, the cooperator group generated by network reciprocity does not, in itself, imply parochialism, because the group members cannot cooperate with other groups simply because they have no chance to interact with each other. If they did have this chance, they would cooperate with out-group members. Network reciprocity is therefore different than parochialism, although both mechanisms lead to the observed absence (or rarity) of cooperation with out-group members.

- 8.
Eshel et al. (1998) focus on two strategic types, altruists and egoists. They also show equivalency of evolutionary dynamics with more general pairs of strategy types, which includes so-called “Hooligans.”

- 9.
In simulations, Choi and Bowles (2007) showed that group conflict, such as war, can be used strategically to promote the co-evolution of parochialism and altruism. Gao et al. (2015) found that intergroup conflict can promote the evolution of parochial altruism. Qin et al. (2014) showed that ethnic harmony in a multi-ethnic society can be promoted by popularizing civic identity.

- 10.
The modifier “weakly” is valid when the parochialist is a

*direct*(i.e., adjacent) neighbor of an altruist, or is an*indirect*(i.e., non-adjacent) neighbor of an egoist. - 11.
This is consistent with Dunbar’s Law, which stipulates (among other things) that players in social networks can maintain only a relatively small number of connections (Dunbar 1993).

- 12.
Some authors (e.g., Qin et al. 2014) distinguish parochial altruism from parochial nonaltruism, based on whether a player is willing to sacrifice to benefit insiders. In our model, there is no need to distinguish between these two sub-types of parochialism, because cooperation and sacrificing (i.e., contributing to a locally shared public good) are identical in the simple game in Table 1.

- 13.
Henceforth, we will omit “spatial” if there is no chance of confusion.

- 14.
Total fitness is the mathematical expectation of total payoffs rather than realized ex post total payoffs in the expression above. This substitution can be justified in two ways. First, each period can be thought of as an infinite number of discrete draws from the profile of probabilities of meeting others in the society, with total payoffs representing the arithmetic mean across all these instances (by the law of large numbers). A second justification would be to anticipate that our analytic expressions involving fitness comparisons correspond to steady states. Replacing realized random total payoffs with their expected values could be appropriate if each agent had observed a large number of similar periods and could compute multi-period averages when comparing fitness by type, as required every time they decide whether to continue choosing their same type.

- 15.
It appears to be unrealistic to assume that each player can observe her neighbors’ genotypes as well as phenotypes, but it is quite reasonable that she can infer the genotypes of her neighbors if she can observe all their choices toward all their own neighbors.

- 16.
In a gene-trait transmission process, players with a certain genotype produce themselves more if and only if their fitness is higher. In our model, there is no reproduction process.

- 17.
If the interaction radius is more than 2, this lexicographic ordering is not guaranteed to hold and becomes vulnerable to violations especially when

*p*is close to 1. - 18.
Note that for any \(i, j\in N\), either \(F(s_i )>F(s_j )\) or \(F(s_i )<F(s_j )\) in this model, except for the case that \(p\ne \frac{1}{2}\), unless they have exactly the same neighbor-phenotype pairs.

- 19.
In general, if the cooperation radius is

*r*and \(p^{t}\) is the probability that \(C_r\) and*C*meet, where the distance between them is*t*, then \(C_r\) performs equally well as*C*if \(t\le r\) and performs better than*C*for any \(t>r\). - 20.
Various stable distributions can be reached, depending on initial distributions. Note that a type which was initially absent or disappeared in the evolution process cannot reappear insofar as no mutations are assumed. Due to this reducible nature of the Markov process under imitation dynamics, the stationary distribution is not unique.

- 21.
The sequence of phenotypes of

*C*’s neighbors in adjacent and non-adjacent positions, respectively, is (*C*,*D*), while the sequence of phenotypes of \(C_1\)’s neighbors is (*C*,*C*). Thus, lexicographic ordering in fitness implies that \(F(C_1 )>\langle F(C)\rangle \).

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## Appendices

### Appendix 1: Results of Eshel et al. (1998)

In Eshel et al. (1998), the assumptions that \(R=1\) and \({\bar{t}}=1\) are maintained throughout. Eshel et al. (1998) identify two trivial stable distributions: one in which all players are cooperators (i.e., altruists) and one in which all players are defectors (i.e., egoists). They also identify a nontrivial stable distribution in which strings of defectors are of length two and strings of cooperators are of length three or longer. The following spatial distribution provides a typical example of a nontrivial stable distribution in their model: \((\cdots , C, C, C, D, D, C, C, C, C, \cdots )\).

Intuition for the string-length conditions that characterize Eshel el al. (1998)’s nontrivial stable distributions is as follows. A *C*-type in the interior of a string of *C*s can survive. A *C*-type on the boundary of a string of *C*s (necessarily between an adjacent *C*-type and an adjacent *D*-type in Eshel et al. (1998)’s two-type model) can survive if the *D*-type neighbor has another *D*-type neighbor and if the *C*-type neighbor has another *C*-type neighbor. The *C*-type at the boundary will not switch her type and therefore survive under the imitation dynamics, because *D*’s payoff is *b*, and the average payoff of *C*-types is \(\langle F(C)\rangle =\frac{b-2c+2b-2c}{2}=\frac{3}{2}b - 2c >b\) if \(\frac{b}{c}>4\). Similarly, a boundary *D*-type will always survive (freeriding on the adjacent *C*-type) since her fitness (*b*) is greater than her *C*-type neighbor’s fitness, (\(b-c\)). Hence, this distribution (a cluster of two *D*-types surrounded by larger clusters of *C*-types) is stable because no player will change her type.

### Proposition 4

(Eshel et al. 1998) If \(\frac{b}{c}>4\), then any distribution in which the defector string is of length 2 and the cooperator string is of length greater than 2 is stable in the sense that it is an absorbing state.

Note that a defector string of length three or more cannot be in an absorbing state. To see this, consider that, in a defector string of length three or more, the average fitness of a boundary *D*-type’s neighboring *D*-type (interior to the *D*-string) is *b*/2. The condition for her to survive is that \(\frac{b}{2}>b-c\), i.e., \(c>2b\), which is contradictory to the condition for a boundary *C*-type to survive.

The main conclusions of Eshel et al. (1998) are twofold. First, cooperation can survive evolutionarily if players interact locally (which means they play the prisoner’s dilemma only with nearby neighbors rather than with everyone in their society). For survival, it is essential that cooperators are clustered (i.e., comprising a *string*, which refers to multiple same-type players next to each other with two boundary players at either end of a string). Second, the proportion of defectors in the population cannot exceed 40%, which means that the proportion of cooperators must be substantially larger than the proportion of defectors in any nontrivial stable distribution. Otherwise, cooperators would invade defectors and the distribution would not be stable.

### Appendix 2: Lemmas and proofs

In this appendix, several lemmas are put in order to help characterize absorbing states. Also, we will provide the proofs for the lemmas and our main propositions.

### Lemma 2

In an absorbing state, an isolated type cannot be located between two strings of other types of length two or more.

### Proof

(i) The result follows from Lemma 1 if the adjacent strings are of the same type and of length more than two. (ii) Suppose that the adjacent strings are of different types with length more than two. If the isolated type is *C* or *D*, then the result follows from Lemma 1, because *C* would be invaded, and *D* would invade the adjacent strings. The only nontrivial case is that the isolated type is \(C_1\) with neighboring strings *C* and *D*. Straightforward computations show that \(C_1\) is invaded by *D*, because \(F(C_1 )=b-2c+pb <b+pb =F(D)\). (iii) Finally, we consider the case in which the adjacent strings are of length two. If the isolated type is *C* or *D*, then the result, once again, follows from Lemma 1 that the isolated type either invades or is invaded by its neighbors. If the isolated type is \(C_1\) and one of the adjacent strings is type *D*, then simple computations show that \(C_1\) is invaded by *D*. If the isolated type is \(C_1\) and one of the adjacent strings is *C*, then \(C_1\) invades the *C* string. \(\square \)

### Lemma 3

Generically, no isolated type can exist in any absorbing state.

### Proof

Suppose two isolated types *x* and *y* \((x\ne y)\) are adjacent. If \((x,y)=(C, D)\), the other direct neighbor of \(x=C\) (\(y=D\), respectively) is *D* or \(C_1\) (*C* or \(C_1\), respectively), because *x* and *y* are isolated. If direct neighbors of \(x=C\) are both *D*, we have \(F(D)>F(C)\) by the domination argument in Footnote 18 and the extra cost *c* that a *C* type must incur in a prisoner’s dilemma game. If the other direct neighbor of \(x=C\) is \(C_1\), the phenotypes of direct neighbors of \(x=C\) are *C* and *D*, while the counterparts of \(y=D\) are both *C*. Again, by the domination argument in Footnote 18, \(F(x)\ne F(y)\) generically. If \((x,y)=(D, C_1 )\), we can show similarly that \(F(x)\ne F(y)\) generically. If \((x,y)=(C, C_1 )\), it is possible that \((\cdots , D, D, C, C, C_1 , D, D, \cdots )\). In this case, any one of the neighbor pairs does not *t*-dominate the other’s neighbor pairs for any \(t=1, 2\). But it is easy to see from direct comparison that \(F(C_1 )>F(C)\) because \(F(C)=b-2c -2cp <F(C_1 )=b-2c\). Therefore, if two isolated types *x* and *y* \((x\ne y)\) are adjacent, then generically either \(F(x)>F(y)\) or \(F(y)>F(x)\). This implies that either *x* imitates *y* or *y* imitates *x*, implying that the distribution cannot be an absorbing state. Also, by Lemma 2, there cannot exist a single isolated type between two other strings. This completes the proof. \(\square \)

### Proof of Proposition 1

Consider a population distribution in which a \(C_1\)-string meets a *C*-string, as follows: \((\cdots , C, C, C, C, C_1 , C_1 , x)\). We can show that it cannot be an absorbing state.

We have:

where \(C^B\) is the boundary *C*-type and \(C^I\) is an interior *C*-type, which implies that \(\langle F(C)\rangle =2(b-c)+p(b-2c)\). On the other hand, the fitness of the boundary \(C_1\)-type depends on the type of *x*. If \(x=C\), then the fitness calculation for the \(C_1\)-type is:

It is clear from the earlier argument regarding the lexicographic order by fitness that \(\langle F(C)\rangle <F(C_1 )\).^{Footnote 21} Accordingly, the boundary *C*-type will choose to imitate \(C_1\) and therefore cannot survive. If \(x=C_1\) or *D*, then the fitness of the boundary \(C_1\)-type is \(F(C_1 )=2(b-c)+pb\), which is still higher than \(\langle F(C)\rangle = 2(b-c)+p(b-2c)\).

If the length of the *C*-string is less than four, then the average fitness of *C* will be lower, which implies that a *C*-string cannot be adjacent to a \(C_1\) string. \(\square \)

### Proof of Proposition 2

Consider a distribution consisting only of \(C_1\)- and *D*-types, for example: \((\cdots , C_1 , C_1 , C_1 , C_1 , D, D, x, z)\), where “*z*” is another place holder representing one of the three types.

(i) It suffices to show that \(x\ne D\). The fitness of the boundary \(C_1\)-type (preceding the first *D* in the distribution above) is \(b-2c\), and the fitness of her neighboring (i.e., preceding) \(C_1\)-type is \(2b-2c\). Note that both of them choose strategy *D* against their neighbors at distance 2. Thus, the average fitness of the boundary \(C_1\)-type is:

If \(x=D\), then this boundary *D*-type’s fitness is *b*. Because the survival condition for \(C_1\) is \(\langle F(C_1 )\rangle >F(D)\), the \(C_1\)-type can survive if:

i.e., if \(\frac{b}{c}>4\). On the other hand, the survival condition for the boundary *D*-type is \(\langle F(D)\rangle >F(C_1 )\). Because we have \(F(D^B )=b\) and \(F(D^I )=0\), the average fitness of the boundary *D*-type is:

if \(z=D\) or \(C_1\). Thus, the survival condition for *D* is that:

i.e., \(\frac{b}{c}<4\). Note that equations (2) and (3) contradict each other. Therefore, \(x=D\) is impossible, which implies that any *D*-string’s length cannot exceed two.

(ii) If \(x=C_1\), it must be that \(z=C_1\) as well, because \(C_1\) cannot be isolated (according to Lemma 2). Then, the survival condition for \(C_1\) is the same as given by (2). On the other hand, since \(F(D^B )=F(D^I )=b\) and \(\langle F(D)\rangle =b\), the survival condition for *D* is that:

which holds trivially. Therefore, this distribution is an absorbing state if \(\frac{b}{c}>4\). \(\square \)

### Proof of Proposition 3

Consider the length of any *D*-string in a population distribution in which it is adjacent to a *C*-string, \((\cdots C, C, C, D, D, x,C,C,C)\), where “*x*” is a placeholder for a player of any type except \(C_1\) due to Lemma 2. Consider the case of \(x=C\). The boundary *C*-type can survive if:

that is, if \(c<\frac{1-2p}{4(1+p)}b\). If \(p>\frac{1}{2}\), then this inequality cannot be satisfied. But if \(p<\frac{1}{2}\), then *C* can survive. Since \(F(D^B ) =b+2bp > \langle F(C)\rangle =-2c +(b-2c)p\), the boundary *D*-type survives without any condition. Such a distribution is therefore an absorbing state if \(\frac{b}{c}> \frac{4(1+p)}{1-2p}\). \(\square \)

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Berg, N., Kim, JY. & Lee, K.M. Why is parochialism prevalent?: an evolutionary approach.
*J Econ Interact Coord* **16, **769–796 (2021). https://doi.org/10.1007/s11403-021-00328-3

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### Keywords

- Altruism
- Parochialism
- Egoist
- Conditional cooperation
- Local interaction
- Homophily

### JEL Classification Codes

- C73
- D64