## Abstract

The ethnically homogeneous middleman groups (EHMGs), which are informal trading networks, are ubiquitous in less-developed economies where the legal infrastructure for contract enforcement is not well developed. This paper develops a formal model of social interaction among members of the EHMG as well as in more general situations in a multi-ethnic or multi-cultural society consisting of identifiable ethnic or linguistic groups. Behavioral patterns are transmitted between generations and altered via imitation in social contacts. The model demonstrates how different discriminatory behavioral patterns can evolve and persist over time. One result is that the trust between such groups can increase due to a higher frequency of inter-group contacts. In concluding the paper, we speculate about how the model can be expanded to include changes in legal structures, especially contract law. This could lead to an increase in the trust between the different groups.

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## Notes

This aspect of our model is similar to models of statistical discrimination; see, for instance, Aigner and Glen (1977).

In the language of evolutionary theory, we could say that behavior is determined by inheritance (of phenotype), environmental influence, intentional choices, and social or legal norms.

This seems to us to be the natural assumption given the asymmetry of the situation. Note that the interactions are not in the form of a traditional coordination game. A simple coordination game (kissing a person on the right or left cheek when meeting) of course is symmetric and would have \(u_{lr}=u_{rl}\) (in both cases a kiss on the mouth would result, which might be against accepted conventions: \(u_{ll}=u_{rr}>u_{lr}=u_{rl}\)). This does not capture the asymmetry of the situation. If I am gullible and cheated, I am bound to lose more than if I am prepared to be cheated. Similarly, if I cheat a trusting person, I should gain more than if I trust my opposite.

Using expectations, we here appeal to the law of large numbers, assuming the size of the different groups to be sufficiently large.

This assumption is, strictly speaking, not necessary. A sufficiently small leakage from these encounters would alter the “trusting”

*G*-equilibrium (see below) into a fully mixed one, moving it towards the right in Fig. 2. The substance of the discussion would not be affected, however. The elaboration of the formal model would be made considerably more onerous, though.The two functions measure the size of the “shock” due to the “surprise” behavior, and in that way influence the probability of the individual mutating to another state. An alternative interpretation would be to look at the opportunity costs of the

*a priori*actual behavior in comparison to the best response to the two different behavior patterns of the partner, \(u_{cc}-u_{tc}\) and 0, as well as the opportunity costs of adopting the behavior of the partner in comparison to the best response, 0 and \(u_{ct}-u_{tt}\). Intuitively, this contradicts the limited-rationality spirit of the model, which takes its departure in a*status quo*and models behavioral change as the result of external*stimuli*. The formal analysis would in both interpretations remain the same, however.This assumption is, of course, implicitly a combination of two different assumptions, one comparing \(c_{w}(x)\) and \(c_{b}(x)\) for all

*x*, for instance assuming \(c_{w}(x)\ge c_{b}(x) \forall x\), and one other comparing \(\Delta u_{w}\) and \(\Delta u_{b}\), for instance assuming \(\Delta u_{w}\ge \Delta u_{b}\). The chosen formulation is the minimum needed, in order to analyze the non-trivial cases of our model.All equilibria discussed are fixed points of expression 3.3, that is stationary points of Eq. 3.4, the equations for the dynamics of the system. For the sake of simplicity we will talk about “equilibria” in the text and not in every instant specify that we are dealing with asymptotic fixed points of the dynamic system 3.7.

Due to continuity, there also exists at least one more fixed point of the system. This, however, is an unstable equilibrium. In general, we cannot exclude the existence of further fixed points leading to more equilibria, stable and unstable. In our analysis, however, we will limit the analysis to the case with the minimum number of possible fixed points.

Assuming

*r*to be less than one. If \(r=1\) the*G*equilibrium is independent of \(\pi _{1}\) and equal to the set given by the vertical axis, see Proposition 4.1.The curve describing the loci of repellents is in this case implicitly given by the equation \(n(1-p_{3})c_{b}=n(p_{3})c_{w}\). This, of course, has a unique solution, say \(p_{3}^{*}\), such that for an initial value \(p_{3}^{0}\) of \(p_{3}\), \(p_{3}^{0}>p_{3}^{*}\) implies that (0, 0, 1) is an attractor and \(p_{3}^{0}<p_{3}^{*}\) implies that (0, 1, 0) is an attractor, independently of the initial values of \(p_{1}\) and \(p_{2}\).

This is shown in a more precise manner in Appendix A.

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## Acknowledgements

Earlier versions of this paper were presented by Bengt-Arne Wickström at the Universities of Haifa and Bergen, as well as at the Max Planck Institute in Jena. We want to thank the participants at these meetings for many interesting comments. We are also grateful to two anonymous referees for their constructive suggestions. Part of the work of Bengt-Arne Wickström was carried out in the Research group “Economics and language” in Berlin, which is receiving funding from the European Union’s Seventh Framework Program (Project MIME-Grant Agreement 613344). This support is gratefully acknowledged.

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## Appendices

### Appendix A: The dynamics in a phase diagram

In this appendix we analyze how *r* and \(\pi _{1}\) influence the equilibrium values of the *p*’s. We find the dynamics in the \(p_{2}-p_{3}\)-dimensions of the simplex \(p_{1}+p_{2}+p_{3}=1\).

### 1.1 A.1 The dynamics of \(p_{3}\)

The stationary points of \(p_{3}\) are found by solving the equation \(\dot{p}_{3}=0\). It is clear from the equations of motion, 3.7, that \(p_{3}=0\) defines a set of stationary points for \(p_{3}\). It is also readily seen that these points are locally attracting, by evaluating \(\dot{p}_{3}/p_{3}\) at \(p_{3}=0\):

Under Assumption 3.11, this is negative and, hence, \(p_{3}=0\) is a stable value. If \(p_{3}=1\), the equation of motion reduces to:

This is negative, unless \(\pi _{1}=0\). The stable equilibrium \(p_{3}=1\) in the case of no interaction is thus altered if in the outside world there is a positive probability of meeting an individual that can be trusted (\(\pi _{1}>0\)), which is stated in proposition 4.2.

If there are other stationary points of \(p_{3}\) than \(p_{3}=0\), then they have to solve equation:

For a given value of \(p_{2}\) and a sufficiently small \(\pi _{1}\), this equation has at least two roots under Assumptions 3.8 and 3.11. For a sufficiently large \(\pi _{1}\) and a sufficiently small *r*, it might have no root, and \(\dot{p}_{3}<0\) everywhere. The case with two roots has been pictured in the phase diagram, Fig. 2. The lines \(\left( A_{3}^{*},A_{3}^{o}\right) \) and \(\left( A_{3}^{**},A_{3}^{oo}\right) \) are the loci of stationary points for \(p_{3}\). The first one is a repellent and the second one an attractor. The \(p_{3}\) coordinates of points \(A_{3}^{*}\) and \(A_{3}^{**}\), \(p_{3}^{*}\) and \(p_{3}^{**}\), solve equation:

The \(p_{2}\) coordinate is, of course, zero. The \(p_{3}\) coordinates of \(A_{3}^{o}\) and \(A_{3}^{oo}\), \(p_{3}^{o}\) and \(p_{3}^{oo}\), solve equation:

The \(p_{2}\) coordinates are \(1-p_{3}^{o}\) and \(1-p_{3}^{oo}\), respectively.

The behavior of these stationary points can be visualized in simple diagrams. Define \(f^{*}(p_{3})\), \(f^{o}(p_{3})\), and *h* as:

We note that:

That is, \(\vert f^{o}(p_{3})\vert \ge \vert f^{*}(p_{3})\vert \) and they are equal if \(p_{3}=1\) or if the expression in the square brackets is zero. In Figs. 3, 4, and 6 the stationary points of \(p_{3}\) are indicated for different values of *h*. Figure 3 corresponds to the situation in the phase diagram in Fig. 2 with three distinct regions of relevance for the dynamics of \(p_{3}\), one with an increasing and two with a decreasing \(p_{3}\).

In Fig. 4 the situation is pictured when \(\pi _{1}\) approaches zero or *r* approaches one. Now, \(p_{3}^{**}\) and \(p_{3}^{oo}\) approach one and \(p_{3}^{*}\) and \(p_{3}^{o}\) become identical, in fact, the curve (\(A_{3}^{*}\), \(A_{3}^{o}\)) becomes a straight line and the curve (\(A_{3}^{**}\), \(A_{3}^{oo}\)) degenerates to the point \(p_{3}=1\). The equilibrium in dominant strategies in a world without interaction is a limiting case of an equilibrium in a world with vanishing interaction. The resulting phase diagram with two regions for the dynamics of \(p_{3}\) is pictured in Fig. 5.

In Fig. 6 \(p_{3}^{*}\) and \(p_{3}^{**}\) disappear and eqilibrium *D* disappears, too, Fig. 7, or becomes globally very unstable, Fig. 8, depending on the size of \(\pi _{1}\).

### 1.2 A.2 The dynamics of \(p_{2}\)

The equation of motion for \(p_{2}\) is:

If \(p_{2}=0\), it reduces to:

This is positive if \(p_{3}<1\). For \(p_{2}=1\) we similarly find:

This is negative, unless \(\pi _{1}=0\). Hence, due to continuity, there is a region in the phase diagram for sufficiently small values of \(p_{2}\), where \(p_{2}\) is growing and a region for sufficiently high values of \(p_{2}\), where \(p_{2}\) is getting smaller. Along the edges of the phase diagram, for \(p_{3}=0\) and \(p_{2}+p_{3}=1\), we can find expressions for the limits of these regions. If \(p_{3}=0\), we find:

and for \(p_{2}+p_{3}=1\):

The stationary points on the \(p_{3}=0\) axis are given by the solution of:

Under Assumption 3.10, this equation has a unique solution, \(p_{2}^{*}\), which does not depend on *r* and which is monotone in \(\pi _{1}\) and equal to one if \(\pi _{1}\) is zero and zero if \(\pi _{1}\) is one. The stationary points for \(p_{2}+p_{3}=1\) are at \(p_{2}=0\) or given as the solutions to the equation:

This can be rewritten as:

If \(\pi _{1}\) is sufficiently small and *r* sufficiently big, this equation will have at least two roots, which are the \(p_{3}\) coordinates of the points \(A_{2}^{o}\) and \(A_{2}^{oo}\). It is readily seen that these points will lie to the left of the points \(A_{3}^{o}\) and \(A_{3}^{oo}\) in the phase diagram in Fig. 2, since the expression in the square brackets will have to have the opposite sign in the expressions for the stationarity of \(p_{2}\) and \(p_{3}\).

In Figs. 9 and 10 we have indicated, how the stationary points can be found. Figure 9 corresponds to the phase diagram in Fig. 2. Similarly, in Fig. 10 the limiting situation with \(h\rightarrow 0\), corresponding to the phase diagram in Fig. 5 is illustrated.

### 1.3 A.3 Global stability

An inspection of the various phase diagrams clearly indicates the smaller is *r* the more attractive globally becomes equilibrium *G*. For sufficiently small values of *r* the *D* equilibrium even disappears. Furthermore, for sufficiently large values of *h* (small *r* and/or large \(\pi _{1}\)), the phase diagram consists of only two regions and *G* is the only equilibrium, the location of which is determined by the value of \(\pi _{1}\), see the phase diagram in Fig. 11. Hence, contact feeds trust.

### Appendix B: Feasible interaction equilibria

We will show the existence of fixed points of the mappings in Sect. 4.2.1. We show several lemmata.

### Lemma 5.1

The mapping \(g(\gamma (p))\) has at least three fixed points, \(p=0\), \(p=1\), and \(p=\hat{p}\in (0,1)\).

### Proof

First two are trivial. For the third one, we note that the function *g* is monotone and implicitly defined by

and \(\gamma \) by

The function *g* has a fixed point \(\check{p}\) given by \(n(\check{p} )c_{b}=n\left( 1-\check{p}\right) c_{w}\). In view of Assumption , this point exists, and it is unique if *n*(*p*) is strictly monotone. By the same argument we find the fixed point \(\check{\pi }\) of \( \gamma \). If \(1>p>\check{p}\), then \(g(p)<p\), and if \(0<p<\check{p}\), then \( g(p)>p\). The corresponding result holds for \(\gamma \) and \(\check{\pi }\). Combining these results, we find that if \(1>p>\max \{\check{p},\check{\pi }\}\) , then \(g(\gamma (p))<p\), and if \(0<p<\min \{\check{p},\check{\pi }\}\), then \( g(\gamma (p))>p\). By continuity, the fixed point \(\hat{p}=g\left( \hat{\pi } \right) \) exists and \(\min \{\check{p},\check{\pi }\}\le \hat{p}=g\left( \hat{\pi }\right) \le \max \{\check{p},\check{\pi }\}\), and by the same token \(\hat{\pi }=\gamma (\hat{p})\) also satisfies \(\{\check{p},\check{\pi }\}\le \hat{\pi }=\gamma (\hat{p})\le \max \{\check{p},\check{\pi }\}\). \(\square \)

### Remark B.1

If the two groups are identical, \(\hat{p}\) is unique and equal to \(\check{p}= \check{\pi }\).

### Lemma B.2

The mapping \(g(\delta _{1}(p))\) has at least one fixed point, \(p=0\).

### Proof

Trivial. \(\square \)

### Lemma B.3

The mapping \(d_{1}(\delta _{1}(p))\) has at least one fixed point, \(p=0\).

### Proof

Trivial. \(\square \)

There exist then at least the six feasible equilibria listed in Sect. 4.2.1.

### Appendix C: Proof of proposition 4.4

We linearize the system 4.2 at the various equilibria and calculate the eigenvalues of the determinants of the matrices of partial derivatives of the equations of motion. The matrices of partial derivatives are given in Appendix D. The eigenvalues associated with the various equilibria are:

*Equilibrium (i)*:

where \(a=a\left( 1\right) \), \(\alpha =\alpha \left( 1\right) \),\(\ b=b\left( 0\right) \),\(\ \beta =\beta \left( 0\right) \). Since \(a\left( 1\right)>b\left( 0\right) >0\) and \(\alpha \left( 1\right)>\beta \left( 0\right) >0\) , all four eigenvalues are real and negative. Hence, equilibrium (1) is locally stable.

*Equilibrium (iii)*:

where \(a=a\left( 0\right) \), \(\alpha =\alpha \left( 0\right) \),\(\ b=b\left( 1\right) \),\(\ \beta =\beta \left( 1\right) \). Since \(b\left( 1\right)>a\left( 0\right) >0\) and \(\beta \left( 1\right)>\alpha \left( 0\right) >0\) , all four eigenvalues are real and negative. Hence, equilibrium (3) is locally stable.

*Equilibrium (iv)*:

where \(a=a\left( 0\right) \), \(\alpha =\alpha \left( 0\right) \),\(\ b=b\left( 1\right) \),\(\ \beta =\beta \left( 1\right) \), and \(w:=b\left( 1-\rho \right) +\beta \). Since \(b\left( 1\right)>a\left( 0\right) >0\) and \(\beta \left( 1\right)>\alpha \left( 0\right) >0\), all four eigenvalues are real and negative. Hence, equilibrium (4)—and, by symmetry, equilibrium (5)—is locally stable.

*Equilibrium (vi)*:

where \(a=a\left( 0\right) \), \(\alpha =\alpha \left( 0\right) \),\(\ b=b\left( 1\right) \),\(\ \beta =\beta \left( 1\right) \), \(\omega =\left( 1-r\right) \left( 1-\rho \right) \), and \(w:=b\left( 1-\rho \right) +\beta \left( 1-r\right) \). Since \(b\left( 1\right)>a\left( 0\right) >0\) and \(\beta \left( 1\right)>\alpha \left( 0\right) >0\), all four eigenvalues are real and negative. Hence, equilibrium (6) is locally stable.

*Equilibrium (ii)*:

Here, in order to make the problem tractable, we have to make the assumption that both groups have identical functions describing their behavior. That is, \(a(x)=\alpha (x)\), \(b(x)=\beta (x)\), etc., which, of course, implies that \(\hat{p}=\hat{\pi }\). Then the eigenvalues are:

where \(d:=\hat{p}\left( 1-\hat{p}\right) \left( 1-r\right) \left[ \frac{ \partial n\left( 1-\hat{p}\right) }{\partial \left( 1-p\right) }c_{w}+\frac{ \partial n\left( \hat{p}\right) }{\partial p}c_{b}\right] \). The last inequality follows from the fact that, by Assumption 3.10, \(a( \hat{p})>d\).

The positive eigenvalue *d* is associated with the eigenvector

Hence, a perturbation increasing (or decreasing) \(p_{1}\) and \(\pi _{1}\) by a small amount, holding \(p_{1}+p_{2}\) and \(\pi _{1}+\pi _{2}\) constant (which is a movement in the two simplices) will cause a movement away from the equilibrium (2). Equilibrium (2) is thus unstable.

### Appendix D: The matrices of partial derivatives of the equations of motion

In this appendix, we give the matrices of the partial derivatives of the equations of motion, Eq. 4.2. For the definitions of the various parameters, we refer to the discussion in the main text.

For equilibrium (i), (1, 0, 0; 1, 0, 0), we have:

for equilibrium (ii), \((\hat{p},1-\hat{p},0;\hat{\pi },1-\hat{\pi },0)\):

where

for equilibrium (iii), (0, 1, 0; 0, 1, 0):

for equilibrium (iv), (0, 1, 0; 0, 0, 1):

and, finally, for equilibrium (vi), (0, 0, 1; 0, 0, 1):

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Wickström, BA., Landa, J.T. Interethnic relations, informal trading networks, and social integration: imitation, habits, and social evolution.
*J Bioecon* **20**, 263–286 (2018). https://doi.org/10.1007/s10818-018-9278-y

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DOI: https://doi.org/10.1007/s10818-018-9278-y

### Keywords

- Cultural transmission units
- Ethnic networks
- Identity
- Imitation
- Inter-ethnic interaction
- Multi-cultural/multi-ethnic society
- Trust