## Abstract

In this paper, we propose a model of the marriage market in which individuals meet potential partners either directly or through their friends. When socialization is exogenous, a higher arrival rate of direct meetings also implies more meetings through friends. When individuals decide how much to invest in socialization, meetings through friends are first increasing and then decreasing in the arrival rate of direct offers. Hence, our model helps in rationalizing the negative correlation between the advent of online dating and the decrease in marriages through friends observed in the USA over the past decades.

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## Availability of Data and Materials

The data used for generating Figs. 2 and 3 in this article are the “How Couples Meet and Stay Together 2017” (HCMST 2017) dataset [27]. These data are freely available to users who register with SSDS/Stanford Libraries. See https://data.stanford.edu/hcmst2017.

## Notes

As we discuss in Sect. 3, assuming everybody is initially married simplifies our analysis, but it does not qualitatively change the main predictions of our model. Indeed, singles would invest more in friendship than married people, as the first would rely more than the latter on their friends to find a partner. However, the predictions regarding how marriage rates through friends change as online dating becomes more prevalent would be the same.

We focus on friends of the same gender as we want to capture strategic considerations in investments in friendship in a non-repeated setting. Indeed, friendship across genders can be strategic only if the favor of introducing a potential partner can be reciprocated in the future, which requires to model interactions that are repeated in time. Furthermore, this is consistent with empirical findings that same-gendered peers have a stronger relationship and are more important for a variety of outcomes, at least among younger individuals [25, 33].

In the residual category, there are meetings at church or religious activity, non-religious voluntary associations, military service, primary of secondary school, college, customer-client relationship, bar or restaurant, private party, public space, vacation, business trip, blind date and non-online dating services.

For instance, [28] document that the increase in online meetings is accompanied by a decrease in meetings through friends. However, the authors mention that it is unclear whether overall online meetings could constitute a complement to real-life meetings. Thus, we contribute by showing how the endogenous effort in socialization can help rationalizing that the substitution effect dominates. While also other factors might explain the displacement of meetings through friends, such as stigma around online dating, our model suggests that strategic considerations in socialization may be an important factor.

We assume that men and women are completely symmetric. This choice is motivated by the findings in Fig. 2, where we show that there are not substantial gender differences when it comes to the way individuals meet their partner. This is probably due tot the fact that we observe only realized couples and not how each partner searched for a partner before finding one. A more comprehensive analysis of heterogeneity in how search behavior differs across genders would require information on this search behavior, which does not exist. For this reason, we focus our analysis considering heterogeneity only in education levels.

To keep the model symmetric, we focus on heterosexual relationships. Homosexual relationships would be modeled in a similar fashion as in [15].

Note that there always exists an equilibrium where individuals do not invest in the network, i.e., \(s_i=0\) for all \(i\in \mathcal {N}\). If \(c\ge ad(1-a)(1-d)\), this equilibrium is the only symmetric equilibrium. Otherwise, the best response dynamics after a perturbation around the equilibrium reveals that this equilibrium is unstable.

Note that also in this model an equilibrium with zero socialization levels always exists.

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## Acknowledgements

We would like to thank Paolo Pin for comments and suggestions. The usual disclaimers apply.

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Financial support from the Research Foundation - Flanders (FWO) through Grant G026619N is gratefully acknowledged.

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This article is part of the topical collection “Dynamic Games and Social Networks” edited by Ennio Bilancini, Leonardo Boncinelli, Paolo Pin and Simon Weidenholzer.

## Appendix A: Proofs

### Appendix A: Proofs

### Proof of Proposition 1

To prove the proposition, first note that

If individuals are homogeneous and \(s_i=s\) for all \(i\in N\), the network marriage rate is

Hence,

It is trivial to see that \(\partial \Psi (s,a)/\partial a >0\), while \(\Upsilon (s)\) is independent of *a*. Finally,

which is null when \(a\rightarrow 0\), equal to \((1-s)(1-e^{-sd})\) when \(a\rightarrow 1\). Furthermore, \(\partial ^2 m(s,a)/\partial a\) goes to \((1-d)(1-e^{-sd})(1+1/d)>0\) when \(a\rightarrow 0\), and to \(-d-(1-d)e^{-sd}+(1-\Psi (s,a))<0\) when \(a\rightarrow 1\). So, *m*(*s*, *a*) in first increasing and then decreasing in *a*. This concludes the proof of Proposition 1. \(\square \)

### Proof of Proposition 2

Suppose an interior equilibrium exists. Consider a profile \(\mathbf {s}\) where \(s_{j}=s\), \(\forall j\ne i\). Note that \(\Upsilon (s_i,s)\) does not depend on \(s_i\), while \(\Psi (s_i,s)\) can be written as

Then, given (1), an interior equilibrium \(s^*\) solves:

in large marriage markets, \(s^{*}\) must solve (2). The LHS of this expression is decreasing in \(s^{*}\) because both \(\left( 1-e^{-s^{*}d}\right) /s^{*}\) and \(e^{-\frac{a(1-d)}{d}\left( 1-e^{-s^{*}d}\right) }\) are decreasing in \(s^{*}\). Furthermore, when \(s^{*}\) goes to 0, the LHS converges to \(a d(1-a)(1-d)\), while when \(s^{*}\) goes to infinity the LHS converges to 0. Since marginal returns are continuous in \(s_{i}\), it follows that an interior symmetric equilibrium exists if and only if \(c<a d(1-a)(1-d)\), in which case there is only one symmetric interior equilibrium. This concludes the proof of Proposition 2. \(\square \)

### Proof of Proposition 3

We first prove part 1. We derive \(\partial s^{*}/\partial a\) by implicit differentiation of (2) and obtain

First, the denominator is positive. Indeed, its term in the square parenthesis is strictly increasing in *a* and is equal to \((1-e^{-s^*d})/s-de^{-s^*d}\) when \(a=0\). Taking the derivative of this with respect to *d*, we get \(sde^{-s^*d}\), which is also strictly positive. Finally, the limit of \((1-e^{-s^*d})/s-de^{-s^*d}\) as \(d\rightarrow 0\) is 0. Hence, the denominator is positive.

This implies that the sign of this derivative depends on the sign of the numerator, we see that the term in square parenthesis, which we denote by

is 1 when \(a=0\), while it is equal to \(-1\) when \(a=1\). Defining \(x=(1-d)\left( 1-e^{-s^{*}d}\right) /d\), we rewrite the expression (A-2) as \(1-2a-a(1-a) x\). While \(1-2a-a(1-a) x=0\) admits two solutions, only \((2 + x - \sqrt{4 + x^2})/(2 x)\in [0,1]\). Hence, there is a unique \(\bar{a}\) such that if \(a\le \bar{a}\), (A-1) is positive, while if \(a>\bar{a}\), (A-1) is negative. This concludes the first part of the proof of Proposition 3.

As for the second part of Proposition 3, the change in the matching rate \(\Psi \) when *a* changes is described by:

While the first term in the square parenthesis is always positive, the sign of the second one depends on the sign of \(\partial s^{*}/\partial a\). Using part 1, \(\partial s^{*}/\partial a\) is positive when *a* is sufficiently low, and hence \(\partial \Psi /\partial a\) would also be positive. When *a* tends to 1, inspecting (A-1), we see that the numerator goes to \(-(1-e^{-s^{*}d})<0\) when \(a\rightarrow 1\), while the denominator goes to 0 from above (as we have just shown that it is always positive). Hence, if *a* is sufficiently large, *s* goes to zero, so that the first term in the square brackets of (A-3) goes to zero, while the second part is negative; hence (A-3) is negative.

The change in the matching rate \(\Upsilon \) when *a* changes is described by:

Following a similar argument as for (A-3), we can see that (A-4) is positive for a low enough *a* and negative for *a* is large enough.

Given that \(m(s,a)=\Upsilon (s,a)+\Psi (s,a)\), also *m*(*s*, *a*) is increasing in *a* for a low enough *a* and decreasing for *a* is large enough. This concludes the proof of Proposition 3. \(\square \)

### Proof of Proposition 5

To prove existence in the socialization effort game in a large society with heterogeneous individuals, we first analyze separately on *low types* and *high types*, respectively. We derive conditions for an interior equilibrium to exists in each scenario, and then we combine these results in order to prove existence in the market with two types. We focus on symmetric equilibria in which \(s_i=s_h\) for all \(i\in M_H\), and \(s_j=s_l\) for all \(j\in M_L\).

**Part 1: Low types** Taking the first order conditions of the utility function with respect to \(s_i\) for \(i\in M_L\), and focusing on the symmetric equilibrium by setting \(s_i=s_l\), we obtain the following:

Taking the limits of \(\partial MU_{i,l}/\partial s_{i,l}\) for \(s_l\rightarrow 0\) and \(s_l\rightarrow \infty \) we obtain the following:

The LHS of equation (A-5), \(\partial MU_{i,l}/\partial s_{i,l}\), converges to \((a-1) a (d-1) Y e^{-d s_h} \left( e^{d s_h}-1\right) /s_h\) when taking the limit for \(s_l\rightarrow 0\). Note that, if \(h\rightarrow 0\), the problem for the low types becomes the same as with one type (see proof of Proposition 2). Since the marginal returns are always positive and continuous in \(s_l\), a symmetric interior equilibrium for the low types exists if and only if \(c<\bar{c}_l=(a-1) a (d-1) Y e^{-d s_h} \left( e^{d s_h}-1\right) /s_h\), which is decreasing in \(s_h\).

**Part 2: High types** As for individuals of education level *h*, we consider the maximization problem of an individual \(i\in M_H\), keeping \(s_h\) and \(s_l\) fixed. Taking the first order conditions of the utility function with respect to \(s_i\), and setting \(s_i=s_h\) we obtain the following:

Note that \(s_h>0\) and \(s_l>0\), LHS of equation A-6, \(\partial MU_{i,h}/\partial s_{i,h}\), is always positive and continuous in both \(s_h\) and \(s_l\). Moreover:

Hence, we can conclude that there exists a threshold \(\bar{c}_h\) such that for any \(c<\bar{c}_h\) a solution \(s_h\) of \(\partial MU_{i,h}/\partial s_{i,h}=c\) in a symmetric equilibrium exists.

**Part 3**: From part 1 of the proof, we have obtained that a symmetric interior equilibrium for low types exists if and only if \(c<\bar{c}_l\), and that \(\bar{c}_l\) is decreasing in \(s_h\). Additionally, from part 2 of the proof, we have obtained that a symmetric interior equilibrium for high types exists if and only if \(c<\bar{c}_h\). Since \(\partial MU_{i,l}/\partial s_{i,l}\) and \(\partial MU_{i,h}/\partial s_{i,h}\) in a symmetric equilibrium are both continuous in \(s_h\) and \(s_l\), we conclude that there exists a threshold \(\bar{c}\) such that for all \(c<\bar{c}\) both (A-5) and (A-5) are simultaneously satisfied, i.e., at least one interior equilibrium exists. This concludes the proof of Proposition 5. \(\square \)

### Proof of Proposition 4

We report for convenience the equations for the network matching rates.

Inspecting these expressions, it is immediate to see that, given any positive value of \(s_l\) and \(s_h\), the network matching rates are increasing in the arrival rate *a*. This concludes the proof of Proposition 4. \(\square \)