# Assortative Mixing Equilibria in Social Network Games

## TL;DR

This paper uses game theory to analyze why social groups tend to prefer homophily, showing that certain utility functions lead to stable equilibria of either complete homophily or heterophily, regardless of group sizes.

## Contribution

It introduces a game-theoretic model of assortative mixing, revealing conditions under which social groups adopt homophily or heterophily as equilibrium strategies.

## Key findings

- Perfect homophily is the only equilibrium when utility is degree centrality.
- Adding inter-group cooperation rewards yields only homophily or heterophily equilibria.
- Results are independent of minority-majority ratios.

## Abstract

It is known that individuals in social networks tend to exhibit homophily (a.k.a. assortative mixing) in their social ties, which implies that they prefer bonding with others of their own kind. But what are the reasons for this phenomenon? Is it that such relations are more convenient and easier to maintain? Or are there also some more tangible benefits to be gained from this collective behaviour? The current work takes a game-theoretic perspective on this phenomenon, and studies the conditions under which different assortative mixing strategies lead to equilibrium in an evolving social network. We focus on a biased preferential attachment model where the strategy of each group (e.g., political or social minority) determines the level of bias of its members toward other group members and non-members. Our first result is that if the utility function that the group attempts to maximize is the degree centrality of the group, interpreted as the sum of degrees of the group members in the network, then the only strategy achieving Nash equilibrium is a perfect homophily, which implies that cooperation with other groups is harmful to this utility function. A second, and perhaps more surprising, result is that if a reward for inter-group cooperation is added to the utility function (e.g., externally enforced by an authority as a regulation), then there are only two possible equilibria, namely, perfect homophily or perfect heterophily, and it is possible to characterize their feasibility spaces. Interestingly, these results hold regardless of the minority-majority ratio in the population. We believe that these results, as well as the game-theoretic perspective presented herein, may contribute to a better understanding of the forces that shape the groups and communities of our society.