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Games on Endogenous Networks

Evan Sadler, Benjamin Golub

TL;DR

This paper develops a framework for network games in which players jointly choose actions and links, extending Nash equilibrium and pairwise stability to endogenous networks. By imposing ordinal single-crossing conditions—specifically action-link complements/substitutes and spillovers—the authors show that stable network structures are highly restricted to either nested split graphs or ordered overlapping cliques. The theory yields concrete predictions, including how network structure interacts with individual incentives to form groups, and provides polished analyses of notable applications such as endogenized peer effects (Carrel et al. 2013) and status-based clique formation. Existence results are established in key cells of the taxonomy, and the paper discusses extensions to dynamics, multiplex networks, and stochastic payoffs, illustrating the practical relevance for understanding perverse effects of group design and for grounding group-matching models in endogenous network formation. Overall, the approach yields tractable, testable predictions about how structural and behavioral forces co-evolve in small, vertically heterogeneous communities.

Abstract

We study network games in which players choose both the partners with whom they associate and an action level (e.g., effort) that creates spillovers for those partners. We introduce a framework and two solution concepts, extending standard approaches for analyzing each choice in isolation: Nash equilibrium in actions and pairwise stability in links. Our main results show that, under suitable order conditions on incentives, stable networks take simple forms. The first condition concerns whether links create positive or negative payoff spillovers. The second concerns whether actions are strategic complements to links, or strategic substitutes. Together, these conditions yield a taxonomy of the relationship between network structure and economic primitives organized around two network architectures: ordered overlapping cliques and nested split graphs. We apply our model to understand the consequences of competition for status, to microfound matching models that assume clique formation, and to interpret empirical findings that highlight unintended consequences of group design.

Games on Endogenous Networks

TL;DR

This paper develops a framework for network games in which players jointly choose actions and links, extending Nash equilibrium and pairwise stability to endogenous networks. By imposing ordinal single-crossing conditions—specifically action-link complements/substitutes and spillovers—the authors show that stable network structures are highly restricted to either nested split graphs or ordered overlapping cliques. The theory yields concrete predictions, including how network structure interacts with individual incentives to form groups, and provides polished analyses of notable applications such as endogenized peer effects (Carrel et al. 2013) and status-based clique formation. Existence results are established in key cells of the taxonomy, and the paper discusses extensions to dynamics, multiplex networks, and stochastic payoffs, illustrating the practical relevance for understanding perverse effects of group design and for grounding group-matching models in endogenous network formation. Overall, the approach yields tractable, testable predictions about how structural and behavioral forces co-evolve in small, vertically heterogeneous communities.

Abstract

We study network games in which players choose both the partners with whom they associate and an action level (e.g., effort) that creates spillovers for those partners. We introduce a framework and two solution concepts, extending standard approaches for analyzing each choice in isolation: Nash equilibrium in actions and pairwise stability in links. Our main results show that, under suitable order conditions on incentives, stable networks take simple forms. The first condition concerns whether links create positive or negative payoff spillovers. The second concerns whether actions are strategic complements to links, or strategic substitutes. Together, these conditions yield a taxonomy of the relationship between network structure and economic primitives organized around two network architectures: ordered overlapping cliques and nested split graphs. We apply our model to understand the consequences of competition for status, to microfound matching models that assume clique formation, and to interpret empirical findings that highlight unintended consequences of group design.

Paper Structure

This paper contains 41 sections, 10 theorems, 47 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Framework
  3. Solution concepts
  4. Separable network games
  5. More general ordinal conditions
  6. The structure of stable graphs
  7. General characterization
  8. Separable network games
  9. Perverse consequences of group design
  10. Payoffs
  11. The structure of stable outcomes
  12. The importance of intermediate-ability types: An illustration
  13. Three illustrative squadrons
  14. Discussion
  15. Other applications
  16. ...and 26 more sections

Key Result

Lemma 1

Suppose a network game with network formation has consistent and aligned linking incentives. Then, at any outcome $(G,\mathbf{s})$, there exist weak orders $\succeq_{\text{in}}$ and $\succeq_{\text{out}}$ on the players such that (again omitting arguments $(G,\mathbf{s})$) That is, if $k \succeq_{\text{in}} j$, then every player who (strictly) wants to link with $j$ (strictly) wants to link with

Figures (2)

  • Figure 1: Examples of the two types of networks that are characterized by our main result. Players' actions (effort levels) are ordered according to their vertical position in the diagram.
  • Figure 2: An illustration of the stable outcomes for the three squadrons. Ability levels $b_i$ appear inside each node, while equilibrium actions $s_i$ are next to the node.

Theorems & Definitions (37)

  • Definition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 2
  • Definition 3
  • Remark 4
  • Definition 4
  • Definition 5
  • Lemma 1
  • ...and 27 more