Endogenous Link-Formation Game

• Endogenous link-formation games are strategic models where agents simultaneously choose connections and behaviors, generating network externalities like peer effects and contagion.
• They employ equilibrium concepts such as pairwise stability and k-player Nash stability to analyze both local adjustments and global network dynamics.
• Analytical insights reveal how interventions can amplify outcomes, with theoretical methods supporting empirical estimation under uncertainty.

An endogenous link-formation game is a strategic, multi-agent framework in which each agent’s utility depends not only on an explicitly chosen set of links to other agents, but also (potentially) on actions, behaviors, or attributes chosen endogenously, and on network externalities driven by the realized network structure itself. This class of models encompasses the simultaneous or joint determination of network topology and agent-level decisions, capturing a broad array of phenomena such as peer effects, behavioral homophily, coordination, contagion, and strategic externalities. Formally, the link formation process and agent choices are interdependent and typically analyzed through equilibrium notions ranging from pairwise stability to generalized Nash equilibrium and refined k-player Nash stability.

1. Formal Model Frameworks

Endogenous link-formation games are commonly framed as follows:

• Player set: A finite population $N = \{1, ..., n\}$.
• Strategy space: Each agent $i$ selects a vector of outgoing links $g_{ij} \in \{0,1\}$ (or link strengths), alongside possibly an endogenous attribute $a_i \in A$ (such as behavior or effort).
• Network representation: Directed or undirected adjacency matrices encode the evolving network.
• Payoff specification: Agent $i$'s utility $u_i(S, X)$ depends on personal actions, network links, and externalities—often decomposing into intrinsic benefits, local peer effects (reciprocation, homophily), aggregative externalities, and network motifs (such as triangles for transitivity or rivalry). For example, the canonical form in Badev (Badev, 2017) is:

$$u_i(S, X) = a_i v(X_i) + a_i h \sum_{j \neq i} a_j + \sum_{j \neq i} a_i a_j g_{ij} g_{ji} + \sum_{j \neq i} g_{ij} w(X_i, X_j) + \sum_{j \neq i} g_{ij} g_{ji} m(X_i, X_j) + q \sum_{j,k \neq i} g_{ij} g_{jk} g_{ki}$$

• Dynamics: Network states may evolve under local (sequential) or global (simultaneous) best-response dynamics, possibly subject to random utility shocks.

2. Equilibrium Concepts

Endogenous link-formation games typically invoke equilibrium concepts that generalize traditional Nash and stability notions. Central types include:

• Pairwise stability: No single link can be added or removed to the mutual benefit of both involved parties (Jackson–Wolinsky).
• k-player Nash stability (Badev, 2017): For any subset $I_k \subset N$ of size $k$, the subgame in which only those $k$ agents re-wire their mutual links and re-select their actions admits no joint deviation making every member strictly better off. This forms a hierarchy: $$S_2^* \supseteq S_3^* \supseteq \ldots \supseteq S_n^*$$, interpolating from local to global stability.
• Potential game structure: Many endogenous link-formation models (e.g., those with triangulated or homophily motifs (Betancourt, 13 Oct 2025, Betancourt, 2023)) admit a global potential function $P(S)$ where local best-response changes coincide with local differences in $P$.
• Stationary distributions: Under random-utility dynamics, the unique stationary law is typically a Gibbs measure over network states, ranking equilibria cardinally and ordinally by their potential values.

Table: Equilibrium Notions

Notion	Locality	Joint Deviations
Pairwise Stability	Single link/action	2 agents
k-player Nash stability	k-agent subgroups	k agents ($2 \leq k \leq n$)
Nash Equilibrium	Global	All agents

3. Analytical Properties and Existence

Under broad conditions, endogenous link-formation games admit rich equilibrium structures:

• Existence: When the payoff gradient with respect to single-player or single-link deviations matches the gradient of a potential function, at least one k-player Nash-stable state exists for every $k$ (Badev, 2017, Betancourt, 13 Oct 2025).
• Explicit characterization: For certain payoff forms, equilibria can be classified:
  • Nested split graphs (NSG), ordered overlapping cliques (OOC) arise under separable positive/negative spillovers and complement/substitute structures, with succinct network architectures (Sadler et al., 2021).
  • Community structure, clustering bounds: Equilibria can be sharply constrained (e.g., lower bounds on clustering coefficient, fragmentation into cliques, or triangles generated by triadic closure) (Borgs et al., 2010).

4. Comparative Statics, Welfare, and Policy Implications

Endogenous link-formation games exhibit extreme sensitivity of equilibria—and system-level outcomes—to exogenous parameters:

• Amplification of interventions: In applications to adolescent smoking, allowing endogenous re-formation of friendship links magnifies policy effects—e.g., a price increase on tobacco amplifies reduction in equilibrium smoking via peer network reconfiguration (Badev, 2017).
• Contagion and epidemic persistence: In stochastic epidemic networks, strategic selection of link intensity results in a structural reversal of classic epidemiological thresholds. The epidemic always persists in equilibrium, even at arbitrarily low spreading rates, contrasting sharply with exogenously fixed networks (Xu, 2016).
• Efficiency metrics: The price of anarchy and price of stability can be computed explicitly; for heterogeneous link costs, equilibrium and social optimum may diverge quantitatively and structurally (Govindaraj, 2016).
• Phase transitions: As motif parameters are varied in motif-based potential games, the equilibrium network density, clustering, and community structure can jump discontinuously—first-order phase transitions—reflecting the nonlinear dependence of global structure on local incentives (Betancourt, 13 Oct 2025, Betancourt, 2023).

5. Estimation, Identification, and Empirical Methodology

Modern work operationalizes endogenous link-formation models for empirical estimation:

• Bayesian structural estimation: Network formation games can be represented as exponential random graph models (ERGM), where motif weights (links, triangles, homophily) correspond to marginal payoffs in the potential function. Parameters are identified and inferred via exchange algorithms, exploiting reversibility and detailed balance in formation dynamics (Gaonkar et al., 2021).
• Inference under link misclassification: Moment conditions and maximum-likelihood with robust confidence regions enable estimation even when network data are subject to classification errors, controlling for homophily and network externalities (Candelaria et al., 2018).
• Structural models for social preferences and information: Logit-best-response and quantal response dynamics facilitate identification of altruism, reciprocity, and inertia parameters in collaborative networks, enabling predictive counterfactual simulations and behavioral intervention assessment (Solimine et al., 2021).

6. Extensions and Critiques

Several important extensions and caveats shape the modern understanding of endogenous link-formation games:

• Action-spillover structure: When payoffs depend on local averages (rather than additive spillovers), claimed clique or locally-complete characterizations of stable networks may fail—degree dependence and denominator effects can render monotonicity and clique structure non-robust (Golub et al., 2023). Structural results must be tailored to the explicit form of payoff interdependence.
• Strategic manipulation and learning: Models with private information and strategic misrepresentation demonstrate rich patterns—multiple Nash equilibria, discontinuity in utility shifts, and polynomial-time learning protocols for strategic agents—highlighting complex incentive and information landscapes (Jalan et al., 2024).
• Large-population and continuum limits: Graphon-based and piecewise-constant models capture the equilibrium formation process in large populations, bridging finite agent games with continuous models via forward-backward stochastic differential equations and variational principles (Dayanikli et al., 5 Aug 2025).

7. Representative Applications and Implications

Endogenous link-formation games are foundational across domains:

• Education, adolescent social norms: Smoking and peer selection (Badev, 2017), study group fragmentation (Sadler et al., 2021).
• Collaborative networks and voluntary contributions: Reciprocity and positive intervention (Solimine et al., 2021).
• Epidemics, financial networks: Endogenous contagion, immunization, persistent risk (Xu, 2016, Jalan et al., 2024).
• Information sharing, social platforms: Complementarity, heterogeneity, and welfare (Alaa et al., 2015, Zhang et al., 2011).
• Interfirm ties, venture capital: Homophily, transitivity, ERGM estimation (Gaonkar et al., 2021).

Endogenous link-formation games, via their nuanced incorporation of strategic behavior, network externalities, and motif-driven aggregation, provide a theoretically robust and empirically tractable foundation for analysis and policy in a diverse array of networked systems.