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Credible Nash Bargaining Solution for Bilateral Trading Networks

Kang Rong, Qianfeng Tang

TL;DR

The paper addresses surplus division in network-constrained bilateral trading by introducing a credible bargaining solution (CBS) that anchors each matched pair's Nash bargaining split to credible outside options drawn from stable outcomes of the submarket after removing their link. It establishes general properties, proves existence via a symmetric-compromise construction, and provides a complete unit-surplus characterization emphasizing essential links. The approach refines stability-based predictions without deriving a specific bargaining protocol, yielding sharp implications for payoffs on essential links and offering a graph-theoretic method (Edmonds–Gallai decomposition) to identify them. This framework advances understanding of how network structure shapes feasible and credible surplus divisions in assignment-type markets with transfers. The results have potential normative and predictive value for decentralized matching in economics and related applications where outside options are endogenously determined by submarket outcomes.

Abstract

We study surplus division in network constrained bilateral matching markets with transferable utility. We introduce a new solution concept, the credible bargaining solution, which refines stability by requiring that, for each matched pair of buyer and seller, surplus be divided according to the Nash bargaining solution with respect to credible outside options, defined as their payoffs in some stable outcome of the submarket obtained by removing their link. We establish general properties of the credible bargaining solution, prove existence, and provide a complete characterization in the unit-surplus case based on the notion of essential links.

Credible Nash Bargaining Solution for Bilateral Trading Networks

TL;DR

The paper addresses surplus division in network-constrained bilateral trading by introducing a credible bargaining solution (CBS) that anchors each matched pair's Nash bargaining split to credible outside options drawn from stable outcomes of the submarket after removing their link. It establishes general properties, proves existence via a symmetric-compromise construction, and provides a complete unit-surplus characterization emphasizing essential links. The approach refines stability-based predictions without deriving a specific bargaining protocol, yielding sharp implications for payoffs on essential links and offering a graph-theoretic method (Edmonds–Gallai decomposition) to identify them. This framework advances understanding of how network structure shapes feasible and credible surplus divisions in assignment-type markets with transfers. The results have potential normative and predictive value for decentralized matching in economics and related applications where outside options are endogenously determined by submarket outcomes.

Abstract

We study surplus division in network constrained bilateral matching markets with transferable utility. We introduce a new solution concept, the credible bargaining solution, which refines stability by requiring that, for each matched pair of buyer and seller, surplus be divided according to the Nash bargaining solution with respect to credible outside options, defined as their payoffs in some stable outcome of the submarket obtained by removing their link. We establish general properties of the credible bargaining solution, prove existence, and provide a complete characterization in the unit-surplus case based on the notion of essential links.
Paper Structure (14 sections, 9 theorems, 17 equations, 8 figures)

This paper contains 14 sections, 9 theorems, 17 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Stable outcome
  5. Balanced outcome
  6. Credible bargaining solution
  7. Definition
  8. Illustrative examples
  9. Discussion on recursivity
  10. Results
  11. General results
  12. Existence
  13. Unit-surplus case
  14. Conclusion

Key Result

Lemma 1

Suppose $ij \in \mu$ for some optimal matching $\mu$ of the market $(G, v)$. If $(\mu^{\prime},x^{\prime})$ is a stable outcome of the submarket $(G_{-ij},v)$, then

Figures (8)

  • Figure 1: Four agents connected by a line
  • Figure 2: Balanced outcome payoffs
  • Figure 3: The graph $G_{-12}$ and stable outcome payoffs
  • Figure 4: Credible bagaining solution payoffs
  • Figure 5: Credible bargaining solution payoffs
  • ...and 3 more figures

Theorems & Definitions (27)

  • Definition 1
  • Example 1
  • Definition 2: Rochford84; CookYamagishi92
  • Example 2
  • Definition 3
  • Definition 4
  • Example 3
  • Example 4
  • Example 5
  • Lemma 1
  • ...and 17 more