Abstract

One of the most important stability concepts for network formation is pairwise stability. We develop a homotopy algorithm that is effective in computing pairwise stable networks for a generic network formation problem. To do so, we reformulate the concept of pairwise stability as a Nash equilibrium of a non-cooperative game played by the links in the network and adapt the linear tracing procedure for non-cooperative games to the network formation problem. As a by-product of our main result, we obtain that the number of pairwise stable networks is generically odd. We apply the algorithm to the connections model and obtain a number of novel insights.

Highlights

  • Networks are at the forefront of research in economics, operations research, and computer science as powerful tools to model social and economic interactions

  • We develop a homotopy algorithm that is effective in computing pairwise stable networks for a generic network formation problem

  • This paper studies the problem of computing a pairwise stable network as introduced in Bich and Morhaim (2020), extending work by Jackson and Wolinsky (1996)

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Summary

Introduction

Networks are at the forefront of research in economics, operations research, and computer science as powerful tools to model social and economic interactions. Bich and Morhaim (2020) prove that pairwise stable networks exist if all agents have quasi-concave and continuous utility functions. For normal-form games, the Nash equilibrium selected by the linear tracing procedure can be computed by the homotopy algorithm of Herings and Peeters (2001). Our algorithm adapts the linear tracing procedure of Harsanyi and Selten (1988) to select a pairwise stable network. We show that for a generic network formation problem, there is a unique path that transforms arbitrary prior beliefs of agents to beliefs that are compatible with a pairwise stable network.

Pairwise Stable Networks
The Linear Tracing Procedure for Networks
The Structure of the Linear Tracing Procedure
A Differentiable Approach
An Example for the Connections Model
Numerical Implementation
Conclusion
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