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Decentralized Convergence to Equilibrium Prices in Trading Networks

Edwin Lock, Benjamin Patrick Evans, Eleonora Kreacic, Sujay Bhatt, Alec Koppel, Sumitra Ganesh, Paul W. Goldberg

TL;DR

This work proposes a decentralized market model in which agents can negotiate bilateral contracts, and designs a best response dynamic intended to capture such negotiations between market participants, and assumes fully substitutable preferences for market participants.

Abstract

We propose a decentralized market model in which agents can negotiate bilateral contracts. This builds on a similar, but centralized, model of trading networks introduced by Hatfield et al. in 2013. Prior work has established that fully-substitutable preferences guarantee the existence of competitive equilibria which can be centrally computed. Our motivation comes from the fact that prices in markets such as over-the-counter markets and used car markets arise from decentralized negotiation among agents, which has left open an important question as to whether equilibrium prices can emerge from agent-to-agent bilateral negotiations. We design a best response dynamic intended to capture such negotiations between market participants. We assume fully substitutable preferences for market participants. In this setting, we provide proofs of convergence for sparse markets (covering many real world markets of interest), and experimental results for more general cases, demonstrating that prices indeed reach equilibrium, quickly, via bilateral negotiations. Our best response dynamic, and its convergence behavior, forms an important first step in understanding how decentralized markets reach, and retain, equilibrium.

Decentralized Convergence to Equilibrium Prices in Trading Networks

TL;DR

This work proposes a decentralized market model in which agents can negotiate bilateral contracts, and designs a best response dynamic intended to capture such negotiations between market participants, and assumes fully substitutable preferences for market participants.

Abstract

We propose a decentralized market model in which agents can negotiate bilateral contracts. This builds on a similar, but centralized, model of trading networks introduced by Hatfield et al. in 2013. Prior work has established that fully-substitutable preferences guarantee the existence of competitive equilibria which can be centrally computed. Our motivation comes from the fact that prices in markets such as over-the-counter markets and used car markets arise from decentralized negotiation among agents, which has left open an important question as to whether equilibrium prices can emerge from agent-to-agent bilateral negotiations. We design a best response dynamic intended to capture such negotiations between market participants. We assume fully substitutable preferences for market participants. In this setting, we provide proofs of convergence for sparse markets (covering many real world markets of interest), and experimental results for more general cases, demonstrating that prices indeed reach equilibrium, quickly, via bilateral negotiations. Our best response dynamic, and its convergence behavior, forms an important first step in understanding how decentralized markets reach, and retain, equilibrium.

Paper Structure

This paper contains 26 sections, 10 theorems, 8 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main Contributions.
  3. Model
  4. Preferences
  5. Market Dynamic
  6. The Emergence of Equilibria
  7. Experiments
  8. Market Topologies.
  9. Experimental Results
  10. BS Networks.
  11. BIS Networks.
  12. General Networks.
  13. Shocks
  14. Results Summary
  15. Discussion and Conclusion
  16. ...and 11 more sections

Key Result

Proposition 5

Markets consisting of a single trade converge to equilibrium after $O(V)$ BRs. Two-agent FSMs with two trades converge to equilibrium after $O(V^2)$ BRs.

Figures (10)

  • Figure 1: An illustration of a market with two producers (coffee farms), one intermediary (a roaster), and two buyers (coffee shop and supermarket).
  • Figure 2: An example trade between two parties. Rather than trades having a single price, each trade has two offers, the buyer's offer and the seller's offer. Offers are initially arbitrary, and iteratively updated through best responses.
  • Figure 3: Illustrating the proof of \ref{['proposition:convergence-guarantees']}.
  • Figure 4: Convergence paths by market topology. The $x$-axis shows the number of iterations (e.g., number of best response updates), and the $y$-axis shows the proportion of satisfied agents.
  • Figure 5: Social welfare (mean $u_i$) by agent composition in the BS network. The $x$-axis varies the proportion of buyers, and the $y$-axis displays the resulting welfare.
  • ...and 5 more figures

Theorems & Definitions (28)

  • Example 1
  • Definition 2
  • Example 3
  • Definition 4
  • Proposition 5
  • Example 6
  • Conjecture 7
  • Definition 8
  • Proposition 9
  • Theorem 10
  • ...and 18 more