[Paper Review] Fast Convergence in the Double Oral Auction

Sepehr Assadi, Sanjeev Khanna+2 more|arXiv (Cornell University)|Oct 1, 2015

Auction Theory and Applications4 citations

TL;DR

This paper proposes a decentralized mechanism that simulates the Double Oral Auction (DOA) and proves it converges to a Walrasian equilibrium in polynomial time, even under private information, strategic uncertainty, and arbitrary bid/offer recognition order. The mechanism ensures all Walrasian equilibria are reachable via better-response dynamics, resolving Friedman's conjecture on DOA convergence.

ABSTRACT

A classical trading experiment consists of a set of unit demand buyers and unit supply sellers with identical items. Each agent's value or opportunity cost for the item is their private information and preferences are quasi-linear. Trade between agents employs a double oral auction (DOA) in which both buyers and sellers call out bids or offers which an auctioneer recognizes. Transactions resulting from accepted bids and offers are recorded. This continues until there are no more acceptable bids or offers. Remarkably, the experiment consistently terminates in a Walrasian price. The main result of this paper is a mechanism in the spirit of the DOA that converges to a Walrasian equilibrium in a polynomial number of steps, thus providing a theoretical basis for the above-described empirical phenomenon. It is well-known that computation of a Walrasian equilibrium for this market corresponds to solving a maximum weight bipartite matching problem. The uncoordinated but rational responses of agents thus solve in a distributed fashion a maximum weight bipartite matching problem that is encoded by their private valuations. We show, furthermore, that every Walrasian equilibrium is reachable by some sequence of responses. This is in contrast to the well known auction algorithms for this problem which only allow one side to make offers and thus essentially choose an equilibrium that maximizes the surplus for the side making offers. Our results extend to the setting where not every agent pair is allowed to trade with each other.

Motivation & Objective

• To resolve the second part of Friedman's conjecture on why DOA converges to Walrasian equilibrium despite private information and strategic uncertainty.
• To design a mechanism that captures the four key properties of DOA: two-sided trading, private information, strategic freedom, and arbitrary recognition order.
• To prove that every Walrasian equilibrium is reachable through better-response dynamics, unlike prior auction algorithms that favor one side.
• To extend convergence guarantees to restricted trade networks, under mild conditions on bid/offer adjustments.

Proposed method

• Uses a potential function Φ = ΣP(B) + ΣP(S) over buyers and sellers to track progress toward equilibrium.
• Employs a random recognition order for bids and offers, ensuring fairness and distributability.
• Applies a bounded step size ε for bid/offer changes, ensuring incremental convergence.
• Analyzes convergence via stochastic processes: random walks on Φ and active pair dynamics.
• Uses a two-phase analysis: (1) active pairs drive u^t (unmatched agents) down, (2) in absence of active pairs, Φ drifts to bounds, forcing u^t to decrease.
• Applies concentration bounds and union bounds to show high-probability convergence within O((n³/ε²)log n) steps.

Experimental results

Research questions

• RQ1 Can a decentralized mechanism simulate the DOA and converge to a Walrasian equilibrium in polynomial time?
• RQ2 Does the mechanism ensure all Walrasian equilibria are reachable under better-response dynamics?
• RQ3 Can convergence be guaranteed under arbitrary recognition order and private information?
• RQ4 What conditions on bid/offer adjustments ensure convergence in restricted trade networks?

Key findings

• The mechanism converges to a Walrasian equilibrium in O((n³/ε²)log n) steps with high probability.
• Every Walrasian equilibrium is reachable via some sequence of better responses, unlike one-sided auction algorithms.
• Convergence is guaranteed under random recognition and bounded bid/offer adjustments, even in restricted trade networks.
• The number of unmatched agents u^t decreases in expectation when active pairs exist, driven by recognition probability ≥ 2/n.
• When no active pairs exist, the potential function Φ performs a random walk on a bounded grid, reaching bounds in O((n/ε)² log n) steps with high probability.
• The analysis uses a union bound over O(n) steps, ensuring convergence with probability 1 - O(1/n).