A single-item continuous double auction game

TL;DR

This paper introduces a model of a single-item continuous double auction with private valuations, analyzing equilibrium strategies and showing that competitive behavior is not an equilibrium, with implications for market efficiency.

Contribution

It provides a formal analysis of Bayesian Nash equilibria in a double auction with linear supply and demand, connecting to existing market models.

Findings

Existence of at most one Bayesian Nash equilibrium for linear supply and demand.

Competitive behavior is not an equilibrium in this auction model.

Total expected profit at equilibrium matches that of competitive market behavior.

Abstract

A double auction game with an infinite number of buyers and sellers is introduced. All sellers posses one unit of a good, all buyers desire to buy one unit. Each seller and each buyer has a private valuation of the good. The distribution of the valuations define supply and demand functions. One unit of the good is auctioned. At successive, discrete time instances, a player is randomly selected to make a bid (buyer) or an ask (seller). When the maximum of the bids becomes larger than the minimum of the asks, a transaction occurs and the auction is closed. The players have to choose the value of their bid or ask before the auction starts and use this value when they are selected. Assuming that the supply and demand functions are known, expected profits as functions of the strategies are derived, as well as expected transaction prices. It is shown that for linear supply and demand functions, there exists at most one Bayesian Nash equilibrium. Competitive behaviour is not an equilibrium of the game. For linear supply and demand functions, the sum of the expected profit of the sellers and the buyers is the same for the Bayesian Nash equilibrium and the market where players behave competitively. Connections are made with the ZI-C traders model and the $k$-double auction.