Extension of Excess Demand Ascending Auction to Multi-Demand Model by Discrete Convex Analysis Approach

TL;DR

This paper extends ascending auction algorithms for finding minimal Walrasian equilibrium prices to multi-demand, multi-unit auction models using discrete convex analysis, specifically L-natural-convex functions.

Contribution

It generalizes the excess-demand ascending auction approach to multi-demand models by leveraging discrete convexity properties of the Lyapunov function.

Findings

The equilibrium prices are characterized as minimizers of an L-natural-convex Lyapunov function.

A generalized excess-demand concept is defined within the L-natural-convex function framework.

The approach provides a theoretical foundation for auction algorithms in multi-demand multi-unit settings.

Abstract

We consider the problem of finding the (unique) minimal Walrasian equilibrium price in multi-item, multi-unit auction models: there are multiple indivisible items for sale, with several units of each item, and a bidder may be interested in buying more than one copy of each item. In its special case with unit-demand bidders, where each bidder demands at most one unit of any item, Andersson, Andersson, and Talman (2013) proposed a general framework of ascending auction algorithms based on the concept of excess-demand item set. This paper extends this approach to the multi-unit case by exploiting the discrete convexity of the Lyapunov function associated with the auction model. In particular, we make use of the facts that (i) the equilibrium price vectors are characterized as the minimizers of the Lyapunov function, (ii) the Lyapunov function is an instance of an L-natural-convex function, and (iii) a concept generalizing ``excess-demand item set'' can be defined in L-natural-convex function minimization in general.