An Optimal Decision Rule for a Multiple Selling Problem with a Variable Rate of Offers

Georgy Sofronov

doi:10.3390/math8050690

Abstract

An asset selling problem is one of well-known problems in the decision making literature. The problem assumes a stream of bidders who would like to buy one or several identical objects (assets). Offers placed by the bidders once rejected cannot be recalled. The seller is interested in an optimal selling strategy that maximizes the total expected revenue. In this paper, we consider a multi-asset selling problem when the seller wants to sell several identical assets over a finite time horizon with a variable number of offers per time period and no recall of past offers. We consider the problem within the framework of the optimal stopping theory. Using the method of backward induction, we find an optimal sequential procedure which maximizes the total expected revenue in the selling problem with independent observations.

Highlights

Assume we sequentially observe independent identically distributed random variables y1, y2, . . . , y N, where the value yn can be interpreted as the value of an asset at time n
We have K identical objects, which we want to sell within the finite time horizon N, and at each time n we may receive no more than Cn offers, and we are not allowed to recall past offers
We have considered one of well-known problems in the decision making literature, an asset selling problem

Summary

Introduction

Assume we sequentially observe independent identically distributed random variables y1 , y2 , . . . , y N , where the value yn can be interpreted as the value of an asset (for example, a house) at time n. A problem of selling several identical objects with a finite horizon and with one offer per time period, Cn = 1, and no recall of past offers is discussed in [18]. This problem is generalised in [1], where we consider a model with a fixed number of offers per time period, Cn = C. The methodology of optimal multiple stopping rules allows us to solve problems with different types of actions, for example, buying and selling. In [28], the authors propose a non-parametric approach to a buying-selling problem with the finite time horizon.

Definitions

Finite Case

Main Result

Uniform Distribution

Exponential Distribution

Discussion