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Dynamic Realization Games in Newsvendor Inventory Centralization

M. Dror, Luis A. Guardiola, Ana Meca, Justo Puerto

TL;DR

The paper models newsvendor centralization as a pair of cooperative games: an expected-cost game $c_E$ and a realization-cost game $c_R$, highlighting that $E$-games are balanced while $R$-games can be unbalanced. It introduces a dynamic, online cost allocation framework by applying Lehrer’s repeated allocation rules to dynamic realization games (DR-games), proving almost-sure convergence of the diagonal allocation sequences to a least-squares value or to the core of the expected game. The results extend to general dynamic cost games and accommodate relaxing independence assumptions to strong stationarity, preserving convergence toward equitable long-run allocations. This work provides principled, provable methods for cost-sharing in centralized inventory systems with uncertain, time-varying demand, offering practical guidance for sustaining cooperation under realism-driven dynamics.

Abstract

Consider a set N of n (>1) stores with single-item and single-period nondeterministic demands like in a classic newsvendor setting with holding and penalty costs only. Assume a risk-pooling single-warehouse centralized inventory ordering option. Allocation of costs in the centralized inventory ordering corresponds to modelling it as a cooperative cost game whose players are the stores. It has been shown that when holding and penalty costs are identical for all subsets of stores, the game based on optimal expected costs has a non empty core (Hartman et. al., 2000, Muller \textit{et. al.}, 2002). In this paper we examine a related inventory centralization game based on demand realizations that has, in general, an empty core even with identical penalty and holding costs (Hartman and Dror, 2005). We propose a repeated cost allocation scheme for dynamic realization games based on allocation processes introduced by Lehrer (2002a). We prove that the cost subsequences of the dynamic realization game process, based on Lehrer's rules, converge almost surely to either a least square value or the core of the expected game. We extend the above results to more general dynamic cost games and relax the independence hypothesis of the sequence of players' demands at different stages.

Dynamic Realization Games in Newsvendor Inventory Centralization

TL;DR

The paper models newsvendor centralization as a pair of cooperative games: an expected-cost game and a realization-cost game , highlighting that -games are balanced while -games can be unbalanced. It introduces a dynamic, online cost allocation framework by applying Lehrer’s repeated allocation rules to dynamic realization games (DR-games), proving almost-sure convergence of the diagonal allocation sequences to a least-squares value or to the core of the expected game. The results extend to general dynamic cost games and accommodate relaxing independence assumptions to strong stationarity, preserving convergence toward equitable long-run allocations. This work provides principled, provable methods for cost-sharing in centralized inventory systems with uncertain, time-varying demand, offering practical guidance for sustaining cooperation under realism-driven dynamics.

Abstract

Consider a set N of n (>1) stores with single-item and single-period nondeterministic demands like in a classic newsvendor setting with holding and penalty costs only. Assume a risk-pooling single-warehouse centralized inventory ordering option. Allocation of costs in the centralized inventory ordering corresponds to modelling it as a cooperative cost game whose players are the stores. It has been shown that when holding and penalty costs are identical for all subsets of stores, the game based on optimal expected costs has a non empty core (Hartman et. al., 2000, Muller \textit{et. al.}, 2002). In this paper we examine a related inventory centralization game based on demand realizations that has, in general, an empty core even with identical penalty and holding costs (Hartman and Dror, 2005). We propose a repeated cost allocation scheme for dynamic realization games based on allocation processes introduced by Lehrer (2002a). We prove that the cost subsequences of the dynamic realization game process, based on Lehrer's rules, converge almost surely to either a least square value or the core of the expected game. We extend the above results to more general dynamic cost games and relax the independence hypothesis of the sequence of players' demands at different stages.
Paper Structure (7 sections, 9 theorems, 38 equations)

This paper contains 7 sections, 9 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Inventory centralization in newsvendor environments
  3. Repeated allocation processes
  4. Allocation processes for the dynamic newsvendor realization game
  5. Final comments and remarks
  6. Allocation processes for dynamic cost games
  7. Removing the independence hypothesis

Key Result

Proposition 2.1

Consider an $E$-game $(N,c_{E})$ and its corresponding $R$-game $(N,c_{R}).$ Then, $E\left[ c_{R}(S)\right] =c_{E}(S)$ for all $\emptyset \neq S\subseteq N.$

Theorems & Definitions (9)

  • Proposition 2.1
  • Lemma 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 5.1
  • Theorem 5.2
  • Lemma 5.3
  • Theorem 5.4
  • Theorem 5.5