Dynamic Inventory Management with Mean-Field Competition

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Dynamic Inventory Management with Mean-Field Competition ⋆

Ryan Donnellya, Zi Lia

aDepartment of Mathematics, King’s College London,

Strand, London, WC2R 2LS, United Kingdom

Abstract

Agents attempt to maximize expected proﬁts earned by selling multiple units of a perishable

product where their revenue streams are aﬀected by the prices they quote as well as the

distribution of other prices quoted in the market by other agents. We propose a model which

captures this competitive eﬀect and directly analyze the model in the mean-ﬁeld limit as

the number of agents is very large. We classify mean-ﬁeld Nash equilibrium in terms of the

solution to a Hamilton-Jacobi-Bellman equation and a consistency condition and use this to

motivate an iterative numerical algorithm. Convergence of this numerical algorithm yields

the pricing strategy of a mean-ﬁeld Nash equilibrium. Properties of the equilibrium pricing

strategies and overall market dynamics are then investigated, in particular how they depend

on the strength of the competitive interaction and the ability to oversell the product.

Keywords: mean-ﬁeld game, dynamic pricing, optimal control

1. Introduction

This paper considers an optimal price setting model in which agents attempt to liquidate a

given product inventory over a ﬁnite time horizon. Agents quote prices to potential buyers

which aﬀects both the revenue made on individual sales and the intensity at which sales

are made. In addition, the intensity of sales is aﬀected by the distribution of prices quoted

across all agents, meaning the revenue rate of an individual agent is impacted by the eﬀects

of competition. When the ﬁnite time horizon is reached, each agent that has unsold units of

the perishable good recovers a salvage cost for their remaining inventory, essentially selling it

with an imposed penalty. We employ a mean-ﬁeld game approach to the model which lowers

⋆The authors would like to thank Matt Loring and Ronnie Sircar for their helpful comments on an earlier

draft.

Email addresses: ryan.f.donnelly@kcl.ac.uk (Ryan Donnelly), zi.2.li@kcl.ac.uk (Zi Li)

arXiv:2210.17208v2 [q-fin.TR] 15 Apr 2025

the complexity of numerical computation of optimal pricing strategies compared to the ﬁnite

agent setting.

Models of dynamic inventory pricing have been used in the past to dictate optimal pricing

strategies for products which have a ﬁnite shelf life, notably fashion garments, seasonal leisure

spaces, and airline tickets (see for example Gallego and Van Ryzin (1994), Anjos et al. (2004),

Anjos et al. (2005), and Gallego and Hu (2014)). Some early work in dynamic inventory

pricing restricts the underlying dynamics to be deterministic (see Jørgensen (1986), Dock-

ner and Jørgensen (1988), and Eliashberg and Steinberg (1991)) which may allow for more

tractability and further analysis of optimal policies, with or without considering competition.

Models with stochastic demand have also been studied (see Gallego and Van Ryzin (1994)

and Zhao and Zheng (2000)), but most results pertain to the case of a monopolistic agent

without competition. The paper Gallego and Hu (2014) considers an oligopolistic market, but

equilibrium with stochastic revenue streams is only classiﬁed in terms of a system of coupled

diﬀerential equations. Instead of trying to analyse the solution to these equations, which is

very computationally intense even for only two agents, the authors show that the stochastic

game is well approximated by a deterministic diﬀerential game under suitable scaling of model

parameters. The mean-ﬁeld setting we consider can be thought of as allowing the number of

agents to grow very large rather than the parameters which control the underlying dynamics.

This retains a level of computational complexity at a level similar to the single agent case

which allows us to investigate the eﬀects of competition.

Our model is a generalization of the single agent model considered in Gallego and Van Ryzin

(1994), which we brieﬂy summarize and refer to as the reference model when discussing the

mean-ﬁeld case. Speciﬁcally, agents hold a positive integer number of units of an asset and

quote a selling price for each unit continuously through time. Sales occur at random times

with an intensity that depends on the price quoted by the agent such that higher prices result

in less frequent trades, creating a trade-oﬀ between large but infrequent revenue of quoting

high prices, versus small but frequent revenue of low prices. Additionally, competition between

agents is modelled by specifying that the sell intensity also depends on the distribution of

prices quoted by all agents. Thus, an agent’s selling rate increases if other agents begin to

quote higher prices. Our model setting is quite similar to the one in Yang and Xia (2013),

especially in regards to the dynamics of agents’ inventory levels. The main diﬀerence is that

our model allows prices to be continuous rather than being selected from a ﬁnite set of either

high price or low price. Additionally, as a mean-ﬁeld extension of one of the models presented

in Dockner and Jørgensen (1988), Chenavaz et al. (2021) has a similar sell intensity to our

work that depends on the distribution of prices quoted by all agents. Our work mainly diﬀers

from Chenavaz et al. (2021) in that their model considers agents with states determined by

a continuous quantity which changes deterministically, whereas in our work a representative

agent has a discrete inventory level which changes stochastically.

By working in a mean-ﬁeld setting, our conditions for equilibrium do not require the solution

to a coupled system of diﬀerential equations. Instead, equilibrium is classiﬁed by a single

diﬀerential equation and a consistency condition. This lends itself to an eﬃcient iterative

algorithm which upon convergence yields a mean-ﬁeld Nash equilibrium. We are unable

to prove the existence of a mean-ﬁeld Nash equilibrium, but we have conducted extensive

numerical experiments with the iterative algorithm which always converge to the same result

within a small numerical tolerance. The low dimension of the system of equations which

classify equilibrium allows us to easily demonstrate the resulting pricing strategies, and hence

investigate how prices under the eﬀects of competition compare to those of the reference

model.

The tractability oﬀered by a mean-ﬁeld game framework over ﬁnite agent models has also led

to their use in studying competition in other types of markets. In Chan and Sircar (2017)

and Ludkovski and Yang (2017), the eﬀects of competition through mean-ﬁeld interaction are

incorporated into models of energy production and commodity extraction. In Donnelly and

Leung (2019), agents compete for a reward in an R&D setting within a mean-ﬁeld framework,

in which earlier success yields greater rewards for the expended eﬀort. In Li et al. (2024)

agents expend eﬀort to mine cryptocurrency, where an agent’s rate of mining depends on

their hash rate as well as that of the entire population of miners. Our modeling framework

has some similarities to these papers which allows us to employ a nearly direct adaptation of

relevant numerical methods to compute equilibrium in our model.

A novel focus of our work is regarding how overall market behaviour is aﬀected by features

describing individual agents. In particular, we investigate how the magnitude of competitive

interaction, the ability to oversell the asset (with penalty), and price caps aﬀect the total

wealth transferred from consumers in the market, the average price paid per unit asset, and

the probability that a particular consumer will end up empty handed due to overselling of

the asset. The dependence of market behaviour on these phenomena could be used to guide

regulatory framework with the goal of achieving desired levels of various measurements of

economic welfare.

The rest of the paper is organized as follows: in Section 2 we give an overview of the reference

model, which is the single-agent equivalent to the mean-ﬁeld setting we cover in more detail.

In Section 3 we specify our model that incorporates the eﬀects of competition, including

the deﬁnition of equilibrium which we consider. In Section 4 we show several examples of

numerically solving for equilibrium and investigate the eﬀects of competition and other market

phenomena. Section 5 concludes, and longer proofs are contained in the Appendix.

1.1. General Notation

Here we introduce the general notation which is used throughout the remainder of the work.

(Ω,F,{Ft}0≤t≤T,P) A ﬁltered probability space.

XAIndicator function of the event A.

Et,s,x,q[Y] Conditional expectation of Ygiven St=s,Xt−=xand Qt−=q.

Q, Q Finite upper and lower bounds of inventory.

TFinite terminal time horizon.

S= (St)t∈[0,T ]Reference price process.

AThe set of admissible controls.

δi, δfSpread process of agent i, or induced by feedback control f.

δ, δfMean spread posted by all agents, or induced by the feedback control f.

Nλ(δ), Nλ(δi,δ)

iCounting process of the number of items sold.

Qδ, Qδi,δ

iRemaining inventory held by an agent.

Xδ, Xδi,δ

iThe cash generated by an agent selling inventory.

λ(δ), λ(δ, δ) The intensity function which determines the rate of inventory sales.

f(t, q) Feedback form of a Markov strategy.

Pf,M

q, P f

qProportion process in the M-player setting and in the mean-ﬁeld limit.

2. Single Agent Reference Model

In this section, we introduce a reference model where we only consider one agent. Many

aspects of the dynamics we consider are equivalent to those found in Gallego and Van Ryzin

(1994) with some modiﬁcations made to the price process and agent’s performance criterion.

This style of model which relates intensity to price has also been used frequently in the

literature on algorithmic trading. See for example Gu´eant et al. (2012), Gu´eant and Lehalle

(2015), and Cartea and Jaimungal (2015).

We work on a probability space (Ω,F,P) which we assume supports all random variables

and stochastic processes deﬁned below. We consider an agent who has to liquidate a ﬁnite

quantity Qof a given product within a ﬁnite time horizon of length T. The reference price

process of the product is denoted by S= (St)t∈[0,T ]with dynamics

dSt=σ dWt,(1)

where σ > 0 is a constant and W= (Wt)t∈[0,T ]is a Brownian motion.

We denote the agent’s spread process above the reference price by δ= (δt)t∈[0,T ], so that she

continuously quotes her selling price at St+δt−1at every time point, and is committed to sell

one item2with the quoted price. Once the agent clears her inventory, she will stop trading.

Given a non-negative bounded process λ= (λt)t∈[0,T ], her number of items sold follows a

counting process denoted by Nλ= (Nλ

t)t∈[0,T ]deﬁned as follows: let {un}∞

n=1 be a sequence

of independent standard uniform random variables. Deﬁne

τ0= 0 ,

τn= inf{t≥τn−1:e−Rt

τn−1λudu ≤un},

Nλ

t= sup{n≥0 : t≥τn}.

Then Nλis a doubly stochastic Poisson process with intensity process λ(see Lando (1998)),

and the sequence of times at which Nλjumps is {τn}∞

n=1. Subsequently, we will let the

intensity process depend on her spread through the relation λt=λ(δt) for a function to be

speciﬁed later, and we will write Nλ(δ)to denote the number of items sold when the agent

quotes a spread according to δ= (δt)t∈[0,T ]. We denote the indicator function of any event A

by XA. Thus, the agent’s inventory Qδ= (Qδ

t)t∈[0,T ]satisﬁes

dQδ

t=−XQδ

t−>0dNλ(δ)

with initial value Qδ

0=Q. As a consequence of her trades, the agent accumulates cash denoted

by Xδ= (Xδ

t)t∈[0,T ]with dynamics given by

dXδ

t=XQδ

t−>0(St+δt−)dNλ(δ)

with given initial cash Xδ

0=x0.

The agent’s goal is to maximize her expected P&L at time Twith an inventory penalty.

Speciﬁcally, her value functional is

J(δ) = EXδ

T+Qδ

TST−α Qδ

T−ϕZT

0Qδ

u2du,

where αand ϕare positive constants. The objective functional consists of three parts.

The ﬁrst term in the expectation Xδ

Tis the amount of cash at time T. The second term

Qδ

TST−α Qδ

Tcorresponds to the salvage value of unsold items remaining at time T, and

1Here by t−, we mean the left limit to time t.

2Note that selling one item may be understood as selling a block of units of the product, each block being

of the same size.

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摘要：

DynamicInventoryManagementwithMean-FieldCompetition⋆RyanDonnellya,ZiLiaaDepartmentofMathematics,King’sCollegeLondon,Strand,London,WC2R2LS,UnitedKingdomAbstractAgentsattempttomaximizeexpectedprofitsearnedbysellingmultipleunitsofaperishableproductwheretheirrevenuestreamsareaffectedbythepricestheyquote...

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Dynamic Inventory Management with Mean-Field Competition

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