# Dynamic Inventory Management with Mean-Field Competition 111The authors
would like to thank Matt Loring and Ronnie Sircar for their helpful comments
on an earlier draft.

Ryan Donnelly<EMAIL_ADDRESS>Zi Li<EMAIL_ADDRESS>Department
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 to compute 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

††journal: TBA

##  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 allows for more tractability in
the analysis and computation of optimal pricing strategies compared to
previous works.

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), Dockner 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 most
of the computational tractability of 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, agent’s 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.

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. The tractability of the
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.

This style of model which relates intensity to price has also been used
frequently in the literature on algorithmic trading. See for example Guéant et
al. (2012), Guéant and Lehalle (2015), and Cartea and Jaimungal (2015). The
main modelling differences in these works compared to those of dynamic
inventory management already mentioned are the existence of an exogenous
reference price process, relative to which optimal selling prices are quoted,
and positive risk-aversion of the agent, both of which we also include in our
model. In the financial setting of algorithmic trading, it is natural to give
agents the ability to both buy and sell. This two-sided setting has also been
heavily studied with similar intensity models, such as in Avellaneda and
Stoikov (2008), Guéant et al. (2013), Cartea et al. (2014), and Cartea et al.
(2017). These models are frequently applied in over-the-counter markets which
are somewhat of a financial analogue to the consumer market setting which we
envision in this work.

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 with mean-field, in which
earlier success yields greater rewards for the expended effort. In Li et al.
(2019) 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, Section 6.

##  2 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. Similarities can also be found with
models pertaining to algorithmic trading, such as Guéant et al. (2012).

We work on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with
filtration $\\{\mathcal{F}_{t}\\}_{0\leq t\leq T}$ satisfying the usual
conditions, and assume that all random variables and stochastic processes are
defined on the filtered probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{0\leq t\leq T},\mathbb{P})$. We
consider an agent who has to liquidate a certain quantity $\overline{Q}$ of a
given product within a finite time horizon of length $T$. The reference price
process of the product is denoted by $S=(S_{t})_{t\in[0,T]}$ with dynamics

$\displaystyle dS_{t}$ $\displaystyle=\sigma\,dW_{t}\,,$ (1)

where $\sigma>0$ is a constant, and $W=(W_{t})_{t\in[0,T]}$ is a Brownian
motion.

We denote the agent’s spread process above the reference price by
$\delta=(\delta_{t})_{t\in[0,T]}$, so that she continuously quotes her selling
price at $S_{t}+\delta_{t^{-}}$ at every time point, and is committed to sell
one item222Note that selling one item may be understood as selling a block of
units of the product, each block being of the same size. with the quoted
price. Once the agent clears her inventory, she will stop trading. Her number
of items sold follows a counting process denoted by
$N^{\delta}=(N_{t}^{\delta})_{t\in[0,T]}$. The superscript of $N^{\delta}$
denotes the dependence of the instantaneous intensity of this process
$\lambda(\delta)$ on the spread process $\delta$. Thus, the agent’s inventory
$Q^{\delta}=(Q_{t}^{\delta})_{t\in[0,T]}$ satisfies

$\displaystyle
dQ_{t}^{\delta}=-\mathcal{X}_{Q_{t^{-}}^{\delta}>0}\,dN_{t}^{\delta},$

with initial value $Q_{0}^{\delta}=\overline{Q}$. As a consequence of her
trades, the agent accumulates cash denoted by
$X^{\delta}=(X_{t}^{\delta})_{t\in[0,T]}$ with dynamics given by

$\displaystyle
dX^{\delta}_{t}=\mathcal{X}_{Q_{t^{-}}^{\delta}>0}\,\left(S_{t}+\delta_{t^{-}}\right)dN_{t}^{\delta},$

with given initial cash $X_{0}^{\delta}=x_{0}$.

The agent’s goal is to optimize her expected P&L at time $T$ with an inventory
penalty. Specifically, her value functional is

$\displaystyle
J(\delta)=\mathbb{E}\left[X_{T}^{\delta}+Q_{T}^{\delta}\,\left(S_{T}-\alpha
Q_{T}^{\delta}\right)-\phi\int^{T}_{0}\left(Q_{u}^{\delta}\right)^{2}du\right]\,,$

where $\alpha$ and $\phi$ are positive constants. The objective functional
consists of three parts. The first term in the expectation $X_{T}^{\delta}$ is
the amount of cash at time $T$. The second term $Q_{T}^{\delta}(S_{T}-\alpha
Q_{T}^{\delta})$ corresponds to the salvage value of unsold items remaining at
time $T$, and the parameter $\alpha$ represents a penalty for failing to sell
all inventory by the end of the trading period. The running inventory penalty
term $\phi\int^{T}_{0}(Q_{u}^{\delta})^{2}du$ penalizes deviation from zero
during the entire trading horizon, so it can be interpreted as an urgency
penalty, and the parameter $\phi$ acts as a risk control. Thus, the agent’s
dynamic value function is

$\displaystyle H(t,s,x,q)=\sup_{(\delta_{u})_{t\leq u\leq
T}\in\mathcal{A}}\mathbb{E}\left[X_{T}^{\delta}+Q_{T}^{\delta}\,\left(S_{T}-\alpha
Q_{T}^{\delta}\right)-\phi\int^{T}_{0}\left(Q_{u}^{\delta}\right)^{2}du\right]\,,$
(2)

where the set of admissible controls $\mathcal{A}$ consists of predictable
processes on $[0,T]$ bounded from below by some large negative constant, $B$.

To solve the optimal control problem described above, we consider the
associated Hamilton-Jacobi-Bellman (HJB) equation along with terminal
condition

$\displaystyle\begin{split}&\partial_{t}H+\frac{1}{2}\,\sigma^{2}\,\partial_{ss}H-\phi\,q^{2}+\sup_{\delta\geq
B}\lambda(\delta)\left[H(t,s,x+s+\delta,q-1)-H(t,s,x,q)\right]\mathcal{X}_{q>0}=0\,,\\\
&H(T,s,x,q)=x+q\,(s-\alpha\,q)\,,\end{split}$ (3)

where $\mathcal{X}_{q>0}$ is the indicator function that the inventory is non-
zero. To solve the HJB equation, we use the excess value ansatz for $H$ given
by

$\displaystyle H(t,s,x,q)=x+q\,s+h_{q}(t)\,.$ (4)

The first term $x$ is the current cash in hand, the second term $qs$ accounts
for the reference value of the current inventory, and the last term
$h_{q}(t;\overline{\delta})$ represents the expected profit or loss from
liquidating $q$ items with both the terminal penalty and the running inventory
penalty. Substituting the ansatz into HJB equation gives a system of ODEs with
terminal condition given by

$\displaystyle\begin{split}&\partial_{t}h_{q}-\phi\,q^{2}+\sup_{\delta\geq
B}\lambda(\delta)\left[\delta+h_{q-1}(t)-h_{q}(t)\right]\mathcal{X}_{q>0}=0\,,\\\
&h_{q}(T)=-\alpha q^{2}\,.\end{split}$ (5)

To solve for the optimal feedback controls in terms of $h_{q}(t)$, we assume
that the intensity function $\lambda$ follows

$\displaystyle\lambda(\delta)=A\exp{\\{-\kappa\,\delta\\}}\,$ (6)

where $A,\kappa\geq 0$. Here $\kappa$ is the sensitivity of the demand rate of
the product with respect to the agent’s quoted price333Other forms of the
intensity function can be used, but a particular property which is desired is
that there is a unique maximizer of $\delta\,\lambda(\delta)$. We choose this
exponential form for tractability reasons.. This means that the lower the
quote is, the faster the item will be sold.

We delay a proof of existence and uniqueness of solution to Equation (5) until
we have developed the control problem that incorporates multiple agents. The
solution to Equation (5) is a special case of the solution to Equation (15)
which appears in Proposition 3. A verification that solutions to the HJB
equation (3) yield the value function defined in (2) is also postponed to the
more general setting when we include competition (see Theorem 4).

######  Proposition 1 (Optimal Feedback Control)

The optimal feedback controls of the HJB equation are given by

$\displaystyle\delta^{*}(t,q)=\max\left\\{\frac{1}{\kappa}+h_{q}(t)-h_{q-1}(t),B\right\\}\,,\quad
q\neq 0\,.$

  * Proof

Substituting the intensity function from (6) into Equation (5), we have

$\displaystyle\partial_{t}h_{q}-\phi\,q^{2}+\sup_{\delta\geq
B}A\,\exp{\\{-\kappa\,\delta\\}}\left[\delta+h_{q-1}(t)-h_{q}(t)\right]\mathcal{X}_{q>0}=0\,.$

Apply first order conditions to the $\sup$ term. It is an easy task to check
that the second order derivative of the sup term at $\delta^{*}$ is negative.

To intuitively understand the optimal feedback control form of the quote, note
that the first term $\frac{1}{\kappa}$ maximizes the rate of expected incoming
profit, without any consideration for risk or terminal penalties. The quantity
$h_{q-1}(t)-h_{q}(t)$ is the change of the expected future risk-adjusted P&L
due to selling one item. Thus, the negative of this value,
$h_{q}(t)-h_{q-1}(t)$, represents the amount that the agent is willing to
adjust her quote, positively or negatively, depending on the change of future
risks and potential future profits by having one fewer item to sell.

Figure 1: Optimal pricing strategy in reference model as a function of $t$ for
various $Q$. Other parameters are $T=10$, $\overline{Q}=5$, $\alpha=0.1$,
$\kappa=1$, $\phi=0.03$, $A=1$, and $B=-10$.

Figure 1 shows the optimal quotes when Equation $\eqref{eqn:HJBSingle}$ is
numerically solved after substituting the optimal feedback controls in
Proposition 1. We observe that the optimal quotes are decreasing in inventory
level monotonically. This is sensible as the agents with more inventory have
more urgency to sell the products and avoid the terminal penalty and
uncertainty of the underlying price. In addition, the agent would lower her
quotes in the early time interval to sell fast, and then increase her quotes
near the terminal time $T$. This is because at final time, the agent is more
willing to benefit from a trading opportunity through proposing a higher quote
and receiving a lower probability to sell.

We can also see that optimal quotes could be negative sometimes. This happens
when there is a need to liquidate products at a very fast speed to avoid
terminal penalty or underlying price risk. To avoid this situation, we could
set the lower bound of admissible strategies to $0$, or in the algorithmic
trading context, the agent could use market order instead of limit order when
negative quotes appear.

##  3 Competition Model

In this section we introduce a model with multiple agents, specify the
dynamics of each agent’s inventory and wealth processes based on the prices
they quote, and specify the optimization problem that each agent attempts to
solve. Additionally, and distinct from the previous section with only one
agent, we give the dynamics of the distribution of inventory across all agents
in the mean-field limit, and finally we define the corresponding notion of
equilibrium. This model is similar to that of Gallego and Hu (2014), and while
we initially consider a finite number of agents we will only search for
equilibrium in the mean-field limit of an infinite number of agents because
this allows for more tractability and further analysis of the solution.

We consider a finite collection of agents indexed by $i\in\\{1,\dots,M\\}$
aiming to liquidate shares of a given product. The reference price process is
the same as Equation (1).

###  3.1 Inventory, Wealth, and Performance Criterion

Each agent chooses a spread process $\delta^{i}=(\delta^{i}_{t})_{t\in[0,T]}$,
but only actively posts a price when they hold non-zero inventory. Once their
inventory reaches zero, their inventory state remains constant until time $T$,
effectively leaving the market. The mean spread chosen by all agents in the
market is denoted by $\overline{\delta}=(\overline{\delta}_{t})_{t\in[0,T]}$.
Each agent has an associated inventory and wealth process denoted by
$Q^{\delta^{i},\overline{\delta}}_{i}=(Q^{\delta^{i},\overline{\delta}}_{i,t})_{t\in[0,T]}$
and
$X^{\delta^{i},\overline{\delta}}_{i}=(X^{\delta^{i},\overline{\delta}}_{i,t})_{t\in[0,T]}$,
respectively, and we assume all agents begin at time $t=0$ with the same
maximum inventory $Q^{\delta^{i},\overline{\delta}}_{i,0}=\overline{Q}$. The
superscripts of $\delta^{i}$ and $\overline{\delta}$ are to indicate that the
inventory and wealth processes of a single agent depend not only on their own
strategy, but also on the mean strategy across all agents444We assume that
dynamics depend on the distribution of spreads only through the mean spread,
but other dependencies on the distribution of spreads across agents could also
be implemented.. Keeping in mind that spreads are only posted by agents with
non-zero inventory, the mean spread at time $t$ is equal to

$\displaystyle\overline{\delta_{t}}$
$\displaystyle=\frac{\sum_{i=1}^{M}\delta^{i}_{t}\,\mathcal{X}_{Q_{i,t}^{\delta^{i},\overline{\delta}}>0}}{\sum_{i=1}^{M}\mathcal{X}_{Q_{i,t}^{\delta^{i},\overline{\delta}}>0}}\,.$

The inventory and wealth of agent $i$ will changed based on the arrivals of a
counting process denoted by
$N^{\delta^{i},\overline{\delta}}_{i}=(N^{\delta^{i},\overline{\delta}}_{i,t})_{t\in[0,T]}$
which has instantaneous intensity given by
$\lambda(\delta^{i}_{t},\overline{\delta}_{t})$ for a function $\lambda$ to be
specified later. Thus, the inventory of agent $i$ changes according to the
dynamics

$\displaystyle
dQ_{i,t}^{\delta^{i},\overline{\delta}}=-\mathcal{X}_{Q_{i,t^{-}}^{\delta^{i},\overline{\delta}}>0}\,dN^{\delta^{i},\overline{\delta}}_{i,t}\,.$

When a transaction is executed with agent $i$ at time $t$ it occurs at a price
of $S_{t}+\delta^{i}_{t^{-}}$ so that their wealth changes according to

$\displaystyle
dX_{i,t}^{\delta^{i},\overline{\delta}}=\mathcal{X}_{Q_{i,t^{-}}^{\delta^{i},\overline{\delta}}>0}\,(S_{t}+\delta^{i}_{t^{-}})\,dN_{i,t}^{\delta^{i},\overline{\delta}}\,.$

We assume that, conditional on each spread process, the jump processes
$N_{i}^{\delta^{i},\overline{\delta}}$ are independent. For a single agent
with index $i$, given fixed strategies for each other agent, their performance
criterion is given by

$\displaystyle
J_{i}(\delta^{i};\delta^{-i})=\mathbb{E}\biggl{[}X_{i,T}^{\delta^{i},\overline{\delta}}+Q_{i,T}^{\delta^{i},\overline{\delta}}\,(S_{T}-\alpha\,Q_{i,T}^{\delta^{i},\overline{\delta}})-\phi\int_{0}^{T}(Q_{i,t}^{\delta^{i},\overline{\delta}})^{2}\,dt\biggr{]}\,.$
(7)

###  3.2 Mean-Field Population Dynamics

We now proceed to the mean-field limit $M\rightarrow\infty$. Because the
agents are homogeneous with respect to their inventory dynamics, wealth
dynamics, and performance criteria, it is expected in equilibrium that they
will employ Markov strategies with the same feedback form. Thus, we may
consider a representative agent for all subsequent computations which allows
us to suppress the index $i$ from all quantities. Suppose a representative
agent chooses their spread according to the function $f$ such that

$\displaystyle\delta^{f}_{t}$
$\displaystyle=f(t,Q_{t}^{\delta^{f},\overline{\delta}^{f}})\,,$ (8)

where $\overline{\delta}^{f}$ represents the mean spread posted by all agents
in the mean-field limit. An important quantity to track is the proportion of
all agents that hold any particular value of inventory. For the inventory
level $q\in\\{0,1,\dots,\overline{Q}\\}$, denote this proportion by
$P_{q}^{f}=(P_{q,t}^{f})_{t\in[0,T]}$ which is equal to

$\displaystyle P_{q,t}^{f}$
$\displaystyle=\lim_{M\rightarrow\infty}\frac{1}{M}\sum_{i=1}^{M}\mathcal{X}_{Q_{i,t}^{\delta^{i},\overline{\delta}}=q}\,.$

Because all agents use the same feedback strategy $f$, the process $P^{f}$ is
equivalent to the law of the inventory process through the relation

$\displaystyle\mathbb{P}(Q^{\delta^{f},\overline{\delta}^{f}}_{t}=q)$
$\displaystyle=P^{f}_{q,t}\,,$

and then the mean spread $\overline{\delta}^{f}$ may be written as a weighted
average

$\displaystyle\overline{\delta}^{f}_{t}$
$\displaystyle=\frac{\sum_{q=1}^{\overline{Q}}f(t,q)\,P^{f}_{q,t}}{1-P^{f}_{0,t}}\,.$
(9)

Further, any single agent with inventory $q$ leaves that state with intensity
$\lambda(f(t,q),\overline{\delta}_{t}^{f})$, which means there is a flow of
agents from state $q$ to state $q-1$ at a rate of
$\lambda(f(t,q),\overline{\delta}_{t}^{f})\,P^{f}_{q,t}$. Thus, the dynamics
of each $P^{f}_{q}$ can be written (see Yang and Xia (2013) and Sun (2006)) as

$\displaystyle
dP^{f}_{\overline{Q},t}=-\lambda(f(t,\overline{Q}),\overline{\delta}_{t}^{f})\,P^{f}_{\overline{Q},t}\,dt\,,$
(10a) $\displaystyle
dP^{f}_{q,t}=\biggl{(}\lambda(f(t,q+1),\overline{\delta}_{t}^{f})\,P^{f}_{q+1,t}-\lambda(f(t,q),\overline{\delta}_{t}^{f})\,P^{f}_{q,t}\biggr{)}\,dt\,,\qquad
q\neq 0,\,\overline{Q}\,,$ (10b) $\displaystyle
dP^{f}_{0,t}=\lambda(f(t,1),\overline{\delta}_{t}^{f})\,P^{f}_{1,t}\,dt\,,$
(10c) and because each agent begins with maximum inventory $\overline{Q}$ the
initial conditions are given by $\displaystyle\hskip
113.81102ptP^{f}_{\overline{Q},0}=1\,,\quad P^{f}_{q,0}=0\,,\qquad
q\neq\overline{Q}\,.$ (10d)

###  3.3 Mean-Field Performance Criterion and Equilibrium

Each agent still wishes to maximize their expected terminal wealth subject to
all other agents strategies being fixed. Inspection of Equations (9) and (10)
show that for a fixed $f$, the process $\overline{\delta}^{f}$ is
deterministic, and in this case the dynamics of a representative agent’s
wealth and inventory do not have explicit dependence on $P^{f}$, but only on
$\overline{\delta}^{f}$. Thus, when specifying the performance criterion of a
representative agent, it also will not depend on the entire flow of the
population distribution $P$, but only on the mean spread $\overline{\delta}$.
For a given deterministic mean spread, we write the performance criterion as

$\displaystyle
J(\delta^{f};\overline{\delta})=\mathbb{E}\biggl{[}X_{T}^{\delta^{f},\overline{\delta}}+Q_{T}^{\delta^{f},\overline{\delta}}\,(S_{T}-\alpha\,Q_{T}^{\delta^{f},\overline{\delta}})-\phi\int_{0}^{T}(Q_{t}^{\delta^{f},\overline{\delta}})^{2}\,dt\biggr{]}\,.$

######  Definition 2 (Markov Mean-Field Nash Equilibrium)

A Markov mean-field Nash equilibrium is a function $f$ which satisfies

$\displaystyle J(\delta^{f};\overline{\delta}^{f})\geq
J(\delta^{g};\overline{\delta}^{f}),$

for all function $g\neq f$.

The interpretation of the equilibrium given in Definition 2 is that we seek a
function $f$ which determines the strategy of all agents through the relation
(8) such that no particular agent can increase their performance by deviating
from that strategy. The analytical tractability of this problem over the
finite-agent case is due to the fact that in the mean-field setting, no single
agent can affect the dynamics of the whole population, as determined by
Equation (10) by changing their own strategy. Finding the mean-field Nash
equilibrium thus comes down to solving a single optimization problem along
with checking a consistency condition. Specifically, we will take a mean
spread process $\overline{\delta}$ as given and solve for the optimal strategy
for a representative agent in feedback form. Then we will check for the
consistency relation that if every agent uses the computed strategy then the
mean spread process, as determined by Equation (9) is given by
$\overline{\delta}$, the same process initially given. If this holds then the
feedback control is an equilibrium.

###  3.4 Mean-Field Optimization Problem

The intensity function $\lambda(\delta,\overline{\delta})$ for the
representative agent states that her sales depend on both her own quote and
the mean quote of all agents. We assume that $\lambda$ is twice differentiable
in both inputs. According to Dockner and Jørgensen (1988), the intensity
$\lambda$ should satisfy the following assumptions:

$\displaystyle\frac{\partial\lambda}{\partial\delta}$ $\displaystyle<0\,,$
(11a) $\displaystyle\frac{\partial\lambda}{\partial\overline{\delta}}$
$\displaystyle>0\,,$ (11b)
$\displaystyle\frac{\partial\lambda}{\partial\delta}+\frac{\partial\lambda}{\partial\overline{\delta}}$
$\displaystyle<0\,,$ (11c)
$\displaystyle\frac{\partial^{2}\lambda}{\partial\delta\partial\overline{\delta}}$
$\displaystyle\leq 0\,,$ (11d)
$\displaystyle\lambda\frac{\partial^{2}\lambda}{\partial\delta^{2}}$
$\displaystyle<2\left(\frac{\partial\lambda}{\partial\delta}\right)^{2}\,.$
(11e)

Condition (11a)-(11d) are standard assumptions in competition theory.
Condition (11a) is equivalent to assuming a downward-sloping demand curve,
that is, with a fixed mean quote a lower price quoted from the representative
agent leads to higher demand. Condition (11b) states that an increase in the
mean quote causes the sales of the representative agent to rise. Condition
(11c) implies that if all agents raise their quotes by the same amount, their
sales will decrease simultaneously. Condition (11d) implies that the higher
the mean quote, the more difficult it is for the representative agent to
increase her probability of selling by reducing her own quote. Conversely, if
the mean quote is higher, the representative agent will lose market share more
quickly when she raises her quote. Condition (11e) is a technical condition,
originating from the strict concavity of the Hamiltonian with respect to quote
(i.e., the control variable).

We assume that the instantaneous intensity $\lambda$ is of the following form
similar to (Chenavaz et al., 2021, (4.2))

$\displaystyle\begin{split}\lambda(\delta,\overline{\delta})&=A\,\exp\left\\{-\kappa\,\delta+\beta\,(\overline{\delta}-\delta)\right\\}\\\
&=A\,\exp\left\\{-(\kappa+\beta)\,\delta+\beta\,\overline{\delta}\right\\},\end{split}$
(12)

for constants $A,\kappa,\beta\geq 0$. This assumption satisfies all the
conditions stated in Equation (11). Compared to the single-agent intensity
function defined in Equation (6), there is an extra term in the exponent that
represents the competitiveness in the market whose strength is characterized
by $\beta$. Under this assumption, there are two drivers for faster execution:
lower quoted price of one’s own, and higher mean quoted price of other market
participants. This is consistent with the expected sales loss to competitors
when one’s price becomes higher relative to the market price for the same
product.

We briefly investigate the effect of the mean quote $\overline{\delta}$ on the
total sales of the market. First we consider the finite $M$-player game, and
then take $M$ to infinity. According to Equation (12), the intensity
$\lambda^{i}$ for the $i$-th agent is

$\displaystyle\lambda^{i}$
$\displaystyle=A\exp{\left\\{-(\kappa+\beta)\delta^{i}+\frac{\beta}{M}\sum_{j=1}^{M}\delta^{j}\right\\}}$
$\displaystyle=A\exp{\left\\{-\left(\kappa+\frac{\beta(M-1)}{M}\right)\delta^{i}+\frac{\beta}{M}\sum_{j\neq
i}\delta^{j}\right\\}}\,.$

Suppose a representative agent, denoted by index $i$, fixes her quote
$\delta^{i}$, and other agents increase their quotes by the same amount. The
change in total market intensity then depends on the sum of the derivatives of
individual intensities with respect to quotes of all other agents, which is

$\displaystyle\sum^{M}_{j=1}\sum_{k\neq
i}\frac{\partial\lambda^{j}}{\partial\delta^{k}}$ $\displaystyle=\sum_{j\neq
i}\frac{\partial\lambda^{i}}{\partial\delta^{j}}+\sum_{j\neq i}\sum_{k\neq
i}\frac{\partial\lambda^{j}}{\partial\delta^{k}}$
$\displaystyle=\frac{\beta(M-1)}{M}\lambda^{i}+\sum_{j\neq
i}\left(-\left(\kappa+\frac{\beta(M-1)}{M}\right)\lambda^{j}+\frac{\beta(M-2)}{M}\lambda^{j}\right)$
$\displaystyle=\beta\left(1-\frac{1}{M}\right)\lambda^{j}-\left(\kappa+\frac{\beta}{M}\right)\sum_{j\neq
i}\lambda^{j}$
$\displaystyle\leq\beta\left(1-\frac{1}{M}\right)\lambda^{j}-\left(\kappa\left(M-1\right)+\beta\left(1-\frac{1}{M}\right)\right)\min_{j\neq
i}\lambda^{j}\,.$

For sufficiently large $M$, this quantity is negative. This means that if
every agent except for the representative agent raises their quotes by the
same amount, although the probability of selling for the representative agent
increases, the total sales across all agents decrease, which is economically
sensible.

For a given deterministic function $\overline{\delta}$, the representative
agent optimizes

$\displaystyle H(t,s,x,q;\overline{\delta})=\sup_{(\delta_{u})_{t\leq u\leq
T}\in\mathcal{A}}\mathbb{E}_{t,s,x,q}\left[X_{T}^{\delta,\overline{\delta}}+Q_{T}^{\delta,\overline{\delta}}\,(S_{T}-\alpha\,Q_{T}^{\delta,\overline{\delta}})-\phi\,\int^{T}_{t}(Q_{u}^{\delta,\overline{\delta}})^{2}du\right],$
(13)

and the set of admissible controls $\mathcal{A}$ is defined as

$\displaystyle\mathcal{A}=\left\\{\delta\,\Big{|}\,\delta\,\,\text{is}\,\,\mathcal{F}\text{-predictable,}\quad\delta\geq\underline{B}\right\\}\,,$

where $\underline{B}$ is a non-negative constant lower bound. By the dynamic
programming principle, we introduce the HJB equation with terminal condition
associated to this optimization problem

$\displaystyle\begin{split}&\partial_{t}H+\frac{1}{2}\,\sigma^{2}\,\partial_{ss}H-\phi\,q^{2}+\sup_{\delta\geq\underline{B}}\lambda(\delta,\overline{\delta})\left[H(t,s,x+s+\delta,q-1;\overline{\delta})-H(t,s,x,q;\overline{\delta})\right]\mathcal{X}_{q>0}=0\,,\\\
&H(T,s,x,q;\overline{\delta})=x+q\,(s-\alpha\,q)\,.\end{split}$ (14)

We use a similar ansatz as Equation (4) to solve the HJB equation.

######  Proposition 3 (Solution to HJB Equation)

The HJB equation (14) admits the ansatz
$H(t,s,x,q;\overline{\delta})=x+q\,s+h_{q}(t;\overline{\delta})$, where $h$
satisfies

$\displaystyle\partial_{t}h_{q}-\phi\,q^{2}+\sup_{\delta\geq\underline{B}}A\,\exp{\left\\{-(\kappa+\beta)\,\delta+\beta\,\overline{\delta}\right\\}}\left[\delta+h_{q-1}(t;\overline{\delta})-h_{q}(t;\overline{\delta})\right]\mathcal{X}_{q>0}=0\,,$
(15)

subject to $h_{q}(T;\overline{\delta})=-\alpha\,q^{2}$. The optimum in
Equation (15) is achieved at

$\displaystyle\delta^{*}(t,q;\overline{\delta})=\max\left\\{\frac{1}{\kappa+\beta}+h_{q}(t;\overline{\delta})-h_{q-1}(t;\overline{\delta})\,,\,\,\underline{B}\right\\}\,,\quad
q\neq 0\,.$ (16)

Furthermore, Equation (15) has an unique classical solution.

  * Proof

See Appendix A.

We see in Proposition 3 that the feedback form of the optimal strategy is
similar in form to the single agent case. There are two differences, one being
a change based on immediate effects of competition which sees $\kappa$
replaced $\kappa+\beta$, and the other is the fact that the change in expected
future risk-adjusted P&L, represented by
$h_{q-1}(t;\overline{\delta})-h_{q}(t;\overline{\delta})$, will also be
different from the corresponding term in Proposition 1. The terminal condition
$h_{q}(T;\overline{\delta})=-\alpha q^{2}$ determines the optimal quotes at
time $T$ regardless of equilibrium considerations, and inspection shows that
these quotes will be smaller than the single agent case, as would be expected
with the effect of competition. The same cannot necessarily be said for all
$t<T$ without solving for equilibrium.

To show that the solution to the HJB equation is really the solution to the
control problem, we prove the following verification theorem. In particular,
this establishes that the strategy computed in the iterative algorithm of
Section 4.1 are optimal given an assumed mean quote process.

######  Theorem 4 (Verification Theorem)

Given continuous $\overline{\delta}:[0,T]\rightarrow\mathbb{R}$, let
$h_{q}(t;\overline{\delta})$ be the solution to Equation (15). Then
$H(t,s,x,q;\overline{\delta})=x+q\,s+h_{q}(t;\overline{\delta})$ is the value
function to the agent’s control problem (13) and the optimal controls are
given by Equation (16) in feedback form.

  * Proof

See Appendix B.

In the statement of Theorem 4, we make the technical assumption that the given
mean quote, $\overline{\delta}$, is continuous. In fact, we only require that
it is bounded, for technical reasons needed in the proof. However, this is not
a restrictive assumption because any optimal strategy with a feedback form
given by (16) will be continuous, and therefore bounded, and the resulting
$\overline{\delta}$ arising from all agents using this feedback control will
also have these properties. Thus, the assumption is without loss of
generality.

##  4 Numerical Experiments

In this section, we numerically solve for the equilibrium, and illustrate the
behaviour of the optimal strategy $\delta^{*}$, the mean spread
$\overline{\delta}$, the inventory distribution across agents $P$, and other
quantities of interest.

###  4.1 Algorithm

Inspired by Li et al. (2019), we use the following algorithm to numerically
find an equilibrium. First, for a given mean quote we solve the control
problem, the optimality of which is ensured by our verification theorem. Then,
we use the optimal individual quote to find the population dynamics. With
these two solutions, we calculate the corresponding new mean spread and repeat
this process until convergence. The details are as follows:

  1. 1.

We divide the entire time horizon $T$ into equidistant time grid. Initialize
with a mean quote $t\to\overline{\delta}^{(0)}_{t}$ for every time point.

  2. 2.

Given any mean quote, $\overline{\delta}^{(n)}$, we solve the optimal control
problem numerically. This is done with a fully explicit backward finite
difference method to solve Equation (15) starting from time $T$. This
simultaneously gives the optimal control for the given mean quote,
$\delta^{*}(\overline{\delta}^{(n)})$, in feedback form through Equation (16).

  3. 3.

With the optimal control $\delta^{*}(\overline{\delta}^{(n)})$ given in
feedback form, we compute the population process $\mathbf{P}^{\delta^{*}}$
using Equation (10). This is also done numerically using a fully explicit
forward finite difference method.

  4. 4.

We introduce a learning rate parameter $\gamma\in[0,1)$ to update the mean
quote process. To reduce oscillations in searching for the equilibrium, we
choose $\gamma$ to be a small number, and we update the mean quote price
according to

$\displaystyle\overline{\delta}^{(n+1)}_{t}=\gamma\overline{\delta}^{(n)}_{t}+(1-\gamma)\frac{\sum_{q=1}^{\overline{Q}}\delta^{*}(t,q;\overline{\delta}^{(n)})P^{\delta^{*}}_{q,t}}{1-P_{0,t}^{\delta^{*}}}.$

These steps are repeated until convergence of the sequence
$\\{\overline{\delta}^{(n)}\\}_{n\geq 0}$ to within a specified tolerance555In
the following numerical examples, we identify the algorithm as converged if
the standard error of $\overline{\delta}^{(n+1)}$ and
$\overline{\delta}^{(n)}$ is not more than $10^{-12.5}$. We do not claim that
equilibrium is guaranteed to exist or is unique, but for all sets of
parameters considered, convergence was attained and was found not to depend on
the initializing $\overline{\delta}$. More numerical details and explanations
can be found in Appendix C., and we drop the counting index in the final mean
quote process.

Once convergence is attained, we will have found a function $f$ which
satisfies the conditions of equilibrium given in Definition 2, given by
$f(t,q)=\delta^{*}(t,q;\overline{\delta})$.

###  4.2 Optimal Quotes in Equilibrium

Figure 2: Optimal quotes in equilibrium as a function of $t$ for various $Q$.
Other parameters are $T=10$, $\overline{Q}=5$, $\alpha=0.1$, $\kappa=1$,
$\phi=0.03$, $A=1$, $\beta=0.3$, $\gamma=0.1$, and $\underline{B}=-10$.

The optimal quotes $\delta^{*}$ are presented in Figure 2, as well as the mean
quote $\overline{\delta}$ in Figure 3 and the dynamics of the proportion
process $P$ in Figure 4. From Figure 2, we observe that the optimal quotes in
mean-field case shares the same decreasing pattern against inventory level
with ones in one-agent case in Figure 1. In addition, for lower inventory
levels ($Q\leq 3$) the optimal quotes are decreasing with time, whereas for
higher inventory levels ($Q\geq 4$) the optimal quotes slowly increase early
in the time interval and then decrease. This effect is caused by the changing
nature of competition embedded in the mean spread $\overline{\delta}$ (see
Figure 3). At early times, everyone is posting small spreads, but as some
agents begin to sell the mean spread quickly increases. For an agent that
remains in a high inventory state, this increase in average price in the
market creates less competition, and they begin to benefit themselves by
quoting slightly higher prices.

Figure 3: Mean quote in equilibrium as a function of $t$. Other parameters are
$T=10$, $\overline{Q}=5$, $\alpha=0.1$, $\kappa=1$, $\phi=0.03$, $A=1$,
$\beta=0.3$, $\gamma=0.1$, and $\underline{B}=-10$.

In Figure 3, because each agent begins with maximum inventory $\overline{Q}$,
the mean quote begins at the optimal quote for that inventory level. It
quickly rises as agents begin to sell inventory and move to lower inventory
states, causing them to increase their quoted price. At later times the mean
quote begins to decrease due to two effects. First, as time approaches the end
of the trading period agents are incentivized to quote smaller prices to avoid
the liquidation penalty. Second, more and more agents end up fully liquidating
their positions, removing their large quoted prices from the market and
thereby decreasing the mean. Figure 4 shows the distribution, $P$, of the
agents’ inventories through time. For this set of parameters, the majority of
agents sell all of their inventory by time $T$, indicated by the fact aht
$P_{0,T}>0.5$.

Figure 4: Dynamics of the population proportion process, $P_{q,t}$, in
equilibrium as a function of $t$. Other parameters are $T=10$,
$\overline{Q}=5$, $\alpha=0.1$, $\kappa=1$, $\phi=0.03$, $A=1$, $\beta=0.3$,
$\gamma=0.1$, and $\underline{B}=-10$.

###  4.3 Comparison with Single-Agent Reference Model

Figure 5: Optimal quotes in mean-field equilibrium (solid curves) compared to
single-agent reference model (dotted curves). Other parameters are $T=10$,
$\overline{Q}=5$, $\alpha=0.1$, $\kappa=1$, $\phi=0.03$, $A=1$, $\beta=0.3$,
$\gamma=0.1$, and $\underline{B}=-10$.

Figure 5 compares the optimal quotes in the monopoly and mean-field cases. For
lower inventory levels $Q\leq 4$, the optimal quotes in multi-agent case are
consistently lower than the ones in single-agent case. However, for the
maximum inventory level $Q=5$, the optimal quotes in the multi-agent case are
sometimes higher than in the single-agent case. This trend is because the
optimal quotes corresponding to higher inventory levels tend to be lower than
the mean quote in the market. Thus, the items in multi-agent case have a
higher probability to be executed compared to the single-agent case as per
Equation (12). As a result, the agent can be more aggressive by setting a
higher quote in the multi-agent case.

###  4.4 Effect of Competitiveness on Equilibrium

Figure 6: Optimal quotes in equilibrium for different values of competition
parameter $\beta$. The solid curves represent a low level of competitive
interaction ($\beta=0.3$) and the dotted curves represent a high level of
competitive interaction ($\beta=0.9$). Other parameters are $T=10$,
$\overline{Q}=5$, $\alpha=0.1$, $\kappa=1$, $\phi=0.03$, $A=1$, $\gamma=0.1$,
and $\underline{B}=-10$.

Figure 6 shows how the competitiveness parameter, $\beta$, affects the optimal
quotes. In general, higher competitiveness of the market brings the optimal
quotes of different inventories closer together. However, note that higher
competitiveness does always move the optimal quotes of different inventory
levels in the same direction. For lower inventory levels, larger $\beta$ leads
to lower optimal quotes, but for higher inventory levels the direction of
price change depends on time. From figure 7, the mean quote is decreasing in
$\beta$ at all points in time. This means more intense competition brings
lower market quote price, which is economically sensible. In addition, we see
that a larger proportion of agents end up selling their whole inventory when
the competitiveness parameter is larger.

Figure 7: Mean quotes and the distribution of inventory across agents at
terminal time in equilibrium with different values of competition parameter
$\beta$. Other parameters are $T=10$, $\overline{Q}=5$, $\alpha=0.1$,
$\kappa=1$, $\phi=0.03$, $A=1$, $\gamma=0.1$, and $\underline{B}=-10$.

To illustrate the effects of competitiveness on the consumers and agents, we
define cumulative cost $C(t)$, cumulative revenue $R(t)$ and cumulative volume
$V(t)$ up to time $t$ as

$\displaystyle C(t)$
$\displaystyle=\sum_{q=1}^{\overline{Q}}\int_{0}^{t}f(u,q)\,P^{f}_{q,u}\,\lambda(f(u,q),\overline{\delta}_{u})\,du\,,$
$\displaystyle R(t)$
$\displaystyle=\sum_{q=1}^{\overline{Q}}\left(\int_{0}^{t}f(u,q)\,P^{f}_{q,u}\,\lambda(f(u,q),\overline{\delta}_{u})\,du-\alpha\,P^{f}_{q,t}\,q^{2}\right)\,,$
$\displaystyle V(t)$
$\displaystyle=\sum_{q=1}^{\overline{Q}}\int_{0}^{t}\,P^{f}_{q,u}\,\lambda(f(u,q),\overline{\delta}_{u})\,du\,.$

Cumulative cost $C(t)$ corresponds to the total wealth paid by consumers up to
time $t$, and cumulative revenue $R(t)$ represents the profits made by agents
up to time $t$. Theoretically, cumulative revenue $R(t)$ should be equal to
cumulative cost $C(t)$ until terminal time when the penalty proportional to
$\alpha$ is realized, but to ensure $R(t)$ is continuous, we preemptively
charge this penalty in the form of $\alpha\,P^{f}_{q,t}\,q^{2}$, as if all
agents stopped trading at time $t$. With the above functions, we can compute
the corresponding average transaction cost $K(t)$ by

$\displaystyle K(t)=\frac{C(t)}{V(t)}\,,$

and the instantaneous average transaction cost $\overline{K}(t)$ by

$\displaystyle\overline{K}(t)=\frac{\partial C(t)}{\partial t}/\frac{\partial
V(t)}{\partial
t}=\frac{\sum_{q=1}^{\overline{Q}}f(t,q)P^{f}_{q,t}\lambda(f(t,q),\overline{\delta}_{t})}{\sum_{q=1}^{\overline{Q}}P^{f}_{q,t}\lambda(f(t,q),\overline{\delta}_{t})}\,.$

The average transaction cost $K(t)$ represents the average price consumers
have paid for one unit of products up to time $t$, and the instantaneous
average transaction cost $\overline{K}(t)$ is the the average price to buy one
unit at time $t$.

Figure 8: Cumulative cost (top left), average transaction cost (top right) and
instantaneous average cost (bottom) with different values of competition
parameter $\beta$. The solid curves represent a low level of competitive
interaction ($\beta=0.3$) and the dotted curves represent a high level of
competitive interaction ($\beta=0.9$). Other parameters are $T=10$,
$\overline{Q}=5$, $\alpha=0.1$, $\kappa=1$, $\phi=0.03$, $A=1$, $\gamma=0.1$,
and $\underline{B}=-10$.

Figure 9: Cumulative revenue and volume with different values of competition
parameter $\beta$. The solid curves represent a low level of competitive
interaction ($\beta=0.3$) and the dotted curves represent a high level of
competitive interaction ($\beta=0.9$). Other parameters are $T=10$,
$\overline{Q}=5$, $\alpha=0.1$, $\kappa=1$, $\phi=0.03$, $A=1$, $\gamma=0.1$,
and $\underline{B}=-10$.

In Figure 8, we can see that for this set of parameters, a higher value of
competitiveness $\beta$ leads to lower cost, average cost and instantaneous
average cost. This implies that consumers indeed benefit from a more
competitive market with lower purchase prices. The instantaneous average cost
$\overline{K}(t)$ shares a similar concave shape with the mean quote
$\overline{\delta}$ in Figure 7. This means that the best purchasing time for
consumers is either the start or the end of the time horizon, consistent with
the early bird price and closing sale in reality. Additionally, for the larger
value of $\beta$, the maximum of the curves $\overline{K}(t)$ and
$\overline{\delta}$ occur at earlier times. This means that the vendors will
decrease price earlier in a more competitive market, and consumers indeed
enjoy this lower market price earlier. In Figure 9, we can see that for this
set of parameters, a higher value of competitiveness $\beta$ leads to a higher
volume, but lower revenue near the terminal time. One interesting phenomenon
is that, before the terminal time, we enter into a win-win situation for both
agents and consumers with a higher value of competitiveness $\beta$. We
observe that for this time period, with higher demand of the market, a higher
level of competitive interaction can boost the volume and cumulative revenue
for agents and decrease the transaction cost for consumers simultaneously.

###  4.5 Effect of Upper Bound on Equilibrium

In this section, we impose an upper bound on admissible strategies. In
reality, this price control is sometimes enforced by the government on the
necessities, and we discuss how the behaviour of optimal strategy
$\delta^{*}$, the mean spread $\overline{\delta}$, and the inventory
distribution $P$ changes under an additional price ceiling.

Suppose we define a new set of admissible controls by

$\displaystyle\hat{\mathcal{A}}=\left\\{\delta\,\Big{|}\,\delta\,\,\text{is}\,\,\mathcal{F}\text{-predictable,}\quad\underline{B}\leq\delta\leq\overline{B}\right\\}\,,$

where $\overline{B}$ is the non-negative upper bound. Then the optimal
feedback control becomes

$\displaystyle\delta^{*}(t,q;\overline{\delta})=\min\left\\{\overline{B}\,,\,\,\max\left\\{\frac{1}{\kappa+\beta}+h_{q}(t;\overline{\delta})-h_{q-1}(t;\overline{\delta})\,,\,\,\underline{B}\right\\}\right\\}\,,\quad
q\neq 0\,.$

The proof of uniqueness of the solution to HJB equation and verification
theorem still stands with minor adjustments.

Figure 10: Optimal quotes in equilibrium with (dotted curves) and without
(solid curves) an upper bound. Other parameters are $T=10$, $\overline{Q}=5$,
$\alpha=0.1$, $\kappa=1$, $\phi=0.03$, $A=1$, $\beta=0.3$, $\gamma=0.1$,
$\underline{B}=-10$, and $\overline{B}=1$.

In Figure 10, we can observe that optimal quotes with the upper bound decrease
for all inventory levels. This is consistent with our expectation that the
competition becomes more intense with an upper bound on price, as the agents
with initially lower prices will also try to keep their market share while
others are lowering their prices. For lower inventory levels $Q\leq 2$, the
differences are more significant. This is due to the fact that agents with
lower holdings tend to propose a higher quote, and that is where the
restriction on pricing comes in. The optimal quotes converge at the terminal
time, because the agents lower their price as much as possible for quick
sales, at the level lower than the constraint.

Figure 11: Mean quote and the distribution of inventory across agents at
terminal time in equilibrium with and without an upper bound. Parameters are
$T=10$, $\overline{Q}=5$, $\alpha=0.1$, $\kappa=1$, $\phi=0.03$, $A=1$,
$\beta=0.3$, $\gamma=0.1$, $\underline{B}=-10$, and $\overline{B}=1$.

In Figure 11, mean quote with the upper bound is always lower. At the
beginning, the two lines are very close. As time goes by, more agents come to
lower inventory levels and receive price restrictions, thus resulting in
bigger differences between the mean quotes. Closer to the terminal time $T$,
the two lines approach each other again. This is because the distribution of
inventory levels and the corresponding optimal quotes are similar near
terminal time in these two scenarios, as shown in Figure 10 and Figure 11.

Figure 12: Cumulative revenue (top left), volume (top right), average
transaction cost (bottom left) and instantaneous average cost (bottom right)
with (dotted curves) and without (solid curves) an upper bound. Parameters are
$T=10$, $\overline{Q}=5$, $\alpha=0.1$, $\kappa=1$, $\phi=0.03$, $A=1$,
$\beta=0.3$, $\gamma=0.1$, $\underline{B}=-10$, and $\overline{B}=1$.

Figures 12 shows that the price limit boosts the sales volume but reduces
revenue for the agents. We can see that consumers buy products at a lower
price with price ceiling, which means the restriction on high prices is
effective in reducing consumer expenditure under our setting666In reality,
price ceiling does not always work out as intended. If it leads to a severe
imbalance between supply and demand, this can in turn cause shortages and
underground markets..

###  4.6 Effect of Overselling

In this section, we study the effects of overselling where agents are allowed
to sell more volume of inventory than they have physically available during
the trading period, and they must pay extra cost to clear out the negative
inventory at terminal time. This phenomenon frequently happens in the tourism
industry. For example, airlines oversell air tickets anticipating some
cancellations or no-shows of the customer bookings. If everyone shows up,
airlines may refuse boarding to certain passengers but must pay compensation
depending on jusrisdiction.

We set the admissible inventory level set as
$\\{\underline{Q},\underline{Q}+1,\cdots,\overline{Q}\\}$, and $\underline{Q}$
is a non-positive constant. The new model under overselling setting is the
same as the one in Section 3, except for the lower bound of inventory and the
new value function

$\displaystyle H(t,s,x,q;\overline{\delta})=\sup_{(\delta_{u})_{t\leq u\leq
T}\in\mathcal{A}}\mathbb{E}_{t,s,x,q}\biggl{[}$ $\displaystyle
X_{T}^{\delta,\overline{\delta}}+Q_{T}^{\delta,\overline{\delta}}\,S_{T}-\left(\alpha_{1}\,\mathcal{X}_{Q_{T}^{\delta,\overline{\delta}}\geq
0}+\alpha_{2}\,\mathcal{X}_{Q_{T}^{\delta,\overline{\delta}}<0}\right)\left(Q_{T}^{\delta,\overline{\delta}}\right)^{2}$
$\displaystyle-\int^{T}_{t}\left(\phi_{1}\,\mathcal{X}_{Q_{u}^{\delta,\overline{\delta}}\geq
0}+\phi_{2}\,\mathcal{X}_{Q_{u}^{\delta,\overline{\delta}}<0}\right)\left(Q_{u}^{\delta,\overline{\delta}}\right)^{2}du\biggl{]}\,,$

where $\alpha_{1}<\alpha_{2}$ and $\phi_{1}<\phi_{2}$ are all positive
constants. Compared to Equation (13), we set different parameters $\alpha_{1}$
and $\alpha_{2}$ of terminal penalty for positive and negative inventory
respectively. Here, the difference
$\left(\alpha_{2}-\alpha_{1}\right)(Q_{T}^{\delta,\overline{\delta}})^{2}$
represents the extra expected compensation paid by agents in the case that
consumers make a claim on the oversold units. Since overselling brings more
inventory risk, we set parameters of running inventory penalty
$\phi_{1}<\phi_{2}$ to discourage this behavior. Then the associated HJB
equation is given by

$\displaystyle\begin{split}&\partial_{t}H+\frac{1}{2}\,\sigma^{2}\,\partial_{ss}H-\left(\phi_{1}\mathcal{X}_{q\geq
0}+\phi_{2}\mathcal{X}_{q<0}\right)\,q^{2}\\\ &\hskip
20.77051pt+\sup_{\delta\geq\underline{B}}\lambda(\delta,\overline{\delta})\left[H(t,s,x+s+\delta,q-1;\overline{\delta})-H(t,s,x,q;\overline{\delta})\right]\mathcal{X}_{q>\underline{Q}}=0\,,\\\
&H(T,s,x,q;\overline{\delta})=x+q\,s-\left(\alpha_{1}\mathcal{X}_{q\geq
0}+\alpha_{2}\mathcal{X}_{q<0}\right)\,q^{2}\,.\end{split}$

We use the same ansatz in Proposation 3 to solve the above equations, and get
the following ODEs

$\displaystyle\partial_{t}h_{q}+\left(\phi_{1}\mathcal{X}_{q\geq
0}+\phi_{2}\mathcal{X}_{q<0}\right)\,q^{2}$ $\displaystyle\hskip
21.05501pt+\sup_{\delta\geq\underline{B}}A\,\exp{\left\\{-(\kappa+\beta)\,\delta+\beta\,\overline{\delta}\right\\}}\left[\delta+h_{q-1}(t;\overline{\delta})-h_{q}(t;\overline{\delta})\right]\mathcal{X}_{q>\underline{Q}}=0\,,$
$\displaystyle h_{q}(T;\overline{\delta})=-\left(\alpha_{1}\mathcal{X}_{q\geq
0}+\alpha_{2}\mathcal{X}_{q<0}\right)\,q^{2}\,.$

As overselling does not affect the $\sup$ term in the ODEs, the feedback form
of the optimal control is still the same as Equation (16). We can show that
the above ODEs of $h$ still has a unique solution under overselling setting,
and the proof of verification theorem still stands with minor adjustments.

Figure 13: Optimal quotes in equilibrium with overselling (dotted curves) and
without overselling (solid curve). The dotted curve on the left figure is with
lower $\alpha_{2}=0.2$, $\phi_{2}=0.06$, and the one on the right figure is
with higher $\alpha_{2}=0.9$, $\phi_{2}=0.15$. Other parameters are $T=10$,
$\overline{Q}=5$, $\underline{Q}=-2$, $\alpha_{1}=0.1$, $\kappa=1$,
$\phi_{1}=0.03$, $A=1$, $\beta=0.3$, $\gamma=0.1$, $\underline{B}=-10$, and
$\overline{B}=20$.

Figure 13 shows the change in optimal quotes when the ability to oversell is
introduced, and it also indicates how the size of terminal penalty parameter
$\alpha_{2}$ and running inventory penalty parameter $\phi_{2}$ for negative
inventory affects the change in quotes. As expected, the optimal quotes under
overselling setting are still decreasing in inventory level monotonically. On
the left of Figure 13, with smaller $\alpha_{2}$ and $\phi_{2}$, we can see
that for all positive inventory levels, the optimal quotes with overselling
are generally lower than the ones without overselling. On the right panel of
Figure 13, with larger values of $\alpha_{2}$ and $\phi_{2}$, the quotes for
positive inventory still generally decrease when overselling is allowed, but
the effect of this feature is less pronounced. This indicates that higher
value of $\alpha_{2}$ and $\phi_{2}$ leads to higher optimal quotes across all
inventory levels with overselling. Additionally, with stronger penalties for
overselling, prices at early times are similar to prices when overselling is
prohibited, but over time as more agents reach zero and negative inventory
levels, the quoted prices become very large because these agents are only
willing to sell at high prices to recoup the large overselling penalty.

Figure 14: Mean quotes in equilibrium and the distribution of inventory at
terminal time without overselling (solid curve) and with overselling (dotted
and dash-dotted curves). The dotted curve with overselling is with lower
$\alpha_{2}=0.2$, $\phi_{2}=0.06$, and the dash-dotted curve with overselling
is with higher $\alpha_{2}=0.9$, $\phi_{2}=0.15$. Other parameters are $T=10$,
$\overline{Q}=5$, $\underline{Q}=-2$, $\alpha_{1}=0.1$, $\kappa=1$,
$\phi_{1}=0.03$, $A=1$, $\beta=0.3$, $\gamma=0.1$, $\underline{B}=-10$, and
$\overline{B}=20$.

The left of Figure 14 shows the change in mean quotes when the ability to
oversell is introduced. The solid line stands for the mean quote without
overselling, and the dotted and dash-dotted curves represent overselling for
different values of $\alpha_{2}$ and $\phi_{2}$. The effect of overselling
depends also heavily on the penalty parameters for negative inventory levels.
The crossing of mean quote without overselling and mean quote with overselling
with low penalty is consistent with the result about optimal quotes in the
left of Figure 13. In the beginning, agents who can oversell quote lower
prices, thus resulting in lower mean quote. As time goes by, the optimal
quotes with overselling get closer to the ones without overselling. In the
meantime, more agents achieve negative inventory levels with much higher
quotes, so the mean quote with overselling exceeds the one without overselling
closer to terminal time. For mean quote without overselling and mean quote
with overselling with high penalty, we can see that if the punishment for
overselling is high enough, the mean quote with overselling can always be
higher than the one without overselling. The right of Figure 14 compares the
distributions of inventory at terminal time. The distribution of inventory at
time $T$ with overselling follows a bell-shaped curve. With overselling, most
agents stay at zero inventory, and there are more agents with negative
inventory than the ones with positive inventory. With overselling, we can
observe that higher terminal penalty parameter $\alpha_{2}$ and running
inventory penalty parameter $\phi_{2}$ for shorting indeed reduces the
proportion of agents ending at negative inventory levels.

Figure 15: Cumulative revenue (top left), volume (top right), average
transaction cost (bottom left) and instantaneous average cost (bottom right)
without overselling (solid curve) and with overselling (dotted and dash-dotted
curves). The dotted curve is with lower $\alpha_{2}=0.2$, $\phi_{2}=0.06$, and
the dash-dotted curve is with higher $\alpha_{2}=0.9$, $\phi_{2}=0.15$. Other
parameters are $T=10$, $\overline{Q}=5$, $\underline{Q}=-2$, $\alpha_{1}=0.1$,
$\kappa=1$, $\phi_{1}=0.03$, $A=1$, $\beta=0.3$, $\gamma=0.1$,
$\underline{B}=-10$, and $\overline{B}=20$.

The cumulative cost $C(t)$, cumulative revenue $R(t)$, and cumulative volume
$V(t)$ are slightly modified when overselling is introduced, and we write them
as

$\displaystyle C(t)$
$\displaystyle=\sum_{q=\underline{Q}+1}^{\overline{Q}}\int_{0}^{t}f(u,q)\,P^{f}_{q,u}\,\lambda(f(u,q),\overline{\delta}_{u})\,du\,,$
$\displaystyle R(t)$
$\displaystyle=\sum_{q=\underline{Q}+1}^{\overline{Q}}\left(\int_{0}^{t}f(u,q)\,P^{f}_{q,u}\,\lambda(f(u,q),\overline{\delta}_{u})\,du-\alpha_{1}\,P^{f}_{q,t}\,q^{2}\,\mathcal{X}_{q>0}-\alpha_{2}\,P^{f}_{q,t}\,q^{2}\,\mathcal{X}_{q<0}\right)\,,$
$\displaystyle V(t)$
$\displaystyle=\sum_{q=\underline{Q}+1}^{\overline{Q}}\int_{0}^{t}\,P^{f}_{q,u}\,\lambda(f(u,q),\overline{\delta}_{u})\,du\,.$

Figure 15 shows that regardless of the level of penalty, cumulative cost near
the terminal time with overselling is higher. From the top right panel we see
that overselling leads to higher traded volume, with smaller penalties leading
to higher total volume traded. In the bottom left panel, the average
transaction cost without overselling lies between the ones with overselling.
When overselling with high penalties, overselling leads to higher average
transaction cost. However, with low penalty, consumers indeed pay less on
average to buy more because of overselling. This indicates that with the
appropriate levels of penalty parameters $\alpha_{2}$ and $\phi_{2}$ for
overselling, agents can sell more products, while consumers pay almost the
same average transaction cost at terminal time compared to the non-overselling
case. In the bottom right panel, we can observe that the instantaneous average
cost with overselling is always high close to the terminal time. This is due
to the fact that the instantaneous fraction of oversold products is higher
near the terminal time, thus raising up the unit price. It is generally more
beneficial for consumers to purchase earlier to avoid higher price due to
overselling.

Figure 16: Cumulative revenue without overselling (solid curve) and with
overselling (dotted and dash-dotted curves). The dotted curve is with lower
$\alpha_{2}=0.2$, $\phi_{2}=0.06$, and the dash-dotted curve is with higher
$\alpha_{2}=0.9$, $\phi_{2}=0.15$. Other parameters are $T=10$,
$\overline{Q}=5$, $\underline{Q}=-2$, $\alpha_{1}=0.1$, $\kappa=1$,
$\phi_{1}=0.03$, $A=1$, $\beta=0.3$, $\gamma=0.1$, $\underline{B}=-10$, and
$\overline{B}=20$.

Perhaps counterintuitive is the result that a higher overselling penalty can
result in greater total revenue by agents, as seen in Figure 16. However, as
the discussion of Figure 13 has shown, when overselling penalties are large,
agents spend much of the time interval quoting prices as if they are not
allowed to oversell. These prices are higher than the situation with weak
overselling penalties, because those agents wish to lower prices and
accelerate the rate of liquidating their inventory. With a large penalty, when
most agents have sold off their positive positions, the market moves into an
overselling regime where prices are much higher. This is why we see a
deviation of the solid (no overselling) and dash-dotted (highly penalized
overselling) curves in Figures 14 (left panel) and 15 (all panels) which
becomes apparent around $t=T/2$.

Our final investigation is on the effect of overselling on consumers from the
perspective of cancellation. If more units of the product are sold than are
physically available, then some consumers will end up empty handed. Depending
on the industry, the compensation policy, and the consumer herself, she may be
indifferent between having the product versus the compensation, but
nevertheless the probability of being in this situation is of interest.

For one particular consumer at terminal time, we define event $E$ as the
scenario of her product being cancelled due to overselling, and events $E_{q}$
as the scenario of her buying from agents who oversold $q$ share, $\forall
q\in\mathbb{Z}^{+}$. We assume that at the terminal time, agents who oversold
randomly cancel orders due to shortage uniformly across all consumers they
transacted with, so the cancellation probability for a particular consumer is
given by

$\displaystyle\mathbb{P}(E)=\sum_{q=1}^{-\underline{Q}}\mathbb{P}(E_{q})\,\mathbb{P}(E|E_{q})=\sum_{q=1}^{-\underline{Q}}\frac{P^{f}_{-q,T}}{1-P^{f}_{\overline{Q},T}}\,\frac{q}{\overline{Q}+q}\,.$

Figure 17: Cancellation probability with different values of competition
parameter $\beta$ and different values of negative inventory penalty
parameters $\alpha_{2}$ and $\phi_{2}$ (we fix the ratio between these
parameters by $\phi_{2}=0.3\,\alpha_{2}$). In both figures, the other fixed
parameters are given by $T=10$, $\overline{Q}=5$, $\underline{Q}=-2$,
$\beta=0.3$, $\kappa=1$, $A=1$, $\alpha_{1}=0.1$, $\phi_{1}=0.03$,
$\gamma=0.1$, $\underline{B}=-10$, and $\overline{B}=20$.

The left of Figure 17 shows how the value of competitiveness parameter $\beta$
effects this cancellation probability. We can see that the cancellation
probability is increasing with competitiveness. With higher competitiveness,
agents will generally lower their quotes, so the proportion of agents ending
at negative inventory levels will increase. In the right panel of Figure 17,
we can observe that higher penalty for overselling leads to a lower
probability of cancellation. This is consistent with the result on the right
of Figure 14 where we demonstrated that a higher terminal penalty parameter
$\alpha_{2}$ and running inventory parameter $\phi_{2}$ reduces the proportion
of agents who oversold, thus decreases the cancellation probability. For
consumers, there is a trade-off between the cancellation probability and the
average transaction cost and instantaneous average cost, which is increasing
in penalty parameters as per Figure 15.

##  5 Conclusion

We have formulated a model for dynamic inventory pricing which accounts for
the effects of competition through a mean-field interaction. First, we
introduce a reference model with one agent looking to liquidate a significant
number of certain product. We expand our model into the case of infinite
players. In our model, the realized sales of each agent is described by a
doubly stochastic Poisson process, whose intensity depends on both one’s own
quoted price and the distribution of quoted prices among all agents. Through
the frequency of individual sales, agents compete with each other. This mean-
field game system consists of two equations: the dynamic programming equation
describing the optimality of representative agent, and the Kolmogorov forward
equation governing the dynamics of the distribution of inventory across
agents. The two equations are coupled by the consistency condition, which
enables us to find the equilibrium by fixed point iterations. We numerically
find an unique equilibrium. As expected, competition leads to more sales and
lower mean quoted prices of the market as a whole. Interestingly, when market
competition’s level increases, the optimal quoted prices for all inventory
levels do _not_ necessarily decrease all the time.

##  6 Appendix

###  A Proof of Proposition 3

  * Proof

Applying the first order conditions directly to $\sup$ term in Equation (15)
leads to the optimizer result. It can be checked that the second order
derivative of the sup term at $\delta^{*}$ is negative.

Equation (15) is of the form $\partial_{t}\mathbb{h}=\mathbb{F}(\mathbb{h})$.
To show existence and uniqueness of the solution to this equation, by the
Picard-Lindelof theorem, the function $\mathbb{F}$ need to be Lipschitz
continuous.

The Lipschitz continuity property of $f$ implies the same for $\mathbb{F}$,
where $f$ satifies

$\displaystyle
f(x,y)=\sup_{\delta\geq\hat{B}}A\,\exp{\left\\{-(\kappa+\beta)\,\delta+\beta\,\overline{\delta}\right\\}}\,(\delta+x-y)\,.$

The Lipschitz continuity property of $f$ will be a result of showing that all
directional derivatives of $f$ exist and are bounded for all
$(x,y)\in\mathbb{R}^{2}$.

The supremum is attained at $\delta^{*}$ in Equation (16). Thus, two separate
domains for $f$ must be considered:
$x-y<-\underline{B}+\frac{1}{\kappa+\beta}$ and
$x-y\geq-\underline{B}+\frac{1}{\kappa+\beta}$. First, consider
$x-y<-\underline{B}+\frac{1}{\kappa+\beta}$, so that
$\delta^{*}=\frac{1}{\kappa+\beta}+y-x$. Substituting this into the expression
for $f$ yields

$\displaystyle
f(x,y)=\frac{A}{\kappa+\beta}\,\exp{\left\\{(\kappa+\beta)\,(x-y)+\beta\,\overline{\delta}-1\right\\}}\,.$

Taking partial derivatives of $f$ in this domain gives us

$\displaystyle\partial_{x}f(x,y)=-\partial_{y}f(x,y)=A\,\exp{\left\\{(\kappa+\beta)\,(x-y)+\beta\,\overline{\delta}-1\right\\}}\,,$

and this expression is bounded in
$\left(0,A\,\exp{\left\\{-\underline{B}\,(\kappa+\beta)+\beta\,\overline{\delta}\right\\}}\right)$.
Thus, $\partial_{x}f$ and $\partial_{y}f$ are bounded in this domain, and so
directional derivatives exist and are also bounded everywhere in the interior
of the domain. On the boundary, directional derivatives exist and are bounded
if the direction is towards the interior of the domain.

Now consider $x-y\geq-\underline{B}+\frac{1}{\kappa+\beta}$, which implies
$\delta^{*}=\underline{B}$. The expression of $f(x,y)$ in this domain is

$\displaystyle
f(x,y)=A\,\exp{\left\\{-\underline{B}\,(\kappa+\beta)+\beta\overline{\delta}\right\\}}\,(\underline{B}+x-y)\,.$

Partial derivatives of $f$ are given by

$\displaystyle\partial_{x}f(x,y)=-\partial_{y}f(x,y)=A\,\exp{\left\\{-\underline{B}(\kappa+\beta)+\beta\,\overline{\delta}\right\\}}\,.$

So similarly to the first domain, directional derivatives exist and are
bounded in the interior. On the boundary, they exist and are bounded in the
direction towards the interior of the domain. Thus, we have existence and
boundedness on the boundary towards every of the two domains. The directional
derivative on the boundary is zero when the direction is parallel to the
boundary. Existence and boundedness of directional derivatives for all
$(x,y)\in\mathbb{R}^{2}$ allows us to show the Lipschitz continuity condition
easily:

$\displaystyle\begin{split}\left|f(x_{2},y_{2})-f(x_{1},y_{1})\right|&=\left|\int_{V}\triangledown
f(x,y)\cdot d\overset{\to}{r}\right|\leq\int_{V}\left|\triangledown
f(x,y)\right|\,ds\\\
&\leq\int_{V}R\,ds=R\,\left|(x_{2},y_{2})-(x_{1},y_{1})\right|\,\end{split}$

where $V$ is the curve which connects $(x_{1},y_{1})$ to $(x_{2},y_{2})$ in a
straight line and $R$ is a uniform bound on the gradient of $f$. This proves
that there exists a unique solution $h$ to Equation (15).

###  B Proof of Theorem 4

  * Proof

We define a candidate value function
$\hat{H}(t,s,x,q)=x+qs+h_{q}(t;\overline{\delta})$. From Ito’s lemma we have

$\displaystyle\begin{split}\hat{H}(T,S_{T^{-}},X_{T^{-}}^{\delta,\overline{\delta}},Q_{T^{-}}^{\delta,\overline{\delta}};\overline{\delta})&=\hat{H}(t,s,x,q;\overline{\delta})+\int_{t}^{T}\partial_{u}h_{Q^{\delta,\overline{\delta}}_{u}}(u)du+\sigma\int^{T}_{t}Q^{\delta,\overline{\delta}}_{u}\,dW_{u}\\\
&+\int^{T}_{t}\left(\delta_{u}+h_{Q^{\delta,\overline{\delta}}_{u^{-}}-1}(u)-h_{Q^{\delta,\overline{\delta}}_{u^{-}}}(u)\right)dN_{u}\,.\end{split}$

Let $\delta=(\delta_{t})_{0\leq t\leq T}$ be an arbitrary admissible control
and let $\epsilon>0$ be arbitrary. Then since $h$ satisfies Equation (15), the
following inequality holds almost surely for every $t$

$\displaystyle\partial_{t}h_{Q_{t}}-\phi\,Q_{t}^{2}+A\,\exp{\left\\{-(\kappa+\beta)\,\delta_{t}+\beta\,\overline{\delta}_{t}\right\\}}\left[\delta_{t}+h_{Q_{t}-1}(t;\overline{\delta}_{t})-h_{Q_{t}}(t;\overline{\delta}_{t})\right]<\epsilon\,.$

Thus, taking an expectation of
$\hat{H}(T,S_{T^{-}},X_{T^{-}}^{\delta,\overline{\delta}},Q_{T^{-}}^{\delta,\overline{\delta}};\overline{\delta})$,
we have

$\displaystyle\begin{split}&\mathbb{E}_{t,s,x,q}\left[\hat{H}(T,S_{T^{-}},X_{T^{-}}^{\delta,\overline{\delta}},Q_{T^{-}}^{\delta,\overline{\delta}};\overline{\delta})\right]\\\
&=\hat{H}(t,s,x,q;\overline{\delta})+\mathbb{E}_{t,s,x,q}\bigg{[}\int_{t}^{T}\partial_{u}h_{Q^{\delta,\overline{\delta}}_{u}}(u)du+\sigma\int^{T}_{t}Q^{\delta,\overline{\delta}}_{u}\,dW_{u}\\\
&\hskip
113.81102pt+\int^{T}_{t}\left(\delta_{u}+h_{Q^{\delta,\overline{\delta}}_{u^{-}}-1}(u)-h_{Q^{\delta,\overline{\delta}}_{u^{-}}}(u)\right)dN_{u}\bigg{]}\\\
&\leq\hat{H}(t,s,x,q;\overline{\delta})+\epsilon\,(T-t)+\mathbb{E}_{t,s,x,q}\left[\phi\int_{t}^{T}(Q_{u}^{\delta,\overline{\delta}})^{2}du\right]\,,\end{split}$

where
$\mathbb{E}_{t,s,x,q}\left[\int^{T}_{t}(\delta_{u}+h_{Q^{\delta,\overline{\delta}}_{u^{-}}-1}(u)-h_{Q^{\delta,\overline{\delta}}_{u^{-}}}(u))dN_{u}\right]$
exists due to the boundedness of $\delta$. Therefore $\hat{H}$ satisfies

$\displaystyle\begin{split}\hat{H}(t,s,x,q;\overline{\delta})+\epsilon\,(T-t)&\geq\mathbb{E}_{t,s,x,q}\left[\hat{H}(T,S_{T^{-}},X_{T^{-}}^{\delta,\overline{\delta}},Q_{T^{-}}^{\delta,\overline{\delta}};\overline{\delta})-\phi\int_{t}^{T}(Q_{u}^{\delta,\overline{\delta}})^{2}du\right]\\\
&=\mathbb{E}_{t,s,x,q}\left[X_{T}^{\delta,\overline{\delta}}+Q_{T}^{\delta,\overline{\delta}}(S_{T}-\alpha
Q_{T}^{\delta,\overline{\delta}})-\phi\int^{T}_{t}(Q_{u}^{\delta,\overline{\delta}})^{2}du)\right]\end{split}$

This inequality holds for the arbitrarily chosen control $\delta_{t}$,
therefore

$\displaystyle\hat{H}(t,s,x,q;\overline{\delta})+\epsilon\,(T-t)\geq\sup_{(\delta_{u})_{t\leq
u\leq
T}\in\mathcal{A}}\mathbb{E}_{t,s,x,q}\left[X_{T}^{\delta,\overline{\delta}}+Q_{T}^{\delta,\overline{\delta}}(S_{T}-\alpha
Q_{T}^{\delta,\overline{\delta}})-\phi\int^{T}_{t}(Q_{u}^{\delta,\overline{\delta}})^{2}du)\right]\,,$

and letting $\epsilon\to 0$ we finally obtain

$\displaystyle\hat{H}(t,s,x,q;\overline{\delta})\geq
H(t,s,x,q;\overline{\delta})\,.$

Now let $\delta^{*}$ be the control process defined as (16), then we have

$\displaystyle\begin{split}&\mathbb{E}_{t,s,x,q}\left[\hat{H}(T,S_{T^{-}},X_{T^{-}}^{\delta^{*},\overline{\delta}},Q_{T^{-}}^{\delta^{*},\overline{\delta}};\overline{\delta})\right]\\\
&=\hat{H}(t,s,x,q;\overline{\delta})+\mathbb{E}_{t,s,x,q}\bigg{[}\int_{t}^{T}\partial_{u}h_{Q^{\delta^{*},\overline{\delta}}_{u}}(u)du+\sigma\int^{T}_{t}Q^{\delta^{*},\overline{\delta}}_{u}\,dW_{u}\\\
&\hskip
113.81102pt+\int^{T}_{t}\left(\delta_{u}+h_{Q^{\delta^{*},\overline{\delta}}_{u^{-}}-1}(u)-h_{Q^{\delta^{*},\overline{\delta}}_{u^{-}}}(u)\right)dN_{u}\bigg{]}\\\
&\geq\hat{H}(t,s,x,q;\overline{\delta})+\mathbb{E}_{t,s,x,q}\left[\phi\int_{t}^{T}(Q_{u}^{\delta^{*},\overline{\delta}})^{2}du\right]\,,\end{split}$

and so $\hat{H}$ satisfies

$\displaystyle\begin{split}\hat{H}(t,s,x,q;\overline{\delta})&\leq\mathbb{E}_{t,s,x,q}\left[\hat{H}(T,S_{T^{-}},X_{T^{-}}^{\delta^{*},\overline{\delta}},Q_{T^{-}}^{\delta^{*},\overline{\delta}};\overline{\delta})-\phi\int_{t}^{T}(Q_{u}^{\delta^{*},\overline{\delta}})^{2}du\right]\\\
&=\mathbb{E}_{t,s,x,q}\left[X_{T}^{\delta^{*},\overline{\delta}}+Q_{T}^{\delta^{*},\overline{\delta}}(S_{T}-\alpha
Q_{T}^{\delta^{*},\overline{\delta}})-\phi\int^{T}_{t}(Q_{u}^{\delta^{*},\overline{\delta}})^{2}du)\right]\,,\end{split}$

Therefore,

$\displaystyle\begin{split}\hat{H}(t,s,x,q;\overline{\delta})&\leq\sup_{(\delta_{u})_{t\leq
u\leq
T}\in\mathcal{A}}\mathbb{E}_{t,s,x,q}\left[X_{T}^{\delta,\overline{\delta}}+Q_{T}^{\delta,\overline{\delta}}(S_{T}-\alpha
Q_{T}^{\delta,\overline{\delta}})-\phi\int^{T}_{t}(Q_{u}^{\delta,\overline{\delta}})^{2}du)\right]\\\
&=H(t,s,x,q;\overline{\delta})\,.\end{split}$

Combining the above results, we prove that

$\displaystyle\hat{H}(t,s,x,q;\overline{\delta})=H(t,s,x,q;\overline{\delta})\,.$

###  C Numerical Stability and Uniqueness of Equilibrium

We run the algorithm described in Section 3 100 times, each with a randomized
initial value of mean quote, and check the converged value of mean quote
$\\{\overline{\delta}^{1},\overline{\delta}^{2},\cdots,\overline{\delta}^{100}\\}$
obtained from the experiments. Over all 100 simulations, the average values of
the mean quote at any time $t\in\\{t_{j}\\}_{j=0}^{N}$,
$\frac{1}{100}\sum_{i=1}^{100}\overline{\delta}^{i}_{t}$, is of order
$10^{-2}$ to $1$, while the corresponding standard errors are of order
$10^{-16}$ to $10^{-15}$, which is smaller than the tolerance used in the
algorithm. Thus we conclude that the final value of $\overline{\delta}$ is
always the same, which shows that the algorithm is robust with respect to
initial point, and numerically supports our argument that the equilibrium
exists and is unique.

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