# Game-theoretic Model Predictive Control for Modelling Competitive Supply
Chains

Sophie Hall  Laura Guerrini  Florian Dörfler  Dominic Liao-McPherson
Automatic Control Laboratory, ETH Zürich, Switzerland, (e-mail: {shall,
laurague, doerfler}@control.ee.ethz.ch).  University of British Columbia,
Vancouver, Canada, (e-mail<EMAIL_ADDRESS>
###### Abstract

Supply chains transform raw materials into finished goods and distribute them
to end consumers. The vast majority of products we use daily are supplied to
us through complex global supply chains. This paper proposes a modelling
methodology for dynamic competitive supply chains based on game theory and
model predictive control. We model each manufacturer in the supply chain as a
rational utility maximizing agent that selects their actions by finding an
open-loop generalized Nash equilibrium of a multi-stage game. To react to
competitors and the state of the market, every agent re-plans their actions in
a receding horizon manner based on estimates of market and supplier parameters
thereby creating an approximate closed-loop equilibrium policy. We demonstrate
through numerical simulations that this modelling approach is computationally
tractable and generates economically interpretable behaviors in a variety of
settings such as demand spikes, supply shocks, and information asymmetry.

###### keywords:

Game theory, Model Predictive Control, Supply Chains

††thanks: This work is supported by the Swiss National Science Foundation via
NCCR Automation (Grant Number 180545) and the Natural Sciences and Engineering
Research Council of Canada (Reference Number RGPIN-2023-03257).

##  1 Introduction

Supply chains are complex socio-technical systems that transform raw materials
into finished products and deliver these products to consumers. In modern
market economies, supply chains are interconnected networks of suppliers,
manufacturers, retailers, wholesalers, and consumers, each selfishly seeking
to maximize their utility. Modelling and control of supply chains is crucial
for tackling a variety of real-world challenges such as disruptions to product
availability caused by supply shocks and labour shortages (e.g., due to the
pandemic or geopolitical conflicts). Supply chains are challenging to model
since they are highly dynamic, subject to a variety of constraints, and
comprised of multiple interlinked competitive actors. As a result, supply
chains present a unique and challenging setting for control in socio-technical
systems.

Control engineering principles have been successfully applied to modelling and
control of supply chains in previous works. Typical examples include inventory
control, modelling industrial dynamics, and policy design for avoiding the
Bullwhip Effect (Vassian, 1955; Towill, 1982, 1996), mainly using frequency
domain tools. Recent literature capitalizes on advances in optimal control,
predominantly model predictive control (MPC), to tackle problems in real-time
inventory management and supply chain optimization (Braun et al., 2003; Perea-
Lopez et al., 2003; Subramanian et al., 2014), remanufacturing (Li et al.,
2021), and production and marketing decisions (Pekelman, 1974; Levin et al.,
2008). A thorough survey can be found in Ivanov et al. (2018); Sarimveis et
al. (2008).

The control-oriented works primarily focus on vertically integrated supply
chains, where all agents are working toward a common goal, potentially in a
decentralized way. However, in reality, there is competition at every stage of
the chain starting from resource extraction, through production, wholesaling,
distribution, and retail. Game theory is a powerful framework for modelling
and analyzing supply chains with multiple agents that have conflicting
interests (Cachon and Netessine, 2006). Network equilibrium models
conceptualize supply chains as a network of interconnected producers,
suppliers, and consumers (Nagurney et al., 2002; Nagurney, 2006; Anderson and
Bao, 2010) and solve for (typically Nash) equilibria of network games by
solving variational inequalities. These equilibria can computed for large
networks using distributed algorithms (Belgioioso et al., 2022) and are
flexible enough to model a wide variety of scenarios, e.g., they have been
used to study the impact of waste from electric and electronic equipment
disposal regulations (Hammond and Beullens, 2007) and the consequences of
labour constraints caused by the Covid-19 pandemic (Nagurney, 2021).

Dynamic game-theory (Başar and Olsder, 1998) has been used to study a variety
of scenarios involving dynamic multi-agent competition including optimal
pricing and inventory replenishment policies (Netessine et al., 2006; Kirman
and Sobel, 1974; Bernstein and Federgruen, 2004), the impacts of technology
acquisition (Gaimon, 1989), and capacity investment decisions based on
reputational effects (Taylor and Plambeck, 2007). Dynamic games involve multi-
period decision-making and typically require solving for feedback Nash
equilibria (Nash equilibria over policies) which are very challenging to
compute, limiting these methods to simple settings to remain analytically
tractable. Instead, receding horizon games compute open-loop equilibria and
introduce feedback by re-planning at every time step. Thus, harnessing the
benefits of a feedback policy while remaining computationally tractable even
for complex models with many agents (Hall et al., 2022).

In this paper, we propose a method for modelling dynamic supply chains using
receding horizon games. Specifically, we combine multi-period network
equilibrium models with model predictive control to numerically construct
approximate feedback-equilibrium policies. These policies then serve to form a
closed-loop model of the supply chain. This approach is flexible and allows us
to consider complex scenarios involving dynamics and delays, constraints,
information asymmetry, external forcing, and forecasts that cannot be captured
by either network equilibrium or traditional dynamic game theory approaches.
We illustrate the efficacy of our method by studying demand and supply shocks,
and the impact of information asymmetry. With out methodology we are able to
replicate a variety of economically relevant behaviours: (i) we show that
dynamic pricing can be used to manage demand and inventory, (ii) we show how
dynamic effects propagate through the supply chain, and lastly (iii) that
information offers a competitive advantage.

Notation: We denote the normal cone operator of a closed convex set
$\Omega\subseteq\mathbb{R}^{n}$ by
$\mathcal{N}_{\Omega}:\Omega\rightrightarrows\mathbb{R}^{n}$. For two
matrices, $(A,B)$ denotes vertical concatenation. For an index set
$\mathcal{A}=\\{1,\ldots,M\\}\subset\mathbb{Z}$ and a collection of
matrices/vectors $\\{A^{v}\\}$ the stacking operator is
$(A^{v})_{v\in\mathcal{A}}=(A^{1},\ldots,A^{M})$ and
$A^{-v}=(A^{v})_{v\in\mathcal{A}\setminus\\{v\\}}$ and
$\text{blkdiag}\\{A^{v}\\}_{v\in\mathcal{A}}$ denotes the block diagonal
matrix with $A^{1},...,A^{M}$ on the main diagonal. For the generalized (i.e.,
constrained) game

$\displaystyle\forall
v\in\mathcal{V}:~{}\min_{u^{v}}\left\\{J^{v}(u^{v},u^{-v})~{}|~{}(u^{v},u^{-v})\in\mathcal{C}\right\\},$
(1)

with cost $J^{v}$, coupling constraint set $\mathcal{C}$ and where
$\mathcal{V}=\\{1,\ldots,m\\}$, a vector
$\bar{u}=(\bar{u}^{v})_{v\in\mathcal{V}}$ is a Generalized Nash Equilibrium
(GNE) of the game (1) if it satisfies

$\displaystyle
J^{v}(\bar{u}^{v},\bar{u}^{-v})\leq\min_{u^{v}}\left\\{J^{v}(u^{v},\bar{u}^{-v})~{}|~{}(u^{v},\bar{u}^{-v})\in\mathcal{C}\right\\},\;\forall
v\in\mathcal{V}.$

##  2 Problem Setting

Consider a competitive supply chain consisting of $n_{s}>0$ suppliers
$\mathcal{S}=\\{1,\ldots,n_{s}\\}$, and $n_{m}>0$ manufacturers
$\mathcal{M}=\\{1,\ldots,n_{m}\\}$ serving a common market, as illustrated in
Figure 1. The manufacturers transform raw material purchased from the
suppliers into a finished product and sell it to consumers in the market. In
this paper, we consider a product based on a single raw material.

Figure 1: A supply chain consisting of two upstream suppliers, three
manufacturers, and a competitive downstream market.

The demand $d_{t}^{v}$ on day $t$ for each manufacturers $v\in\mathcal{M}$
product is assumed elastic and set by the market according to the linear
demand model (Choi, 1991)

$d_{t}^{v}=w_{t}^{v}-\beta^{vv}p_{t}^{v}+\sum_{\footnotesize{j\in\mathcal{M}\backslash\\{v\\}}}\beta^{vj}p_{t}^{j},$
(2)

where $\beta^{v}=(\beta^{vj})_{j\in\mathcal{M}}\geq 0$ are the coefficients of
the demand model, $p^{v}_{t}\geq 0$ is the price chosen by manufacturer
$v\in\mathcal{M}$, and $w_{t}^{v}$ is the base (or inelastic) demand. We
observe that the demand for manufacturer $v$’s product falls when they
increase their price $p^{v}$ and rises when their competitors increase their
prices $p^{-v}$.

###### Remark 1

The demand model (2) can be rearranged:

$d^{v}_{t}=w_{t}^{v}-\underbrace{\Big{(}\beta^{vv}-\sum_{\scriptscriptstyle
j\in\mathcal{M}\backslash\\{v\\}}\beta^{vj}\Big{)}p_{t}^{v}}_{\text{marginal
consumer}}+\underbrace{\sum_{\scriptscriptstyle
j\in\mathcal{M}\backslash\\{v\\}}\beta^{vj}(p^{j}_{t}-p^{i}_{t})}_{\text{switching
consumer}}$

for $\beta^{vj}\to 0$ we only have marginal consumers, i.e., the product is
completely differentiated and not substitutable and for
$(\beta^{vv}-\sum_{j\not=v}\beta^{vj})$ there are only switching consumers,
thus the products are substitutable and there is perfect competition (Anderson
and Bao, 2010). $\hfill\square$

To meet consumer demand, manufacturer $v\in\mathcal{M}$ orders $o_{t}^{vs}$
units of raw material from supplier $s\in\mathcal{S}$ and produces

$O_{t}^{v}=\sum_{s\in\mathcal{S}}o_{t}^{vs}$ (3)

units of product. They maintain an inventory $\xi_{t}^{v}$ of finished product
in a warehouse and sell $d^{v}_{t}$ units of product to the common market at a
price $p^{v}_{t}$. Each agent maintains enough inventory on hand to satisfy
the demand (this will be enforced as a constraint later) and that both
production and shipping are subject to a one day delay. This leads to the
inventory balance equation

$\xi_{t+1}^{t}=\alpha^{v}\xi_{t}^{v}+O_{t-1}^{v}-d^{v}_{t-1},$ (4)

where $\alpha^{v}\in(0,1]$ is the decay rate of the product ($\alpha^{v}=1$
for non-perishable products) (Schwager and Etzkorn, 2016) and inventory is
subject to the constraints $0\leq\xi_{t}^{v}\leq\Xi^{v}$ with maximum
inventory $\Xi^{v}>0$. Inventory models of this type are common in supply
chain analysis and control e.g., (Braun et al., 2003; Spiegler et al., 2012).

Each supplier $s\in\mathcal{S}$ can supply a maximum of $\bar{O}^{s}$ units
per day, leading to the constraint

$\sum_{v\in\mathcal{M}}o_{t}^{vs}\leq\bar{O}^{s},$ (5)

and sells raw materials to manufacturer $v$ at the price

$\rho^{s}(o_{t}^{vs},o_{t}^{-vs})=\rho_{0}^{s}+\rho_{1}^{s}\sum_{v\in\mathcal{M}}o_{t}^{vs}.$
(6)

This model of the wholesale price can be understood as an inverse form of the
commonly used linear supply-price curve (Schwager and Etzkorn, 2016). The
first term is a constant minimum price set by each supplier where
$\rho_{0}^{s}\geq 0$. The second term introduces a dependency on the aggregate
number of orders that can model a variety of commercial strategies from the
suppliers. E.g., setting $\rho_{1}^{s}>0$ indicates the supplier feels they
are in a position to raise prices when demand is high while conversely,
setting $\rho_{1}^{s}<0$ indicates the supplier is willing to share reductions
in unit costs due to economies of scale.

Inserting (2) and (3) into (4) and augmenting the state by the delay states,
we find that the dynamics of each manufacturer are a linear time-invariant
(LTI) system of the form

$x^{v}_{t+1}=A^{v}x^{v}_{t}+\sum_{j\in\mathcal{M}}B^{vj}u^{j}_{t}+D^{v}w^{v}_{t}$
(7a) where $x^{v}_{t}=(\xi_{t}^{v},O_{t-1}^{v},d_{t-1}^{v})$ and
$u^{v}_{t}=(o_{t}^{v1},\ldots,o_{t}^{vn_{s}},p^{v}_{t})$ is the state and
control input of manufacturer $v$ and the base demand $w^{v}_{t}$ is the
exogenous input acting as a disturbance. The system matrices are the following
$\displaystyle A^{v}=\begin{bmatrix}\alpha^{v}&1&-1\\\ 0&0&0\\\
0&0&0\end{bmatrix},\;B^{vv}=\begin{bmatrix}0&\cdots&0&0\\\ 1&\cdots&1&0\\\
0&\cdots&0&-\beta^{vv}\end{bmatrix},\;D^{v}=\begin{bmatrix}0\\\ 0\\\
1\end{bmatrix}$ $\displaystyle\text{ and
}B^{vj}=\begin{bmatrix}0&\cdots&0&0\\\ 0&\cdots&0&0\\\
0&\cdots&0&\beta^{vj}\end{bmatrix}\text{ for
}j\in\mathcal{M}\backslash\\{v\\}.$ (7b)

Figure 2: A block diagram of the inventory dynamics for manufacturer $M^{1}$
with $z$ denoting the forward shift operator.

The dynamic model of the inventory balance (7) for a single manufacturer $v$
is summarized in Figure 2. Collecting the inventory dynamics (7), we see that
the global supply chain is governed by the following LTI system:

$x_{t+1}=Ax_{t}+Bu_{t}+Dw_{t}$ (8a) with
$A=\text{blkdiag}\\{A^{v}\\}_{v\in\mathcal{M}},~{}~{}B=\begin{bmatrix}B^{11}&\cdots&B^{1n_{m}}\\\
\vdots&\ddots&\vdots\\\ B^{n_{m}1}&\cdots&B^{n_{m}n_{m}}\end{bmatrix},$ (8b)

$x_{t}=(x^{v}_{t})_{v\in\mathcal{M}}$, $u_{t}=(u^{v}_{t})_{v\in\mathcal{M}}$,
$w_{t}=(w_{t}^{v})_{v\in\mathcal{M}}$, and
$D=\text{blkdiag}\\{D^{v}\\}_{v\in\mathcal{M}}$.

###### Remark 2

In this paper, we focus on a simple supply chain to illustrate the methodology
and simplify interpretation of the results. Single-agent MPC has been applied
to more complex multi-product/multi-layer networks see e.g., (Braun et al.,
2003; Subramanian et al., 2014). Our approach can be generalized in the same
manner. $\hfill\square$

##  3 Game-theoretic MPC-based modelling

We want to model competitive dynamic supply chains. Therefore, we model each
manufacturer as a rational agent seeking to minimize the following stage cost
function:

$\ell^{v}(x^{v},u^{v},u^{-v},w^{v})=\underbrace{\gamma^{v}\cdot(\xi^{v}-\bar{\xi}^{v})^{2}}_{\text{security
stock}}\\\ -\underbrace{\left(p^{v}\cdot
d^{v}(p^{v},p^{-v},w^{v})-\sum_{s\in\mathcal{S}}\rho^{s}(o^{vs},o^{-vs})\cdot
o^{vs}\right)}_{\text{net cash flow}}.$ (9)

The first term with penalizes deviations from the safety inventory
$\bar{\xi}^{v}$, which is a reference term used in inventory control laws
(e.g., (Spiegler et al., 2012)) to ensure that it is always possible to
fulfill demand where $\gamma^{v}>0$, and the second term is net cash flow of
manufacturer $v$.

The manufacturers incorporate the dynamics of the supply chain into their
decision-making process by minimizing their incremental cost function over a
planning horizon $N>0$. This yields the set of coupled optimal control
problems (OCP) $\mathcal{P}_{N}(\theta,\mathbf{x},w):$ for all
$v\in\mathcal{M}$

$\displaystyle\min_{x^{v}_{k},u^{v}_{k}}~{}$
$\displaystyle\sum_{k=0}^{N}\ell^{v}(x_{k}^{v},u_{k}^{v},u_{k}^{-v},w_{k}^{v})$
(10a) $\displaystyle\mathrm{s.t.}$ $\displaystyle
x^{v}_{k+1}=A^{v}x^{v}_{k}+\sum_{j\in\mathcal{M}}B^{vj}u_{k}^{j}+D^{v}w^{v}_{k},$
(10b) $\displaystyle x_{0}^{v}=\mathbf{x}^{v},$ (10c) $\displaystyle
x_{k}^{v}\in\mathcal{X}^{v},~{}~{}u_{k}^{v}\in\mathcal{U}^{v},$ (10d)
$\displaystyle(u^{v}_{k},u^{-v}_{k})\in\mathcal{C}$ (10e) where
$\mathbf{x}^{v}$ is the initial condition,
$\mathcal{U}^{v}=\\{(p^{v},o^{vs})~{}|~{}0\leq p^{v},~{}0\leq
o^{vs}~{}~{}\forall s\in\mathcal{S}\\}$,
$\mathcal{X}^{v}=\\{(\xi,O,d)~{}|~{}0\leq\xi\leq\Xi^{v}\\}$ are the local
constraints and
$\mathcal{C}=\left\\{o^{vs}~{}\bigg{|}~{}\sum_{v\in\mathcal{M}}o^{vs}\leq\bar{O}^{s}~{}~{}\forall
s\in\mathcal{S}\right\\}$ (10f)

encodes the coupling constraint (5). The parameters of the market, suppliers,
and manufacturers are collected in
$\theta=(\rho^{s},\bar{O}^{s},\beta^{v},\alpha^{v},\gamma^{v},\Xi^{v})_{v\in\mathcal{M},s\in\mathcal{S}}$
and the demand forecasts of each agent over the prediction horizon is
$w^{v}=(w^{v}_{k})_{k=0}^{N}$. Together, the $n_{p}$ coupled OCPs
$\mathcal{P}_{N}$ constitute a game.

In reality, there is information asymmetry (i.e., not all manufacturers have
accurate information on the local parameters of others). To model these
effects, we assume that each manufacturer is strategic, in the sense that they
maintain models of their competitors. Specifically:

  * •

The form of the model is fixed and known and all agents plan over an $N$ step
horizon;

  * •

Each manufacturer maintains estimates of the the parameters of the game
$\hat{\theta}^{v}$ and demand forecasts for all agents
$\hat{w}^{v}=(\hat{w}^{vj})_{j\in\mathcal{M}}$ where
$\hat{w}^{vj}=(\hat{w}^{vj}_{k})_{k=0}^{N}$ denotes the forecast of agent $v$
for agent $j$ over the prediction horizon $N$;

  * •

The state of each agent $x^{v}_{t}$ is measurable (or can be estimated).

To compensate for the finite planning horizon and inaccurate information about
other, agents re-plans their actions each day. Specifically, they find a Nash
equilibrium $(\bar{u}^{v}_{k},\bar{u}^{-v}_{k})_{k=0}^{N}$ of
$\mathcal{P}_{N}(\hat{\theta}^{v},x_{t},\hat{w}^{v}_{t})$ based on the
measured state $x_{t}$, their parameter estimates $\hat{\theta}^{v}$ and their
forecast $\hat{w}^{v}_{t}$ of the base demand over the interval
$[t~{}~{}t+N]$, and they apply the first element of their planned control
sequence $u^{v}_{t}=\bar{u}^{v}_{0}(\hat{\theta}^{v},x_{t},\hat{w}^{v}_{t})$.
For each agent, this re-planning procedure constitutes a model predictive
controller where they use game-theory to model how their competitors will
react to their own actions (e.g., if I play $u^{v}_{0}$ it would be rational
for my competitors to play $u^{-v}_{0}$). Similar ideas are used in multi-
robot systems e.g., drone racing and autonomous driving (Wang et al., 2021;
Spica et al., 2020; Fridovich-Keil et al., 2020) and energy management
applications (Hall et al., 2022).

To make this more precise, we write (10) in a compact form by substituting the
dynamics (10b) into the cost (10a) and constraints (10d)-(10e). This results
in a condensed game of the form $\forall v\in\mathcal{M}$:

$\displaystyle\min_{u^{v}}~{}~{}$ $\displaystyle
J^{v}(u^{v},u^{-v},\theta,\mathbf{x},w^{v})$ (11a)
$\displaystyle~{}\mathrm{s.t.}$ $\displaystyle G(\theta)u\leq
g(\theta,\mathbf{x},w)$ (11b)

where $u^{v}=(u^{v}_{k})_{k=0}^{N-1}$, $u=(u^{v})_{v\in\mathcal{M}}$,
$\mathbf{x}=(\mathbf{x}^{v})_{v\in\mathcal{M}}$, and
$w=(w^{v})_{k\in\mathcal{M}}$. The feedback law $\kappa^{v}$ is given by

$u^{v}_{t}=\kappa^{v}(x_{t},\hat{\theta}^{v},\hat{w}^{v}_{t})=\Psi^{v}S(x_{t},\hat{\theta}^{v},\hat{w}^{v}_{t})$
(12)

where $S(\mathbf{x},\theta,w)=\\{u~{}|~{}u\text{ is a v-GNE of
}\eqref{eq:condensed-game}\\}$ is the solution map of (11) and $\Psi^{v}$ is a
matrix selecting $u^{v}_{0}$.

Our global supply chain model is given by the dynamics of the supply chain (8)
in closed-loop with the feedback policies (12). Combining them we arrive at
the closed-loop dynamics of the entire supply chain

$x_{t+1}=Ax_{t}+B\kappa(x_{t},\hat{\theta},\hat{w}_{t})+Dw_{t}$ (13)

where
$\kappa(x_{t},\hat{\theta},\hat{w}_{t})=(\kappa^{v}(x_{t},\hat{\theta}^{v},\hat{w}_{t}^{v}))_{v\in\mathcal{M}}$,
and the collected parameter estimate and demand forecasts are denoted by
$\hat{w}_{t}=(\hat{w}_{t}^{v})_{v\in\mathcal{M}}$ and
$\hat{\theta}=(\hat{\theta}^{v})_{v\in\mathcal{M}}$ respectively.

Our approach is algorithmic in the sense that we evaluate $\kappa^{v}$ by
solving for v-GNEs of (11) numerically. Since the cost $J^{v}$ is quadratic
and convex in $u^{v}$ for each agent if $\rho_{1}^{s}\geq 0$, v-GNEs of the
condensed game (11) can be computed by solving an affine variational
inequality (AVI) (Facchinei and Kanzow, 2010). Solutions of AVI’s are locally
unique under mild conditions and a variety of algorithms exist for solving
them. In this paper we use an algorithm based on complimentary functions and a
non-smooth Newton’s method (Liao-McPherson et al., 2018) to solve for the
v-GNEs of (11). An important property of v-GNEs is that they “fairly”
distribute the cost of enforcing the coupling constraints (11b) (Belgioioso et
al., 2022).

###### Remark 3

More details on how we evaluate the feedback law $\kappa^{v}$, including
explicit definitions for $J^{v},G$, $g$, and the AVI for (11), along with
conditions for uniqueness are included in Appendix A. $\hfill\square$

##  4 Numerical studies

We investigate the behaviour of the dynamic supply chain model in three
different settings to demonstrate that our methodology can capture interesting
and interpretable competitive behaviours. In this paper, we only investigate
scenarios where all manufacturers know the parameters perfectly (so that
$\hat{\theta}^{v}=\theta$) and only consider information asymmetry in the
demand forecasts $\hat{w}^{v}$. This can be relaxed in the future by e.g.,
having the manufacturers estimate each others parameters using game-theoretic
inference (Maddux et al., 2023; Le Cleac’h et al., 2021).

The parameter values used in the simulations are listed in Table 1. The
prediction horizon is chosen as $N=15$ and the safety inventory and inventory
limit is set to $\bar{\xi}^{v}=25$ and $\Xi^{v}=50,\;\forall v\in\mathcal{M}.$
The initial condition for all simulations is
$\mathbf{x}^{v}=\begin{bmatrix}30&0&0\end{bmatrix}^{\top},\;\forall
v\in\mathcal{M}.$

$\alpha^{v}$ | $\beta^{v}$ | $\gamma^{v}$ | $\rho_{0}^{s}$ | $\rho_{1}^{s}$
---|---|---|---|---
$\alpha^{1}=0.9$ | $\beta^{1}=[0.7,0.3,0.3]$ | $\gamma^{1}=0.1$ | $\rho_{0}^{1}=1$ | $\rho_{1}^{1}=0.1$
$\alpha^{2}=0.7$ | $\beta^{2}=[0.3,0.8,0.3]$ | $\gamma^{2}=0.1$ | $\rho_{0}^{2}=1.5$ | $\rho_{1}^{2}=0.15$
$\alpha^{3}=0.5$ | $\beta^{3}=[0.3,0.3,0.6]$ | $\gamma^{3}=0.1$ |  |
Table 1: Parameter values for all case studies.

###  4.1 Demand spike

Figure 3: Sudden spike in base demand $w_{t}^{v}$ by a factor of two for
manufacturers $M^{1}$ and $M^{2}$ for $t=10,\dots,18$. The manufacturers have
no preview of the demand spike.

Consider a scenario where base demand suddenly spikes, as e.g., during a
pandemic. We simulate the spike by doubling the base demand $w_{t}^{v},\,$ for
$v=\\{1,2\\}$ for days $t=10,\dots,18$ without giving agents preview. The
results are shown in Figure 3, we observe that:

  * •

The manufacturers $M^{1}$ and $M^{2}$ exploit the demand spike to increase
their prices and thus also increase their revenue (second, third, and fourth
plot).

  * •

The suppliers also get a share of “the pie” as they can increase their
wholesale price $\rho(o)$ with the increase in orders (bottom blot) and, as
expected in a supply chain, the increased demand trickles-down.

  * •

$M^{3}$ who does not have a demand spike also increases the prices to
compensate for higher raw material cost. $M^{3}$ can increase prices without
fearing loss of demand as others increased their prices.

Overall, as expected, even other manufacturers without the demand spike
benefit economically from the increased consumer base.

###  4.2 Supply shock

As a second case study, we investigate the global supply chain behaviour in
the presence of sudden shortages in supply of input materials. This may occur
if e.g., supply routes are disrupted by geopolitical conflicts, labour
shortages, or natural disasters such as droughts or flooding.

As in the previous case study, the manufacturers are given no warning of the
sudden supply shock. The coupling constraint limit for supplier one
$\bar{O}^{1}$ is dropped on day 10 by 70%, as displayed in Figure 4. We make
the following key observations:

  * •

As manufacturers hit the supply limit $\bar{O}^{1}$ for the cheaper supplier
1, they instantly order more from the other available supplier.

  * •

To compensate for the increase in raw material cost, they increase their
prices, experiencing a drop in demand as the number of marginal consumers
decreases.

  * •

Manufacturers also use prices to control demand (lower it) such that they do
not run out of inventory as they can not order enough raw materials to keep up
with demand.

Figure 4: Sudden drop in supply from the cheaper supplier 1 by $70\%$ between
days $t=10,\dots,18$.

###  4.3 Information asymmetry

In the real world, not every manufacturer would have an accurate demand
forecast and also does not know what information is available to other
manufacturers. We consider the same demand spike as in Subsection 4.1 but for
two agents and now investigate what happens when:

  1. 1)

$M^{1}$ has prefect forecast of $w_{t}$, and $M^{2}$ has no forecast;

  2. 2)

$M^{1}$ and $M^{2}$ both have no forecast of $w_{t}$.

The results are given in Figure 5 and we observe that:

Figure 5: Information asymmetry: Case 1) $M^{1}$ has perfect forecast of the
demand spike and $M^{2}$ has no forecast; Case 2) None of the manufacturers
have forecast.

  * •

$M^{1}$ exploits the preview they have in case 1 in order to increase the
price and therefore the net cash flow.

  * •

$M^{1}$ uses the price inputs to control their inventory which is cheaper than
ordering more.

  * •

$M^{2}$ suffers more when $M^{1}$ has the forecast due to the price-demand
coupling between the manufacturers.

Overall, from the stage cost (bottom plot) we notice that $M^{1}$ gets a
competitive advantage from having preview leading to better inventory control
and higher profits.

Summarizing the three case studies we observe the following economically
relevant behaviours in our dynamic supply chain model: (i) Agents use prices
to control demand and to manage their inventory; (ii) increased demand and
profits are passed throughout the supply chain; (iii) information is a major
competitive advantage; (iv) agents without information suffer from information
asymmetry.

###  4.4 Turnpike of the open-loop trajectories

Besides the economically relevant behaviours, we also observe interesting
system properties of the game-theoretic MPC such as the turnpike property
exhibited by the open-loop trajectories, see Figure 6. The turnpike property
is a well known phenomena in economic MPC where an optimal trajectory
approaches an equilibrium state, stays close to it for a while, and then turns
away from it again (Damm et al., 2014). This observation calls for a detailed
study of the system properties of receding horizon games in future works.

Figure 6: All open-loop trajectories over the simulation horizon of
$t=0,\dots,30$ days exhibit a turnpike effect.

##  5 Conclusion

We present a dynamic supply chain model to study the competition between
manufacturers that share raw material suppliers and a common consumer base.
The manufacturers compete to maximize their profit by setting prices and
managing their inventory levels. We formalized a dynamic supply chain model as
a game-theoretic MPC where the approximate equilibrium policies serve as a
closed-loop model for the supply chain. We study the behaviour of the supply
chain in a variety of scenarios such as demand spikes and supply shocks and
demonstrate that economically intuitive behaviours emerge from our model of
dynamic supply chain competition. Future work includes: game-theoretic
inference for validation and parameter estimation, studying recursive
feasibility under information asymmetry, understanding the impact of varying
prediction horizons, as well as studying a variety of supply chain scenarios
such as the design of market regulations and carbon reduction incentives.

##  6 Acknowledgements

We we would like to thank Efe Balta and Giuseppe Belgioioso for their support
in the early conception of the idea and and their support throughout the
project.

## Appendix A

Variational GNE’s of the condensed game (11) can be found by solving the
following AVI (we have suppressed all dependencies on $\hat{\theta}^{v}$ for
notational simplicity):

$\displaystyle\begin{bmatrix}H&G^{T}\\\ -G&0\end{bmatrix}\begin{bmatrix}u\\\
\lambda\end{bmatrix}+\begin{bmatrix}f(\mathbf{x})\\\
g(\mathbf{x})\end{bmatrix}+\begin{bmatrix}\\{0\\}\\\
\mathcal{N}_{\Lambda}(\lambda)\end{bmatrix}\ni 0$ (14a) where
$\mathcal{N}_{\Lambda}$ is the normal cone mapping of the set
$\Lambda=\\{\lambda~{}|~{}\lambda\geq 0\\}$ and
$Hu+f(\mathbf{x})=(\nabla_{u^{v}}J^{v}(u^{v},u^{-v},\hat{\theta}^{v},\mathbf{x}))_{v\in\mathcal{M}}$
(14b)

is the pseudo-gradient map. The AVI (14) is of the same form as the KKT
conditions for a quadratic program and can be readily solved using the
algorithm in (Liao-McPherson et al., 2018) with some small modifications to
account for the asymmetry of $H$. Solutions of AVI’s are locally unique under
mild conditions111Let $(\bar{u},\bar{\lambda})$ satisfy (14) and
$\mathcal{A}=\\{i~{}|~{}G_{i}\bar{u}=g_{i}(\bar{x})\\}$ denote the active
constraints at $\bar{u}$. Then if $y^{T}Hy>0$ for all
$y\in\\{u~{}|~{}G_{i}u=0,~{}i\in\mathcal{A}\\}$ and rank
$((G_{i})_{i\in\mathcal{A}})=|\mathcal{A}|$ then $(\bar{u},\bar{\lambda})$ is
a locally unique v-GNE of (11) (Dontchev et al., 2009, Theorem 2E.6)..

The objects $H$ and $f$ can be constructed as follows:

$\displaystyle H=H_{1}+H_{2}$ $\displaystyle
H_{1}=\left[\begin{smallmatrix}2R^{11}&\cdots&R^{1n_{m}}\\\
\vdots&\ddots&\vdots\\\
R^{n_{m}1}&\cdots&2R^{n_{m}n_{m}}\end{smallmatrix}\right],$ $\displaystyle
R^{vv}=I_{N}\otimes\left[\begin{smallmatrix}\text{blkdg}(\rho_{1}^{s})&0&\\\
0&\beta^{vv}\end{smallmatrix}\right],R^{vj}=I_{N}\otimes\left[\begin{smallmatrix}\text{blkdg}(\rho_{1}^{s})&0&\\\
0&-\beta^{vj}\end{smallmatrix}\right],\quad$

and

$\displaystyle
H_{2}=\;\left[\begin{smallmatrix}2(\tilde{B}^{11})^{\top}Q^{1}\tilde{B}^{11}&\ldots&(\tilde{B}^{11})^{\top}Q^{1}\tilde{B}^{1n_{m}}\\\
\vdots&\ddots&\vdots\\\
(\tilde{B}^{n_{m}1})^{\top}Q^{n_{m}}\tilde{B}^{n_{m}1}&\ldots&2(\tilde{B}^{n_{m}1})^{\top}Q^{n_{m}}\tilde{B}^{n_{m}n_{m}}\end{smallmatrix}\right],$
$\displaystyle Q^{v}=I_{N}\otimes\left[\begin{smallmatrix}\gamma^{v}&0&0\\\
0&0&0\\\ 0&0&0\\\ \end{smallmatrix}\right]$

and further

$\displaystyle
f^{v}(\mathbf{x}^{v})=(r^{v}+(\tilde{B}^{vv})^{\top}Q^{v}(\tilde{A}^{v}\,\mathbf{x}^{v}+\tilde{D}^{v}\,w^{v})+(\tilde{B}^{vv})^{\top}q^{v})$

with

$\displaystyle
r^{v}=(r^{v}_{k})_{k=0}^{N-1},\;r_{k}^{v}=\left[\begin{smallmatrix}\rho_{0,k}^{\top}&-w^{v}_{k}\end{smallmatrix}\right]^{\top},q^{v}=\mathds{1}_{N}\otimes\left[\begin{smallmatrix}(-2\gamma^{v}\,\bar{\xi}^{v})&0&0\end{smallmatrix}\right]^{\top}$

where $f(\mathbf{x})=(f^{v}(\mathbf{x}^{v}))_{v\in\mathcal{M}}$ and
$\tilde{A}^{v}$, $\tilde{B}^{vv}$, $\tilde{B}^{vj}$ and $\tilde{D}^{v}$ are
the $N$ step free-response, forced response and disturbance response matrices
respectively of the LTI system (7) of agent $v$. Same as in (2), $w^{v}$ is
the base demand of agent $v$.

The constraint are encoded in $G$ and $g$ which can be constructed as follows:

$\displaystyle G=\left[\begin{smallmatrix}G_{1}\\\ G_{2}\tilde{B}\\\ G_{3}\\\
\end{smallmatrix}\right],\quad g(\mathbf{x})=\left[\begin{smallmatrix}g_{1}\\\
g_{2}-G_{2}(\tilde{A}\mathbf{x}+\tilde{D}\,w)\\\
g_{3}\end{smallmatrix}\right]$

with

$\displaystyle G_{1}=I_{N\cdot
n_{m}}\otimes\left[\begin{smallmatrix}-I_{n_{s}}&0\\\
\mymathbb{0}_{n_{s}}&-1\\\ \end{smallmatrix}\right],\;g_{1}=\mathds{1}_{N\cdot
n_{m}}\otimes\left[\begin{smallmatrix}\mymathbb{0}_{n_{s}}\\\ 0\\\
\end{smallmatrix}\right],$ $\displaystyle G_{2}=I_{N\cdot
n_{m}}\otimes\left[\begin{smallmatrix}1&0&0\\\ -1&0&0\\\
\end{smallmatrix}\right],\;g_{2}=(g_{2}^{v})_{v\in\mathcal{M}},\;g_{2}^{v}=\mathds{1}_{N}\otimes\left[\begin{smallmatrix}\Xi^{v}\\\
0\\\ \end{smallmatrix}\right],$ $\displaystyle
G_{3}=\mymathbb{1}_{n_{m}}^{\top}\otimes(I_{N}\otimes\left[\begin{smallmatrix}I_{n_{s}}&\mymathbb{0}_{n_{s}}\end{smallmatrix}\right]),\;g_{3}=\mathds{1}_{N}\otimes[\bar{O}^{1},\;\bar{O}^{2}]^{\top},$

where $\tilde{A}^{v}=\text{blkdiag}\\{\tilde{A}^{v}\\}_{v\in\mathcal{M}}$,
$\tilde{D}^{v}=\text{blkdiag}\\{\tilde{D}^{v}\\}_{v\in\mathcal{M}}$ and

$\displaystyle\tilde{B}=\left[\begin{smallmatrix}\tilde{B}^{11}&\cdots&\tilde{B}^{1n_{m}}\\\
\vdots&\ddots&\vdots\\\
\tilde{B}^{n_{m}1}&\cdots&\tilde{B}^{n_{m}n_{m}}\end{smallmatrix}\right].$

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