# A fundamental Game Theoretic model and approximate global Nash Equilibria
computation for European Spot Power Markets

Ioan Alexandru Puiu  Raphael Andreas Hauser

###### Abstract

Spot electricity markets are considered under a Game-Theoretic framework,
where risk averse players submit orders to the market clearing mechanism to
maximise their own utility. Consistent with the current practice in Europe,
the market clearing mechanism is modelled as a Social Welfare Maximisation
problem, with zonal pricing, and we consider inflexible demand, physical
constraints of the electricity grid, and capacity-constrained producers. A
novel type of non-parametric risk aversion based on a defined worst case
scenario is introduced, and this reduces the dimensionality of the strategy
variables and ensures boundedness of prices. By leveraging these properties we
devise Jacobi and Gauss-Seidel iterative schemes for computation of
approximate global Nash Equilibria, which are in contrast to derivative based
local equilibria. Our methodology is applied to the real world data of Central
Western European (CWE) Spot Market during the 2019-2020 period, and offers a
good representation of the historical time series of prices. By also solving
for the assumption of truthful bidding, we devise a simple method based on
hypothesis testing to infer if and when producers are bidding strategically
(instead of truthfully), and we find evidence suggesting that strategic
bidding may be fairly pronounced in the CWE region.

###### keywords:

Spot Electricity Market , Game-theoretic Model , Numerical Algorithms , Risk
Aversion , Network Constraints , Real-world Case Study

###### MSC:

74S99 , 90C20 , 91A10 , 91B26

[inst1]organization=Mathematical Institute, University of
Oxford,addressline=Andrew Wiles Building, Woodstock Road, city=Oxford,
postcode=OX26GG, state=Oxfordshire, country=United Kingdom

##  1 Introduction

Electricity spot markets throughout the world have undertaken strong
deregulation in the last two decades, with the end goal of encouraging
investments and increasing economic efficiency [1]. Due to the complexity of
electricity distribution and difficulty of storage, spot electricity markets
are generally structured as Day Ahead Auctions, which are usually defined by a
market clearing mechanism that allocates production and consumption
quantities, and determines prices. However, even today, many challenges are
yet to be overcome, and extensive research was performed to attempt ensuring
these Market Clearing Mechanisms (MCMs) satisfy desirable properties.
Nevertheless, the negative result of Myerson and Satterwaite [2] shows, albeit
for a greatly simplified case, that the four desirable properties they define
cannot be satisfied simultaneously by any mechanism. This reveals the
complexity of these markets and indicates that desirable economic properties
cannot simply be assumed to hold, but have to be guaranteed through the MCM.
In the game theoretic study of [3], the authors recognise that transmission
constraints and market concentration may prevent power markets from being
fully competitive. In other words, truthful bidding (that fully reflects cost
or benefit structure) is not guaranteed, and participants may place strategic
orders. Other properties such as economic efficiency, budget balance, or
incentive for producers to participate may also not be guaranteed. As a result
of these difficulties, a large number of Market Clearing Mechanisms were
devised, according to different policy requirements of different countries.
These mechanisms can have widely different rules and properties, and thus
theoretical or simulation results from one market cannot be simply generalised
to another. Nevertheless, a thorough understanding of each particular market
is of great interest for all entities involved in this market: policy makers,
market participants grid operators, and researchers.

To the best of our knowledge, not much attention was given to fundamental
game-theoretic studies of market clearing mechanisms that are directly
applicable to Day Ahead Auctions for European Power Markets. We here propose a
game-theoretic model based on a market clearing mechanism that is consistent
with the methodology of European spot power markets, and with European Power
Exchange clearing mechanism in particular. By applying our model to the real
world case of what was the Central Western European spot electricity market
over the period of 2019-2020 (now extended to a 12 country market), we show
good representation of the true price series. We further show that based on a
simple hypothesis testing approach, the assumption of truthful bidding, which
is usually employed by practitioners and researchers for the CWE market, may
be greatly overused. This is because for the period considered, we can assert
with high confidence that players were bidding strategically for a much larger
fraction of times, than we can assert that they were bidding truthfully [4].

###  1.1 Brief literature survey and motivation

Game theoretic models for electricity markets have been extensively studied.
These are often more realistic than purely competitive economic equilibrium
models, due to market complexity and the fairly small number of producers in a
market. Early competition models applied to spot electricity markets are based
on what is known as Cournot [5] or Bertrand competition. In Cournot
competition, it is conjectured that each player assumes that opponents do no
change their offered if the player changes the strategy, while Betrand
competition conjectures that offered price of opponents is unaffected.
However, these models are not realistic as in practice market participants
choose neither the offered quantity nor the offered price. Instead, (proposed)
price-quantity pairs, referred to as orders, are submitted to the market
clearing mechanism, and these are accepted or rejected based on their
feasibility and economic merit [6]. The strategy of each player is then not
the quantity or price offered but rather orders defined by price-quantity
pairs. Game theoretic models of this type are often referred to as supply
function equilibrium models.

We concern ourselves with this type of more realistic models. These are also
well investigated in the literature, but are not directly applicable to the
European Market Clearing Mechanism, such as EPEX spot. This is because most
literature studies use a market clearing mechanism based on Optimal Power Flow
(OPF) grid modelling, and Locational Marginal Prices (LMP). By contrast,
European markets model the grid via the Flow Based Market Coupling (FBMC)
approach, which are explained in detail in [7], [8], and [9]. Further, Zonal
Marginal Pricing (ZMP) is used, in which a uniform price for each zone
(country) is set according to the highest marginal price of accepted asks.
This yields a very different mechanism, and as stated in [3], many studies
show that market peculiarities can have a great impact on market power, and
therefore on market outcome.

Some of the most related research includes the work of [3], where a game-
theoretic oligopolistic model based on a Market Clearing Mechanism with OPF
grid model with LMP pricing is considered. Bids and offers are parametrised
via linear marginal prices with respect to cleared quantity, and the demand is
considered as flexible. A leader-follower (Stackelberg) model is considered,
where only one or a few participants manipulate prices, while the rest naively
assume that they cannot impact prices. As generally the case with these
models, players solve a profit maximisation problem that depends on the MCM
outcome, which is itself given by an optimisation problem. This results in bi-
level optimisation for each player, that is often reformulated as a
Mathematical Program with Equilibrium Constraints (MPEC), and the collection
of these for all players results in an Equilibrium Program with Equilibrium
Constraints (EPEC) [3], [10]. The increased realism obtained by modelling the
MCM and players’ strategy space as price-quanitity pairs comes at the expense
of increased mathematical complexity and computational cost. A very similar
game-theoretic model, based on OPF-LMP market clearing is analysed in [10],
where conditions for existence of pure Nash Equilibria are derived, under the
same linear parametrisation of orders. Further, diagonalisation algorithms
based on Gauss-Seidel and Jacobi approaches are given for numerical
computation of equilibria. However, the existence conditions are only proved
for the OPF-LMP case.

It is generally recognized by researchers that solving EPECs is very
challenging and computationally intense [11]. This can be observed by the fact
that real world case studies based on these models are rather scarce or over-
simplified. A large part of the computational complexity can be attributed to
the Market Clearing Mechanism, which is generally a difficult and large scale
optimisation problem, with many and difficult constraints. Certainly, if non-
convexities of the market such as start-up costs, ramping times, block orders
and others are considered, simply obtaining the global optimum of the MCM
problem is extremely challenging as this is a Mixed Integer Programming [4].
The extra layer of complexity added by considering strategy adjustment via a
game-theoretic framework drastically further increases the computational
complexity. Alternative approaches are proposed in [12], where a Neural
Network is used to approximate the MCM, and in [13], where a no-regret
algorithm is used to obtain coarse-correlated equilibria in a stochastic
environment with partial information. However, these alternative approaches
are subject to typical pitfalls of data-driven methods.

The aforementioned research on game-theoretic models is not directly
applicable to European spot electricity markets, and to the best of our
knowledge, the literature of such game-theoretic models is very scarce.
Further, obtaining numerical solutions to such models is generally very
computationally intense, and detailed numerical results on real world case
studies are generally not available. The work in this paper aims to address
these problems.

###  1.2 Main contributions

By contrast with the literature, we therefore consider a Market Clearing
Mechanism based on FBMC with Zonal Marginal Pricing, but similar to [10] and
[3], we use a linear parametrisation of marginal prices as a function of
quantity. However, instead of forming an EPEC, which is difficult to solve, we
perform (diagonalised) grid search in the strategy space for each player, and
solve the MCM exactly via a Convex Quadratic Program formulation. We consider
inflexible demand and introduce a novel type of non-parametric risk aversion
that bounds and reduces the dimensionality of the strategy space, and thus
facilitates efficient grid search. By contrast to the typical EPEC solutions,
our approach leverages problem-specific structure to reduce computational
effort. Further, solutions to EPECs generally only guarantee weak notions of
equilibrium such as local Nash Equilibria, or stationary Nash Points [10].
Such concepts are only useful if players are not prepared to drastically
change their strategies over limited time periods. However, by contrast, our
approach offers a global pure Nash Equilibrium in a discretised strategy space
(if such a point exists), and we refer to this as approximate global Nash
Equilibrium, since the grid can in principle be refined arbitrarily well. The
main contributions of our work are then as following:

  1. 1.

We introduce a MCM consistent with European spot electricity markets, and
model strategic player behaviour under a game-theoretic framework. We devise a
computationally efficient approach to obtain approximate global Nash
Equilibrium points, if such points exist.

  2. 2.

We introduce a novel type of non-parametric risk aversion which we call Robust
Strategy Selection (RSS). By contrast to mean-variance risk aversion
approaches [14], RSS bounds the strategy space, ensuring boundedness of prices
even under inflexible demand. Further, by contrast to other works such as [15]
and [3], where dimensionality of strategy space is forcefully reduced by
holding one parameter constant, our dimensionality reduction occurs naturally
due to the risk aversion. A further advantage of this approach is the non-
parametric form that requires no assumption over the distribution of possible
outcomes. The properties of RSS facilitate efficient (diagonalised) grid
search for equilibrium strategies.

  3. 3.

We apply our model to the real world case of Central Western European Day
Ahead Auction over the period of 2019-2020 and show that the model offers a
good representation of the historical time series of prices.

  4. 4.

Finally, we challenge the un-tested assumption that producers are bidding
truthfully in the CWE market. In particular, based on the MCM used in the CWE,
there is no theoretical evidence that participants are incentivised to bid
truthfully. To this end, we devise a simple method based on hypothesis testing
to infer if and when players are bidding strategically instead of truthfully.
This approach requires the outputs of our Game Theoretic model and a truthful
bidding model which simply solves the MCM based on the true cost structure
instead of strategic orders. We show evidence that strategic bidding may be
fairly pronounced, in contrast to what researchers, practitioners and policy
makers usually assume for the CWE market.

###  1.3 Discussion of Model features

The complexity of our approach is a result of the attempt to realistically
model strategic bidding in European Day Ahead Auctions. Admittedly, ignoring
non-convexities of power plants and assuming linear marginal prices for orders
is a simplification, but these are very often used [10], [3]. However, we do
not make other unrealistic assumptions that are commonly employed in the
literature such as: (i) price taking, (ii) the ability of players to directly
control their accepted quantities, (iii) ignoring network constraints, (iv)
ignoring capacity constraints, (v) purely flexible demand, or (vi) using over-
simplified pricing schemes. In an oligopoly environment such as spot
electricity markets, market players are aware that they influence prices, and
the assumption of controlling the traded quantity is simply not true. However,
the removal of these two commonly used assumptions greatly increases the
difficulty of the problem, as the KKT conditions of both the MCM and producer
optimisation problems become Nonlinear Complementarity Problems (NCP). These
cannot be easily solved or converted to a Quadratic Program, as is the case
for the Linear Complementarity Problems (LCP) resulting when using assumptions
(i), (ii), such as in [14], [5], where the combined KKT conditions can be
transformed to a Quadratic Program. Assumptions (i) and (ii) thus facilitate
theoretical analysis and faster computations at the expense of reduced
realism. By contrast, our approach focuses on retaining realistic features and
remaining computationally tractable, albeit theoretical analysis of the
resulting model is much more challenging, and is only briefly addressed here
for a further simplified MCM. This brief theoretical analysis reveals some
issues concerning the CWE market clearing mechanism, and motivates the
introduction of our novel non-parametric risk aversion.

###  1.4 Background information

We briefly review the concept of auction mechanism, the model of market
clearing and network constraints used in the CWE market, the idea of strategic
bidding, role of Game Theory in our analysis and the concept of Nash
Equilibrium.

In contrast to typical spot financial markets, spot electricity markets are
organised mostly in Day-Ahead Auctions due to the complexity of transmission
coupled with difficulty of storage. As a result, Day-Ahead electricity markets
only allow for placing limit orders once per day, before the market gets
cleared for the next day, once and for all. The market is usually cleared
separately, for each 15 minutes or 1 hour interval in the next day, although
inter-temporal constraints between these slots are sometimes considered. This
type of market clearing is required to ensure balance of supply and demand
simultaneously with other market specific feasibility constraints.

A market clearing mechanism then usually has an objective, which could be for
example Social Welfare Maximisation [7], Cost Minimisation, or any other
objective usually defined by the regulators with the goal of achieving a
desirable economic outcome. Feasibility constraints are also enforced by the
MCM, ensuring balance of supply and demand, feasibility of electricity
transmission, feasibility of production levels and their changes over time
[14], and others. As a result, any market clearing mechanism can be expressed
as a mathematical optimisation problem. This mechanism must include a quantity
allocation and a pricing mechanism, and to maintain mathematical simplicity,
price determination is usually performed in a second optimisation step, once
the first step of determining optimal quantity allocation is complete [16],
but this does not have to be the case.

As we shall see in the next section, the CWE market is split in multiple zones
(usually countries), each containing many production and consumption nodes,
and transmission lines. The corresponding MCM uses what are known as Flow
Based Market Coupling (FBMC) network constraints, that are essentially a set
of linear constraints applied on the net power exports on each zone
$\hat{y}_{z}$. The linear constraints then have the form
$M_{p}\hat{y}\leq\hat{b}_{p}$. The matrix $M_{p}$ is the collection of what
are known as Power Transmission Distribution Factors, while the $\hat{b}_{p}$
vector is obtained using what are known as the Remaining Available Margins
(RAMs) [7]. We discuss how the constraint set $(M_{p},\hat{b}_{p})$ is
obtained for the real world case in more detail in Section 5.2, where we apply
our model to the real world case of CWE. However, since available historical
data of these constraints is very sparse [17], a complete explanation of how
the full set of constraints is obtained for each time index from publicly
available data is beyond the scope of this work, but this is the topic of our
paper in [17]. Nevertheless, for the purpose of this work we only require
using that $M_{p}\hat{y}\leq\hat{b}_{p}$ is the set of linear constraints
ensuring feasibility of transmission, often referred to as network
constraints.

In simple terms, the CWE DAA Market Clearing Mechanism determines the market
outcome by first solving a (primal) optimisation problem corresponding to
maximising social welfare subject to supply and demand balance, network
constrained and other constraints. A pricing problem is then solved. This uses
the the dual variables of the primal problem to account for the economic merit
of transmission, and a rather arbitrary objective that ensures the price is
”intuitive” [7].

It is not obvious based on the definition of this mechanism, what is the
resulting player behaviour and market outcome. A simple assumption allowing
for outcome analysis is that participants bid truthfully, according to their
true cost or benefit structure. However, especially in an oligopolistic market
such as electricity markets, there may be strong incentives to bid
strategically. Game Theory is the standard mathematical tool for fundamental
analysis of (fully rational) strategic behaviour. In this framework, each
player $i$ has an utility $u_{i}:\mathcal{S}_{i}\to\mathbb{R}$ to maximise,
over a strategy space $\mathcal{S}_{i}$, given other players’ strategies
denoted as $\sigma_{-i}\in\mathcal{S}_{-i}$. Perhaps the single most important
concept is that of a Nash Equilibrium, which is a strategy set
$\sigma^{*}=\\{\sigma_{1}^{*},...,\sigma_{i}^{*},...\sigma_{n}^{*}\\}$ such
that no player can unilaterally deviate and increase their utility, that is,
for every player $i$ we have that

$u_{i}(\sigma_{i}^{*};\sigma_{-i}^{*})\geq
u_{i}(\sigma_{i};\sigma_{-i}^{*}),\hskip
2.84526pt\forall\sigma_{i}\in\mathcal{S}_{i}.$ (1)

In auction design, market clearing mechanisms are then analysed under a game-
theoretic framework to verify if the MCM attains desirable economic
properties. By contrast, while we provide some simple analysis of interest to
auction design, the focus of our paper is on fundamental estimation of market
outcomes when considering strategic behaviour, and the application of the
resulting game-theoretic model to the real world case of CWE market. To our
knowledge, this was not done before to the same level of detail on real world
data.

The rest of the paper is structured as follows. Section 2 introduces our
proposed model representing the CWE market clearing while Section 3 is
concerned with the strategic behaviour of market participants under a game-
theoretic framework. In Section 3 we further reveal some (undesirable)
theoretical properties of the CWE clearing mechanism, and propose a novel type
of player risk aversion. In Section 4 we present numerical approaches for
obtaining approximate global Nash Equilibria, while in Section 5 we are
concerned with the application of our game-theoretic model to the real world
case of CWE spot power market.

##  2 Social Welfare Maximisation for Market Clearing

We now present a market clearing model initially introduced by us in [18],
that models the practice of market clearing procedure in the Central Western
European (CWE) power market. To our knowledge the analysis and application of
optimisation models directly reflecting the CWE market is scarce in the
literature.

We consider multiple zones $z\in\mathcal{Z}$, with $\mathcal{P}_{z}$ producers
per zone. We take the demand in each zone denoted as $d_{z}$ as inflexible,
which facilitates efficient computation of the zonal prices and avoids the
issue of paradoxically accepted orders [18]. To maintain computational
tractability we neglect block order bids, and similarly to [10], [3] and
others, we assume that producers bid linear marginal ask prices $\lambda_{i}$,
at sold quantity $x_{i}$, of the form

$\lambda_{i}(x_{i})=m_{i}x_{i}+a_{i},$ (2)

where $m_{i}>0$ and $a_{i}>0$ are order constants. This takes the same form as
their cost structure, assumed to be $C_{i}(x_{i})$ given by

$\frac{\partial}{\partial x_{i}}C_{i}(x_{i})=c_{i}x_{i}+b_{i}.$ (3)

Considering inflexible demand and the form of the bids we assumed, the social
welfare objective to be maximised becomes

$\displaystyle F_{SWM}$
$\displaystyle=-\sum_{i}\int_{0}^{x_{i}}(m_{i}\chi_{i}+a_{i})d\chi_{i}$ (4)
$\displaystyle=-\sum_{i}\left[\frac{1}{2}m_{i}x_{i}^{2}+a_{i}x_{i}\right]$
$\displaystyle=-\frac{1}{2}x^{T}D_{m}x-a^{T}x,$

where $D_{m}$ is a diagonal matrix with $(D_{m})_{i,i}=m_{i}$. We consider
supply and demand balance constraints, capacity constraints and network
constraints. Let $x$ be the vector of allocated productions s.t. $x_{i}$ is
the production allocated to player $i$. The supply and demand balance then
becomes $1^{T}x=d$ where $d=\sum_{z\in\mathcal{Z}}d_{z}$ is the total demand.
We do not allow any producer $i$ to be net short or exceed its production
capacity $Q_{i}$, and therefore we have that each $x_{i}$ is bounded by $0\leq
x_{i}\leq Q_{i}$.

Finally, we consider network constraints. Let
$E\in\mathbb{R}^{|\mathcal{Z}|\times|\mathcal{P}|}$ be the matrix that maps
production quantities $x$ to zonal production quantities vector $y$, where
$y_{z}=\sum_{i\in\mathcal{P}_{z}}x_{i}$ is the total production of zone $z$.
Thus $y=Ex$ and further, let $d^{Z}$ be the zonal production vector, such that
$d^{Z}_{z}=d_{z}$. Since the network constraints are imposed for the net
production at zonal level, we have that the network constraints are given by

$M_{p}(y-d^{z})\leq\hat{b}_{p},$ (5)

which can be re-written as

$M_{p}Ex\leq\hat{b}_{p}+M_{p}d^{z}=:b_{p}.$ (6)

By collecting all the constraints and using that $\min_{w}f(w)=\max_{w}-f(w)$,
the SWM optimisation problem can then be written as

$\displaystyle\min_{x}$
$\displaystyle\sum_{i}\left(\frac{1}{2}m_{i}x_{i}^{2}+a_{i}x_{i}\right)=\frac{1}{2}x^{T}D_{m}x+a^{T}x$
(SWM) $\displaystyle s.t.$ $\displaystyle\hskip 5.69054ptM_{p}Ex\leq b_{p}$
$\displaystyle\hskip 5.69054ptx_{i}\leq Q_{i},\hskip 2.84526pt\forall i$
$\displaystyle\hskip 5.69054ptx_{i}\geq 0,\hskip 8.53581pt\forall i$
$\displaystyle\hskip 5.69054pt1^{T}x=d.$

Note that there are no price variables in this problem, and for inflexible
demand, these can be computed once the optimal solution $x^{*}$ is obtained,
as [18]:

$v_{z}=\max_{k\in\mathcal{P}_{z}}\\{m_{k}x_{k}^{*}+a_{k}:x_{k}^{*}>0\\}$ (7)

where $v_{z}^{*}$ is the price in zone $z$ at optimality of (SWM).
Fortunately, since $m_{i}>0$ $\forall i$ and the constraints are linear, our
SWM model is strictly convex quadratic program (CQP), and this can be solved
very effectively numerically. We solve the (SWM) optimisation problem via the
quadprog package in Python 3.

Under the particular case of one zone, and by ignoring the network constraints
the market clearing given by the Social Welfare Maximisaiton problem becomes

$\displaystyle\min_{x}$
$\displaystyle\sum_{i\in\mathcal{P}}\frac{1}{2}m_{i}x_{i}^{2}+a_{i}x_{i},$
(SWM-s) $\displaystyle s.t.$ $\displaystyle\hskip
2.84526pt\sum_{i\in\mathcal{P}}x_{i}=d;\hskip 8.53581ptx_{i}\geq 0\hskip
2.84526pt\forall i.$

In the next section we consider strategic bidding and we analyse some of the
theoretical properties of the simplified (SWM-s).

##  3 Strategic Bidding and the Game-Theoretic Model

In the CWE market, it is often assumed that market participants place orders
purely based on their true costs. However, market orders are each
participant’s choice and need not be given exclusively by the cost structure.
Since each player has its own interest, they may strategize accordingly, and
thus the market mechanism needs to incentivise players to bid truthfully, but
this is not a guaranteed property [2]. In fact, achieving this and other
desirable properties is a challenging auction design task [2], [19], [16]. For
this reason, considering strategic bidding offers a more descriptive model of
the electricity spot market. We consider the case when players bid
strategically to maximise their own profit knowing that the market is cleared
by (SWM). We assume complete information, that is, each player knows the true
values of $c$, $b$, $M_{p}$, $b_{p}$ and $d$, and focus on one time period
only.

We next introduce producer’s optimisation problem given by strategic bidding
in subsection 3.1, obtain some theoretical results on a simplified clearing
model in subsection 3.2, and introduce a novel non-parametric risk aversion in
subsection 3.3.

###  3.1 Producer’s optimisation problem

Each producer submits a supply curve given by $\sigma_{i}=(m_{i},a_{i})$, but
we assume that the true capacity $Q_{i}$ of each producer is common knowledge.
Each producer aims to maximise its own profit, which can be expressed as the
total revenue minus the total cost:

$\pi_{i}(\sigma_{i};\sigma_{-i})=v_{z_{i}}x_{i}-\frac{1}{2}c_{i}x_{i}^{2}-b_{i}x_{i},\hskip
2.84526pt\forall i\in\mathcal{P},$ (8)

where $z_{i}:=\\{z:i\in\mathcal{P}_{z}\\}$ is the zone of player $i$. This is
subject to production constraints $0\leq x_{i}\leq Q_{i}$, but can be ignored
here, since this is enforced at auction level in (SWM), and the players do not
directly choose $x_{i}$. The optimisation problem then becomes

$\begin{aligned}
\max_{\sigma_{i}=(m_{i},a_{i})}&\pi_{i}(\sigma_{i};\sigma_{-i})=v_{z_{i}}x_{i}-\frac{1}{2}c_{i}x_{i}^{2}-b_{i}x_{i},\hskip
2.84526pt\forall i\in\mathcal{P}\end{aligned}.$ (9)

Note that the strategy variables $\sigma_{i}$ do not explicity appear in the
$\pi_{i}(\sigma_{i};\sigma_{-i})$ expression (8), but they change $\pi_{i}$
since both $v_{z_{i}}$ and $x_{i}$ are functions of $\sigma_{i}$ (and
$\sigma_{-i}$), as defined by (SWM). To estimate their profit for a chosen
strategy $\sigma_{i}$, each player then has to estimate the strategies of all
other players, $\sigma_{-i}$, their capacity constraints $Q_{-i}$, the network
constraints $(M_{p},b_{p})$ and demand level $d$, and solve the market
clearing problem (SWM). However, estimating other players’ strategies is
particularly challenging especially since historical bids data is not
available. Game theory then provides a way for such an estimation, by assuming
that each player is perfectly rational and maximises their own profit
functions, and knowing that all other players behave identically.

We can now see that solving the game theoretic model is a bilevel optimisation
problem in which each player solves (SWM) once for each new strategy set
considered, to obtain $v_{z}$ and $x_{i}$ and thus a profit value, and the
outer optimisation problem (9) requires the maximum value of this profit. A
common way to solve this model is to write the KKT conditions of (SWM) and
combine them with the KKT conditions of (9) for each player $i$. This results
in what is known as an Equilibrium Problem with Equilibrium Constraints
(EPEC), but is generally hard to solve [3], [10].

However, by leveraging the structure of our problem and the risk aversion
introduced in subsection 3.3, we propose a more computationally efficient
method in Section 5.2, that also gives approximate global NE instead of local
stationary Nash points obtained by solving EPECs [10].

###  3.2 Characterisation of Equilibria for simplified MCM

We next show that for a simplified case of a one zone market with no capacity
constraints, if producers are perfectly rational and aim to maximise their own
profit, there exist infinitely many local Nash equilibria, if demand is
inflexible and full information is available. Further, we argue that
equilibrium prices could be arbitrarily high.

Intuitively, this may occur because the mapping from the strategy set to the
market outcome given by (SWM) is not injective, and in particular for a player
$i$, multiple strategies may yield the same profit level meaning that the
player is indifferent to the choice between these strategies. The next results
reveal some characteristics of the equilibria.

######  Result 1.

Consider a one zone market with inflexible demand $d>0$ and producer set
$\mathcal{P}$. Assume producers have no capacity constraints and that there
are no network constraints. Assume that the submitted orders have $m_{i}>0$,
$\forall i$, and that the market clearing takes the form (SWM-s). Then the
optimal quantities allocated to active players in set $\mathcal{P}^{A}$ are

$\begin{aligned}
x_{i}^{*}&=\frac{d+\sum_{j\in\mathcal{P}^{A}\setminus\\{i\\}}m_{j}^{-1}(a_{j}-a_{i})}{m_{i}\sum_{j\in\mathcal{P}^{A}}\frac{1}{m_{j}}},\hskip
5.69054pti\in\mathcal{P}^{A}\\\ \end{aligned},$ (10)

and $x_{i}=0$ if $i\notin\mathcal{P}^{A}$. Further, the marginal prices for
any player in the active set $\mathcal{P}^{A}$ gives the zonal price, that is
$v^{*}=\lambda_{i}=m_{i}x_{i}^{*}+a_{i}$.

###### Proof.

See A for the full proof. ∎

This result gives us a way to track the impact of each player’s strategy on
their (and others’) quantity allocation. We next show that if $m$ is fixed and
known by all players, strategising over $a$ only is well behaved, yielding a
unique equilibrium.

######  Theorem 1.

Consider a one zone market with inflexible demand $d>0$ and active producer
set $\mathcal{P}$. Assume producers have no capacity constraints and that
there are no network constraints. Further assume that demand $d$ and cost
structure $\\{c_{i},b_{i}\\}_{i\in\mathcal{P}}$ is common knowledge, and that
each producer is aiming to maximise its profit,
$\pi_{i}(m_{i},a_{i},m_{-i},a_{-i})=vx_{i}-\frac{1}{2}c_{i}x_{i}^{2}-b_{i}x_{i}$,
with $v$ the zonal clearing price. Let the market clearing be given by
(SWM-s). Finally, assume that for every $i$ we have $c_{i}>0$,
$m_{i}^{+}\geq\frac{1}{2}c_{i}$ is held fixed and that $a_{i}$ is the only
strategy variable, and that this is common knowledge. Then for any vector
$m^{+}\in\mathcal{M}$ there exists a unique Nash equilibrium
$a^{+}(m^{+})\in\mathcal{A}$, which is defined by the (full rank) linear
system

$\theta_{i}\sum_{j\in\mathcal{P}\setminus\\{i\\}}\frac{a_{i}^{+}}{m_{j}}-\sum_{j\in\mathcal{P}\setminus\\{i\\}}\frac{a_{j}^{+}}{m_{j}}=d+(\theta_{i}-1)\sum_{j\in\mathcal{P}\setminus\\{i\\}}\frac{b_{j}}{m_{j}},$
(11)

where $\theta_{i}=\frac{2-k_{i}K_{-i}^{2}}{1-k_{i}K_{-i}^{2}}$,
$k_{i}=2m_{i}-c_{i}$,
$K_{-i}^{2}=\frac{\sum_{j\in\mathcal{P}\setminus\\{i\\}}m_{j}^{-1}}{m_{i}\sum_{j\in\mathcal{P}}m_{j}^{-1}}$.

###### Proof.

See A for the full proof. ∎

######  Remark 1.

The existence of a unique equilibria is beneficial for computation of the
market outcome, but does not imply that the market outcome is purely
competitive. In other words, it may be the case that even when $m^{+}=c$,
$a_{i}^{+}>b_{i}$ for some or all $i$, and that $v^{*}>v_{0}$, where $v_{0}$
is the price when players bid $(c,b)$. In fact, numerical simulations suggest
that this is generally the case.

Although the result of Theorem 1 may appear to be a positive result in
isolation, as we show next, in combination with allowing strategising over
$m$, an unsatisfactory result characterises (SWM-s): infinitely many
equilibria can be found, and potentially yielding arbitrarily high prices. Our
results are consistent with [3] that observes that ”multiple equilibria are
likely” under $(m,a)$ parametrisation, and with [20] that states for the same
case that ”multiple Nash Equilibria are almost ineviatble”. We quantify and
prove this in Theorem 2.

######  Theorem 2.

Consider identical conditions to Theorem 1, with the exception that $m_{i}$
are not held fixed, but rather part of the strategy choice for each player
$i$. Then for each $m^{+}\in\mathcal{M}$, there exists an $a^{+}(m^{+})$,
computed according to Theorem 1 by taking $m^{+}$ as fixed, such that
$(m^{+},a^{+})$ is a local Nash equilibrium in the
$(m,a)\in\mathcal{M}\times\mathcal{A}$ space. In other words, since $m^{+}$ is
chosen from an infinite set, there are infinitely many local equilibria.
Further, the local equilibria form a continuous subspace $\mathcal{MA}^{+}$.

###### Proof.

See A for the full proof. ∎

######  Remark 2.

Theorem 2 suggests that by considering $m$ values as part of the strategy,
competition between players does not increase. However, the extra parameter of
choice may improve the ability to influence the market, potentially resulting
in less competitive outcomes.

######  Remark 3.

While these results are proved for a very simplistic case of one zone, the
same result could be expected for multiple zones, since when zones become
decoupled, they can be treated individually.

######  Remark 4.

By continuity of the local Nash Equilibria, players can gradually change their
strategies along these equilibria by small increments, and eventually reach an
equilibrium that is most desirable for everyone, if such a point exists.

The infinitely many equilibria are clearly a problem for the purpose of
outcome estimation and for ensuring high social welfare and reasonable prices.
We further speculate that in this equilibria subspace ($\mathcal{MA}^{+}$),
there may exist points $(m^{+},a^{+})$ that give arbitrarily high zonal
clearing price $v$. However, this is hard to prove as values in $a^{+}$ become
negative for increasingly large $m^{+}$, making it difficult to place bounds
on $\lambda_{i}=m_{i}^{+}x_{i}^{*}+a_{i}^{+}$. Nevertheless, via numerical
experimentation we observed cases where $v$ at equilibrium point
$\left(m(k),a^{+}(m(k))\right)$ increases linearly with increasing $k$
defining $m(k)=kc$. This problem is expected, since Theorem 2 essentially
tells us that there is too much freedom in the strategy choice, and the price
is not invariant to these choices. This makes the market un-competitive, and
players can use this freedom to push the clearing price towards higher levels.
This problem is generally ignored in the literature, and often $m$ is taken as
fixed, for example in the works of [10], [3], [15] or other artificial
parametrisations are considered, as discussed in [20].

In [15], it is implied that the choice of parametrisation is not critical to
the outcome, because each player $i$ can perform trade-offs between slope and
intercept and obtain the same optimal outcome. However this is not sufficient
to ensure outcome invariance to parametrisation choice between $a$ and
$(m,a)$. We can see by comparing the results of Theorem 1 and Theorem 2, that
the choice of parametrisation is indeed critical. The reason for the
difference between $a$-parametrisation and $(m,a)$-parametrisation is subtle
but important: while it is true that given a fixed strategy
$\sigma_{-i}=(m_{-i}^{+},a_{-i}^{+})$ of opponents and an optimal response
$\sigma_{i}^{+}$, changing the strategy to any other optimal response
$\sigma_{i}^{\prime}$ will not change the profit, as shown in the proof of
Theorem 2, simultaneous change of $\sigma$ for all players, can result in an
equilibrium with different market outcome. The existence of an infinite subset
of optimal responses facilitates the existence of multiple equilibria and also
provides a way to alter these equilibria over time in practice.

However, all this analysis assumes availability of perfect information, and
risk neutral players. In reality, players are likely risk averse and only have
incomplete information. To maintain computational tractibility we do not
consider incomplete information here. Further, mixed (stochastic) strategies
[21] are also not considered for the same reason. Note that even under perfect
information, for the $(m,a)$-parametrisation, each player $i$ is still
presented with a dilemma: which equilibrium strategy will the other players
play. Numerical experimentation reveals that if different players
simultaneously play different equilibrium strategies, negative profits for one
or even all participants are possible. On the other hand, bidding
$m_{i}\geq\frac{1}{2}c_{i}$ and $a_{i}\geq b_{i}$ guarantees non-negative
profit. Thus, in the presence of risk aversion, even under perfect
information, acknowledging the possibility of multiple outcomes requires
discrimination between previously equivalent best responses. In the next
subsection we introduce a sensible approach that rational players could likely
employ to circumvent the issue of multiple equilibria.

###  3.3 Risk Aversion: Robust Strategy Selection

It is clear that the multiplicity of equilibria comes from the problem that
players have ”too much choice”. This is obvious by contrasting Theorem 1 with
Theorem 2. Strategising over $(m,a)$ instead of only $a$ increases the
strategy space but does not intensify the competition accordingly. A rational
player may recognize that due to multiplicity of equilibria, playing a Nash
Equilibria strategy does not guarantee that the outcome will correspond to
this particular equilibrium point. Since negative profits are possible when
players play different Nash Equilibria strategies, a risk averse player may
take extra precautions in selecting the strategy played.

First, he could observe that playing strategies only of the form
$\mathcal{S}_{R}(i)=\\{(m_{i},a_{i}):m_{i}\geq\frac{1}{2}c_{i},\hskip
2.84526pta_{i}\geq b_{i}\\}$, would guarantee that the profit is non-negative
as per Result 2

######  Result 2.

. Under the market clearing given by (SWM), player $i$ is guaranteed non-
negative profit by playing any
$\sigma_{i}\in\mathcal{S}_{R}^{\prime}(i):=\\{(m_{i},a_{i}):m_{i}\geq\frac{1}{2}c_{i},\hskip
2.84526pta_{i}\geq b_{i}\\}$, for any $(m_{-i},a_{-i})$.

###### Proof.

See A. ∎

Next, the player could assign probabilities to every possible combination of
equilibrium strategies that a player may play, and maximise the expected (risk
adjusted) profit. However, estimating such probabilities is a very difficult
task and wrong estimation could have a detrimental impact on the actual
outcome. Thus, instead of considering a mean-variance approach that adjusts
the utility function to account for risk, a conservative player may wish to
first ensure a minimum profit level in a worst case scenario. Note that
according to Result 2, such a guarantee with non-negative profit is always
possible. If multiple equivalent such strategies exist, then the player may
use this choice to maximise his actual profit (as opposed to the worst case).
Further, let us assume that all players behave in this way, and that this is
common knowledge. The worst case optimisation problem is then

$\begin{aligned}
\max_{\sigma_{i}\in\mathcal{S}_{R}(i)}\min_{\sigma_{-i}\in\mathcal{S}_{R}(-i)}\pi_{i}(\sigma_{i};\sigma_{-i})\end{aligned},$
(WCP(i))

and let us call the set of all solutions to (WCP(i)) $\mathcal{S}_{WCP}(i)$.
The advantage of this approach is the lack of parameters or required
estimates, at least when assuming complete information. Unfortunately (WCP(i))
is very difficult to solve since an analytical form for $\pi_{i}$ as a
function of strategy vector $\sigma$ cannot generally be obtained under the
(SWM) market clearing model. Due to (SWM) yielding the value of $\pi_{i}$,
(WCP(i)) is a tri-level optimisation problem. Solving this is clearly very
challenging. However, players may be willing to simplify (WCP(i)) to enable
for progress, to at least estimate strategies that perform well under the
worst case. One such simplification would be to consider (SWM-s) instead of
(SWM). While this ignores the capacity and network constraints, as we shall
see, it allows for an analytic closed form for the resulting optimal set of
strategies $\mathcal{S}_{WCP}(i)$.

Let us now look at how this simplifies the problem. To begin with, under
(SWM-s) we have a unique price that satisfies

$v=m_{i}x_{i}+a_{i},\hskip 2.84526pt\forall i:x_{i}>0$ (12)

By taking derivatives of $\pi_{i}$ with respect to $a_{j}$ and $m_{j}$ for any
$j\neq i$ we get that

$\displaystyle\frac{\partial\pi_{i}}{\partial a_{j}}=$
$\displaystyle\left[(2m_{i}-c_{i})x_{i}+(a_{i}-b_{i})\right]\frac{\partial
x_{i}}{\partial a_{j}}$ (13) $\displaystyle\frac{\partial\pi_{i}}{\partial
m_{j}}=$
$\displaystyle\left[(2m_{i}-c_{i})x_{i}+(a_{i}-b_{i})\right]\frac{\partial
x_{i}}{\partial m_{j}}$

where

$\displaystyle\frac{\partial x_{i}}{\partial a_{j}}$
$\displaystyle=\frac{1}{m_{j}}\geq 0$ (14) $\displaystyle\frac{\partial
x_{i}}{\partial a_{j}}$
$\displaystyle=\frac{m_{i}x_{i}+a_{i}-a_{j}}{m_{j}^{2}m_{i}\sum_{k}\frac{1}{m_{j}}}\geq
0$

where the inequality on the second line holds since
$m_{i}x_{i}+a_{i}=v=m_{j}x_{j}+a_{j}>a_{j}$ if player $j$ is active i.e.
$x_{j}>0$. If player $j$ is inactive, then $\frac{\partial x_{i}}{\partial
a_{j}}=0$. At the boundary between active and inactive, we have that
$a_{j}=v$. Either way, $\frac{\partial x_{i}}{\partial m_{j}}\geq 0$ and
$\frac{\partial x_{i}}{\partial a_{j}}\geq 0$, and strict inequalities hold if
player $j$ is active. This means that $\pi_{i}$ is increasing function of
$a_{j}$ and $m_{j}$, for any $j\neq i$. Thus, the worst case for player $i$,
regardless of its strategy is when all other players bid their minimum
acceptable values, i.e. $a_{j}=b_{j}$ and $m_{j}=\frac{1}{2}c_{j}$ for
$\forall j\neq i$. This essentially removes the $\min$ optimisation problem
from (WCP(i)). Further, the (SWM-s) optimisation can be solved analytically,
and thus we are left with the following maximisation problem for player $i$:

$\max_{(m_{i},a_{i})\in\mathcal{S}_{R}(i)}\pi_{i}(m_{i},a_{i};\frac{1}{2}c_{-i},b_{-i})$
(15)

Using Result 1 we obtain that

$\frac{\partial\pi_{i}}{\partial
a_{i}}=\left[(2m_{i}-c_{i})x_{i}+(a_{i}-b_{i})\right]\frac{-\sum_{k\neq
i}m_{k}^{-1}}{m_{i}\sum_{k}m_{k}^{-1}}+x_{i},$ (16)

and in the proof of Theorem 2, we have shown that
$\frac{\partial\pi_{i}}{\partial m_{i}}=x_{i}\frac{\partial\pi_{i}}{\partial
a_{i}}$, and this can be verified by direct calculation. Thus, the optimal
strategies form a set, as when setting $\frac{\partial\pi_{i}}{\partial
a_{i}}=0$ we obtain one equation with two variables, which can be written as

$\left[(2m_{i}-c_{i})x_{i}+a_{i}-b_{i}\right]\frac{-\sum_{j\neq
i}m_{j}^{-1}}{m_{i}\sum_{j}m_{j}^{-1}}+x_{i}=0,$ (17)

where $x_{i}$ can be written exclusively in terms of $a$, $m$ and $d$ as per
Result 1, and $m_{j}=c_{j}$ for $j\neq i$. Equation (17) gives a non-linear
relationship between $a_{i}$ and $m_{i}$ that ensures best response under the
worst case for the simplified (SWM-s) market clearing. However, this can be
simplified further by noting that the number of players in the real market is
generally fairly large, and therefore

$\frac{\partial x_{i}}{\partial a_{i}}=-\frac{\sum_{j\neq
i}m_{j}^{-1}}{\sum_{j}m_{j}^{-1}}\frac{1}{m_{i}}\approx-1\cdot\frac{1}{m_{i}},$
(18)

and this transforms (17) to

$a_{i}=v^{WCP(i)}-\frac{v^{WCP(i)}-b_{i}}{c_{i}}m_{i},$ (19)

where $v^{WCP(i)}=m_{i}x_{i}(m_{i},a_{i};c_{-i},b_{-i})+a_{i}$. By using that
$x_{i}=\frac{v-a_{i}}{m_{i}}$ to express $x_{i}$ in
$\pi_{i}=vx_{i}-\frac{1}{2}c_{i}x_{i}^{2}-b_{i}x_{i}$ and assuming price-
taking for player $i$, i.e. $\frac{\partial v}{\partial a_{i}}=0$ gives the
same result as (19). In fact, approximation (18) is perfectly equivalent to
price taking. This means that $v^{WCP(i)}$ does not change along the (19)
curve, and therefore there is a linear relationship between $a_{i}$ and
$m_{i}$. The value of $v^{WCP(i)}$ can be computed by fixing any feasible
value $m_{i}^{W}\geq\frac{1}{2}c_{i}$ and first computing the resulting
$a_{i}^{W}$ by equation (17) with $x_{i}$ given as per Result 1, with the
approximation in (18). The price $v^{WCP(i)}$ can then be computed by solving
(SWM-s) or using Result 1, and since $v^{WCP(i)}$ is constant along (19), the
optimal response strategy set is now fully defined. However, we observed in
practice that obtaining $v^{WCP(i)}$ by solving (SWM) gives better results.
Since each player requires solving (SWM) once, computational cost can be
reduced by solving (SWM) only once for $m=\frac{1}{2}c$, $a=b$, yielding
$v^{WCP(i)}=v^{WCP}_{z}$ to be the same for each player in a zone $z$. We use
this approach in practice and therefore in this case the optimal strategy set
is given by

$S_{WCP}(i):=\left\\{\begin{aligned}
(m_{i},a_{i}):&a_{i}=v^{WCP}_{z_{i}}-\frac{v^{WCP}_{z_{i}}-b_{i}}{c_{i}}m_{i},\\\
&m_{i}\geq\frac{1}{2}c_{i},\hskip 2.84526pta_{i}\geq
b_{i}\end{aligned}\right\\}.$ (20)

We can clearly see that so long as $v^{WCP}_{z_{i}}>b_{i}$, $S_{WCP}(i)$ is a
one dimensional set with bounds $m_{i}\in[\frac{1}{2}c_{i};c_{i}]$, and
$a_{i}\in[b_{i};\frac{1}{2}v_{z_{i}}^{WCP}+\frac{1}{2}b_{i}]$ and the choice
over set $S_{WCP}(i)$ can be used to maximise actual profit. Note that
$a_{i}=b_{i}$ and $m_{i}=c_{i}$ is a point in $S_{WCP}(i)$, thus this approach
is guaranteed to be superior for player’s $i$ profit when compared to simply
taking $a_{i}=b_{i}$, $m_{i}=c_{i}$ because the extra choice can be used to
play a game of profit maximisation.

If $v^{WCP}_{z_{i}}=b_{i}$, the choice is unique, while if
$v^{WCP}_{z_{i}}<b_{i}$ the set $S_{WCP}(i)$ is empty and player $i$ is
inactive, but player $i$ can play $(c_{i},b_{i})$ which ensures that in the
case of unexpected outcome the profit is non-negative. Generally, $S_{WCP}(i)$
is a one dimensional set, and since this set is always bounded, we devise a
method to compute efficient (approximate) global Nash Equilibria in Section 4.
Before presenting the computational approach, we comment on our proposed RSS
idea.

###  3.4 A comment on Robust Strategy Selection

The core idea of RSS is to use the extra dimensionality in the strategy space
to essentially make player $i$ play two simultaneous games: (i) one of profit
maximisation in worst case scenario, and (ii) another of profit maximisation
given that all players play game (i). There are clearly many possible choices
of such secondary games (or optimisation problems). However, not all these are
compatible with the profit maximisation problem (since the resulting system
may have no feasible solution). Further, compatible optimisation problems may
yield very difficult problems to solve.

Although we make fairly strong assumptions to proceed with RSS, the merit of
our approach is finding a compatible objective, representing possible risk
aversion behaviour, that can be solved for analytically, and thus reducing the
dimensionality of the strategy space for the main game to one (bounded)
dimension for each player. This facilitates obtaining approximate global Nash
Equilibria instead of purely local ones (or simply stationary points) obtained
using EPECs [3], [10]. Our obtained equilibria may not only be much faster to
compute, but also more representative of the market due to the global nature
of the equilibria.

While admittedly RSS has its limitations, another merit of our work is perhaps
more important: we provided a more general idea for research, by trying to use
the excess choice in strategy to represent risk aversion of players and obtain
simplified strategy sets to consider for the main game, instead of arbitrarily
reducing the strategy space.

##  4 Numerical Solution and Equilibria Computation

The approach we use for finding Nash Equilibria differs from the one used in
[10], where EPECs are solved because: (i) the approach used in [10] only finds
stationary points (also known as stationary Nash points), which are weaker
than local Nash Equilibria [10], and here we aim to compute (approximate)
global Nash Equilibria, and (ii) Robust Strategy Selection reduces the
strategy space to one (bounded) dimension for each player, making it much
cheaper to compute best responses using grid search.

Algorithm 1 Compute approximate best response function
$\hat{B}_{i}(m_{i};\omega_{i},N_{pts},\mathcal{M})$ under RSS (a-BRF-RSS)

input $(m_{k},a_{k},Q_{k})$ $\forall
k\in\cup_{z}\mathcal{J}_{z}\setminus\\{i\\}$, $(c_{i},b_{i})$, $N_{pts}$

solution method $\mathcal{M}$ to solve (SWM)

if $b_{i}>s_{z_{i}}^{min}$ then

return $(c_{i},b_{i})$

end if

for $l=1,2,...,N_{pts}$ do

get player’s $i$ strategy
$m_{i}(l)=\frac{1}{2}c_{0}+\frac{1}{2}\frac{l-1}{N_{pts}-1}c_{0}$

compute $a_{i}(l)$ using $(m_{i}(l),a_{i}(l))\in\mathcal{S}_{WCP}(i)$

solve (SWM) using $\mathcal{M}$, and obtain $x^{*}$, $v^{*}$ and $y^{*}$

compute $\Pi_{i}(x_{i}^{*},m_{i}(l),a_{i}(l),c_{i},b_{i})$

end for

$m_{i}^{B}\leftarrow\textrm{arg}\max_{m_{i}(l);l\in\overline{1,N_{pts}}}\Pi_{i}(x_{i}^{*},m_{i}(l),a_{i}(l),c_{i},b_{i})$

return $(m_{i}^{B},a_{i}^{B})$

Instead of computing derivatives and updating an estimate of a local Nash
point until convergence, we consider iteratively improving each player’s best
response function given other players’ strategies. Let us denote the best
response function for player $i$, given other players’ strategies
$(m_{-i},a_{-i})$ as $B_{i}(m_{-i},a_{-i};\omega_{i})$ where the problem
parameters for player $i$ are $\omega_{i}=(c_{i},b_{i},Q,d,M_{p},b_{p})$.
Since $(m_{i},a_{i})\in\mathcal{S}_{WCP}(i)$, we drop $a_{i}$ and only denote
the strategy of player $i$ by $m_{i}$, with the implicit corresponding $a_{i}$
value given by (19). Further, we have that $\frac{1}{2}c_{i}\leq m_{i}\leq
c_{i}$ and therefore the feasible strategy space is bounded when using RSS.
Thus, to compute an approximate best response function,
$\hat{B}_{i}(m_{i};\omega_{i},N_{pts})$ we discretise the domain in $N_{pts}$
equispaced points over $[\frac{1}{2}c_{i},c_{i}]$, and for each strategy point
solve (SWM) using a CQP solver (quadprog in Python 3). Algorithm 1 formalises
the computation of the best response. Note that our idea for computation of
(approximate) global Nash Equilibria remains the same regardless of the market
clearing mechanism used.

Given an algorithm to compute an (approximate) best response function
$\hat{B}_{i}(m_{i};\omega_{i},N_{pts})$, one can now obtain an (approximate)
global Nash Equilibrium point by iteratively updating strategies until no
further improvement is possible for any player. Similar to [10], we propose
two types of iterative updates: (i) a Gauss-Seidel type of approach (GS),
where each player uses the most recent information about other players’
strategies, and (ii) a Jacobi type of approach (JS), where each player is
updated simultaneously using the last strategy point with synchronised
information. Both and are relevant here since each approach has its strengths
and limitations. The main advantage of GS is that the most recent information
is always used, potentially requiring fewer iterations, but paralellisation
can only be done across $N_{pts}$ different tasks, as each player has to wait
for the previous one’s best response function to be updated. On the other
hand, JS does not make best use of the last available iterations, and thus may
require more iterations, but it has the big advantage that it can be
parallelised across $(i,l)\in\mathcal{P}\times\overline{1,N_{pts}}$, i.e
across both the player set and discretisation points indepedently.

Algorithm 2 Compute strategy update by JS (C-JS)

input strategies $m_{i}$ $\forall i$, $N_{pts}$

solution method $\mathcal{M}$ to solve (SWM)

for $(i,l)\in\mathcal{P}\times\\{1,2,...,N_{pts}\\}$ do

get player’s $i$ strategy
$m_{i}(l)=\frac{1}{2}c_{0}+\frac{l-1}{N_{pts}-1}\frac{1}{2}c_{0}$

compute $a_{i}(l)$ using $(m_{i}(l),a_{i}(l))\in\mathcal{S}_{WCP}(i)$

solve (SWM) using $\mathcal{M}$, for $(m_{i}(l),m_{-i})$ and obtain $x^{*}$,
$v^{*}$ and $y^{*}$

compute $\Pi_{i}(x_{i}^{*},m_{i}(l),a_{i}(l),c_{i},b_{i})$

end for

for $i\in\\{1,2,...N_{p}\\}$ do

$m_{i}^{B}\leftarrow\textrm{arg}\max_{m_{i}(l);l\in\overline{1,N_{pts}}}\Pi_{i}(x_{i}^{*},m_{i}(l),a_{i}(l),c_{i},b_{i})$

end for

return $\\{(m_{i}^{B},a_{i}^{B})\\}_{i\in\mathcal{P}}$

Thus, $N_{p}N_{pts}$ tasks can be solved simultaneously, instead of just
$N_{pts}$ and assuming identical computational time for each task, and
sufficiently many cores to perform independent tasks, it can take $N_{p}$
times more iterations when compared to GS to achieve the same computational
time, albeit at a larger energy usage. Performing a detailed numerical
analysis is beyond the scope of this paper and we just use this simple
analysis to conclude that JS is more appropriate when sufficiently many cores
are available, while GS might be preferred when running on a machine with
fewer cores ($\leq 32$). The ideas of the two approaches for computing a
strategy update for all players are formalised in Algorithm 2 and 3.

Algorithm 3 Compute strategy update by GS (C-GS)

initial strategies $m_{i}$ $\forall i$, $N_{pts}$

market clearing solution method $\mathcal{M}$

for $i\in\\{1,2,...N_{p}\\}$ do

compute $(m_{i}^{B},a_{i}^{B})$ using
$\hat{B}_{i}(m_{i};\omega_{i},N_{pts},\mathcal{M})$

$m_{i}\leftarrow m_{i}^{B}$

end for

return $\\{(m_{i}^{B},a_{i}^{B})\\}_{i\in\mathcal{P}}$

Finally, one can now obtain a Nash equilibrium by repeatedly using C-GS or
C-JS using the previous strategy output as input, until convergence, as shown
in Algorithm 4. To perform intensive computations on the real world data in
Section 5.2 we use DescartesLabs’s cloud computing services, while the less
intensive computations are performed on a machine with 8 cores.

Algorithm 4 Compute Nash Equilibrium Point (CNEP)

initial strategies $m_{i}^{(0)}$ $\forall i$, $N_{pts}$, maximum number of
cycle $N_{max}^{C}$, and tolerance $\delta_{NE}\geq 0$

market clearing solution method $\mathcal{M}$

for $k\in\\{1,2,...N_{max}^{C}\\}$ do

compute $\\{(m_{i}^{(k)},a_{i}^{(k)})\\}_{i\in\mathcal{P}}$ using C-GS or C-JS
using $\\{(m_{i}^{(k-1)},a_{i}^{(k-1)})\\}_{i\in\mathcal{P}}$ as input
strategies

if $m_{i}^{(k-1)}=m_{i}^{(k)}$ (or
$\|m_{i}^{(k-1)}-m_{i}^{(k-1)}\|_{2}<\delta_{NE}$) then

break

end if

end for

return $\\{(m_{i}^{(k)},a_{i}^{(k)})\\}_{i\in\mathcal{P}}$

When running CNEP using the cloud services, we presolve by using CNEP with
much coarser $N_{pts}$ and rather small $N_{max}^{C}$. Under the limited runs
performed, we found this approach to give smaller number of iterations
performed as well as shorter computational time.

##  5 Numerical Results

In this section we briefly review some numerical experiments based on
synthetic data in subsection 5.1 to confirm our analytical results and claims
in subsection 3.2. The main focus in on the application of the game-theoretic
model (and the model assuming truthful bidding) to the real world case of
Central Western European power market over the year commencing on 1 January
2019. Further, we challenge the often employed assumption that producers bid
truthfully in the CWE. We devise a simple method to uncover strategic bidding
and show that this is fairly pronounced.

###  5.1 Numerical Experiments on Synthetic Data

In this subsection we briefly show numerical results for a synthetic problem
under the (SWM-s) market clearing, that confirm our findings and claims in
subsection 3.2.

First, we show that even under the conditions of Theorem 1, the existence and
uniqueness of equilibrium does not imply that the outcome is purely
competitive, as mentioned in Remark 1. To this end we consider a simulation
exercise with one zone and $5$ producers, $d=1$, and sample $c_{i}\sim
U(1,10)$ and $b_{i}\sim U(0.5,2)$ for $\forall i$ independently. We solve
$N=100,000$ such problems and for each problem we solve (SWM-s) for two cases:
(i) $m_{0}=c$ and $a_{0}=b$, and (ii) $m^{+}=c$, $a^{+}(m^{+})$ given as per
Theorem 1, and obtain the resulting prices $v_{0}$ and $v^{*}$. We plot the
empirical distribution of the ratio $r_{v}=v^{*}/v_{0}$ in Figure 1.

Figure 1: Empirical distribution of $r_{v}$ ratio for $N=100,000$ samples of
$c$ and $b$.

The idea is to reveal how a NE under conditions of Theorem 1, compares with
the outcome under truthful bidding. We can clearly see that the price $v^{*}$
is significantly larger than $v_{0}$ for the vast majority of times, and the
mean of the price ratio is well above $1$. An aware reader may observe that
there is a second mode with the mean below and $r_{v}$ value of $1$. We
observe this mode to become more pronounced when $b_{i}$ values are drawn from
a much wider range, when comparing to the range for $c_{i}$. However, it is
clear that under this setup, the outcome of strategising is generally very
different than the outcome corresponding to truthful bidding. Further,
reasonable value pairs for $(c,b)$ are observed to generally give higher
prices than in the case of truthful bidding.

We next consider a specific problem with $c^{T}=[2.65,1.5,2,1.8,2.2,2.1]$ and
$b^{T}=[0.5,2,1,1.1,0.6,1]$ as an example, and look at the profit landscape of
the first player when all other players bid $m_{-i}=c_{-i}$,
$a_{-i}=a_{-i}^{+}(c_{-i})$ and plot the results in Figure 2.

Figure 2: Best response of player $i=0$ for the (SWM-s) problem on synthetic
data.

First, we can observe that the profit becomes $0$ for values of $a_{i}$ above
a certain threshold. This is because $x_{i}^{*}=0$ is given by (SWM-s). Next,
we can clearly see that for a fixed $m_{i}$, $\pi_{i}(a_{i})$ is a strictly
concave function so long as $x_{i}>0$. We can also see that there exist a
range of values $(m_{i},a_{i})$ that yield the same maximum profit and that
$a_{i}^{+}\sim-C_{m}m_{i}^{+}$ for some $C_{m}>0$. Finally the plot also
confirms our curvature results for derivatives of $\pi_{i}$ with respect to
$m_{i}$ and provides some intuition as to why multiplicity of equilibria
exists.

We finally show evidence that equilibrium prices $v^{*}$ may be arbitrarily
large. To this end, we consider $m^{+}(k)=k\cdot c$ for $k\in[1,K_{max}]$ and
for each $m^{+}(k)$ obtain $a^{+}(m^{+}(k))$ as per Theorem 1. We further
consider perturbing the strategies for all but player $i=0$ as
$\tilde{m}_{-i}=(1-f_{m})m^{+}_{-i}(k)$
$\tilde{a}_{-i}=(1-f_{m})a^{+}_{-i}(m^{+}(k))$. Finally, we also consider the
case of allocating each two consecutive players to a different zone, imposing
capacity and network constraints as per the synthetic example in [18] and
solving (SWM). The resulting prices are plotted in Figure 3.

Figure 3: Equilibrium price for the considered synthetic problem with
$m(k)=kc$ under (SWM-s) (blue and orange curves), and under (SWM) (green
curve).

We can clearly see that the equilibrium price appears to increase linearly
with increasing $k$, albeit the values of $a$ decrease and even become
negative. This suggests that under this simplified setup, there may exist
equilibria with arbitrarily high prices. However, considering network
constraints and capacity constraints appears not to change this result,
although $(m^{+},a^{+})$ are not guaranteed to be equilibrium strategies under
(SWM). Nevertheless, this result requires complete information and assumes
that players are willing to bid even very negative values of their $a_{i}$
parameter. This is rather unlikely in practice and although this reveals
critical limitations of the SWM mechanism, the price is unlikely to become
arbitrarily large as players may view bidding very negative values of $a_{i}$
as very risky, complete information is not available, and the multitude of
equilibria provides another barrier to the realization of these outcomes. As
an attempt to account for these issues, we introduced the Robust Strategy
Selection approach.

###  5.2 Real world case study of the CWE spot market

The CWE market over the period we are concerned with (2019-2020), was composed
of five zones (roughly) corresponding to five countries: Austria, Belgium,
Germany (plus Luxembourg), France and the Netherlands. We evaluate the
suitability of our models by comparing directly with the historical time
series of prices for the corresponding time period, which we hereafter refer
to as target prices. The majority of computations are performed via Descartes
Labs’ computing services [22].

####  5.2.1 Input data

We apply the Game-Theoretic model introduced here, and also solve (SWM) under
the assumption of truthful bidding by solving independently for $2712$
consecutive hourly slots, with the first hour starting on the first hour of 2
Jan 2019. Thus we require the temporal time series for player’s
characteristics $(c(t),b(t),Q(t))$, network constraints
$\\{M_{p}(t),b_{p}(t)\\}$, and zonal demand levels $d^{Z}(t)$. The demand time
series is obtained by using the day ahead forecasted values available at [23].

Obtaining the time series of the full network constraints
$\\{M_{p}(t),b_{p}(t)\\}_{t\in\mathcal{T}}$ set is a challenging task, and
this is obtained by first obtaining the historical time series of the PTDFs
and RAMs as described in our paper in [17]. If we denote the set of line-
scenario combinations [17] that are tracked by the operator as $\mathcal{LS}$,
the concatenation of all these constraints at any time $t$ gives us a form of
network constraints as following

$r^{\mathcal{LS}}\leq PTDF_{t}^{\mathcal{LS}}(y(t)-d^{Z}(t))\leq
R_{t}^{\mathcal{LS}},$ (21)

and the constraint set can be re-written in the form used in (SWM) as

$M_{p}(t)=\begin{bmatrix}-PTDF^{\mathcal{LS}}_{t}\\\
+PTDF^{\mathcal{LS}}_{t}\end{bmatrix};\hskip
2.84526ptb_{p}(t)=\begin{bmatrix}-r^{\mathcal{LS}}_{t}-PTDF^{\mathcal{LS}}_{t}d(t)\\\
+R^{\mathcal{LS}}_{t}+PTDF^{\mathcal{LS}}_{t}d(t)\end{bmatrix}.$ (22)

To ensure that the feasible domain is a (closed) polytope, we further add to
this set box constraints defined by $(1-\Delta_{max})d_{z}(t)\leq
y_{z}(t)\leq(1-\Delta_{max})d_{z}(t)$, $\forall z$, with $\Delta_{max}=0.6$.

Before we obtain producer’s characteristics at each time $t$, we must define
the player set. For simplicity, similar to the work in [14], we define each
player to represent all power plants of a specific type of production in one
zone (e.g. one player for all Gas plants in Austria). We consider all main
production types as per the Installed Capacity per Production Type data
available at ENTSOE [23]. This gives around 11 players per zone, and 55
players in total. However, some of these players have very small capacity
fraction compared to the total capacity in that country, and therefore they
are unlikely to significantly affect the market outcome, but significantly
increase computational cost. Since we require solving our game-theoretic model
for thousands of time indexes and the network constraints have over $600$
rows, the computational cost can grow very large. As a result, we take a very
parsimonious approach, and restrict the player set as an attempt to minimise
computational cost whilst still retaining the dominant fundamental factors

Figure 4: Thresholding process for retaining the top $n_{z}^{*}$ players in
each zone (country). Figure 5: Player types and fraction of capacities covered
for each country. The total capacity for each country is shown at the left
hand side of the graph.

For each zone we apply a thresholding approach and choose the top $n_{z}$
players covering a fraction of at least $88\%$ of the total capacity, that is,
$n_{z}^{*}=\min\\{n_{z}\hskip 0.28453pt:\hskip 2.84526pt\phi_{z}(n_{z})\geq
0.88,n_{z}\geq 5,\hskip 2.84526pt,\hskip 2.84526ptn_{z}\in\mathbb{N}\\}$,
where $\phi_{z}(n_{z})$ is the fraction covered for the top $n_{z}$ players,
and we also choose at least five players to retain most of the diversity of
production types. This thresholding approach is summarised in Figure 4. As a
final step, once $n_{z}^{*}$ is obtained, we combine similar players in one
type where possible to increase the capacity covered but retain low player
number. To this end we combine On- with Off-shore wind, Hydro Pumped Storage
with Hydro Water Reservoir, and Hard Coal with Brown Coal (Lignite). In this
way, we get $28$, players, just over half of what we had initially, greatly
reducing the computational cost. Their types and capacity fractions are shown
Figure 5.

With the player set well defined, we now have to obtain for each player $i$,
the corresponding time series of cost structure $(c_{i}(t),b_{i}(t))$ and
available capacity $Q_{i}(t)$, which may differ from the total installed
capacity (especially for renewables). As per the relevant types in Figure 5,
we construct these time series for the following types: (i) Fossil Gas, (ii)
Coal, (iii) Nuclear, (iv) Solar, (v) Wind, (vi) Hydro Pumped Storage, (vii)
Hydro Water Reservoir and (viii) Hydro Run-of-river and poundage. For
simplicity, for each player, we obtain the two cost parameters for each time
index $t$ by extrapolating from a representative value indicating the
production cost per MWh. This usually (but not always) corresponds to the fuel
cost for one MWh of production. Let this cost for type $\theta$ in zone $z$ at
time $t$ be $k_{\theta,z}(t)$. Note that the double $(\theta,z)$ defines a
player, say $j$. To obtain two parameters we then define the cost function for
player $j$ as

$\lambda_{j,t}(x):=k_{\theta,z}(t)\left(1-\frac{f_{\theta}}{1-f_{\theta}}n_{\theta}\right)+n_{\theta}\frac{k_{\theta,z}}{(1-f_{\theta})Q_{j}}x=b_{j}+c_{j}x\,$
(23)

where $\lambda_{j,t}(x)$ is the price asked at quantity $x$, $f_{\theta}$ is
the fraction of total capacity at which the price asked is exactly
$k_{\theta,z}(t)$, and $n_{\theta}+1$ is the number of times the production
cost is higher at full capacity when compared to the base average level,
$k_{\theta,z}(t)$. We take the cost to be $k_{\theta,z}(t)$ at $50\%$ capacity
level, i.e. $f_{\theta}=0.5$ for all types for simplicity and
$n_{\theta}\in[0,1]$, to guarantee that $b_{j}\geq 0$, with the exact value of
$n_{\theta}$ depending on the fuel type. We are now left with obtaining
$k_{\theta,z}(t)$ and $n_{\theta}$.

Type | Data | $k_{\theta,z}$ | $n_{.}$ | Conversion
---|---|---|---|---
|  |  |  | comments
Gas | spot prices | time series | $0.2$ | DTH
|  TTF [24] | as per data |  |
| dark | time series | $0.4$ | $\frac{\$}{\text{€}}=0.89$
Coal |  spread, | as per data |  | $\frac{MWh}{ton}=\frac{20}{11}$
| proxy |  |  | DTH
| of [25] |  |  |
| estimate |  |  | may sell
Nuclear | value | $13.8\frac{\text{€}}{MWh}$ | $0.8$ | at up to
|  |  |  | 50% loss
Wind and | day ahead |  |  | strongly
Solar | generation | $0.5\frac{\text{€}}{MWh}$ | $0.05$ | variable
| forecast |  |  | $Q_{i}(t)$
| [23] |  |  |
Hydro | base load | time series |  |
Storage | electricity | as per data | $0.2$ |
|  | prices |  |
Hydro run | estimate | $8.45\frac{\text{€}}{MWh}$ | $0.1$ | constant
of river | value |  |  | value
Table 1:  Summary of data, parameters and assumptions used to estimate costs.

How these are obtained is summarised in Table 1, but a more detailed
description is available in out related paper [18], which uses the same data.
Where $k_{\theta,z}$ is stated explicitly it means that it is a constant,
while for the production types it is given by the time series described under
the data column. Finally, Gas and Coal spot daily data is converted to hourly
by assuming constant value during the day and this is denoted by DTH in the
table.

Admittedly, the approach used here to obtain time series of cost parameters
for each player is rather simplistic. However, we use this approach to
facilitate focusing on the formulation, analysis and solution of the market
clearing and our game-theoretic model. Further, in subsection 5.2.2 we
calibrate the cost parameters to improve our rather coarse guess. We show that
with minimial fitting of only two parameters per zone (i.e. 10 in total) we
obtain results that are very representative of the market, capturing well the
main features of the actual price series, with test set errors of only
$24.5\%$ and $25.2\%$ for the truthful bidding (TB) and game-theoretic (GT)
models. The test set is a rather large set of $672$ hours (or 28 days), and
these results demonstrate the robustness of our fundamental approach but also
suggest that the reconstructed network constraints form our work in [17] offer
a satisfactory representation of the true constraints, at least for the
application at hand.

####  5.2.2 Model calibration

Due to the complexity of the game-theoretic model, which has one extra layer
of complexity when compared to the (SWM) solved under the truthful bidding
model, calibrating the game-theoretic model is very difficult. We by-pass this
problem by first calibrating the (SWM) model under the truthful bidding
assumption which we call SWM-TB, and use these parameters for SWM-TB when this
model is applied to the CWE case study. We then perform a simple one step
adjustment of the SWM-TB parameters for the (SWM)-based game theoretic model
(denoted as SWM-GT), and this is explained at the end of this subsection.

The calibration approach for the SWM-TB is very simple and is constrained to
adjusting the scale of the costs for each zone $z$, by two scale parameters
$(s_{z}^{c},s_{z}^{b})$ such that the target prices are represented well by
the costs on some training data. This requires expressing the adjusted costs
as

$c_{i}(t)^{final}=s_{z}^{c}c_{i}(t)^{initial};\hskip
2.84526ptb_{i}(t)^{final}=s_{z}^{b}b_{i}(t)^{initial}\hskip 2.84526pt\forall
i\in\mathcal{P}_{z}.$ (24)

By taking $s_{z}^{c}=1$ and $s_{z}^{b}=1$ $\forall z$ we would simply use the
values obtained in the previous subsection. However, we optimise over these
values to improve our initial guess of the costs, while allowing for unchanged
inter-temporal variability. For this approach, the number of fitted parameters
and computational costs are very low. We aim for a small number of parameters
to fit because in principle, our fundamental model should be representative of
the market without requiring any fitting, if the cost structure and other
inputs are descriptive and estimated accurately. A further advantage of this
approach is computational: $\partial v_{z}/\partial s^{.}_{z}$ can be easily
computed without knowing the set of players that set the price, i.e.
$\\{i:m_{i}x_{i}+a_{i}=v_{z_{i}}\\}$. The optimisation problem has the
objective

$F(s)=\frac{1}{N_{T}}\sum_{t=1}^{N_{T}}\sum_{z}(v_{z,t}-P_{z,t})^{2},$ (25)

where $v_{z,t}$ and $P_{z,t}$ are the model and target prices respectively, at
zone $z$ for time $t$. Since the scale parameters $s=(s^{c},s^{b})$ are not
dependent on time, we have a very strongly over-determined Nonlinear Least
Squares Problem. If we now let

$F_{t}(s)=\sum_{z}(v_{z,t}-P_{z,t})^{2},$ (26)

we then have that

$\frac{\partial F_{t}}{\partial
s_{z}^{c}}=2(v_{z,t}-P_{z,t})c_{k_{z}}x_{k_{z}};\hskip 5.69054pt\frac{\partial
F_{t}}{\partial s_{z}^{b}}=2(v_{z,t}-P_{z,t})b_{k_{z}},$ (27)

which by summation, and collecting for all zones $z$, gives $\nabla_{s}F(s)$.
The second derivatives can be similarly computed as

$\frac{\partial^{2}F_{t}}{\partial^{2}s_{z}^{c}}=c_{k_{z}}^{2}x_{k_{z}}^{2};\hskip
11.38109pt\frac{\partial^{2}F_{t}}{\partial^{2}s_{z}^{b}}=b_{k_{z}}^{2};\hskip
11.38109pt\frac{\partial^{2}F_{t}}{\partial s_{z}^{b}\partial
s_{z}^{c}}=b_{k_{z}}c_{k_{z}}x_{k_{z}},$ (28)

with all other entries in the $\nabla^{2}_{s}F$ being 0. We choose to perform
Gradient Descent over Newton’s Method, with line-search as globalisation
method since the former appears to obtain lower values of the objective
function with better generalisations, although our investigation on this
particular matter is limited. Let us call the obtained parameters
$(s^{c}(TB),s^{b}(TB))$.

The scale parameters for SWM-GT are fit by first using $(s^{c}(TB),s^{b}(TB))$
as input parameters to compute the output price series of our SWM-GT model for
a subset of $\mathcal{T}_{GT}^{0}=\overline{24,672}$ time indices, and
computing the average price ratio for each zone

$\hat{r}_{z}=\frac{1}{|\mathcal{T}_{GT}^{0}|}\sum_{t\in\mathcal{T}_{GT}^{0}}\frac{v_{z,t}^{SWM-
GT}}{P_{z,t}}.$ (29)

The scale parameters $s_{z}^{c}(GT)$ for the SWM-GT models are then updated as
$s_{z}^{c}(GT)\leftarrow s_{z}^{c}(TB)/\hat{r}_{z}$, $\forall z$ only once,
but we do not re-scale $s_{z}^{b}$, i.e. $s_{z}^{b}(GT)\leftarrow
s_{z}^{b}(TB)$. While this approach is very simplistic, we are not aiming to
obtain the parameters that best represent the cost structure given the prices,
but rather simply to improve on our rather coarse initial guess of costs.

The aware reader might observe that by definition $s_{z}^{c}(GT)\neq
s_{z}^{c}(TB)$ in general, and that this cannot be true in practice since the
costs do not depend on the model choice. While this is a valid point, we are
here interested in evaluating the representation capacity of both SWM-TB and
SWM-GT, as well as inferring if and when strategic bidding takes place. Simply
fitting the SWM-TB model and using these parameters for both models biases the
results in favour of the SWM-TB model. This is especially true since strategic
bidding may be assimilated into the truthful bidding model by implying higher
costs. This is indeed what we observe in practice as $(s^{c}(TB),s^{b}(TB))$
are always well above one. While we are prepared to accept that our coarse
estimation of costs is an under-statement, we cannot simply say that all the
error in price estimation is due to the initial cost structure. For this
reason, we aim for a very robust and simple calibration process, where we fit
a limited number of scale parameters with a rather simplistic approach, and we
have to allow for differences in scale parameters for SWM-GT against SWM-TB.
In other words, we can only compare the case where players are bidding
truthfully and the cost structure is the one given by $(s^{c}(TB),s^{b}(TB))$,
with strategic bidding and cost structure given by $(s^{c}(GT),s^{b}(TB))$.
This is due to the uncertainty in cost parameters, and this problem can be by-
passed fully by obtaining very accurate cost estimates via a bottom up
approach, but unfortunately this is outside the scope of this paper. Note that
the approach in this paper remains valid if one obtains such accurate cost
estimates.

We next apply our SWM-TB and SWM-GT models to the real world case of CWE day
ahead electricity market.

####  5.2.3 Model performance

We apply the SWM based truthful bidding model (SWM-TB) and the SWM based game-
theoretic model (SWM-GT) to CWE data representing the first $2688$ consecutive
hours starting on the first hour of 2 Jan 2019. We split the data in a train
set $\mathcal{T}_{train}:=\\{t\in\mathbb{N}:24\geq t<2040\\}$ and a test set
$\mathcal{T}_{train}:=\\{t\in\mathbb{N}:2040\geq t<2712\\}$, where $t$ is a
hourly time index. We start with a high level comparison of the SWM-TB and
SWM-GT models by summarising the relative mean error for both the train set
$\mathcal{T}_{train}$ and test set $\mathcal{T}_{test}$, in Table 2.

| SWM-TB | SWM-TB | SWM-GT | SWM-GT
---|---|---|---|---
| ($\mathcal{T}_{train}$) | ($\mathcal{T}_{test}$) | ($\mathcal{T}_{train}$) | ($\mathcal{T}_{test}$)
AT | $20.3\%$ | $28.8\%$ | $21.9\%$ | $26.7\%$
BE | $14.2\%$ | $22.8\%$ | $18.5\%$ | $25.4\%$
DE | $26.1\%$ | $30.5\%$ | $30.0\%$ | $34.2\%$
FR | $14.5\%$ | $19.6\%$ | $18.2\%$ | $17.6\%$
NL | $13.6\%$ | $18.5\%$ | $17.4\%$ | $17.6\%$
AVG | $17.9\%$ | $24.6\%$ | $21.4\%$ | $25.2\%$
Table 2:  Comparison of relative mean errors for the Truthful Bids and Game
Theoretic models, on train and test data sets. Row AVG denotes the aggregate
results across all zones.

By direct comparison of the relative mean error, we observe that the Truthful
Bids model outperforms the Game Theoretic one. However, the difference is
rather small, and it is worth recalling that while the cost scale parameters
for the SWM-TB were fit via Gradient Descent, only a one step heuristic fit
was performed for the parameters of the SWM-GT model. Further, the summary
statistic presented here does not reveal any details with regards to the
temporal structure, and producers may choose to bid strategically only at
specific times. A potential reason for this is that the regulators require the
producers to bid truthfully, in an attempt to minimise market power exercised
by the supply side, but producers are concerned with their own profit, and
thus participants may attempt to conceal exercising their market power by
bidding strategically only at specific times, with better reward-to-risk
profile.

A detailed view of the model and observed prices time series is shown in
Figure 6, for a better intuition of models’ performance.

Figure 6: Price time series results (Train set): SWM - Truthful Bids (blue)
and SWM - Game Theoretic (black) comparison against observed prices (red).

We can see that the Game Theoretic model still accurately traces the target
price, but while the Truthful Bids model has the tendency to under-predict,
the Game Theoretic one is more likely to over-predict. We can qualitatively
observe that the GT model is often very accurate at peak hours, when compared
to the TB one. This is rather remarkable, given the extremely simple
calibration of the GT model, and suggests that players may not only bid
strategically, but also strategically choose the times at which they do so.
One may argue against the possibility of strategic bidding, on the premise
that no parameter fitting is required for our models, since they are of
fundamental nature.

Figure 7: Price time series results (Test set): SWM - Truthful Bids (blue) and
SWM - Game Theoretic (black) comparison against observed prices (red).

However, the complexity of the players’ cost structure makes it very difficult
to obtain accurate representations without very granular data and analysis, or
calibration. Further note that calibration of the SWM-TB model to the observed
price data could in fact adversely incorporate and conceal gaming. To observe
this, imagine a tacit agreement between all players to bid exactly twice their
costs. In this case, the value of our cost scales would be double, but the
SWM-TB model would perfectly represent the price series. Indeed, for our fit,
we observe our cost scales to be well above one. This issue can be best solved
by having highly granular and reliable cost data and an accurate cost model.
In our case, given the limited information on cost structure, it is precisely
the inconsistencies in the bidding patterns that help us reveal strategic
behaviour. We can observe that there are isolated days where prices are much
higher, while the demand pattern is essentially the same. It is possible but
unlikely for the production costs to change this fast, and thus assuming that
the network constraints remain constant, the only other possibility is a state
change: from truthful bidding to strategic bidding.

Admittedly the price spikes could occur due to fast changes in network
constraints, while our reconstructed data from [17] may change more slowly.
However, since we only use the reconstructed constraints data, without
forecasting into the future, and given the results in Chapter 2, it seems
reasonable to assume that our constraints time series captures the main
features of the real one. From Figure 7 we observe that both models perform
very well on the test set, revealing the robustness of our fundamental models.

Figure 8: Classification of time points to their the most representative
state. The red bars represent $97.5\%$ CI for each axis respectively.

####  5.2.4 Uncovering strategic behaviour

We next identify if and when players are gaming, assuming that our models,
cost stucture and values, constraints and demand inputs are representative. We
achieve this via simple hypothesis testing, and classifying different possible
outcomes. Our analysis is performed based on price level, for each zone and
per time point individually, and no time-dependent structure is considered. To
this end we define two error time series, one for the TB and other for the GT
models:

$e^{TB}_{z,t}=P_{z,t}-v_{z,t}^{SWM-TB};\hskip
5.69054pte^{GT}_{z,t}=P_{z,t}-v_{z,t}^{SWM-GT},$ (30)

where $P_{z,t}$ is the zonal price in zone $z$ at time $t$ and $v_{z,t}^{SWM-
TB}$ and $v_{z,t}^{SWM-GT}$ are the zonal prices given by the SWM-TB and SWM-
GT models respectively.

For each zone $z$, each time point is assigned the coordinates
$\left(e^{TB}_{z,t},e^{GT}_{z,t}\right)$, and hypothesis testing is performed
for each of the two error components individually. The null hypotheses are
that the error at time $t$ comes from the error distribution of the TB (for
$e^{TB}_{z,t}$) and GT (for $[e^{GT}_{z,t}$) models respectively, which are
approximated by Gaussians with parameters set as their empirical estimators.
Let us denote the null hypotheses corresponding to TB and GT as
$\mathcal{N}_{TB}^{0}$ and$\mathcal{N}_{GT}^{0}$ respectively. We reject the
null hypothesis if the corresponding error (for TB or GT) lies outside a
symmetric $97.5\%$ confidence interval (CI) centred in the mean of the
distribution. However, the null hypothesis can be rejected because
$e^{.}_{z,t}$ is either ”very low” or ”very high”, and this gives us different
information about the actual state of market. Thus, there are three regions of
interest when performing hypothesis testing for each of the TB and GT models.
Since we are interested in the cartesian product of the $e^{TB}$ and $e^{GT}$
space, this means that there are $9$ regions in total, and these are annotated
in Figure 8 (f), for visualisation. If the error lies in the low end then the
model over-predicts while a high error indicates under-predicting. The meaning
of each region is then as following:

  1. 1.

If a time point lies in region R5 then neither $\mathcal{N}_{TB}^{0}$ nor
$\mathcal{N}_{GT}^{0}$ can be rejected, and we cannot confidently say if
players were bidding truthfully or strategically. We call this state Null and
denote it as $\mathcal{S}_{0}$.

  2. 2.

If a time point lies in R4 or R6, then $\mathcal{N}_{TB}^{0}$ is rejected, but
not $\mathcal{N}_{GT}^{0}$, meaning that with high probability (w.h.p), the
players were bidding strategically. We call this state GT (w.h.p) and denote
it as $\mathcal{S}_{GT}$.

  3. 3.

If a time point lies in R2 or R8, then $\mathcal{N}_{GT}^{0}$ is rejected, but
not $\mathcal{N}_{TB}^{0}$, meaning that w.h.p the players were bidding
truthfully. We call this state TB (w.h.p) and denote it as $\mathcal{S}_{TB}$.

  4. 4.

If a time point lies in R3, both models under-predict with respect to the
observed price. This suggests that both models are inappropriate in this case.
However, under the assumption of accurate inputs in our model, the only
possibility for this to happen is if the players bid strategically, although
different risk aversions or game setups may be relevant in this case. We also
assign these points to state GT (w.h.p).

  5. 5.

If a time point lies in R7, then both models over-predict, meaning that the
actual price is much lower. This case seems to occur when electricity prices
become negative due to unexpected and high influx of renewable power, and thus
we call these points Expected Anomalies and denote their set as
$\mathcal{S}_{EA}$.

  6. 6.

It is extremely unlikely for a point to lie in R1, as this means that the GT
model severely under-predicts, while the TB model severely over-predicts. This
requires the TB model giving a much higher price than the GT model. Due to
multiple zones, this is not necessarily impossible but extremely unlikely. It
is also fairly unlikely for a point to lie in R9 where the GT model
drastically over-predicts, while the TB model drastically under-predicts. We
do not observe any of these points on our data set but classify these points
as Other Anomalies and denote their set as $\mathcal{S}_{OA}$.

Figure 9: Comparison of price time series (aggregate TD-SD): State Detection
results vs observed prices

We call the method of assigning these states Two Dimensional State Detection
(TD-SD). While the choice of confidence threshold may impact results, this
simple model enables us to gather some insight into whether the players were
bidding truthfully or strategically. Since the analysis was performed on the
price time series for each zone individually, we next aggregate the results
for a final single state at each time $t$ over all $z$ based on the results
obtained for each of the $5$ zones, and this requires further definitions. We
found that performing the same hypothesis testing on total cost, the resulting
price series is less accurate, because price decoupling happens fairly often
and thus gaming may occur only in one or few regions, but this becomes harder
to identify at market level. For this reason, we perform TD-SD independently
for each zone, and define logical rules for aggregation. Let
$\varsigma(t)\in\mathbb{R}^{N_{z}}$ be the vector of states assigned for each
state, then we define the aggregate state $\chi_{t}$ as following:

  1. 1.

If $\varsigma_{z}(t)\in\mathcal{S}_{0}$ $\forall z$, then the aggregate state
is $\chi_{t}\in\mathcal{S}_{0}$.

  2. 2.

If $\varsigma_{z}(t)\in\mathcal{S}_{0}$ $\forall z$, $\exists z$ s.t.
$\varsigma_{z}(t)\in\mathcal{S}_{TB}$ and $\nexists z$ s.t.
$\varsigma_{z}(t)\in\mathcal{S}_{GT}\cup\mathcal{S}_{EA}\cup\mathcal{S}_{OA}$,
then $\chi_{t}\in\mathcal{S}_{TB}$.

  3. 3.

If $\exists z$ s.t. $\varsigma_{z}(t)\in\mathcal{S}_{GT}$ and $\nexists z$
s.t. $\varsigma_{z}(t)\in\mathcal{S}_{EA}\cup\mathcal{S}_{OA}$, then
$\chi_{t}\in\mathcal{S}_{GT}$.

  4. 4.

If $\exists z$ s.t. $\varsigma_{z}(t)\in\mathcal{S}_{EA}$ and $\nexists z$
s.t. $\varsigma_{z}(t)\in\mathcal{S}_{OA}$, then
$\chi_{t}\in\mathcal{S}_{EA}$.

  5. 5.

If $\exists z$ s.t. $\varsigma_{z}(t)\in\mathcal{S}_{OA}$, then
$\chi_{t}\in\mathcal{S}_{OA}$.

While the aggregation rules are mostly a natural extension, there is some
degree of arbitrariness. However, the aggregation of zonal results enable us
to summarise the main observations about the market as a whole, instead of
describing each country individually (and their interaction), which is much
more complex. Nevertheless, the zonal results can also be used if a more
detailed analysis is required. We present the price estimation based on the
aggregated modes for the market in Figure 9.

Our approach gives an indication of when players are bidding strategically,
and we can observe that times tend to be consecutive, usually at peak hours,
and under consecutive days. Strategic bidding around peak hours could be an
approach employed by margin producers to mitigate the risk of being detected
at least with respect to potential gain, since producers’ market power is at
its peak when demand is highest.

Figure 10: Results for the average day: Proportion of each observed state

Since it is very difficult to visualise the results for the whole length of
the series, and ingest the main conclusions in this way, we review the state
profile of an average day. To this end, we count the number of each type of
states for each hour of the day, and plot the results in Figure 10. We observe
that generally, for each zone we can say with high confidence much more often
that the players are bidding strategically, than we can say they are bidding
truthfully. However, for a very large proportion of times we essentially
cannot draw any conclusion. This is due to the fairly large variance of our
model errors, which is dependent on model structure and inputs. This could be
potentially improved if data of higher quality and granularity is available,
as well as by including more market features in the model. However, this
requires more reliable data sources, extensive added labour and increased
computational cost, and for these reasons we confined ourselves to the current
approach. Reducing the confidence requirement will increase the fraction of
times we can classify, but this increases the chance of over-stating results.
Thus, we adopt a conservative approach, requiring $97.5\%$ confidence, and
this still reveals interesting features. Nevertheless, the final general
conclusion appears to remain the same even when varying the confidence
threshold: strategic bidding occurs a significant number of times with high
probability.

Figure 10 reveals that strategic bidding can be observed in each zone for a
significant fraction of times, and this generally tends to be more pronounced
at intervals around peak hours: $8am-1pm$ and $5pm-9pm$. As discussed before,
this could be due to a much more attractive risk-reward profile of strategic
bidding alternative for the producers. We also observe that strategic bidding
seems most pronounced in the Netherlands, and Belgium and least pronounced in
France and Germany. This could be because the Netherlands and Belgium are
fairly small trading zones with a fairly small number of players and therefore
market power can be exercised much more easily. Heavy reliance on gas power
plants and electricity imports also aggravate this issue. On the other hand
France is a large trading zone and much cheap production in the form of
nuclear power plants is available. Further these plants cannot easily withhold
capacity and thus they do not have strong incentives to bid strategically. Out
of the five, Germany is the largest trading zone, with the largest number of
participants, and a large number of coal power plants ready to produce if
price increase due to strategic bidding exceeds the cost of extra emissions.
As a result, we can observe that for Germany we obtain the largest number of
times for which we can say with high confidence that the producers were
bidding truthfully.

Our results suggest that strategic bidding in Austria is also fairly elevated.
While Austria is also a fairly small zone, the results could be somewhat over-
stated due to valuation of Hydro storage power. This is difficult to price and
our approach may be less conservative than the one employed by these plants’
operations manager.

We also observe a fairly pronounced number of times where both SWM-TB and SWM-
GT models over-predict (Expected Outliers). This occurs mostly in early
morning hours, when demand is low and renewable energy production may be
unexpectedly high due to high wind. This appears to be most pronounced in
France, which is expected since nuclear power plants are not flexible to
adjust, and prefer to sell at lower prices or even a loss.

The aggregate results presented in Figure 10 indicate that while we cannot
identify with certainty the bidding behaviour of players for a large number of
times, strategic bidding (in at least one zone at each time) may be fairly
pronounced. In fact, the fraction of times we can conclude with high
confidence that players in all zones were bidding truthfully is very small.
While it is possible that all the unidentified time instances truly belong to
the truthful bidding case, even in this situation, our results reveal that the
fraction of strategic bidding may be much higher than what market
participants, regulators and academic papers generally assume.

##  6 Conclusion

We considered the case of strategic bidding in spot electricity markets,
focusing on the market mechanism used in European markets, which to our
knowledge was not studied before. We take the demand as inflexible, network
and capacity constraints are considered, while submitted orders are restricted
to supply functions with linearly increasing price with respect to the
produced quantity. We present theoretical results on a simplified case
revealing that infinitely many equilibria may exist, if players are risk
neutral, and show numerically that equilibria with arbitrarily high prices may
exist. As a solution, a novel type of non-parametric risk aversion based on a
defined worst case scenario is introduced, and this ensures boundedness of
equilibrium prices and reduces dimensionality of the strategy space. By
leveraging these properties we devise Jacobi and Gauss-Seidel type of
iterative schemes for computation of approximate global Nash Equilibria, which
is in contrast to the derivative based local equilibria usually obtained by
solving Equilibrium Problems with Equilibrum Constraints (EPECs).

For the case of strategic bidding (represented by the game theoretic
framework), as well as truthful bidding we apply our resulting models to the
real world data of Central Western European Day Ahead market during the
2019-2020 period. We show that both our models offer a good representation of
the historical time series of prices. We finally devise a simple method based
on hypothesis testing to infer if and when producers are bidding
strategically, instead of truthfully, and we find evidence that strategic
bidding may be fairly pronounced in the CWE region.

##  Appendix A Proofs of theoretical results

######  Result 1.

Consider a one zone market with inflexible demand $d>0$ and producer set
$\mathcal{P}$. Assume producers have no capacity constraints and that there
are no network constraints. Assume that the submitted orders have $m_{i}>0$,
$\forall i$, and that the market clearing takes the form (SWM-s). Then the
optimal quantities allocated to active players in set $\mathcal{P}^{A}$ are

$\begin{aligned}
x_{i}^{*}&=\frac{d+\sum_{j\in\mathcal{P}^{A}\setminus\\{i\\}}m_{j}^{-1}(a_{j}-a_{i})}{m_{i}\sum_{j\in\mathcal{P}^{A}}\frac{1}{m_{j}}},\hskip
5.69054pti\in\mathcal{P}^{A}\\\ \end{aligned},$ (31)

and $x_{i}=0$ if $i\notin\mathcal{P}^{A}$. Further, the marginal prices for
any player in the active set $\mathcal{P}^{A}$ gives the zonal price, that is
$v=\lambda_{i}=m_{i}x_{i}^{*}+a_{i}$.

###### Proof.

Without loss of generality we can assume that $\mathcal{P}=\mathcal{P}^{A}$,
since all other players are irrelevant. Problem (SWM-s) can be simplified by
re-writing the balance constraint as
$x_{n}=d-\sum_{i\in\mathcal{P}\setminus\\{n\\}}x_{i}$ and plugging this in the
objective to yield

$\displaystyle\min_{x_{i}:i\neq n}$ $\displaystyle\sum_{i\neq
n}\left(\frac{1}{2}m_{i}x_{i}^{2}+a_{i}x_{i}\right)+\frac{1}{2}m_{n}\left(d-\sum_{i\in\mathcal{P}\setminus\\{n\\}}x_{i}\right)^{2}$
(32)
$\displaystyle+a_{n}\left(d-\sum_{i\in\mathcal{P}\setminus\\{n\\}}x_{i}\right)$

which is an unconstrained problem with one degree of freedom corresponding to
$x_{n}$ removed. We can set the gradient of this objective to zero to obtain
the minimum. This gives us the following system of equations

$m_{i}x_{i}+a_{i}=m_{n}\left(d-\sum_{i\in\mathcal{P}\setminus\\{n\\}}x_{i}\right)+a_{n}\hskip
8.53581pt\forall i,$ (33)

which tells us that at optimality the marginal price of all players with
$x_{i}>0$ is the same, and thus $v=\lambda_{i}=m_{i}x_{i}+a_{i}$, $\forall
i:x_{i}>0$. System (33) can be solved by writing it as

$\left(\hat{D}_{m}+m_{n}\hat{e}\hat{e}^{T}\right)\hat{x}=m_{n}d\hat{e}+a_{n}\hat{e}-\hat{a}$
(34)

where $\hat{x}$, $\hat{a}$, $\hat{m}$ represent the vectors of quantities for
all players except $n$, $\hat{e}$ is the vector of all ones of length
$|\mathcal{P}|-1$, and $\hat{D}_{m}=diag(\hat{m})$. This system is full rank
since $m_{i}>0$ $\forall i$, and an analytic solution can be obtained by
applying Woodbury’s matrix identity as an inversion formula which in this case
leverages the diagonal structure of $\hat{D}_{m}$ and the rank one of
$\hat{e}\hat{e}^{T}$.

The optimal value of $x_{i}$ for $i\neq n$ is then

$x_{i}=\frac{d+\sum_{j\neq
i}\frac{a_{j}-a_{i}}{m_{j}}}{m_{i}\sum_{j}\frac{1}{m_{j}}}.$ (35)

However, the above equation does not contain $n$ indices and there was no
specific property of player $n$ used in the derivation. Thus, by symmetry the
equation above must hold for player $n$ as well. ∎

######  Theorem 1.

Consider a one zone market with inflexible demand $d>0$ and active producer
set $\mathcal{P}$. Assume producers have no capacity constraints and that
there are no network constraints. Further assume that demand $d$, cost
structure $\\{c_{i},b_{i}\\}_{i\in\mathcal{P}}$ and network constraints
$(M_{p},b_{p})$ are common knowledge, and that each producer is aiming to
maximise its profit,
$\pi_{i}(m_{i},a_{i},m_{-i},a_{-i})=vx_{i}-\frac{1}{2}c_{i}x_{i}^{2}-b_{i}x_{i}$,
with $v$ the zonal clearing price. Let the market clearing be given by
(SWM-s). Finally, assume that for every $i$ we have $c_{i}>0$,
$m_{i}^{+}\geq\frac{1}{2}c_{i}$ is held fixed and that $a_{i}$ is the only
strategy variable, and that this is common knowledge. Then for any vector
$m^{+}\in\mathcal{M}$ there exists a unique Nash equilibrium
$a^{+}(m^{+})\in\mathcal{A}$, which is defined by the (full rank) linear
system

$\theta_{i}\sum_{j\in\mathcal{P}\setminus\\{i\\}}m_{j}^{-1}a_{i}^{+}-\sum_{j\in\mathcal{P}\setminus\\{i\\}}\frac{a_{j}^{+}}{m_{j}}=d+(\theta_{i}-1)\sum_{j\in\mathcal{P}\setminus\\{i\\}}\frac{b_{j}}{m_{j}},$
(36)

where $\theta_{i}=\frac{2-k_{i}K_{-i}^{2}}{1-k_{i}K_{-i}^{2}}$,
$k_{i}=2m_{i}-c_{i}$,
$K_{-i}^{2}=\frac{\sum_{j\in\mathcal{P}\setminus\\{i\\}}m_{j}^{-1}}{m_{i}\sum_{j\in\mathcal{P}}m_{j}^{-1}}$.

###### Proof.

The proof is obtained by using the optimal form of $x_{i}$ values for the
(SWM-s) problem, as per Result 1, given $\\{m_{i},a_{i}\\}_{i\in\mathcal{P}}$.
Using the expression for each $x_{i}$, the local Nash Equilibria equations are
obtained, and the final part of the proof shows this is a full rank, square
linear system.

We now look at the profit maximisation problem of each player $i$. By using
Result 1, the profit function can be written as

$\pi_{i}=vx_{i}-\frac{1}{2}c_{i}x_{i}^{2}-b_{i}=m_{i}x_{i}^{2}+a_{i}x_{i}-\frac{1}{2}c_{i}x_{i}^{2}-b_{i}x_{i},$
(37)

and by taking the partial derivative with respect to $a_{i}$ we obtain

$\frac{\partial\pi_{i}}{\partial a_{i}}=(k_{i}x_{i}+a_{i}-b_{i})\frac{\partial
x_{i}}{\partial a_{i}}+x_{i},$ (38)

where $k_{i}=2m_{i}-c_{i}$. Using (31), we get that

$\frac{\partial x_{i}}{\partial
a_{i}}=-\frac{\sum_{j\in\mathcal{P}\setminus\\{i\\}}m_{j}^{-1}}{m_{i}\sum_{j\in\mathcal{P}}m_{j}^{-1}}=:-K_{-i}^{2},$
(39)

and we can re-write $x_{i}$ as

$x_{i}=K_{-i}^{1}-K_{-i}^{2}a_{i},$ (40)

where

$K_{-i}^{1}=\frac{d}{m_{i}\sum_{j\in\mathcal{P}}m_{j}^{-1}}.$ (41)

By using these results and imposing the first order condition that
$\frac{\partial\pi_{i}}{\partial a_{i}}=0$ we have that

$\frac{\partial\pi_{i}}{\partial
a_{i}}=(1-k_{i}K_{-i}^{2})x_{i}-K_{-i}^{2}(a_{i}-b_{i})=0,$ (42)

which is equivalent to

$\theta_{i}\sum_{j\in\mathcal{P}\setminus\\{i\\}}m_{j}^{-1}a_{i}-\sum_{j\in\mathcal{P}\setminus\\{i\\}}\frac{a_{j}}{m_{j}}=d+(\theta_{i}-1)\sum_{j\in\mathcal{P}\setminus\\{i\\}}\frac{b_{j}}{m_{j}},$
(43)

where

$\theta_{i}=\frac{2-k_{i}K_{-i}^{2}}{1-k_{i}K_{-i}^{2}}.$ (44)

This system is linear in $a$ given fixed $m$ and can be re-written in vector
form as

$\left(\hat{D}^{m}+ev_{m}^{T}\right)a=\hat{r},$ (45)

where $\hat{D}^{m}$ is a diagonal matrix with
$\hat{D}^{m}_{i,i}=\theta_{i}\sum_{j\in\mathcal{P}\setminus\\{i\\}}m_{j}^{-1}$,
$v_{m}$ is a vector with $(v_{m})_{i}=m_{i}^{-1}$ and
$\hat{r}_{i}=d+(\theta_{i}-1)\sum_{j\in\mathcal{P}\setminus\\{i\\}}\frac{b_{i}}{m_{j}}$.
Since $1+e^{T}\hat{D}^{m}v_{m}=1+\sum_{j}m_{j}^{-1}>1\neq 0$ by Sherman-
Morrison formula, the matrix $\hat{D}^{m}+ev_{m}^{T}$ is indeed invertible
meaning that system (43) has a unique solution. Further, the exact inverse can
be easily computed, albeit this is unnecessary here. Thus, there is a unique
stationary point, and we next show that this is indeed a maximum for each
player, by showing that $\frac{\partial^{2}\pi_{i}}{\partial a_{i}^{2}}<0$.
Differentiating we get that

$\displaystyle\frac{\partial^{2}\pi_{i}}{\partial a_{i}^{2}}$
$\displaystyle=2\frac{\partial x_{i}}{\partial
a_{i}}+k_{i}\left(\frac{\partial x_{i}}{\partial a_{i}}\right)^{2}$ (46)
$\displaystyle=2K_{-i}^{2}\left(-1+\frac{(1-\frac{c_{i}}{2m_{i}})\sum_{j\in\mathcal{P}\setminus\\{i\\}}m_{j}^{-1}}{\sum_{j\in\mathcal{P}}m_{j}^{-1}}\right).$

Since $K_{-i}^{2}>0$, $0<1-\frac{c_{i}}{2m_{i}}<1$ and
$0<\frac{\sum_{j\in\mathcal{P}\setminus\\{i\\}}m_{j}^{-1}}{\sum_{j\in\mathcal{P}}m_{j}^{-1}}<1$
it follows that $\frac{\partial^{2}\pi_{i}}{\partial a_{i}^{2}}<0$ concluding
our proof. ∎

######  Theorem 2.

Consider identical conditions to Theorem 1, with the exception that $m_{i}$
are not held fixed, but rather part of the strategy choice for each player
$i$. Then for each $m^{+}\in\mathcal{M}$, there exists an $a^{+}(m^{+})$,
computed according to Theorem 1 by taking $m^{+}$ as fixed, such that
$(m^{+},a^{+})$ is a local Nash equilibrium in the
$(m,a)\in\mathcal{M}\times\mathcal{A}$ space. In other words, since $m^{+}$ is
chosen from an infinite set, there are infinitely many local equilibria.
Further, the local equilibria form a continuous subspace $\mathcal{MA}^{+}$.

###### Proof.

The idea of the proof has two steps. In the first step we show that any point
$(m,a)$ satisfying $\frac{\partial\pi_{i}}{\partial a_{i}}=0$ also satisfies
$\frac{\partial\pi_{i}}{\partial m_{i}}=0$, meaning that a stationary point
for $\mathcal{A}$ space is also stationary for $\mathcal{M}\times\mathcal{A}$
space. The second part of the proof shows that the hessian of $\pi_{i}$ for
each $i$, $\nabla^{2}\pi_{i}\in\mathbb{R}^{2\times 2}$ has negative trace and
zero determinant at $(m^{+},a^{+})$, meaning that the stationary point is a
local (but not isolated) maximum for each player $i$, and that these lie on a
one dimensional subspace for each player.

We start by looking at $\frac{\partial\pi_{i}}{\partial m_{i}}$ and we get by
differentiation of $x_{i}^{*}$ that

$\frac{\partial x_{i}}{\partial m_{i}}=-\sum_{j\neq
i}\frac{1}{m_{j}}\frac{d+\sum_{j\neq
i}\frac{a_{j}-a_{i}}{m_{j}}}{(m_{i}\sum_{j}\frac{1}{m_{j}})^{2}}=-K_{-i}^{2}x_{i}=x_{i}\frac{\partial
x_{i}}{\partial a_{i}}.$ (47)

We now have that

$\displaystyle\frac{\partial\pi_{i}}{\partial m_{i}}$
$\displaystyle=(k_{i}x_{i}+a_{i}-b_{i})\frac{\partial x_{i}}{\partial
m_{i}}+x_{i}^{2}$ (48) $\displaystyle=(k_{i}x_{i}+a_{i}-b_{i})\frac{\partial
x_{i}}{\partial a_{i}}x_{i}+x_{i}^{2}$
$\displaystyle=x_{i}\left((k_{i}x_{i}+a_{i}-b_{i})\frac{\partial
x_{i}}{\partial a_{i}}+x_{i}\right)$
$\displaystyle=x_{i}\frac{\partial\pi_{i}}{\partial a_{i}}.$

Since this means $\frac{\partial\pi_{i}}{\partial
a_{i}}=0\implies\frac{\partial\pi_{i}}{\partial m_{i}}=0$, the first part of
the proof is now done. Given any $m^{+}$, and obtaining $a^{+}(m^{+})$ as per
Theorem 1, $(m^{+},a^{+})$ satisfies the first order stationarity conditions.
We next show that this stationary point gives a local maximum for each player
$i$. From proof of Theorem 1 we have that $\frac{\partial^{2}\pi_{i}}{\partial
a_{i}^{2}}<0$. Let us now evaluate the sign of
$\frac{\partial^{2}\pi_{i}}{\partial m_{i}^{2}}$ at $(m^{+},a^{+})$. We have
that

$\frac{\partial\pi_{i}}{\partial m_{i}}=x_{i}\frac{\partial\pi_{i}}{\partial
a_{i}},$ (49)

and therefore

$\frac{\partial^{2}\pi_{i}}{\partial m_{i}\partial
a_{i}}=\frac{\partial}{\partial a_{i}}\left(\frac{\partial\pi_{i}}{\partial
m_{i}}\right)=\frac{\partial\pi_{i}}{\partial a_{i}}\frac{\partial
x_{i}}{\partial a_{i}}+x_{i}\frac{\partial^{2}\pi_{i}}{\partial a_{i}^{2}}.$
(50)

Then, using that $\frac{\partial x_{i}}{\partial m_{i}}=x_{i}\frac{\partial
x_{i}}{\partial a_{i}}$, we get that

$\frac{\partial}{\partial m_{i}}\left(\frac{\partial\pi_{i}}{\partial
m_{i}}\right)=\frac{\partial}{\partial
m_{i}}\left(x_{i}\frac{\partial\pi_{i}}{\partial
a_{i}}\right)=x_{i}\frac{\partial x_{i}}{\partial
a_{i}}\frac{\partial\pi_{i}}{\partial
a_{i}}+x_{i}\frac{\partial^{2}\pi_{i}}{\partial m_{i}\partial a_{i}},$ (51)

which can be re-arranged by using (50) to obtain

$\frac{\partial^{2}\pi_{i}}{\partial m_{i}^{2}}=2x_{i}\frac{\partial
x_{i}}{\partial a_{i}}\frac{\partial\pi_{i}}{\partial
a_{i}}+x_{i}^{2}\frac{\partial^{2}\pi_{i}}{\partial a_{i}^{2}}.$ (52)

By definition of $(m^{+},a^{+})$, we have that
$\frac{\partial\pi_{i}}{\partial a_{i}}=0$ at $(m^{+},a^{+})$ and therefore

$\frac{\partial^{2}\pi_{i}}{\partial
m_{i}^{2}}=x_{i}^{2}\frac{\partial^{2}\pi_{i}}{\partial a_{i}^{2}}<0,$ (53)

at $(m^{+},a^{+})$. The trace of the hessian $\nabla^{2}\pi_{i}$ is then

$Tr(\nabla^{2}\pi_{i})=\frac{\partial^{2}\pi_{i}}{\partial
m_{i}^{2}}+\frac{\partial^{2}\pi_{i}}{\partial a_{i}^{2}}<0$ (54)

at the equilibrium point $(m^{+},a^{+})$. The determinant at this point is

$\displaystyle\left|\nabla^{2}\pi_{i}\right|$
$\displaystyle=\frac{\partial^{2}\pi_{i}}{\partial
m_{i}^{2}}\frac{\partial^{2}\pi_{i}}{\partial
a_{i}^{2}}-\left(\frac{\partial^{2}\pi_{i}}{\partial m_{i}\partial
a_{i}}\right)^{2}$ (55) $\displaystyle=\left(2x_{i}\frac{\partial
x_{i}}{\partial a_{i}}\frac{\partial\pi_{i}}{\partial
a_{i}}+x_{i}^{2}\frac{\partial^{2}\pi_{i}}{\partial
a_{i}^{2}}\right)\frac{\partial^{2}\pi_{i}}{\partial
a_{i}^{2}}-\left(\frac{\partial^{2}\pi_{i}}{\partial m_{i}\partial
a_{i}}\right)^{2}$
$\displaystyle=x_{i}^{2}\left(\frac{\partial^{2}\pi_{i}}{\partial
a_{i}^{2}}\right)-\left(0+x_{i}\frac{\partial^{2}\pi_{i}}{\partial
a_{i}^{2}}\right)^{2}$ $\displaystyle=0$

where the second line follows from equation (52), the third line follows by
using equation (50) and that $\frac{\partial\pi_{i}}{\partial a_{i}}=0$ at
$(m^{+},a^{+})$. Simple linear algebra tells us that the two eigenvalues of
$\nabla^{2}\pi_{i}$ satisfy

$\begin{aligned} \lambda_{1}\lambda_{2}&=\left|\nabla^{2}\pi_{i}\right|=0\\\
\lambda_{1}+\lambda_{2}&=Tr(\nabla^{2}\pi_{i})<0\end{aligned}\hskip
2.84526pt,$ (56)

which can only happen if $\lambda_{1}<0$ and $\lambda_{2}=0$ (w.l.o.g. we take
$\lambda_{1}\leq\lambda_{2}$. This tells us that the stationary point
$(m^{+},a^{+})$ indeed gives a local maximum for each player $i$, meaning that
$(m^{+},a^{+})$ is a local Nash Equilibrium. Further, since $\lambda_{2}=0$,
unilateral deviations by changing both $m_{i}$ and $a_{i}$ are possible while
still maintaining exactly the same profit level. ∎

######  Result 2.

Under the market clearing given by (SWM), player $i$ is guaranteed non-
negative profit by playing any
$\sigma_{i}\in\mathcal{S}_{R}^{\prime}(i):=\\{(m_{i},a_{i}):m_{i}\geq\frac{1}{2}c_{i},\hskip
2.84526pta_{i}\geq b_{i}\\}$, for any $(m_{-i},a_{-i})$.

###### Proof.

If the player plays strategy $\sigma_{i}\in\mathcal{S}_{R}^{\prime}(i)$ and he
is out of the market ($x_{i}=0$), then $\pi_{i}=0\geq 0$. Let us consider the
case of $x_{i}>0$. By definition of
$v_{z_{i}}=\max\\{m_{j}x_{j}+a_{j}:x_{j}>0;j\in\mathcal{P}_{z_{i}}\\}$ and
since $i\in\mathcal{P}_{z_{i}}$, $x_{i}>0$ we have that

$v_{z_{i}}\geq m_{i}x_{i}+a_{i}$ (57)

The profit of player $i$ can then be lower bounded as

$\pi_{i}=v_{z_{i}}x_{i}-\frac{1}{2}c_{i}x_{i}^{2}-b_{i}x_{i}\geq\left[(2m_{i}-c_{i})x_{i}+(a_{i}-b_{i})\right]x_{i}$
(58)

where the inequality comes by using (57). Since $x_{i}\geq 0$ is guaranteed by
(SWM), and $m_{i}\geq\frac{1}{2}c_{i}$, $a_{i}\geq b_{i}$ by the fact that
$(m_{i},a_{i})\in\mathcal{S}_{R}^{\prime}(i)$, we conclude that $\pi_{i}\geq
0$. The choice of $m_{-i}$ and $a_{-i}$ is absolutely irrelevant for the
argument, and thus this must be the case for any such values. ∎

## Acknowledgements

We would like to thank CGM London Power and Gas desk within Macquarie Group
for partially funding this research, and providing computational resources via
Descartes Labs. Ioan Alexandru Puiu would like to further thank Vincent
Guffens for his support, Descartes Labs for extending the account, and Jeremy
Malczyk and Chris Moulton for offering technical support with the Descartes
Labs platform.

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