# The existence and uniqueness of a power price equilibrium

Miha Troha Mathematical Institute, Oxford University, Andrew Wiles Building,
Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom,
. This author was supported through grants from the Slovene human resources
development and scholarship fund, and the Oxford-Man Institute.
<EMAIL_ADDRESS>Raphael Hauser Mathematical Institute, Oxford University,
Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford
OX2 6GG, United Kingdom, . Associate Professor in Numerical Mathematics, and
Tanaka Fellow in Applied Mathematics at Pembroke College, Oxford. This author
was supported through grant EP/H02686X/1 from the Engineering and Physical
Sciences Research Council of the UK<EMAIL_ADDRESS>
###### Abstract

We propose a term structure power price model that, in contrast to widely
accepted no-arbitrage based approaches, accounts for the non-storable nature
of power. It belongs to a class of equilibrium game theoretic models with
players divided into producers and consumers. The consumers’ goal is to
maximize a mean-variance utility function subject to satisfying an inelastic
demand of their own clients (e.g households, businesses etc.) to whom they
sell the power. The producers, who own a portfolio of power plants each
defined by a running fuel (e.g. gas, coal, oil…) and physical characteristics
(e.g. efficiency, capacity, ramp up/down times…), similarly, seek to maximize
a mean-variance utility function consisting of power, fuel, and emission
prices subject to production constraints. Our goal is to determine the term
structure of the power price at which production matches consumption. In this
paper we show that in such a setting the equilibrium price exists and also
discuss the conditions for its uniqueness.

###### keywords:

term structure, electricity, game theory, mean-variance, optimization, KKT
conditions.

##  1 Introduction

The electricity price has some unique features which are very distinct from
features of stock prices or other commodities. The literature aiming at
modeling the electricity price can be broadly divided into non-structural,
structural, and game theoretic approaches.

Non-structural approaches model electricity price directly, without
considering the underlying reasons that cause the price to change over time. A
good overview of regular patterns and statistical properties of electricity
prices, together with one and multi-factor Ornstein-Uhlenbeck process models
of the spot price, is given in [20]. Since Gaussian distributions are not
suitable for modeling spikes, a combination of Ornstein-Uhlenbeck and pure
jump-processes have been proposed in [17]. Further developments include using
also Levy processes (see [22] for example). Even though such models are good
for modeling the spot price, their application for pricing derivatives is
sometimes rather questionable. As pointed out in [3], a classical buy-and-hold
strategy is not applicable to non-storable commodities. All non-structural
models define the price of a forward contract as the expected price of the
security at the delivery under the risk-neutral measure, conditioned on some
filtration, which contains all available market information. It is often
assumed that all available market information is included in the spot price
(i.e. the spot price is a martingale). This is a good approximation for stock
prices, but definitely false for non-storable commodities. Information about a
huge electricity demand increase in a week time, for example, does not have
any impact on today’s spot price. The correct filtration must hence include
information on weather forecasts, planned power plant outages etc. It can be
argued that all the information is contained in prices of forward contracts
and other derivatives. A one-factor model in [14] and multi-factor term-
structure model in [13] are the first that produce prices that are consistent
with observable forward prices.

Another interesting aspect of electricity is its relation to other fuels.
Noting that coal, gas, oil, etc. can be used to produce electricity,
electricity itself can be treated as a derivative with various fuel and
emission prices as the underlying. [2] produced seminal work that used a
supply and demand stack to model electricity prices. The supply stack was
further extended by [18] and by [9] to include various fuels. Using an
exponential bid stack, they calculated spot electricity price as a function of
demand and spot fuel prices. These models belong to a class of structural
approaches since they try to capture some of the physical properties of the
electricity markets.

The third, game theoretic approach, models the physical properties and
decisions of market participants more closely. As pointed out in [24], models
that include ramp up/down constraints of power plants and study the impact of
long term contracts on the spot electricity prices, are needed in order to
prevent and explain disastrous events an example of which happened in
California in 2001 and cost the state as much as $45 billion. The seminal work
of game theoretic models in electricity markets was produced by [5]: a unique
relation between a forward and a spot price is given in a two-stage market
with one producer and one consumer, who each want to maximize their mean-
variance objective function. This work has been used to study benefits of
derivatives in the electricity market, see [11]. It was extended to a multi-
stage setting with a dynamic equilibrium by [8], and [7]. [15] have extended
the two-stage mean-variance model to any convex risk measure, while also
taking into account liquidity constraints.

The game theoretic approach attempts to model decisions of producers and
consumers that participate in the electricity market explicitly. Since a
widely used strategy for risk management is delta hedging, it is important to
know that the delta hedging strategy can be implemented as a minimum variance
strategy in the mean-variance portfolio framework (see [1] for details).

Our work belongs to a class of game theoretic approaches. It extends the model
of [7] to more than one producer and consumer, who maximize their mean-
variance utility functions. In contrast to other game theoretic models, we
include capacity and ramp up/down constraints of power plants. Following ideas
from structural approaches, the profit of power plants is modeled as a
difference between electricity price and fuel costs (including emissions).
Since we model each power plant directly, we do not have to make any
assumptions on the bid stack, which is in our case determined by the physical
properties of power plants. As in [13], our model is consistent with
observable fuel and emission prices. We do not focus on specific emission
market schemes, but rather use a simplified version of it. For a detailed
treatment of emission market in the game theoretic setting see [10] and [21].

The literature distinguishes between dynamic (i.e. stochastic) and static
models. In dynamic models, players adapt to changing environment (i.e. fuel
prices, demand etc.) by adapting their decisions. In each stage of their
decision making they determine the optimal decisions they have to take now and
also the optimal decisions they will take in the future under all possible
changes in the environment. Since future decisions affect present decisions
this is computationally very demanding. In static models on the other hand,
players assume that they know the future state of the environment exactly and
can thus stick to an initial plan about future decisions, regardless of the
changes in the environment. Such approaches are computationally much more
tractable, but they do not reflect the reality very well. The model we propose
here can be seen as hybrid of both approaches. The initial optimization
problem is static, where players determine all their optimal decisions and
assume they will not alter them in the future. However, as the environment
changes, players may take recursive actions by calculating new optimal
decisions while taking into account the new state of the environment, as well
as their decisions taken under the previous state of the environment. Since in
such setting the recursive actions are not very dependent, we will mainly
focus on each of them separately.

This paper is organized as follows: In Section 2 we give a detailed
mathematical description of the model, and in Section 3 we proof that the
solution to our model exists and develop the conditions under which it is also
unique. We conclude the paper in Section 4.

##  2 Problem description

The model we propose can be used to determine the term structure of
electricity prices. The electricity market is defined by a set of producers
$P$ of cardinality $0<\left|P\right|<\infty$, a set of consumers $C$ of
cardinality $0<\left|C\right|<\infty$, and a hypothetical market agent. Each
of the producers and consumers participates in the electricity market in order
to maximize their profit subject to a risk budget under a mean-variance
optimization framework. Producers own a number of power plants, which can have
different physical characteristics and run on different fuels. The set of all
fuels is denoted by $L$. Sets $R^{p,l}$ represent all power plants owned by
producer $p\in P$ that run on fuel $l\in L$.

We are interested in delivery times $T_{j}$, $j\in
J=\left\\{1,...,T^{\prime}\right\\}$, where power for each delivery time
$T_{j}$ can be traded through numerous forward contracts at times $t_{i}$,
$i\in I_{j}$. The electricity price at time $t_{i}$ for delivery at time
$T_{j}$ is denoted by $\Pi\left(t_{i},T_{j}\right)$. Since contracts with
trading time later than delivery time do not exist, we require
$t_{\max\left\\{I_{j}\right\\}}=T_{j}$ for all $j\in J$. The number of all
forward contracts, i.e. $\sum_{j\in J}\left|I_{j}\right|$, is denoted by $N$.
Uncertainty is modeled by a filtered probability space
$\left(\Omega,\mathcal{F},\mathbb{F}=\left\\{\mathcal{F}_{t},t\in
I\right\\},\mathbb{P}\right)$, where $I=\cup_{j\in J}I_{j}$. The
$\sigma$-algebra $\mathcal{F}_{t}$ represents information available at time
$t$.

The exogenous variables that appear in our model are (a) aggregate power
demand $D\left(T_{j}\right)$ for each delivery period $j\in J$, (b) prices of
fuel forward contracts $G_{l}\left(t_{i},T_{j}\right)$ for each fuel $l\in L$,
delivery period $j\in J$, and trading period $i\in I_{j}$, and (c) prices of
emissions forward contracts $G_{em}\left(t_{i},T_{j}\right)$, $j\in J$, $i\in
I_{j}$. Electricity prices and all exogenous variables are assumed to be
adapted to the filtration $\left\\{\mathcal{F}_{t}\right\\}_{t\in I}$ and have
finite second moments.

Let $v_{k}\in\mathbb{R}^{n_{k}}$, $n_{k}\in\mathbb{N}$, $k\in K$, and
$K=\left\\{1,...,\left|K\right|\right\\}$ be given vectors. For convenience,
we define a vector concatenation operator as

$\left|\right|_{k\in
K}v_{k}=\left[v_{1}^{\top},...,v_{\left|K\right|}^{\top}\right]^{\top}.$

###  2.1 Producer

Each producer $p\in P$ has to decide on the number
$V_{p}\left(t_{i},T_{j}\right)$ of electricity forward contracts and the
number $F_{p,l}\left(t_{i},T_{j}\right)$, $l\in L$ of fuel forward contracts
to buy, at trading time $t_{i}$, $i\in I_{j}$ for delivery at time $T_{j}$,
$j\in J$. By $W_{p,l,r}\left(T_{j}\right)$ we denote the actual production
from fuel $l\in L$ at time $T_{j}$ from power plant $r\in R^{p,l}$, and by
$O_{p}\left(t_{i},T_{j}\right)$ we denote the number of emission forward
contracts purchased at time $t_{i}$ for delivery at time $T_{j}$.

Notation is greatly simplified if decision variables are concatenated into

  * •

electricity trading vectors $V_{p}\left(T_{j}\right)=\left|\right|_{i\in
I_{j}}V_{p}\left(t_{i},T_{j}\right)$ and $V_{p}=\left|\right|_{j\in
J}V_{p}\left(T_{j}\right)$,

  * •

fuel trading vectors $F_{p}\left(t_{i},T_{j}\right)=\left|\right|_{l\in
L}F_{p,l}\left(t_{i},T_{j}\right)$,
$F_{p}\left(T_{j}\right)=\left|\right|_{i\in
I_{j}}F_{p}\left(t_{i},T_{j}\right)$, and $F_{p}=\left|\right|_{j\in
J}F_{p}\left(T_{j}\right)$,

  * •

emission trading vectors $O_{p}\left(T_{j}\right)=\left|\right|_{i\in
I_{j}}O_{p}\left(t_{i},T_{j}\right)$ and $O_{p}=\left|\right|_{j\in
J}O_{p}\left(T_{j}\right)$,

  * •

electricity production vectors $W_{p,l}\left(T_{j}\right)=\left|\right|_{r\in
R^{p,l}}W_{p,l,r}\left(T_{j}\right)$,
$W_{p}\left(T_{j}\right)=\left|\right|_{l\in L}W_{p,l}\left(T_{j}\right)$, and
$W_{p}=\left|\right|_{j\in J}W_{p}\left(T_{j}\right)$,

and finally
$v_{p}=\left[V_{p}^{\top},F_{p}^{\top},O_{p}^{\top},W_{p}^{\top}\right]^{\top}$.

Similarly, we define

  * •

electricity price vectors $\Pi\left(T_{j}\right)=\left|\right|_{i\in
I_{j}}\Pi\left(t_{i},T_{j}\right)$, and $\Pi=\left|\right|_{j\in
J}e^{-\hat{r}T_{j}}\Pi\left(T_{j}\right)$, where $\hat{r}\in\mathbb{R}$ is a
constant interest rate,

  * •

fuel price vectors $G\left(t_{i},T_{j}\right)=\left|\right|_{l\in
L}G_{l}\left(t_{i},T_{j}\right)$, $G\left(T_{j}\right)=\left|\right|_{i\in
I_{j}}G\left(t_{i},T_{j}\right)$, and $G=\left|\right|_{j\in
J}e^{-\hat{r}T_{j}}G\left(T_{j}\right)$,

  * •

emission price vector $G_{em}\left(T_{j}\right)=\left|\right|_{i\in
I_{j}}G_{em}\left(t_{i},T_{j}\right)$, and $G_{em}=\left|\right|_{j\in
J}e^{-\hat{r}T_{j}}G_{em}\left(T_{j}\right)$,

and finally
$\pi_{p}=\left[\Pi^{\top},G^{\top},G_{em}^{\top},\underset{\dim\left(W_{p}\right)}{\underbrace{0,...,0}}\right]^{\top}$,
where number of zeros matches the dimension of vector $W_{p}$.

The profit $P_{p}\left(v_{p},\pi_{p}\right)$ of producer $p\in P$ can be
calculated as

(1) $P_{p}\left(v_{p},\pi_{p}\right)=\sum_{j\in
J}e^{-\hat{r}T_{j}}\left(\sum_{i\in
I_{j}}P_{p}^{t_{i},T_{j}}\left(v_{p},\pi_{p}\right)\right)$

where the profit $P_{p}^{t_{i},T_{j}}\left(v_{p},\pi_{p}\right)$ for each
$i\in I_{j}$ and $j\in J$ can be calculated as

$P_{p}^{t_{i},T_{j}}\left(v_{p},\pi_{p}\right)=-\Pi\left(t_{i},T_{j}\right)V_{p}\left(t_{i},T_{j}\right)-O_{p}\left(t_{i},T_{j}\right)G_{em}\left(t_{i},T_{j}\right)-\sum_{l\in
L}G_{l}\left(t_{i},T_{j}\right)F_{p,l}\left(t_{i},T_{j}\right).$

Ramping up and down constraints for each
$j\in\left\\{1,...,T^{\prime}-1\right\\}$, where $T^{\prime}$ denotes the last
delivery period, $l\in L$ and $r\in R^{p,l}$ can be expressed as

(2) $\triangle\overline{W}_{min}^{p,l,r}\leq
W_{p,l,r}\left(T_{j+1}\right)-W_{p,l,r}\left(T_{j}\right)\leq\triangle\overline{W}_{max}^{p,l,r}.$

For each power plant $r\in R^{p,l}$, $\triangle\overline{W}_{max}^{p,l,r}$ and
$\triangle\overline{W}_{min}^{p,l,r}$ represent maximum rates for ramping up
and down, respectively. Similarly, $\overline{W}_{max}^{p,l,r}$ denotes the
maximum production and thus we can write the capacity constraints for each
power plant $r\in R^{p,l}$ as

(3) $0\leq W_{p,l,r}\left(T_{j}\right)\leq\overline{W}_{max}^{p,l,r}.$

Additionally, for each $j\in J$ the electricity sold in the forward market
must equal the actually produced electricity, i.e.

(4) $-\sum_{i\in I_{j}}V_{p}\left(t_{i},T_{j}\right)=\sum_{l\in L}\sum_{r\in
R^{p,l}}W_{p,l,r}\left(T_{j}\right)$

and sufficient amount of fuel $l\in L$ must have been bought for each $j\in
J$, i.e.

(5) $\sum_{r\in R^{p,l}}W_{p,l,r}\left(T_{j}\right)c^{p,l,r}=\sum_{i\in
I_{j}}F_{p,l}\left(t_{i},T_{j}\right)$

where $c^{p,l,r}>0$ is the efficiency of power plant $r$.

The carbon emission obligation constraint can be written as

(6) $\sum_{j\in J}\sum_{i\in I_{j}}O\left(t_{i},T_{j}\right)=\sum_{j\in
J}\sum_{l\in L}\sum_{r\in R^{p,l}}W_{p,l,r}\left(T_{j}\right)g^{l},$

where $g^{l}>0$ denotes the carbon emission intensity factor for fuel $l\in
L$. This constraint ensures that enough emission certificates have been bought
to cover the electricity production over the whole planning horizon.

We bound the number of fuel forward contracts traded for each $i\in I_{j}$,
$j\in J$, and $l\in L$ as

(7) $-F_{trade}\leq F_{p,l}\left(t_{i},T_{j}\right)\leq F_{trade}$

for some large $F_{trade}>0$ and the number of emission forward contracts
traded as

(8) $-F_{trade}\leq O\left(t_{i},T_{j}\right)\leq F_{trade}.$

Additionally, we also bound the number of electricity forward contracts traded
for each $i\in I_{j}$, and $j\in J$ as

(9) $-V_{trade}\leq V_{p}\left(t_{i},T_{j}\right)\leq V_{trade}$

for some large $V_{trade}>0$. Boundedness of the number of the electricity
forward contracts traded enables us to use some of game theoretic theory that
only applies to compact sets. If $V_{trade}$ is set correctly (see Corollary
12), then the optimal $V_{p}\left(t_{i},T_{j}\right)$ can never lie on the
boundary and hence the constraint does not have any impact on the optimal
solution.

Producers would like to maximize their profit subject to a risk budget. Under
a mean-variance optimization framework they are interested in the mean-
variance utility

$\begin{array}[]{rcl}\Psi_{p}\left(v_{p}\right)&=&\mathbb{E}^{\mathbb{P}}\left[P_{p}\left(v_{p},\pi_{p}\right)\right]-\frac{\lambda_{p}}{2}\text{Var}^{\mathbb{P}}\left[P_{p}\left(v_{p},\pi_{p}\right)\right]\\\
\\\
&=&-\mathbb{E}^{\mathbb{P}}\left[\pi_{p}\right]^{\top}v_{p}-\frac{1}{2}\lambda_{p}v_{p}^{\top}Q_{p}v_{p},\end{array}$

where $\lambda_{p}>0$ is their risk preference and
$Q_{p}:=\mathbb{E}^{\mathbb{P}}\left[\left(\pi_{p}-\mathbb{E}^{\mathbb{P}}\left[\pi_{p}\right]\right)\left(\pi_{p}-\mathbb{E}^{\mathbb{P}}\left[\pi_{p}\right]\right)^{\top}\right]$
an “extended” covariance matrix. Their objective is to solve the following
optimization problem

(10) $\begin{array}[]{rl}\Phi_{p}=\underset{v_{p}}{\text{max
}}&\Psi_{p}\left(v_{p}\right)\end{array}$

subject to (2), (3), (4), (5), (6), (7), (8), and (9).

###  2.2 Consumer

We make the assumption that demand is completely inelastic and that each
consumer $c\in C$ is responsible for satisfying a proportion
$p_{c}\in\left[0,1\right]$ of the total demand $D\left(T_{j}\right)$ at time
$T_{j}$, $j\in J$. Since $p_{c}$ is a proportion clearly $\sum_{c\in
C}p_{c}=1.$

For further argumentation let us define electricity trading vectors
$V_{c}\left(T_{j}\right)=\left|\right|_{i\in
I_{j}}V_{c}\left(t_{i},T_{j}\right)$ and $V_{c}=\left|\right|_{j\in
J}V_{c}\left(T_{j}\right)$.

Consumer’s profit can be calculated as

(11) $P_{c}\left(V_{c},\Pi\right)=\sum_{j\in
J}e^{-\hat{r}T_{j}}\left(\sum_{i\in
I_{j}}-\Pi\left(t_{i},T_{j}\right)V_{c}\left(t_{i},T_{j}\right)+s_{c}p_{c}D\left(T_{j}\right)\right),$

where $\hat{r}\in\mathbb{R}$ denotes a constant interest rate and
$s_{c}\in\mathbb{R}$ denotes a contractually fixed price that consumer $c\in
C$ receives for selling the electricity further to households or businesses.
Demand is expected to be satisfied for each $T_{j}$, i.e.

(12) $\sum_{i\in
I_{j}}V_{c}\left(t_{i},T_{j}\right)=p_{c}D\left(T_{j}\right).$

At the time of calculating the optimal decisions, consumers assume that they
know the future realization of demand $D\left(T_{j}\right)$ precisely. If the
knowledge about the future realization of the demand changes, then players can
take recursive actions by recalculating their optimal decisions with the
updated demand forecast.

Note further that the contractually fixed price $s_{c}$ only affects the
optimal objective value of consumer $c\in C$, but not also his optimal
solution. Since we are primarily interested in optimal solutions, we simplify
the notation and set $s_{c}=0$. The correct optimal value can always be
calculated via post-processing when an optimal solution is already known. This
is sometimes needed for risk management purposes.

We bound the number of the electricity forward contracts for each $i\in I_{j}$
and $j\in J$ a consumer is allowed to trade as

(13) $-V_{trade}\leq V_{c}\left(t_{i},T_{j}\right)\leq V_{trade}$

for some large $V_{trade}>0$ for similar reasons as for producers. We will
determine the appropriate $V_{trade}$ in Corollary 12.

Consumers would like to maximize their profit subject to a risk budget. Under
a mean-variance optimization framework they are interested in the mean-
variance utility

$\begin{array}[]{rcl}\Psi_{c}\left(V_{c}\right)&=&\mathbb{E}^{\mathbb{P}}\left[P_{c}\left(V_{c},\Pi\right)\right]-\frac{\lambda_{p}}{2}\text{Var}^{\mathbb{P}}\left[P_{c}\left(V_{c},\Pi\right)\right]\\\
\\\
&=&-\mathbb{E}^{\mathbb{P}}\left[\Pi\right]^{\top}V_{c}-\frac{\lambda}{2}V_{c}^{\top}Q_{c}V_{c},\end{array}$

where $\lambda_{c}>0$ is their risk preference and
$Q_{c}:=\mathbb{E}^{\mathbb{P}}\left[\left(\Pi-\mathbb{E}^{\mathbb{P}}\left[\Pi\right]\right)\left(\Pi-\mathbb{E}^{\mathbb{P}}\left[\Pi\right]\right)^{\top}\right]$
a covariance matrix. Their objective is to solve the following optimization
problem

(14) $\Phi_{c}=\underset{V_{c}}{\text{max }}\Psi_{c}\left(V_{c}\right)$

subject to (12) and (13).

###  2.3 The hypothetical market agent

Given the electricity $\Pi$, fuel $G$, and emission $G_{em}$ price vectors,
each producer $p\in P$ and each consumer $c\in C$ can calculate their optimal
electricity trading vectors $V_{p}$ and $V_{c}$ by solving (10) and (14),
respectively. However, the players are not necessary able to execute their
calculated optimal trading strategies because they may not find the
counterparty to trade with. In reality each contract consists of a buyer and a
seller, which imposes an additional constraint (also called the market
clearing constraint) that matches the number of short and long electricity
forward contracts for each $i\in I_{j}$ and $j\in J$ as

(15) $\sum_{c\in C}V_{c}\left(t_{i},T_{j}\right)+\sum_{p\in
P}V_{p}\left(t_{i},T_{j}\right)=0.$

The electricity market is responsible for satisfying this constraint by
matching buyers with sellers. The matching is done through sharing of the
price and order book information among all market participants. If at the
current price there are more long contract than short contracts, it means that
the current price is too low and asks will start to be submitted at higher
prices. The converse occurs, if there are more short contracts than long
contracts. Eventually, the electricity price at which the number of long and
short contracts matches is found. At such a price, (15) is satisfied
“naturally” without explicitly requiring the players to satisfy it. They do so
because it is in their best interest, i.e. it maximizes their mean-variance
objective functions.

The question is how to formulate such a “natural” constraint in an
optimization framework. A naive approach of writing the market clearing
constraint as an ordinary constraint forces the players to satisfy it
regardless of the price. We need a mechanism that models the matching of
buyers and sellers as it is performed by the electricity market. For this
purpose, we introduce a hypothetical market agent that is allowed to slowly
change electricity prices to ensure that (15) is satisfied.

Assume first that the feasible set of the hypothetical agent must satisfy

(16)
$-\Pi_{max}\leq\mathbb{E}^{\mathbb{P}}\left[\Pi\left(t_{i},T_{j}\right)\right]\leq\Pi_{max}$

for all $i\in I_{j}$, $j\in J$, and for some large $\Pi_{max}>0$. We write
(16) in matrix notation as

(17) $B_{M}\mathbb{E}^{\mathbb{P}}\left[\Pi\right]\leq b_{M}$

for some $B_{M}\in\mathbb{R}^{2N\times N}$ and $b_{M}\in\mathbb{R}^{2N}$.
Boundedness of prices is a reasonable assumption, because unbounded prices
lead to infinite cash flows, which must result in the bankruptcy of one of the
counterparties involved. Besides limiting the possibility of the bankruptcy of
the market players, the boundedness of prices also allows us to use game
theoretic results that only apply to compact sets. If $\Pi_{max}$ is set large
enough (see Lemma 10), then the optimal
$\mathbb{E}^{\mathbb{P}}\left[\Pi\right]$ can never be on the boundary of the
feasible region. Hence the constraint does not have any practical impact on
the equilibrium electricity price, but, as we will see later, it significantly
simplifies the theoretical analysis.

Let the hypothetical market agent have the following profit function

(18) $P_{M}\left(\Pi,V\right)=\sum_{j\in J}e^{-\hat{r}T_{j}}\left[\sum_{i\in
I_{j}}\Pi\left(t_{i},T_{j}\right)\left(\sum_{c\in
C}V_{c}\left(t_{i},T_{j}\right)+\sum_{p\in
P}V_{p}\left(t_{i},T_{j}\right)\right)\right]$

and the expected profit

(19)
$\Psi_{M}\left(\mathbb{E}^{\mathbb{P}}\left[\Pi\right],V\right)=\mathbb{E}^{\mathbb{P}}\left[P_{M}\left(V,\Pi\right)\right],$

where $V=\left[V_{P}^{\top},V_{C}^{\top}\right]^{\top}$,
$V_{P}=\left|\right|_{p\in P}V_{p}$, and $V_{C}=\left|\right|_{c\in C}V_{c}$
and let the hypothetical market agent attempt to solve

(20)
$\Phi_{M}\left(V\right)=\underset{\mathbb{E}^{\mathbb{P}}\left[\Pi\right]}{\text{max
}}\Psi_{M}\left(\mathbb{E}^{\mathbb{P}}\left[\Pi\right],V\right)$

subject to (17). The KKT conditions for (20) in the matrix notation read

(21) $\begin{array}[]{rcl}\sum_{c\in C}V_{c}+\sum_{p\in
P}V_{p}-B_{M}^{\top}\eta_{M}&=&0\\\ \\\
\left(B_{M}\mathbb{E}^{\mathbb{P}}\left[\Pi\right]-b_{M}\right)^{\top}\eta_{M}&=&0\\\
\\\ B_{M}\mathbb{E}^{\mathbb{P}}\left[\Pi\right]-b_{M}&\leq&0\\\ \\\
\eta_{M}&\geq&0,\end{array}$

where $\eta_{M}$ denotes the dual variables of (17). Since the optimal
$\mathbb{E}^{\mathbb{P}}\left[\Pi\right]$ can never be on the boundary of the
feasible region (see Lemma 10), we can conclude that $\eta_{M}=0$. Optimality
conditions (21) are then simplified to (15). Therefore, we can conclude that
(15) and (20) are equivalent, if $\eta_{M}=0$ truly holds. The relationship
between (15) and (20) will be investigated rigorously in Proposition 11.

Note, that the equivalence of (15) and (20) is a theoretical result that has
to be applied with caution in an algorithmic framework. Formulation (20) is
clearly unstable since only a small mismatch in the market clearing constraint
sends the prices to $\pm\Pi_{max}$. A description of an algorithm for the
computation of the equilibrium electricity price exceeds the scope of this
paper and will be examined separately.

For the further argumentation we define $v_{P}=\left|\right|_{p\in P}v_{p}$
and $v=\left[v_{P}^{\top},V_{C}^{\top}\right]^{\top}$.

###  2.4 Construction of the equilibrium electricity price process

In this subsection we take a closer look at the electricity price process
$\Pi\left(t_{i},T_{j}\right)_{i\in I_{j}}$, where the expectation of the
process is defined internally as a decision of the hypothetical market agent
in order to match supply and demand.

Since electricity prices $\Pi\left(t_{i},T_{j}\right)_{i\in I_{j}}$ are
adapted for any fixed $j\in J$ and
$\left|\mathbb{E}^{\mathbb{P}}\left[\Pi\right]\right|<\infty$ (see Constraint
(16)), they can be uniquely decomposed, using the Doob Decomposition Theorem,
into a sum of a martingale process $M\left(t_{i},T_{j}\right)_{i\in I_{j}}$
and an integrable predictable process $A\left(t_{i},T_{j}\right)_{i\in
I_{j}}$, $A\left(t_{0},T_{j}\right)=0$, such that

$\Pi\left(t_{i},T_{j}\right)=A\left(t_{i},T_{j}\right)+M\left(t_{i},T_{j}\right)$

for every $i\in I_{j}$. Define

$M\left(t_{i},T_{j}\right):=\Pi\left(t_{0},T_{j}\right)+\sum_{k=1}^{i}\left(\Pi\left(t_{k},T_{j}\right)-\mathbb{E}^{\mathbb{P}}\left[\Pi\left(t_{k},T_{j}\right)\left|\mathcal{F}_{k-1}\right.\right]\right)$

and

$A\left(t_{i},T_{j}\right):=\sum_{k=1}^{i}\left(\mathbb{E}^{\mathbb{P}}\left[\Pi\left(t_{k},T_{j}\right)\left|\mathcal{F}_{k-1}\right.\right]-\Pi\left(t_{k-1},T_{j}\right)\right),$

where $\mathcal{F}_{k}:=\mathcal{F}_{t_{k}}$. It is easy to see that
$M\left(t_{i},T_{j}\right)$ is a martingale since

$\mathbb{E}^{\mathbb{P}}\left[M\left(t_{i},T_{j}\right)-M\left(t_{i-1},T_{j}\right)\left|\mathcal{F}_{i-1}\right.\right]=0\quad\text{a.s.}$

Moreover, $A\left(t_{i},T_{j}\right)$ is predictable, that is,
$\mathcal{F}_{i-1}$-measurable.

Note that

$\mathbb{E}^{\mathbb{P}}\left[\Pi\left(t_{i},T_{j}\right)\left|\mathcal{F}_{i-1}\right.\right]=A\left(t_{i},T_{j}\right)+M\left(t_{i-1},T_{j}\right),$

and

(22)
$\mathbb{E}^{\mathbb{P}}\left[\Pi\left(t_{i},T_{j}\right)\left|\mathcal{F}_{0}\right.\right]=\mathbb{E}^{\mathbb{P}}\left[A\left(t_{i},T_{j}\right)\left|\mathcal{F}_{0}\right.\right]+M\left(t_{0},T_{j}\right).$

Let us now allow the hypothetical market agent to choose

$\tilde{A_{i}}\left(T_{j}\right):=\left[\tilde{A}\left(t_{i+1},T_{j}\right),\mathbb{E}^{\mathbb{P}}\left[\tilde{A}\left(t_{i+2},T_{j}\right)\left|\mathcal{F}_{i}\right.\right],...,\mathbb{E}^{\mathbb{P}}\left[\tilde{A}\left(t_{\max\left\\{I_{j}\right\\}},T_{j}\right)\left|\mathcal{F}_{i}\right.\right]\right]^{\top}$

at time $t_{i}$, $i\in I_{j}$ for all $j\in J$. We then model
$\Pi\left(t_{i},T_{j}\right)$ with a new probability measure
$\tilde{\mathbb{P}}$ where $A\left(t_{i},T_{j}\right)$ is defined internally
and not by Doob decomposition of $\Pi\left(t_{i},T_{j}\right)$ as before. More
formally, we define a new probability measure
$\tilde{\mathbb{P}}:\mathcal{F}\times\mathbb{R}^{N}\rightarrow\left[0,1\right]$
such that for any fixed $j\in J$, $i\in I_{j}$, and for all
$D\in\mathcal{F}_{i-1}$

$\begin{array}[]{rcl}\tilde{\mathbb{P}}\left(\Pi\left(t_{i},T_{j}\right)\in
D;\tilde{A}_{i-1}\left(T_{j}\right)\right)&=&\mathbb{P}\left(M\left(t_{i},T_{j}\right)+\tilde{A}\left(t_{i},T_{j}\right)\in
D\right)\\\ \\\
&=&\mathbb{P}\left(M\left(t_{i},T_{j}\right)+A\left(t_{i},T_{j}\right)+\tilde{A}\left(t_{i},T_{j}\right)-A\left(t_{i},T_{j}\right)\in
D\right)\\\ \\\
&=&\mathbb{P}\left(\Pi\left(t_{i},T_{j}\right)+\tilde{A}\left(t_{i},T_{j}\right)-A\left(t_{i},T_{j}\right)\in
D\right)\\\ \\\
&=&\mathbb{P}\left(\Pi\left(t_{i},T_{j}\right)\in\varphi\left(D,\tilde{A}\left(t_{i},T_{j}\right)-A\left(t_{i},T_{j}\right)\right)\right)\end{array}$

where $\varphi:\mathcal{F}\times\mathbb{R}\rightarrow\mathcal{F}$ denotes a
translation of a set, i.e. $\varphi\left(D,\Delta d\right):=\left\\{d:d+\Delta
d\in D\right\\}$. Since

(23)
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{k},T_{j}\right)\left|\mathcal{F}_{i}\right.;\tilde{A_{i}}\left(T_{j}\right)\right]=\mathbb{E}^{\mathbb{P}}\left[\tilde{A}\left(t_{k},T_{j}\right)\left|\mathcal{F}_{i}\right.\right]+M\left(t_{i},T_{j}\right)$

where $k\in\left\\{i+1,...,\max\left\\{I_{j}\right\\}\right\\}$, this
selection can be interpreted as determining the term structure of the expected
power price relative to the current value of $M\left(t_{i},T_{j}\right)$.

At time $t_{0}$, when players calculate their optimal decisions for the first
time, they assume that they will execute their strategies without any future
alterations. However, we allow recourse at a later step but this is not taken
into account at time $t_{0}$. Therefore,

(24)
$\tilde{A}\left(t_{i},T_{j}\right)=\mathbb{E}^{\mathbb{P}}\left[\tilde{A}\left(t_{i},T_{j}\right)\left|\mathcal{F}_{k}\right.\right]$

for all $k\in\left\\{0,...,i-1\right\\}$, $j\in J$, and $i\in I_{j}$. In such
a setting, it is enough to determine $\tilde{A}_{0}\left(T_{j}\right)$, since
all other expected prices can be derived from these using (23) and (24).

The variance under the new probability measure $\tilde{\mathbb{P}}$ can be
calculated as

(25)
$\begin{array}[]{rcl}\text{Var}^{\tilde{\mathbb{P}}}\left(\Pi\left(t_{i},T_{j}\right)\left|\mathcal{F}_{0}\right.;\tilde{A_{i}}\left(T_{j}\right)\right)&=&\text{Var}^{\mathbb{P}}\left(\tilde{A}\left(t_{i},T_{j}\right)+M\left(t_{i},T_{j}\right)\left|\mathcal{F}_{0}\right.\right)\\\
\\\
&=&\text{Var}^{\mathbb{P}}\left(M\left(t_{i},T_{j}\right)\left|\mathcal{F}_{0}\right.\right).\end{array}$

We can see that the variance depends only on the process
$M\left(t_{i},T_{j}\right)$ and can not be influenced by the hypothetical
market agent. Using a similar reasoning as in (25), we can also conclude

(26)
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\left(\pi_{p}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi_{p};\tilde{A}_{0}\right]\right)\left(\pi_{p}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi_{p};\tilde{A}_{0}\right]\right)^{\top};\tilde{A}_{0}\right]=Q_{p}$

and

(27)
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\left(\Pi-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi;\tilde{A}_{0}\right]\right)\left(\Pi-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi;\tilde{A}_{0}\right]\right)^{\top};\tilde{A}_{0}\right]=Q_{c},$

where $\tilde{A}_{0}=\left|\right|_{j\in J}\tilde{A}_{0}\left(T_{j}\right)$.

Without loss of generality, we may set $M\left(t_{0},T_{j}\right)=0$. Then
from (23) and (24),

(28)
$\begin{array}[]{rcl}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i},T_{j}\right)\left|\mathcal{F}_{0}\right.;\tilde{A_{i}}\left(T_{j}\right)\right]&=&\mathbb{E}^{\mathbb{P}}\left[\tilde{A}\left(t_{i},T_{j}\right)\left|\mathcal{F}_{0}\right.\right]\\\
\\\ &=&\tilde{A}\left(t_{i},T_{j}\right)\end{array}$

for all $i\in I_{j}$ and $j\in J$. Allowing the hypothetical market agent to
choose $\tilde{A}\left(t_{i},T_{j}\right)$ is thus the same as allowing it to
choose
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i},T_{j}\right)\left|\mathcal{F}_{0}\right.;\tilde{A}\left(t_{i},T_{j}\right)\right]$.
In the rest of the paper, we simplify the notation by writing
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i},T_{j}\right)\right]$ when
we actually mean
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i},T_{j}\right)\left|\mathcal{F}_{0}\right.;\tilde{A}\left(t_{i},T_{j}\right)\right]$.

The measure $\tilde{\mathbb{P}}$ corresponds to a physical measure and
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\cdot\right]$ corresponds to a real
world expectation. The hypothetical market agent is allowed to choose
$\tilde{A}\left(t_{i},T_{j}\right)$ and consequently the physical measure
$\tilde{\mathbb{P}}$. The aim of the hypothetical market agent could be
interpreted as finding the physical measure $\tilde{\mathbb{P}}$ that is
consistent with the facts (e.g. fuel forward prices, emission prices, demand
forecasts etc.) observable in the real world.

###  2.5 Matrix notation

The analysis of the problem is greatly simplified if a more compact notation
is introduced. In this subsection we will also rewrite equations using the new
probability measure $\tilde{\mathbb{P}}$ instead of $\mathbb{P}$ where
applicable.

The profit of producer $p\in P$ can be written as

$P_{p}\left(v_{p},\pi_{p}\right)=-\pi_{p}^{\top}v_{p}.$

The equality constraints can be expressed as

$A_{p}v_{p}=0$

and the inequality constraints as

$B_{p}v_{p}\leq b_{p}$

for some
$A_{p}\in\mathbb{R}^{\left|J\right|\left(\left|L\right|+1\right)+1\times\dim
v_{p}}$, $B_{p}\in\mathbb{R}^{n_{p}\times\dim v_{p}}$ and
$b_{p}\in\mathbb{R}^{n_{p}}$, where $n_{p}$ denotes the number of all
inequality constraints of producer $p\in P$. Define a feasible set

$S_{p}:=\left\\{v_{p}:A_{p}v_{p}=a_{p}\text{ \text{and }}B_{p}v_{p}\leq
b_{p}\right\\}.$

It is useful to investigate the inner structure of the matrices. By
considering equality constraints (4), (5), and (6) we can see that

(29) $A_{p}=\left[\begin{array}[]{ccc}\hat{A}_{1}&0&\hat{A}_{3,p}\\\
0&\hat{A}_{2}&\hat{A}_{4,p}\end{array}\right]$

where $\hat{A}_{1}\in\mathbb{R}^{\left|J\right|\times
N},\hat{A}_{2}\in\mathbb{R}^{\left(\left|J\right|\left|L\right|+1\right)\times
N\left(\left|L\right|+1\right)},\hat{A}_{3,p}\in\mathbb{R}^{\left|J\right|\times\dim
W_{p}},\hat{A}_{4,p}\in\mathbb{R}^{\left(\left|J\right|\left|L\right|+1\right)\times\dim
W_{p}}$. One can see that matrices $\hat{A}_{1}$ and $\hat{A}_{2}$ are
independent of producer $p\in P$ and matrices $\hat{A}_{3,p}$ and
$\hat{A}_{4,p}$ depend on producer $p\in P$. One can further investigate the
structure of $\hat{A}_{1}$ and see

(30) $\hat{A}_{1}=\left[\begin{array}[]{ccc}1_{1}&&0\\\ &\ddots\\\
0&&1_{T^{\prime}}\end{array}\right],$

where $1_{j}$, $j\in J$ is a row vector of ones of length
$\left|I_{j}\right|$. Similarly,

(31) $\hat{A}_{2}=\left[\begin{array}[]{cccc}\hat{A}_{1}&\cdots&0&0\\\
\vdots&\ddots&\vdots&\vdots\\\ 0&\cdots&\hat{A}_{1}&0\\\
0&\cdots&0&1_{\left|N\right|}\end{array}\right],$

where the number of rows in the block notation above is $\left|L\right|+1$.
The first $\left|L\right|$ rows correspond to (5) and the last row correspond
to (6).

In a compact notation, the mean-variance utility of producer $p\in P$ can be
calculated as

$\begin{array}[]{rcl}\Psi_{p}\left(v_{p},\mathbb{E}^{\mathbb{\tilde{P}}}\left[\Pi\right]\right)&=&\mathbb{E}^{\tilde{\mathbb{P}}}\left[-\pi_{p}^{\top}v_{p}-\frac{1}{2}\lambda_{p}v_{p}^{\top}\left(\pi_{p}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi_{p}\right]\right)\left(\pi_{p}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi_{p}\right]\right)^{\top}v_{p}\right]\\\
\\\
&=&-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi_{p}\right]^{\top}v_{p}-\frac{1}{2}\lambda_{p}v_{p}^{\top}Q_{p}v_{p},\end{array}$

where (26) was used. The inner structure of matrix $Q_{p}$ is the following

(32) $Q_{p}=\left[\begin{array}[]{ccc}\hat{Q}_{1}&\hat{Q}_{2}&0\\\
\hat{Q}_{2}^{\top}&\hat{Q}_{3}&0\\\ 0&0&0\end{array}\right],$

where $\hat{Q}_{1}\in\mathbb{R}^{N\times
N},\hat{Q}_{2}\in\mathbb{R}^{N\times\left(\dim B_{p}+\dim
O_{p}\right)}=\mathbb{R}^{N\times
N\left(\left|L\right|+1\right)},\hat{Q}_{3}\in\mathbb{R}^{N\left(\left|L\right|+1\right)\times
N\left(\left|L\right|+1\right)}$. One can see that $\hat{Q}_{1}$,
$\hat{Q}_{2}$, and $\hat{Q}_{3}$ do not depend on producer $p\in P$. The size
of the larger matrix $Q_{p}$ depends on producer $p\in P$, because different
producers have different number of power plants.

Producer $p\in P$ attempts to solve the following optimization problem

$\Phi_{p}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)=\underset{v_{p}\in
S_{p}}{\text{max
}}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi_{p}\right]^{\top}v_{p}-\frac{1}{2}\lambda_{p}v_{p}^{\top}Q_{p}v_{p}.$

The profit of consumer $c\in C$ can be written as

$P_{c}\left(V_{c},\Pi\right)=-\Pi^{\top}V_{c}.$

Note here that we set $s_{c}=0$, w.l.o.g. The equality constraints can be
expressed as

$A_{c}V_{c}=a_{c}$

and the inequality constraints as

$B_{c}V_{c}\leq b_{c}$

where $A_{c}=\hat{A}_{1}$, $B_{c}\in\mathbb{R}^{2N\times N}$,
$a_{c}\in\mathbb{R}^{\left|J\right|}$ and $b_{c}\in\mathbb{R}^{N}$. Define a
feasible set

$S_{c}:=\left\\{V_{c}\in\mathbb{R}^{N}:A_{c}V_{c}=a_{c}\text{ \text{and
}}B_{c}V_{c}\leq b_{c}\right\\}.$

In a compact notation, the mean-variance utility of a consumer $c\in C$ can be
calculated as

$\begin{array}[]{rcl}\Psi_{c}\left(V_{c},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)&=&\mathbb{E}^{\tilde{\mathbb{P}}}\left[-\Pi^{\top}V_{c}-\frac{1}{2}\lambda_{c}V_{c}^{\top}\left(\Pi-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)\left(\Pi-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)^{\top}V_{c}\right]\\\
\\\
&=&-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{\top}V_{c}-\frac{\lambda}{2}V_{c}^{\top}Q_{c}V_{c},\end{array}$

where (27) was used. Moreover, note that $Q_{c}=\hat{Q}_{1}$ for all $c\in C$.

Consumer $c\in C$ attempts to solve the following optimization problem

$\Phi_{c}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)=\underset{V_{c}\in
S_{c}}{\text{max
}}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{\top}V_{c}-\frac{\lambda}{2}V_{c}^{\top}Q_{c}V_{c}.$

The profit function of the hypothetical market agent can be written in a
compact notation as

(33) $P_{M}\left(\Pi,V\right)=\Pi^{\top}\left(\sum_{c\in C}V_{c}+\sum_{p\in
P}V_{p}\right)$

and the expected utility as

(34)
$\begin{array}[]{rcl}\Psi_{M}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right],V\right)&=&\mathbb{E}^{\tilde{\mathbb{P}}}\left[P_{M}\left(\Pi,V\right)\right]\\\
\\\ &=&\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{\top}\left(\sum_{c\in
C}V_{c}+\sum_{p\in P}V_{p}\right).\end{array}$

Consequently, the hypothetical market agent’s objective is to solve

(35)
$\Phi_{M}\left(V\right)=\underset{\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\in
S_{M}}{\text{max
}}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{\top}\left(\sum_{c\in
C}V_{c}+\sum_{p\in P}V_{p}\right),$

where the feasible set of the hypothetical market agent is defined as

$S_{M}:=\left\\{\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\in\mathbb{R}^{N}:-\Pi_{max}\leq\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i},T_{j}\right)\right]\leq\Pi_{max}\text{
\text{for all }}j\in J,i\in I_{j}\right\\}.$

##  3 Analysis of the model

In this section we analyze the existence and uniqueness of solutions of the
model defined in the previous section.

######  Definition 1.

Competitive Equilibrium (CE)

Decisions $v^{*}$ and $\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*}$
constitute a competitive equilibrium if

  1. 1.

For every producer $p\in P$, $v_{p}^{*}$ is a strategy such that

(36)
$\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*}\right)\leq\Psi_{p}\left(v_{p}^{*},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*}\right)$

for all $v_{p}\in S_{p}$;

  2. 2.

For every consumer $c\in C$, $V_{c}^{*}$ is a strategy such that

(37)
$\Psi_{c}\left(V_{c},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*}\right)\leq\Psi_{c}\left(V_{c}^{*},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*}\right)$

for all $V_{c}\in S_{c}$;

  3. 3.

For each $i\in I_{j}$ and $j\in J$

(38) $0=\sum_{c\in C}V_{c}\left(t_{i},T_{j}\right)+\sum_{p\in
P}V_{p}\left(t_{i},T_{j}\right)$

must hold.

######  Definition 2.

Nash Equilibrium (NE)

Decisions $v^{*}$ and $\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*}$
constitute a Nash equilibrium if

  1. 1.

For every producer $p\in P$, $v_{p}^{*}$ is a strategy such that

(39)
$\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*}\right)\leq\Psi_{p}\left(v_{p}^{*},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*}\right)$

for all $v_{p}\in S_{p}$;

  2. 2.

For every consumer $c\in C$, $V_{c}^{*}$ is a strategy such that

(40)
$\Psi_{c}\left(V_{c},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*}\right)\leq\Psi_{c}\left(V_{c}^{*},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*}\right)$

for all $V_{c}\in S_{c}$;

  3. 3.

Price vector $\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*}$ maximizes
the objective function of the hypothetical market agent, i.e.

(41)
$\Psi_{M}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right],v^{*}\right)\leq\Psi_{M}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*},v^{*}\right)$

for all $\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\in S_{M}$.

For all $p\in P$, the exists vector $v_{p}$ such that $A_{p}v_{p}=a_{p}$ a.s.
and $B_{p}v_{p}<b_{p}$ a.s., for all $c\in C$, there exists vector $V_{c}$
such that $A_{c}V_{c}=a_{c}$ a.s. and $B_{c}V_{c}<b_{c}$ a.s., and the vectors
$V_{p}$ and $V_{c}$ can be chosen so that (38) is satisfied.

###  3.1 Existence of Solution

We start the analysis of the existence of solution with stating the following
well known theorem.

######  Theorem 3.

(Debreu, Glickberg, Fan)

Consider a strategic form game
$\left\langle\mathcal{N},\left(S_{i}\right)_{i\in\mathcal{N}},\left(\Psi_{i}\right)_{i\in\mathcal{N}}\right\rangle$,
where $\mathcal{N}$ denotes a set of players. Let $s_{-i}\in
S_{-i}=\prod_{i\neq j}S_{j}$ denote a vector of actions for all players except
player $i\in\mathcal{N}$, and let $s_{i}\in S_{i}$ denote an action for player
$i\in\mathcal{N}$. If for each $i\in\mathcal{N}$

  1. 1.

feasible set $S_{i}$ is non-empty, compact and convex;

  2. 2.

$\Psi_{i}\left(s_{i},s_{-i}\right)$ is continuous in $s_{-i}$

  3. 3.

$\Psi_{i}\left(s_{i},s_{-i}\right)$ is continuous and concave in $s_{i}$,

then a pure Nash equilibrium exists.

The proof for the above theorem can be found in [16]. It is straightforward to
apply the theorem and show that our problem from Definition 2 has a solution.

######  Corollary 4.

Let Assumption 3 hold. Then there exists a pure NE for Problem 2.

###### Proof.

A set of players $\mathcal{N}$ is composed of all producers $p\in P$,
consumers $c\in C$, and the hypothetical market agent. Due to Assumption 3 it
is clear that the feasible region of each player is non-empty. Since all
constraints are affine functions with non-strict inequalities, it is clearly
also convex and closed. Boundedness is guaranteed by (2), (3), (7), (8), and
(9) for producers, by (13) for consumers, and by (16) for the hypothetical
market agent. The covariance matrix $Q_{p}$ in
$\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$,
$v_{p}\in S_{p}$, $\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\in S_{M}$
is by definition positive semidefinite for all $p\in P$ and thus
$\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$
is concave in $v_{p}$. Similarly, we can also see that the covariance matrix
$Q_{c}$ is positive semidefinite and thus
$\Psi_{c}\left(V_{c},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$,
$V_{c}\in S_{c}$, $\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\in S_{M}$
for all $c\in C$ is concave in $V_{c}$. The function
$\Psi_{M}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right],v\right)$ that
corresponds to the expected utility of the hypothetical market agent is linear
in $\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]$ and thus concave. Utility
functions of producers, consumers, and the hypothetical market agent are all
quadratic and therefore continuous. Thus, a pure NE exists. ∎

###  3.2 Uniqueness of Solution

######  Definition 5.

A power plant $r\in R^{p,l}$, $p\in P$, $l\in L$ is at the upper bound at
delivery time $T_{j}$, $j\in J$, if for every $\epsilon>0$,
$W_{p,l,r}\left(T_{j}\right)+\epsilon$ is infeasible. Similarly, a power plant
$r\in R^{p,l}$, $p\in P$, $l\in L$, $r\in R^{p,l}$ is at the lower bound at
delivery time $T_{j}$, $j\in J$, if for every $\epsilon>0$,
$W_{p,l,r}\left(T_{j}\right)-\epsilon$ is infeasible. Bounds on
$W_{p,l,r}\left(T_{j}\right)$ are defined by constraints (2) and (3).

Let us start the discussion about the uniqueness of the Nash equilibrium with
a direct consequence of Assumption 3.

######  Lemma 6.

If there exists $j\in J$ such that all power plants are at the upper bound
simultaneously, then

$0>\sum_{c\in C}\sum_{i\in I_{j}}V_{c}\left(t_{i},T_{j}\right)+\sum_{p\in
P}\sum_{i\in I_{j}}V_{p}\left(t_{i},T_{j}\right).$

Similarly, if there exists $j\in J$ such that all power plants are at the
lower bound simultaneously, then

$0<\sum_{c\in C}\sum_{i\in I_{j}}V_{c}\left(t_{i},T_{j}\right)+\sum_{p\in
P}\sum_{i\in I_{j}}V_{p}\left(t_{i},T_{j}\right).$

###### Proof.

Using (4) and summing over all producers $p\in P$ we get

(42) $\sum_{p\in P}\sum_{i\in I_{j}}V_{p}\left(t_{i},T_{j}\right)=-\sum_{p\in
P}\sum_{l\in L}\sum_{r\in R^{p,l}}W_{p,l,r}\left(T_{j}\right).$

Similarly, using (12) and the fact that $\sum_{c\in C}p_{c}=1$, we get

(43) $\begin{array}[]{rcl}\sum_{c\in C}\sum_{i\in
I_{j}}V_{c}\left(t_{i},T_{j}\right)&=&\sum_{c\in C}p_{c}D\left(T_{j}\right)\\\
\\\ &=&D\left(T_{j}\right).\end{array}$

Assume now that all power plants are at the upper bound simultaneously for
some $j^{\prime}\in J$. Then by Assumption 3,

(44) $D\left(T_{j^{\prime}}\right)<\sum_{p\in P}\sum_{l\in L}\sum_{r\in
R^{p,l}}W_{p,l,r}\left(T_{j^{\prime}}\right)$

and therefore by (42) and (43),

$0>\sum_{c\in C}\sum_{i\in I_{j}}V_{c}\left(t_{i},T_{j}\right)+\sum_{p\in
P}\sum_{i\in I_{j}}V_{p}\left(t_{i},T_{j}\right).$

A similar argument also holds for the lower bound.∎

None of random variables $\Pi$, $G$, and $G_{em}$ can be written as a linear
combination of the others a.s.

In the rest of this paper we assume that Assumption 3 and Assumption 3.2
always hold.

######  Lemma 7.

The mean-variance objective functions of producers $p\in P$ and consumers
$c\in C$ are strictly concave in $V_{p}$, $F_{p}$, and $O_{p}$ for all $p\in
P$ and in $V_{c}$ for all $c\in C$.

###### Proof.

The objective function of each producer $p\in P$ can be written as

$\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)=-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi_{p}\right]^{\top}v_{p}-\frac{1}{2}\lambda_{p}v_{p}^{\top}Q_{p}v_{p},$

where

$Q_{p}=\left[\begin{array}[]{cc}\hat{Q}&0\\\ 0&0\end{array}\right],$

$\hat{Q}:=\left[\begin{array}[]{cc}\hat{Q}_{1}&\hat{Q}_{2}\\\
\hat{Q}_{2}^{\top}&\hat{Q}_{3}\end{array}\right]$, and
$\pi^{\prime}:=\left[\Pi^{\top},G^{\top},G_{em}^{\top}\right]^{\top}$. Under
Assumption 3.2, we conclude that $\hat{Q}\succ 0$. Define
$v_{p}^{\prime}:=\left[V_{p}^{\top},F_{p}^{\top},O_{p}^{\top}\right]^{\top}$
and $v^{\prime\prime}_{p}:=W_{p}$. Then

$\mathcal{D}_{v_{p^{\prime}}}\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)=-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi^{\prime}\right]^{\top}-\lambda_{p}\hat{Q}v_{p}^{\prime}$

and

$\mathcal{D}_{v_{p^{\prime\prime}}}\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)=0.$

The second derivative
$\mathcal{D}_{v_{p^{\prime}}}^{2}\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)=-\lambda_{p}\hat{Q}\prec
0$ since $0<\lambda_{p}<\infty$. Thus,
$\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$,
$p\in P$ are strictly concave in $V_{p}$, $F_{p}$, and $O_{p}$. The proof for
consumers $c\in C$ is similar. ∎

######  Lemma 8.

Optimal fuel trading strategies $F_{p}$ and emission trading strategies
$O_{p}$ are bounded for all producers $p\in P$.111Hence in there exists some
$M_{1}\in\mathbb{R}$ such that
$\left\|\left[F_{p}^{\top},O_{p}^{\top}\right]^{\top}\right\|\leq
M_{1}<\infty$ for all $p\in P$. We set $M_{1}<F_{trade}$.

###### Proof.

Since
$\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$
is strictly concave and quadratic in $F_{p}$ and $O_{p}$ for all $p\in P$, and
fuel and emission prices have finite expectation and variance, it is clear
that
$\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)\rightarrow-\infty$
as $\left\|F_{p}\right\|\rightarrow\infty$ or
$\left\|O_{p}\right\|\rightarrow\infty$. This is not optimal and thus we
conclude that optimal $F_{p}$ and $O_{p}$ must be bounded. ∎

######  Lemma 9.

Consider optimization problem (10) without constraints (9) for producers $p\in
P$, and optimization problem (14) without constraints (13) for consumers $c\in
C$. Denote by $V_{k}\in\mathbb{R}^{N}$ a vector of optimal volumes of any
player $k\in P\cup C$ for a given vector of expected prices
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\in\mathbb{R}^{N}$. Then,

  1. 1.

if
$\left\|\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right\|\rightarrow\infty$,
then $\left[\sum_{k\in P\cup
C}V_{k}\right]^{\top}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\rightarrow-\infty$,
and

  2. 2.

if for each delivery period $T_{j}$, $j\in J$ there exists at least one power
plant that is not at the upper or the lower boundary, then
$\left\|V_{k}\right\|\rightarrow\infty$ for all $k\in P\cup C$ simultaneously
if and only if
$\left\|\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right\|\rightarrow\infty$.

###### Proof.

The necessary and sufficient conditions for $v_{k}$ to be a global maximizer
of
$\Psi_{k}\left(v_{k},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$
are, due to Assumption 3 that implies the Slater condition, the following

(45)
$\begin{array}[]{rcl}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi_{k}\right]-\lambda_{k}Q_{k}v_{k}-B_{k}^{\top}\eta_{k}-A_{k}^{\top}\mu_{k}&=&0\\\
\\\ \left(B_{k}v_{k}-b_{k}\right)^{\top}\eta_{k}&=&0\\\ \\\
B_{k}v_{k}-b_{k}&\leq&0\\\ \\\ A_{k}v_{k}-a_{k}&=&0\\\ \\\
\eta_{k}&\geq&0.\end{array}$

We are only interested in decision variables $V_{k}$. By Lemma 8, we are
allowed to remove constraints (7) and (8) without affecting the optimal
solution. Then, after removing the inequality constraints (9) for producers
$p\in P$, and inequality constraints (13) for consumers $c\in C$, there is no
inequality constraints that involve variable $V_{k}$. Moreover, there is only
one equality constraint, i.e. (4) for producers and (12) for consumers, for
each delivery period $j\in J$ that involve variable $V_{k}$. Considering the
first equation of (45) for each delivery period $j\in J$ separately, and
neglecting all bounded terms, we obtain the following equivalence

(46)
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(T_{j}\right)\right]+\lambda_{k}\hat{Q}_{1}^{j}V_{k}+\mu_{k,j}1\sim
0$

were $\mu_{k,j}\in\mathbb{R}$ is the dual variable of the equality constraint
(4) for producers and (12) for consumers,
$1\in\mathbb{R}^{\left|I_{j}\right|}$ is a vector of ones, and
$\hat{Q}_{1}^{j}\in\mathbb{R}^{\left|I_{j}\right|\times N}$ contains only
those rows of $\hat{Q}_{1}$ that correspond to
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(T_{j}\right)\right]$ in (45).
In the calculation of (46) we took into account the construction of the
equilibrium price process in Section 2.4, and use the finding that all
variances and correlations are finite and can not be affected by the
hypothetical market agent. Therefore, $\left\|Q_{k}\right\|<\infty$. Moreover,
$\left\|\left[F_{p}^{\top},O_{p}^{\top}\right]^{\top}\right\|<\infty$ due to
Lemma 8.

Assume that
$\left\|\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(T_{j}\right)\right]\right\|\rightarrow\infty$.
Then by (46), $\left\|\mu_{k,j}\right\|\rightarrow\infty$ or
$\left\|V_{k}\right\|\rightarrow\infty$.

  1. 1.

Assume first that $\left\|V_{k}\right\|<\infty$ for all $k\in P\cup C$. Let
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i},T_{j}\right)\right]\rightarrow\infty$
for some $i\in I_{j}$ and $j\in J$. Then $\mu_{k,j}\rightarrow-\infty$. Since
$\left\|V_{k}\left(T_{j}\right)\right\|<\infty$, it follows from (46) that
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(T_{j}\right)\right]\rightarrow\infty$
componentwise and all components of
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(T_{j}\right)\right]$ must be
equal up to a constant. From (4) and the general interpretation of the dual
variables, we can see that as $\mu_{p,j}\rightarrow-\infty$, $p\in P$, a small
increase in $\sum_{l\in L}\sum_{r\in R^{p,l}}W_{p,l,r}\left(T_{j}\right)$
would infinitely improve the objective function
$\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$.
As governed by (5) and (6), an increase in $\sum_{l\in L}\sum_{r\in
R^{p,l}}W_{p,l,r}\left(T_{j}\right)$ would also require that more fuel and
emission certificates are bought. Since the fuel and emission prices have a
finite expectation and variance, a decrease of the objective function due to
the change in fuel and emission buying strategy as
$\mu_{p,j}\rightarrow-\infty$ remains bounded. Thus, if
$\mu_{p,j}\rightarrow-\infty$, the sum of production of all power plants
$\sum_{l\in L}\sum_{r\in R^{p,l}}W_{p,l,r}\left(T_{j}\right)$ of producer
$p\in P$ increases as much as allowed by constraints (2) and (3). Since all
producers share the same electricity, fuel, and emission prices, this holds
for all producers $p\in P$ simultaneously. Then by Lemma 6, $0>\sum_{c\in
C}\sum_{i\in I_{j}}V_{c}\left(t_{i},T_{j}\right)+\sum_{p\in P}\sum_{i\in
I_{j}}V_{p}\left(t_{i},T_{j}\right)$ and thus $\left[\sum_{k\in P\cup
C}V_{k}\left(T_{j}\right)\right]^{\top}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(T_{j}\right)\right]\rightarrow-\infty$.
On the other hand, if
$\left\|\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(T_{j}\right)\right]\right\|<\infty$
then also $\left|\left[\sum_{k\in P\cup
C}V_{k}\left(T_{j}\right)\right]^{\top}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(T_{j}\right)\right]\right|<\infty$.
Therefore, we can conclude that $\left[\sum_{k\in P\cup
C}V_{k}\right]^{\top}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\rightarrow-\infty$.

  2. 2.

Assume now that $\left\|V_{k}\right\|\rightarrow\infty$ for at least one $k\in
P\cup C$. Then, $V_{k}^{\top}\hat{Q}_{1}V_{k}\rightarrow\infty$ since
$\hat{Q}_{1}\succ 0$. Assume now that
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{\top}V_{k}>-\infty$. Then
$\Psi_{k}\left(v_{k},\mathbb{E}^{\mathbb{\tilde{P}}}\left[\Pi\right]\right)\rightarrow-\infty$,
which is clearly not optimal for player $k$. Thus,
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{\top}V_{k}\rightarrow-\infty$,
which concludes the proof of point 1.

We continue with the proof of point 2. Assume that for fixed $j\in J$ not all
power plants are at the upper or lower bound simultaneously. We have already
seen above that if $\left|\mu_{k,j}\right|\rightarrow\infty$ for any $k\in
P\cup C$ and $j\in J$ then, at delivery time $T_{j}$, all power plants are at
the upper or lower bound simultaneously. Thus, $\left|\mu_{k,j}\right|<\infty$
for all $k\in P\cup C$ and $j\in J$. Rewriting (46) without focusing on one
delivery period only and taking any norm, we get

(47)
$\left\|\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right\|\sim\left\|\lambda_{k}\hat{Q}_{1}V_{k}\right\|.$

Since $\left|\lambda_{k}\right|<\infty$ and
$\left\|\hat{Q}_{1}\right\|<\infty$, it immediately follows from (47) that if
$\left\|\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(T_{j}\right)\right]\right\|\rightarrow\infty$
then $\left\|V_{k}\right\|\rightarrow\infty$. Because all producers and
consumers share the same price this must hold for all of them simultaneously.
Since $\hat{Q}_{1}$ is invertible, we can write (46) for all delivery periods
together as

(48)
$\begin{array}[]{rcl}\left\|V_{k}\right\|&\sim&\left\|\lambda_{k}^{-1}\hat{Q}_{1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right\|\end{array}.$

Because $\left|\lambda_{k}^{-1}\right|<\infty$ and
$\left\|\hat{Q}_{1}^{-1}\right\|<\infty$, it follows from (48) that if
$\left\|V_{k}\right\|\rightarrow\infty$ then
$\left\|\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right\|\rightarrow\infty$.
∎

######  Lemma 10.

Expected prices $\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]$ in a NE are
bounded.222Hence there exists some $M_{2}\in\mathbb{R}$ such that
$\left\|\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i},T_{j}\right)\right]\right\|\leq
M_{2}<\infty$ for all $j\in J$ and $i\in I_{j}$. We set $M_{2}<\Pi_{max}$.

###### Proof.

Assume that
$\left\|\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right\|\rightarrow\infty$.
Then by Lemma 9, $\left[\sum_{k\in P\cup
C}V_{k}\right]^{\top}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\rightarrow-\infty$.
Inserting that to (34), we get
$\Psi_{M}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right],V\right)=-\infty$,
which is clearly not a NE. Thus,
$\left\|\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right\|<\infty$. ∎

######  Proposition 11.

A NE, if it exists, is a CE and vice versa.

###### Proof.

The equilibrium conditions for producers and consumers are the same for both
the NE and the CE. The remaining part is to show that Point 3 from the CE
implies Point 3 from the NE and conversely.

  1. 1.

Assume the CE holds. Then (38) can be inserted into (18 \- 20). Then
$\Psi_{M}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right],V\right)=0$ for
all $\mathbb{E}^{\mathbb{P}}\left[\Pi\right]$ and thus (41) is satisfied.

  2. 2.

Assume the NE holds. Assume further that (38) does not hold for some $t_{i}$
and $T_{j}$, $i\in I_{j}$, $j\in J$. Then according to (20) the optimal
$\left\|\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i},T_{j}\right)\right]\right\|\rightarrow\infty$.
This contradicts Lemma 10 and hence such a solution can not be a NE. Thus (38)
holds for all $t_{i}$ and $T_{j}$, $i\in I_{j}$, $j\in J$.

∎

######  Corollary 12.

The number of forward contracts $V_{k}$ in a NE is bounded for all $k\in P\cup
C$.333Hence there exists some $M_{3}\in\mathbb{R}$ such that
$\left\|V_{k}\left(t_{i},T_{j}\right)\right\|\leq M_{3}<\infty$ for $k\in
P\cup C$ and for all $j\in J$ and $i\in I_{j}$. We set $M_{3}<V_{trade}$.

###### Proof.

Assume that there exists a player $k\in P\cup C$ such that
$\left\|V_{k}\right\|\rightarrow\infty$. By Proposition 11 every NE is also a
CE and thus (38) holds. Thus, we can use Lemma 9 and conclude that
$\left\|\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right\|\rightarrow\infty$.
This contradicts Lemma 10. Thus, $\left\|V_{k}\right\|<\infty$ for all $k\in
P\cup C$. ∎

Denote by $\bar{S}_{M}$ the set of prices for which power plants are either
all at the upper or all at the lower bound simultaneously for at least one
delivery period $T_{j}$ $j\in J$. By Lemma 6, we know that the optimal price
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\notin\bar{S}_{M}$.

######  Lemma 13.

Given an expected price vector
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\in\mathbb{R}^{N}\backslash\bar{S}_{M}$,
the decision vectors $V_{p}$ for all $p\in P$ and the decision vectors $V_{c}$
for all $c\in C$, are unique.

###### Proof.

The objective function of each producer $p\in P$ can be written as

$\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)=-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi_{p}\right]^{\top}v_{p}-\frac{1}{2}\lambda_{p}v_{p}^{\top}Q_{p}v_{p}.$

Define $v_{p}^{\prime}:=V_{p}$ and
$v^{\prime\prime}_{p}:=\left[F_{p}^{\top},O_{p}^{\top},W_{p}^{\top}\right]^{\top}$.
Then

$\mathcal{D}_{v_{p}}\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)=\left[\mathcal{D}_{v_{p^{\prime}}}\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)^{\top},\mathcal{D}_{v_{p^{\prime\prime}}}\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)^{\top}\right]^{\top}.$

Due to the strict concavity of the expected utility functions in
$v_{p}^{\prime}$, for any
$\hat{v}_{p}:=\left[\hat{v}_{p}^{\prime\top},\hat{v}_{p}^{\prime\prime\top}\right]^{\top}$
and
$\tilde{v}_{p}:=\left[\tilde{v}_{p}^{\prime\top},\tilde{v}_{p}^{\prime\prime\top}\right]^{\top}$
with $\hat{v}_{p}^{\prime}\neq\tilde{v}_{p}^{\prime}$ the following strict
inequality holds

(49)
$\left(\hat{v}_{p}-\tilde{v}_{p}\right)^{\top}\mathcal{D}_{\tilde{v}_{p}}\Psi_{p}\left(\tilde{v}_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)+\left(\tilde{v}_{p}-\hat{v}_{p}\right)^{\top}\mathcal{D}_{\hat{v}_{p}}\Psi_{p}\left(\hat{v}_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)>0.$

We will continue with a proof by contradiction. Assume that there exist
$\hat{v}_{p}^{\prime}\neq\tilde{v}_{p}^{\prime}$ that are both optimal
solutions for player $p\in P$ given the electricity price
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]$. Then both must satisfy the
KKT conditions, i.e.

(50)
$\begin{array}[]{rcl}\mathcal{D}_{\tilde{v}_{p}}\Psi_{p}\left(\tilde{v}_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)-B_{p}^{\top}\tilde{\eta}_{p}-A_{p}^{\top}\tilde{\mu}_{p}&=&0\\\
\\\ \tilde{\eta}_{p}^{\top}\left(B_{p}\tilde{v}_{p}-b_{p}\right)&=&0\\\ \\\
B_{p}\tilde{v}_{p}-b_{p}&\leq&0\\\ \\\ A_{p}\tilde{v}_{p}-a_{p}&=&0\\\ \\\
\tilde{\eta}_{p}&\geq&0\end{array}$

and

(51)
$\begin{array}[]{rcl}\mathcal{D}_{\hat{v}_{p}}\Psi_{p}\left(\hat{v}_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)-B_{p}^{\top}\hat{\eta}_{p}-A_{p}^{\top}\hat{\mu}_{p}&=&0\\\
\\\ \hat{\eta}_{p}^{\top}\left(B_{p}\hat{v}_{p}-b_{p}\right)&=&0\\\ \\\
B_{p}\hat{v}_{p}-b_{p}&\leq&0\\\ \\\ A_{p}\hat{v}_{p}-a_{p}&=&0\\\ \\\
\hat{\eta}_{p}&\geq&0.\end{array}$

Multiplying the first equation of (50) and (51) by
$\left(\hat{v}_{p}-\tilde{v}_{p}\right)^{\top}$ and
$\left(\tilde{v}_{p}-\hat{v}_{p}\right)^{\top}$, respectively and summing them
up, the following strict inequality

(52)
$\begin{array}[]{rcl}0&=&\left(\hat{v}_{p}-\tilde{v}_{p}\right)^{\top}\mathcal{D}_{\tilde{v}_{p}}\Psi_{p}\left(\tilde{v}_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)+\left(\tilde{v}_{p}-\hat{v}_{p}\right)^{\top}\mathcal{D}_{\hat{v}_{p}}\Psi_{p}\left(\hat{v}_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)\\\
\\\
&&-\left(\hat{v}_{p}-\tilde{v}_{p}\right)^{\top}B_{p}^{\top}\tilde{\eta}_{p}-\left(\tilde{v}_{p}-\hat{v}_{p}\right)^{\top}B_{p}^{\top}\hat{\eta}_{p}\\\
\\\
&&-\left(\hat{v}_{p}-\tilde{v}_{p}\right)^{\top}A_{p}^{\top}\tilde{\mu}_{p}^{\top}-\left(\tilde{v}_{p}-\hat{v}_{p}\right)^{\top}A_{p}^{\top}\hat{\mu}_{p}^{\top}\\\
\\\
&>&-\left(\hat{v}_{p}-\tilde{v}_{p}\right)^{\top}B_{p}^{\top}\tilde{\eta}_{p}-\left(\tilde{v}_{p}-\hat{v}_{p}\right)^{\top}B_{p}^{\top}\hat{\eta}_{p}\\\
\\\
&&-\left(\hat{v}_{p}-\tilde{v}_{p}\right)^{\top}A_{p}^{\top}\tilde{\mu}_{p}-\left(\tilde{v}_{p}-\hat{v}_{p}\right)^{\top}A_{p}^{\top}\hat{\mu}_{p}\end{array}$

is obtained by (49).

Rewriting

$\begin{array}[]{rcl}\left(\hat{v}_{p}-\tilde{v}_{p}\right)^{\top}B_{p}^{\top}\tilde{\eta}_{p}&=&\left(B_{p}\hat{v}_{p}-B_{p}\tilde{v}_{p}+b_{p}-b_{p}\right)^{\top}\tilde{\eta}_{p}\\\
\\\
&=&\left(B_{p}\hat{v}_{p}-b_{p}\right)^{\top}\tilde{\eta}_{p}-\left(B_{p}\tilde{v}_{p}-b_{p}\right)^{\top}\tilde{\eta}_{p}\\\
\\\ &=&\left(B_{p}\hat{v}_{p}-b_{p}\right)^{\top}\tilde{\eta}_{p}\end{array}$

and noting $\tilde{\eta}_{p}\geq 0$ and $B_{p}\hat{v}_{p}-b_{p}\leq 0$, we can
conclude that

(53)
$\left(\hat{v}_{p}-\tilde{v}_{p}\right)^{\top}B_{p}^{\top}\tilde{\eta}_{p}\leq
0.$

Due to the symmetry also

(54)
$\left(\tilde{v}_{p}-\hat{v}_{p}\right)^{\top}B_{p}^{\top}\hat{\eta}_{p}\leq
0.$

Rewriting

(55)
$\begin{array}[]{rcl}\left(\hat{v}_{p}-\tilde{v}_{p}\right)^{\top}A_{p}^{\top}\tilde{\mu}_{p}&=&\left(A_{p}\hat{v}_{p}-A_{p}\tilde{v}_{p}+a_{p}-a_{p}\right)^{\top}\tilde{\mu}_{p}\\\
\\\
&=&\left(A_{p}\hat{v}_{p}-a_{p}\right)^{\top}\tilde{\mu}_{p}-\left(A_{p}\tilde{v}_{p}-a_{p}\right)^{\top}\tilde{\mu}_{p}\\\
\\\ &=&0\end{array}$

and due to the symmetry also

(56)
$\left(\tilde{v}_{p}-\hat{v}_{p}\right)^{\top}A_{p}^{\top}\hat{\mu}_{p}=0.$

Inserting (53), (54), (55), and (56) back to (52) gives a contradiction. Hence
$\hat{v}_{p}^{\prime}=\tilde{v}_{p}^{\prime}$. The proof for consumers $c\in
C$ is similar. ∎

Before we continue with a further analysis of our problem, let us introduce
some definitions that are useful for an analysis of piecewise differential
functions.

######  Definition 14.

([23]) A continuous function
$\tilde{\mathcal{Z}}:S_{1}\rightarrow\mathbb{R}^{N}$ defined on an open set
$S_{1}\subseteq\mathbb{R}^{N}$ is $\mathbb{PC}^{r}$ if for every $x\in S_{1}$
there exists a finite family of $\mathbb{C}^{r}$-functions
$\tilde{\mathcal{Z}}^{i}:S_{2}\rightarrow\mathbb{R}^{N}$, where
$S_{2}\subseteq S_{1}$ is an open neighborhood of $x$ and $i\in\mathcal{I}$,
such that
$\tilde{\mathcal{Z}}\left(z\right)\in\left\\{\tilde{\mathcal{Z}}^{i}\left(z\right):i\in\mathcal{I}\right\\}$
for every $z\in S_{2}$. The $\mathbb{C}^{r}$-functions
$\tilde{\mathcal{Z}}^{i}$, $i\in\mathcal{I}$, are called selection functions
of $\tilde{\mathcal{Z}}$ at $x$.

A selection function $\tilde{\mathcal{Z}}^{i}$ of a $\mathbb{PC}^{r}$\-
function $\tilde{\mathcal{Z}}$ at $x$ is essentially active if

(57) $x\in\text{cl}\,\text{int}\left\\{z\in
S_{2}:\tilde{\mathcal{Z}}^{i}\left(z\right)=\tilde{\mathcal{Z}}\left(z\right)\right\\}.$

If $\left\\{\tilde{\mathcal{Z}}^{i}:i\in\mathcal{I}\right\\}$ is a family of
selection functions for $\tilde{\mathcal{Z}}$ at $x$, then the set of indices
$i$ of essentially active selection functions $\tilde{\mathcal{Z}}^{i}$ at $x$
is denoted $\mathcal{I}^{e}\left(x\right)$.

######  Definition 15.

Let $\tilde{\mathcal{Z}}:S_{1}\rightarrow\mathbb{R}^{N}$ be a continuous
$\mathbb{PC}^{1}$ function defined on an open set
$S_{1}\subseteq\mathbb{R}^{N}$. Clarke’s generalized Jacobian ([12]) of
$\tilde{\mathcal{Z}}$ at $x$ is defined as

(58)
$\mathcal{D^{C}}\tilde{\mathcal{Z}}\left(x\right)=\text{conv}\left\\{\mathcal{D}\tilde{\mathcal{Z}}^{i}\left(x\right):i\in\mathcal{I}^{e}\left(x\right)\right\\}$

where conv denotes a convex hull.

######  Proposition 16.

$\mathcal{D}\tilde{\mathcal{Z}}^{i}\left(x\right)\prec 0$ for all
$i\in\mathcal{I}^{e}\left(x\right)$ and all $x\in S_{1}$ if and only if
$\mathcal{J}\left(x\right)\prec 0$ for all
$\mathcal{J}\left(x\right)\in\mathcal{D^{C}}\tilde{\mathcal{Z}}\left(x\right)$.

###### Proof.

This follows directly from (58). For any
$\mathcal{J}\left(x\right)\in\mathcal{D^{C}}\tilde{\mathcal{Z}}\left(x\right)$
and $x\in S_{1}$, $\mathcal{J}\left(x\right)$ can be written as

(59)
$\mathcal{J}\left(x\right)=\sum_{i\in\mathcal{I}^{e}\left(x\right)}\alpha_{i}\mathcal{D}\tilde{\mathcal{Z}}^{i}\left(x\right)$

for some $\alpha_{i}\geq 0$ such that
$\sum_{i\in\mathcal{I}^{e}\left(x\right)}\alpha_{i}=1$. The result then
follows trivially. ∎

######  Definition 17.

A function $\tilde{\mathcal{Z}}:S_{1}\rightarrow\mathbb{R}^{N}$ defined on an
open set $S_{1}\subseteq\mathbb{R}^{N}$ is strictly decreasing on $S_{1}$ if

(60)
$\left(\tilde{\mathcal{Z}}\left(x\right)-\tilde{\mathcal{Z}}\left(y\right)\right)^{\top}\left(x-y\right)<0$

for all $x,y\in S_{1}$ such that $x\neq y$.

######  Proposition 18.

Let $\tilde{\mathcal{Z}}:S_{1}\rightarrow\mathbb{R}^{N}$ be a
$\mathbb{PC}^{1}$ function defined on an open set
$S_{1}\subseteq\mathbb{R}^{N}$. If
$\mathcal{D}\tilde{\mathcal{Z}}^{i}\left(x\right)\prec 0$ for all
$i\in\mathcal{I}^{e}\left(x\right)$ and $x\in S_{1}$, then
$\tilde{\mathcal{Z}}$ is strictly decreasing on $S_{1}$.

###### Proof.

Assume that $\mathcal{D}\tilde{\mathcal{Z}}^{i}\left(x\right)\prec 0$ for all
$i\in\mathcal{I}^{e}\left(x\right)$. Then by Proposition 16 also
$\mathcal{J}\left(x\right)\prec 0$ for all
$\mathcal{J}\left(x\right)\in\mathcal{D^{C}}\tilde{\mathcal{Z}}\left(x\right)$.
By the extended mean-value theorem (see [19])

(61)
$\tilde{\mathcal{Z}}\left(x\right)-\tilde{\mathcal{Z}}\left(y\right)=\mathcal{J}\left(y+\delta\left(x-y\right)\right)\left(x-y\right)$

for some
$\mathcal{J}\left(y+\delta\left(x-y\right)\right)\in\mathcal{D^{C}}\tilde{\mathcal{Z}}\left(y+\delta\left(x-y\right)\right)$
and $\delta\in\left(0,1\right)$. By multiplying (61) from the left by
$\left(x-y\right)^{\top}$ and using the assumption of the negative
definiteness of $\mathcal{J}\left(x\right)$, we conclude

(62)
$\left(\tilde{\mathcal{Z}}\left(x\right)-\tilde{\mathcal{Z}}\left(y\right)\right)^{\top}\left(x-y\right)<0.$

∎

######  Lemma 19.

Let $k\in P\cup C$. There exists a $\mathbb{PC}^{\infty}$ mapping
$\tilde{\mathcal{Z}}_{k}:\mathbb{R}^{N}\backslash\bar{S}_{M}\rightarrow\mathbb{R}^{N}$
that maps the electricity price vector
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\in\mathbb{R}^{N}\backslash\bar{S}_{M}$
to a volume vector $V_{k}\in\mathbb{R}^{N}$.

###### Proof.

Since the proof for producers and consumers is almost the same we will
explicitly write it only for producers. Set $p=k$ for some $p\in P$. The
optimization problem for a producer $p$ can be written as

$\begin{array}[]{rl}\underset{v_{p}\in
S_{p}}{\text{max}}&\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right).\end{array}$

For convenience we define
$\tilde{\Psi}_{p}:\mathbb{R}^{N}\times\mathbb{R}^{N}\rightarrow\mathbb{R}$ as
$\tilde{\Psi}_{p}\left(V_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right):=\underset{F_{p},O_{p},W_{p}}{\text{max}}\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$
subject to (2), (3), (4), (5), (6), (7), and (8). Using Lemma 13 we know that
volumes $V_{p}$ are unique for a given price
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\in\mathbb{R}^{N}\backslash\bar{S}_{M}$.
Thus there exists a mapping
$\tilde{\mathcal{Z}}_{p}:\mathbb{R}^{N}\backslash\bar{S}_{M}\rightarrow\mathbb{R}^{N}$
such that
$V_{p}=\tilde{\mathcal{Z}}_{p}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$.
Results from the parametric quadratic programming show that
$\tilde{\mathcal{Z}}_{p}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$
is a continuous and piecewise affine function (and thus
$\mathbb{PC}^{\infty}$), see [6] for a strictly convex quadratic objective
function and [4] for a convex quadratic objective function. ∎

######  Theorem 20.

Define a $\mathbb{PC}^{\infty}$ mapping
$\tilde{\mathcal{Z}}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right):=\sum_{p\in
P}\tilde{\mathcal{Z}}_{p}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)+\sum_{c\in
C}\tilde{\mathcal{Z}}_{c}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$.
If $\mathcal{D}\tilde{\mathcal{Z}}^{i}\left(x\right)\prec 0$ for all
$i\in\mathcal{I}^{e}\left(x\right)$ and
$x\in\mathbb{R}^{N}\backslash\bar{S}_{M}$, then electricity prices
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]$ in the NE are unique.

###### Proof.

The market condition (38) requires that

$0=\sum_{c\in C}V_{c}+\sum_{p\in P}V_{p}.$

Using Lemma 19 this can be written as

(63) $\begin{array}[]{rcl}0&=&\sum_{p\in P}V_{p}+\sum_{c\in C}V_{c}\\\ \\\
&=&\sum_{p\in
P}\tilde{\mathcal{Z}}_{p}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)+\sum_{c\in
C}\tilde{\mathcal{Z}}_{c}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)\\\
\\\
&=&\tilde{\mathcal{Z}}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right).\end{array}$

If $\mathcal{D}\tilde{\mathcal{Z}}^{i}\left(x\right)\prec 0$ for all
$i\in\mathcal{I}^{e}\left(x\right)$ and
$x\in\mathbb{R}^{N}\backslash\bar{S}_{M}$, then by Proposition 18
$\tilde{\mathcal{Z}}$ is strictly decreasing on
$\mathbb{R}^{N}\backslash\bar{S}_{M}$. Thus, the mapping $\tilde{\mathcal{Z}}$
must be bijective. By Corollary 4, we know that the solution to the problem
$\tilde{\mathcal{Z}}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)=0$
exists and from the bijectivity of $\tilde{\mathcal{Z}}$, we conclude that it
must be unique. ∎

The remaining problem of this section is to show that
$\mathcal{D}\tilde{\mathcal{Z}}^{i}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)\prec
0$ for all
$i\in\mathcal{I}^{e}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$
and
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\in\mathbb{R}^{N}\backslash\bar{S}_{M}$
indeed holds. From the proof of Lemma 19, we know that
$\tilde{\mathcal{Z}}^{i}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$,
$i\in\mathcal{I}$ are all affine functions and thus
$\mathcal{D}\tilde{\mathcal{Z}}^{i}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$
are all constants. Therefore, it is enough to show that
$\mathcal{D}\tilde{\mathcal{Z}}^{i}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)\prec
0$ for all
$i\in\mathcal{I}^{e}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$
and
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\in\left\\{x:x\in\mathbb{R}^{N}\backslash\bar{S}_{M}\wedge\left|\mathcal{I}^{e}\left(x\right)\right|=1\right\\}$.
For such prices $\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]$,
$\mathcal{D}\tilde{\mathcal{Z}}_{k}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$
exists for all players $k\in P\cup C$. In the rest of this section we simplify
the notation and omit writing the dependence on
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\in\left\\{x:x\in\mathbb{R}^{N}\backslash\bar{S}_{M}\wedge\left|\mathcal{I}^{e}\left(x\right)\right|=1\right\\}$
explicitly. All the statements hold for any such
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]$.

For the further argumentation, we introduce $R\left(\cdot\right)$ and
$N\left(\cdot\right)$ that denote a range and a null space, respectively.

######  Lemma 21.

Let $x\in\mathbb{R}^{N}\backslash\left\\{0\right\\}$ and $c\in C$ be any of
the consumers. Then $x^{\top}\mathcal{D}\tilde{\mathcal{Z}}_{c}x\leq 0$.
Moreover, $x^{\top}\mathcal{D}\tilde{\mathcal{Z}}_{c}x=0$ if and only if $x\in
R\left(\hat{A}_{1}^{\top}\right)$.

###### Proof.

Since the optimal power trading strategy $V_{c}$ can not be at the boundary of
the feasible region $S_{c}$ due to Lemma 12, the necessary and due to
convexity and the Slater condition in Assumption 3 also sufficient KKT
conditions for $V_{c}$ to be a global minimizer given the forward electricity
price reads

(64)
$\begin{array}[]{rcl}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]-\lambda_{c}\hat{Q}_{1}V_{c}-\hat{A}_{1}^{\top}\mu_{c}&=&0\\\
\\\ \hat{A}_{1}V_{c}&=&a_{c}.\end{array}$

One can then solve for $V_{c}$ from the first equation of (64) by multiplying
both sides from the left by $\left(\lambda_{c}\hat{Q}_{1}\right)^{-1}$ and
obtain

(65)
$V_{c}=-\frac{1}{\lambda_{c}}\hat{Q}_{1}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]-\frac{1}{\lambda_{c}}\hat{Q}_{1}^{-1}\hat{A}_{1}^{\top}\mu_{c}.$

By further multiplying both sides of (65) from the left by $\hat{A}_{1}$, we
get

(66)
$\hat{A}_{1}V_{c}=-\frac{1}{\lambda_{c}}\hat{A}_{1}\hat{Q}_{1}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]-\frac{1}{\lambda_{c}}\hat{A}_{1}\hat{Q}_{1}^{-1}\hat{A}_{1}^{\top}\mu_{c}.$

Since $\hat{A}_{1}$ has full row rank
$\left(\hat{A}_{1}\hat{Q}_{1}^{-1}\hat{A}_{1}^{\top}\right)^{-1}$ exists. By
using the second equation of (64), we get

(67)
$\mu_{c}=-\left(\hat{A}_{1}\hat{Q}_{1}^{-1}\hat{A}_{1}^{\top}\right)^{-1}\left(\lambda_{c}a_{c}+\hat{A}_{1}\hat{Q}_{1}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right).$

Inserting that back into (65), we can calculate

(68) $\frac{\partial
V_{c}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]}=-\frac{1}{\lambda_{c}}\hat{Q}_{1}^{-1}+\frac{1}{\lambda_{c}}\hat{Q}_{1}^{-1}\hat{A}_{1}^{\top}\left(\hat{A}_{1}\hat{Q}_{1}^{-1}\hat{A}_{1}^{\top}\right)^{-1}\hat{A}_{1}\hat{Q}_{1}^{-1}.$

Since $\hat{Q}_{1}$ (and consequently also $\hat{Q}_{1}^{-1}$) is positive
definite and symmetric, we can use the Cholesky decomposition to define a
matrix $\tilde{Q_{1}}\in\mathbb{R}^{N\times N}$ such that
$\hat{Q}_{1}^{-1}=\tilde{Q}_{1}^{-1}\tilde{Q_{1}}^{-\top}$. Since
$\tilde{Q_{1}}$ is invertible, we can rewrite (68) as

(69) $\tilde{P}:=-\lambda_{c}\tilde{Q}_{1}\frac{\partial
V_{c}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]}\tilde{Q_{1}}^{\top}=I-\tilde{Q_{1}}^{-\top}\hat{A}_{1}^{\top}\left(\hat{A}_{1}\hat{Q}_{1}^{-1}\hat{A}_{1}^{\top}\right)^{-1}\hat{A}_{1}\tilde{Q_{1}}^{-1}.$

It is trivial to check that $\tilde{P}=\tilde{P}^{2}$ (i.e. $\tilde{P}$ is
idempotent) and symmetric. Thus, $\tilde{P}$ must be a projection matrix. It
is known that every projection matrix is positive semidefinite and thus
$\frac{\partial
V_{c}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]}\preceq 0$.
Moreover, for any $x\in\mathbb{R}^{N}\backslash\left\\{0\right\\}$,
$x^{\top}\tilde{P}x=0$ if and only if $x\in
R\left(\tilde{Q_{1}}^{-\top}\hat{A}_{1}^{\top}\right)$. Therefore, it follows
from (69) that $x^{\top}\frac{\partial
V_{c}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]}x=0$ if and
only if $x\in R\left(\hat{A}_{1}^{\top}\right)$.∎

######  Lemma 22.

Assume that $x^{\top}\mathcal{D}\tilde{\mathcal{Z}}_{p}x\leq 0$ for all $p\in
P$ and all $x\in\mathbb{R}^{N}\backslash\left\\{0\right\\}$. Then
$\mathcal{D}\tilde{\mathcal{Z}}\prec 0$ if and only if $\sum_{p\in
P}\hat{A}_{1}\mathcal{D}\tilde{\mathcal{Z}}_{p}\hat{A}_{1}^{\top}$ has a full
rank.

###### Proof.

We can write

$x^{\top}\mathcal{D}\tilde{\mathcal{Z}}x=\sum_{k\in P\cup
C}x^{\top}\mathcal{D}\tilde{\mathcal{Z}}_{p}x.$

By Lemma 21 and the assumption that
$x^{\top}\mathcal{D}\tilde{\mathcal{Z}}_{p}x\leq 0$ for all $p\in P$ and all
$x\in\mathbb{R}^{N}\backslash\left\\{0\right\\}$, we can see that
$x^{\top}\mathcal{D}\tilde{\mathcal{Z}}_{k}x\leq 0$ for all $k\in P\cup C$.
Thus, $x^{\top}\mathcal{D}\tilde{\mathcal{Z}}x<0$ if and only if for each
$x\in\mathbb{R}^{N}\backslash\left\\{0\right\\}$ there exists at least one
player $k^{\prime}\in P\cup C$ such that
$x^{\top}\mathcal{D}\tilde{\mathcal{Z}}_{k^{\prime}}x<0$. If $x\notin
R\left(\hat{A}_{1}^{\top}\right)$ then
$x^{\top}\mathcal{D}\tilde{\mathcal{Z}}_{k}x<0$ for all $k\in C$. On the other
hand, if $x\in R\left(\hat{A}_{1}^{\top}\right)$ then
$x^{\top}\mathcal{D}\tilde{\mathcal{Z}}_{k}x=0$ for all $k\in C$. Thus, for
all $x\in R\left(\hat{A}_{1}^{\top}\right)$,
$\mathcal{D}\tilde{\mathcal{Z}}\prec 0$ if and only if $\sum_{p\in
P}\hat{A}_{1}\mathcal{D}\tilde{\mathcal{Z}}_{p}\hat{A}_{1}^{\top}$ has a full
rank.∎

######  Lemma 23.

Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$, and
$C\in\mathbb{R}^{m\times m}$ be real matrices such that $A^{-1}$ and
$\left(C-BC^{-1}B^{\top}\right)$ exist. Then

$\left[\begin{array}[]{cc}A^{\top}&B^{\top}\end{array}\right]\left[\begin{array}[]{cc}A&B\\\
B^{\top}&C\end{array}\right]^{-1}=\left[\begin{array}[]{cc}I&0\end{array}\right]$

and consequently

$\left[\begin{array}[]{cc}A^{\top}&B^{\top}\end{array}\right]\left[\begin{array}[]{cc}A&B\\\
B^{\top}&C\end{array}\right]^{-1}\left[\begin{array}[]{c}A\\\
B\end{array}\right]=A.$

###### Proof.

Since $A^{-1}$ and $\left(C-BC^{-1}B^{\top}\right)$ exist, we can use the
block matrix inverse formula to calculate

$\left[\begin{array}[]{cc}A&B\\\
B^{\top}&C\end{array}\right]^{-1}=\left[\begin{array}[]{cc}A^{-1}+A^{-1}B\left(C-B^{\top}A^{-1}B\right)^{-1}B^{\top}A^{-1}&-A^{-1}B\left(C-B^{\top}A^{-1}B\right)^{-1}\\\
-\left(C-B^{\top}A^{-1}B\right)^{-1}B^{\top}A^{-1}&\left(C-B^{\top}A^{-1}B\right)^{-1}\end{array}\right].$

The result then follows trivially.∎

######  Theorem 24.

For all $x\in\mathbb{R}^{N}$ and all $p\in P$,
$x^{\top}\mathcal{D}\tilde{\mathcal{Z}}_{p}x\leq 0$ holds. Moreover, if for
each delivery period $j\in J$ there exist at least one power plant that has a
strictly feasible optimal production, then $\sum_{p\in
P}\hat{A}_{1}\mathcal{D}\tilde{\mathcal{Z}}_{p}\hat{A}_{1}^{\top}$ has a full
rank.

###### Proof.

We can assume w.l.o.g that there exists a producer $p\in P$ who owns a set of
power plants among which at least one has a strictly feasible optimal
production in each delivery periods $j\in J$. In this case, showing that
$\sum_{p\in P}\hat{A}_{1}\mathcal{D}\tilde{\mathcal{Z}}_{p}\hat{A}_{1}^{\top}$
has a full rank simplifies to showing that
$\hat{A}_{1}\mathcal{D}\tilde{\mathcal{Z}}_{p}\hat{A}_{1}^{\top}$ has a full
rank.

Assume that the set of active inequality constraints $B_{p}$ is known.
Constraints (7) and (8) are never active due to Lemma 8. Similarly,
Constraints (9) are never active due to Lemma 12. The KKT conditions can thus
be written as

(70)
$\begin{array}[]{rcl}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]-\lambda_{p}\hat{Q}v_{p}^{\prime}-\hat{A}_{12}^{\top}\mu^{\prime}_{p}&=&0\\\
\\\ -\hat{A}_{p}^{\top}\mu^{\prime}_{p}-B_{p}^{\top}\eta^{\prime}_{p}&=&0\\\
\\\ \hat{A}_{12}v_{p}^{\prime}+\hat{A}_{p}v^{\prime\prime}_{p}&=&0\\\ \\\
B_{p}v^{\prime\prime}_{p}&=&b\end{array}$

where
$v_{p}^{\prime}:=\left[V_{p}^{\top},F_{p}^{\top},O_{p}^{\top}\right]^{\top}$
and $v_{p}^{\prime\prime}:=W_{p}$. The internal structure of the matrices is
the following

(71) $\hat{Q}=:\left[\begin{array}[]{cc}\hat{Q}_{1}&\hat{Q}_{2}\\\
\hat{Q}_{2}^{\top}&\hat{Q}_{3}\end{array}\right],\quad\tilde{A}_{12}:=\left[\begin{array}[]{cc}\hat{A}_{1}&0\\\
0&\hat{A}_{2}\end{array}\right],\quad\hat{A}_{p}:=\left[\begin{array}[]{c}\hat{A}_{3,p}\\\
\hat{A}_{4,p}\end{array}\right].$

Since $\hat{Q}\succ 0$, we can express $v_{p}^{\prime}$ from the first
equation of (70) as

(72)
$v_{p}^{\prime}=-\frac{1}{\lambda_{p}}\hat{Q}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]-\frac{1}{\lambda_{p}}\hat{Q}^{-1}\hat{A}_{12}^{\top}\mu^{\prime}_{p}$

and further $\hat{A}_{12}v_{p}^{\prime}$ as

(73)
$\hat{A}_{12}v_{p}^{\prime}=-\frac{1}{\lambda_{p}}\hat{A}_{12}\hat{Q}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]-\frac{1}{\lambda_{p}}\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\mu^{\prime}_{p}.$

Using the third equation of (70) this reads

(74)
$\hat{A}_{p}v^{\prime\prime}_{p}=\frac{1}{\lambda_{p}}\hat{A}_{12}\hat{Q}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]+\frac{1}{\lambda_{p}}\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\mu^{\prime}_{p}.$

Since $\hat{A}_{1}$ and $\hat{A}_{2}$ both have a full rank, also
$\hat{A}_{12}$ has a full rank and thus
$\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}$ exists. Then

(75)
$\mu^{\prime}_{p}=\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\left(\lambda_{p}\hat{A}_{p}v^{\prime\prime}_{p}-\hat{A}_{12}\hat{Q}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]\right).$

Multiplying both sides with $\hat{A}_{p}^{\top}$ from the left and using the
second equation of (70), we get

(76)
$B_{p}^{\top}\eta^{\prime}_{p}=\hat{A}_{p}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\left(-\lambda_{p}\hat{A}_{p}v^{\prime\prime}_{p}+\hat{A}_{12}\hat{Q}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]\right).$

It is not possible to express $v^{\prime\prime}_{p}$ from (76) because
$\text{rank}\left(\hat{A}_{p}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{p}\right)\leq\left|J\right|\left(\left|L\right|+1\right)+1<\dim
W_{p}$ and thus
$\hat{A}_{p}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{p}$
is not invertible. We can write (76) and the last equation of (70) as the
following system of linear equations

(77)
$\left[\begin{array}[]{cc}\lambda_{p}\hat{A}_{p}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{p}&B_{p}^{\top}\\\
\\\ B_{p}&0\end{array}\right]\left[\begin{array}[]{c}v^{\prime\prime}_{p}\\\
\\\
\eta^{\prime}_{p}\end{array}\right]=\left[\begin{array}[]{c}\hat{A}_{p}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{12}\hat{Q}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]\\\
\\\ b_{p}\end{array}\right].$

Before we attempt to solve (77), let us try to evaluate $\frac{\partial
v_{p}^{\prime}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]}$ with
the currently known facts. We get

$\begin{array}[]{rcl}\frac{\partial
v_{p}^{\prime}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]}&=&-\frac{1}{\lambda_{p}}\hat{Q}^{-1}-\frac{1}{\lambda_{p}}\hat{Q}^{-1}\hat{A}_{12}^{\top}\frac{\partial\mu^{\prime}_{p}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]}\\\
\\\
&=&-\frac{1}{\lambda_{p}}\hat{Q}^{-1}+\frac{1}{\lambda_{p}}\hat{Q}^{-1}\hat{A}_{12}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\left(\hat{A}_{12}\hat{Q}^{-1}-\lambda_{p}\hat{A}_{p}\frac{\partial
v^{\prime\prime}_{p}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]}\right),\end{array}$

where (72) and (75) were used to obtain the first and the second equality,
respectively. We are interested in the rank of
$\hat{A}_{1}\mathcal{D}\tilde{\mathcal{Z}}_{p}\hat{A}_{1}^{\top}$. Define
$\tilde{A}_{1}=\left[\hat{A}_{1}0\right]$, where $0$ is a matrix of zeros of
dimension $\left|J\right|\times\left(N\left|L\right|+1\right)$. Then,

(78)
$\begin{array}[]{rcl}\hat{A}_{1}\mathcal{D}\tilde{\mathcal{Z}}_{p}\hat{A}_{1}^{\top}&=&\tilde{A}_{1}\frac{\partial
v_{p}^{\prime}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]}\tilde{A}_{1}^{\top}\\\
\\\
&=&-\frac{1}{\lambda_{p}}\tilde{A}_{1}\hat{Q}^{-1}\tilde{A}_{1}^{\top}+\frac{1}{\lambda_{p}}\tilde{A}_{1}\hat{Q}^{-1}\hat{A}_{12}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{12}\hat{Q}^{-1}\tilde{A}_{1}^{\top}\\\
\\\
&&-\tilde{A}_{1}\hat{Q}^{-1}\hat{A}_{12}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{p}\frac{\partial
v^{\prime\prime}_{p}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]}\tilde{A}_{1}^{\top}.\end{array}$

Using the internal structure of $\hat{Q}$ and $\hat{A}_{12}$ as defined in
(71), we evaluate

(79)
$\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}=\left[\begin{array}[]{cc}\hat{A}_{1}\hat{Q}_{1}\hat{A}_{1}^{\top}&\hat{A}_{1}\hat{Q}_{2}\hat{A}_{2}^{\top}\\\
\hat{A}_{2}\hat{Q}_{2}^{\top}\hat{A}_{1}^{\top}&\hat{A}_{2}\hat{Q}_{3}\hat{A}_{2}^{\top}\end{array}\right]$

and similarly

(80)
$\tilde{A}_{1}\hat{Q}^{-1}\hat{A}_{12}^{\top}=\left[\begin{array}[]{cc}\hat{A}_{1}\hat{Q}_{1}\hat{A}_{1}^{\top}&\hat{A}_{1}\hat{Q}_{2}\hat{A}_{2}^{\top}\end{array}\right].$

Using Lemma 23, (79), and (80), we can rewrite (78) as

(81)
$\hat{A}_{1}\mathcal{D}\tilde{\mathcal{Z}}_{p}\hat{A}_{1}^{\top}=-\tilde{A}_{1}\hat{Q}^{-1}\hat{A}_{12}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{p}\frac{\partial
v^{\prime\prime}_{p}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]}\tilde{A}_{1}^{\top}.$

In order to evaluate (81), we return back to the system of linear equations
(77). Using the generalized Bott-Duffin constrained inverse (see [25]), we can
express $\frac{\partial
v^{\prime\prime}_{p}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]}$
from (77) as

(82) $\frac{\partial
v^{\prime\prime}_{p}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]}=\left(\check{A}_{p}\right)_{S}^{\left(+\right)}\hat{A}_{p}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{12}\hat{Q}^{-1}+P_{N(\check{A}_{p})\cap
S}\frac{\partial
z_{p}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]},$

where
$\left(\check{A}_{p}\right)_{S}^{\left(+\right)}=P_{S}\left(\check{A}_{p}P_{S}+P_{S^{\perp}}\right)^{\left(+\right)}$,
$\check{A}_{p}:=\lambda_{p}\hat{A}_{p}^{\top}\left(\hat{A}_{12}Q_{P}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{p}$,
$S$ is a null space of $B_{p}$, i.e. $S=N\left(B_{p}\right)$, $P_{S}$ a
projection on $S$, and $z_{p}\in\mathbb{R}^{\dim W_{p}}$ an arbitrary vector.
The Moore–Penrose pseudoinverse is denoted by (+). For every $x\in
N(\check{A}_{p})$,

(83) $\begin{array}[]{rcl}0&=&\check{A}_{p}x\\\ \\\
&=&\hat{A}_{p}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}A_{12}^{\top}\right)^{-1}\hat{A}_{p}x\\\
\\\
&=&x^{\top}\hat{A}_{p}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{p}x\\\
\\\ &=&\tilde{Q}\hat{A}_{p}x\\\ \\\ &=&\hat{A}_{p}x,\end{array}$

where
$\tilde{Q}\in\mathbb{R}^{\left|J\right|\left(\left|L\right|+1\right)+1\times\left|J\right|\left(\left|L\right|+1\right)+1}$
is defined by the Cholesky decomposition such that

(84)
$\tilde{Q}\tilde{Q}^{\top}=\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}.$

Since this holds for every $x\in N(\check{A}_{p})$, it must hold also for
every $x\in N(\check{A}_{p})\cap S$ and therefore

(85)
$-\tilde{A}_{1}\hat{Q}^{-1}\hat{A}_{12}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{p}P_{N(\check{A}_{p})\cap
S}\frac{\partial
z_{p}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]}=0.$

Properties about the Bott-Duffin inverse (see [25]) show that if
$\check{A}_{p}$ is symmetric, then also
$\left(\check{A}_{p}\right)_{S}^{\left(+\right)}$ is symmetric. Moreover, if
$\left(v^{\prime\prime}_{p}\right)^{\top}\check{A}_{p}v^{\prime\prime}_{p}\geq
0$ for all $v^{\prime\prime}_{p}$ such that $B_{p}v^{\prime\prime}_{p}=b_{p}$,
then also
$\left(v^{\prime\prime}_{p}\right)^{\top}\left(\check{A}_{p}\right)_{S}^{\left(+\right)}v^{\prime\prime}_{p}\geq
0$ for all $v^{\prime\prime}_{p}$. Thus, we can see from (81), (82), and (85)
that $x^{\top}\mathcal{D}\tilde{\mathcal{Z}}_{p}x\leq 0$, $p\in P$ for all
$x\in\mathbb{R}^{N}$, which proves the first part of this theorem.

Using (85), (79) and (80), equation (81) reads

(86)
$\hat{A}_{1}\mathcal{D}\tilde{\mathcal{Z}}_{p}\hat{A}_{1}^{\top}=\hat{A}_{3,p}\left(\check{A}_{p}\right)_{S}^{\left(+\right)}\hat{A}_{3,p}^{\top}.$

From [25] we know that
$R\left(\left(\check{A}_{p}\right)_{S}^{\left(+\right)}\right)=R\left(P_{S}\check{A}_{p}\right)$.
Since $\left(\check{A}_{p}\right)_{S}^{\left(+\right)}$ is symmetric and
positive semidefinite there exists $U\in\mathbb{R}^{\dim W_{p}\times\dim
W_{p}}$ such that $\left(\check{A}_{p}\right)_{S}^{\left(+\right)}=UU^{\top}$.
Then,

(87)
$\begin{array}[]{rcl}R\left(P_{S}\check{A}_{p}\right)&=&R\left(\left(\check{A}_{p}\right)_{S}^{\left(+\right)}\right)\\\
\\\ &=&R\left(UU^{\top}\right)\\\ \\\ &=&R(U)\end{array}$

and

(88)
$\begin{array}[]{rcl}\text{rank}\left(\hat{A}_{3,p}\left(\check{A}_{p}\right)_{S}^{\left(+\right)}\hat{A}_{3,p}^{\top}\right)&=&\text{rank}\left(\hat{A}_{3,p}U\right)\\\
\\\ &=&\dim R\left(U\right)-\dim N\left(\hat{A}_{3,p}\right)\cap
R\left(U\right).\end{array}$

The next step is to show that
$R\left(P_{S}\check{A}_{p}\right)=R\left(P_{S}\hat{A}_{p}^{\top}\right)$. We
know that for any matrices $\tilde{A}$ and $\tilde{B}$ of the same size
$R\left(\tilde{A}\right)=R\left(\tilde{B}\right)$ if and only if
$N\left(\tilde{A}^{\top}\right)=N\left(\tilde{B}^{\top}\right)$. Since
$\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}$ is positive
definite and symmetric, there exists an invertible matrix $\tilde{Q}$ as
defined in (84). Then for every $x\in
N\left(\left(P_{S}\check{A}_{p}\right)^{\top}\right)$

(89)
$\begin{array}[]{rcl}0&=&\hat{A}_{p}^{\top}\tilde{Q}\tilde{Q}^{\top}\hat{A}_{p}P_{S}x\\\
\\\
&=&x^{\top}P_{S}\hat{A}_{p}^{\top}\tilde{Q}\tilde{Q}^{\top}\hat{A}_{p}P_{S}x\\\
\\\ &=&\tilde{Q}^{\top}\hat{A}_{p}P_{S}x\\\ \\\
&=&\hat{A}_{p}P_{S}x,\end{array}$

and thus
$R\left(P_{S}\check{A}_{p}\right)=R\left(P_{S}\hat{A}_{p}^{\top}\right)$. It
follows from (87), (88) and (89) that

(90)
$\begin{array}[]{rcl}\text{rank}\left(\hat{A}_{3,p}\left(\check{A}_{p}\right)_{S}^{\left(+\right)}\hat{A}_{3,p}^{\top}\right)&=&\text{rank}\left(\hat{A}_{3,p}P_{S}\left[\begin{array}[]{c|c}\hat{A}_{3,p}^{\top}&\hat{A}_{4,p}^{\top}\end{array}\right]\right)\\\
\\\ &=&\text{rank}\left(\hat{A}_{3,p}P_{S}\right),\end{array}$

where the last equality holds since $P_{S}$ is symmetric.

A projection matrix $P_{S}$ on the subspace $S=N\left(B_{p}\right)$, can be
written as

(91) $P_{S}=I-B_{p}^{\top}\left(B_{p}B_{p}^{\top}\right)^{-1}B_{p}.$

By taking into account the structure of $B_{p}$ and $\hat{A}_{3,p}$ is easy to
see that $\hat{A}_{3,p}P_{S}$ has full rank if and only if for each delivery
period $j\in J$ there exists at least one power plant that has a strictly
feasible optimal production (i.e. one of the power plants does not appear in
the set of active constraints $B_{p}$). ∎

It is possible to establish the following unique closed form relation among
the prices
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i},T_{j}\right)\right]$,
$i\in\left\\{1,2\right\\}$ for any fixed $j\in J$. Since the claim holds for
any fixed $j\in J$ we simplify the notation and avoid writing $T_{j}$.

######  Proposition 25.

If $\Pi\left(t_{1}\right)$ and
$\mathbb{E}_{t_{1}}^{\mathbb{P}}\left[\Pi\left(t_{2}\right)\right]$ denote the
two-stage equilibrium prices at $t_{1}$, then the following equality

(92)
$\begin{array}[]{rcl}\Pi\left(t_{1}\right)&=&\mathbb{E}_{t_{1}}^{\mathbb{\tilde{P}}}\left[\Pi\left(t_{2}\right)\right]-\lambda^{-1}\text{Cov}_{t_{1}}\left[\Pi\left(t_{2}\right),\sum_{p\in
P}C_{p}\left[k_{p},F_{p}\left(t\right),G\left(t\right)\right]-\left(\sum_{c\in
C}s_{c}p_{c}\right)D\right]\end{array}$

where
$\lambda^{-1}=\sum_{p=1}^{P}\frac{1}{\lambda_{p}}+\sum_{c=1}^{C}\frac{1}{\lambda_{c}}$
and $k_{p}=V_{p}\left(t_{1}\right)+V_{p}\left(t_{2}\right)$, must be
satisfied. $C_{p}\left(V,F,G\right)$ denotes the costs of producing $V$ units
of electricity with the fuel trading strategy $F$ at fuel prices $G$.

###### Proof.

Profit of a producer $p\in P$ can in this setting be written as

(93)
$\begin{array}[]{rcl}P_{p}\left(V_{p}\left(t\right),F_{p}\left(t\right),\Pi\left(t\right)\right)&=&-\Pi\left(t_{1}\right)V_{p}\left(t_{1}\right)-\Pi\left(t_{2}\right)V_{p}\left(t_{2}\right)\\\
\\\
&&-C_{p}\left[V_{p}\left(t_{1}\right)+V_{p}\left(t_{2}\right),F_{p}\left(t\right),G\left(t\right)\right]\end{array}$

and profit of a consumer $c\in C$ as

$\begin{array}[]{rcl}P_{c}\left(V_{c}\left(t\right),\Pi\left(t\right)\right)&=&s_{c}p_{c}D\\\
\\\
&&-\Pi\left(t_{1}\right)V_{c}\left(t_{1}\right)-\Pi\left(t_{2}\right)V_{c}\left(t_{2}\right).\end{array}$

Using Lemma 12, we can see that for any producer $p\in P$ there is no
constraints on the small changes of $V_{p}\left(t_{1}\right)$ and
$V_{p}\left(t_{2}\right)$, but there is a capacity constraint on changes in
the sum $V_{p}\left(t_{1}\right)+V_{p}\left(t_{2}\right)$. Thus by setting
$V_{p}\left(t_{1}\right)+V_{p}\left(t_{2}\right)=k_{p}$ for some fixed
$k_{p}\in\mathbb{R}$ for all $p\in P$, constraints can not be violated by
small changes in volumes $V_{p}\left(t_{1}\right)$ or
$V_{p}\left(t_{2}\right)$. By inserting (93) into a mean-variance objective
function and taking a derivative, we obtain the following expression

$\begin{array}[]{rcl}\frac{\partial\Psi_{p}\left(P_{p}\left(V_{p}\left(t\right),F_{p}\left(t\right),\Pi\left(t\right)\right)\right)}{\partial
V_{p}\left(t_{2}\right)}&=&-\Pi\left(t_{1}\right)+\mathbb{E}_{t_{1}}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{2}\right)\right]\\\
\\\
&&+\lambda_{p}\left[\text{Var}_{t_{1}}\Pi\left(t_{2}\right)V_{p}\left(t_{2}\right)+\text{Cov}_{t_{1}}\left(\Pi\left(t_{2}\right),C_{p}\left[k_{p},F_{p}\left(t\right),G\left(t\right)\right]\right)\right].\end{array}$

By setting
$\frac{\partial\Psi_{p}\left(P_{p}\left(V_{p}\left(t\right),B_{p}\left(t\right),\Pi\left(t\right)\right)\right)}{\partial
V_{p}\left(t_{2}\right)}=0$ and expressing $V_{p}\left(t_{2}\right)$, we
obtain

(94)
$V_{p}\left(t_{2}\right)=\frac{\Pi\left(t_{1}\right)-\mathbb{E}_{t_{1}}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{2}\right)\right]}{\lambda_{p}\text{Var}_{t_{1}}\Pi\left(t_{2}\right)}+\frac{\text{Cov}_{t_{1}}\left[\Pi\left(t_{2}\right),C_{p}\left(k_{p},F_{p}\left(t\right),G\left(t\right)\right)\right]}{\text{Var}_{t_{1}}\Pi\left(t_{2}\right)}.$

Similarly, for each consumer $c\in C$

(95)
$V_{c}\left(t_{2}\right)=\frac{\Pi\left(t_{1}\right)-\mathbb{E}_{t_{1}}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{2}\right)\right]}{\lambda_{c}\text{Var}_{t_{1}}\Pi\left(t_{2}\right)}+\frac{\text{Cov}_{t_{1}}\left(D,\Pi\left(t_{2}\right)\right)}{\text{Var}_{t_{1}}\Pi\left(t_{2}\right)}.$

The market condition requires

(96) $0=\sum_{c\in C}V_{c}\left(t_{2}\right)+\sum_{p\in
P}V_{p}\left(t_{2}\right).$

Inserting expressions (94) and (95) into (96), we obtain the result

$\begin{array}[]{rcl}\Pi\left(t_{1}\right)&=&\mathbb{E}_{t_{1}}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{2}\right)\right]-\lambda^{-1}\text{Cov}_{t_{1}}\left[\Pi\left(t_{2}\right),\sum_{p\in
P}C_{p}\left[k_{p},F_{p}\left(t\right),G\left(t\right)\right]-\left(\sum_{c\in
C}s_{c}p_{c}\right)D\right]\end{array}$

where
$\lambda^{-1}=\sum_{p=1}^{P}\frac{1}{\lambda_{p}}+\sum_{c=1}^{C}\frac{1}{\lambda_{c}}$.
∎

The analytical expression obtained in Proposition 25 matches expression in
[5], [8], and [7], but extends results to more than one producer and consumer.
In contrast to the previous work, it also allows capacity constraints.

###  3.3 Mean Maximization

In this subsection we would like to examine a mean maximization problem
instead of the mean-variance maximization, in the context of assumptions and
theorems presented in the previous subsection.

A mean maximization optimization problem for each producer $p\in P$ is defined
as

$\begin{array}[]{rl}\underset{v_{p}}{\max}&-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi_{p}\right]^{\top}v_{p}\\\
\\\ \text{s.t.}&A_{p}v_{p}=a_{p}\\\ \\\ &B_{p}v_{p}\leq b_{p}\end{array}$

and a mean maximization optimization problem for each consumer $c\in C$ as

$\begin{array}[]{rl}\underset{V_{c}}{\max}&-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{\top}V_{c}\\\
\\\ \text{s.t.}&A_{c}V_{c}=a_{c}\\\ \\\ &B_{c}V_{c}\leq b_{c}.\end{array}$

Under Assumption 3, we use Corollary 4 to conclude that there exists a
solution to the mean maximization problem.

However, we are not able to claim the uniqueness of solution. Since
$\mathcal{D}_{v_{p}}\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)=0$
for all $p\in P$ and
$\mathcal{D}_{V_{c}}\Psi_{c}\left(V_{c},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{\top}\right)=0$
for all $c\in C$, expected utility functions of producers and consumers do not
satisfy strict concavity. To get some insight into this problem let us examine
the most simple case where $\left|P\right|=1$, $\left|C\right|=1$,
$T^{\prime}=1$, and $I_{1}=\left\\{1\right\\}$. We simplify the notation by
setting $V:=V_{p_{1}}$. The sensitivity analysis of the linear programming
shows that the optimal value
$V^{*}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)$ for a
given price $\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]$, is an
increasing piecewise constant function. Since we are looking for a price
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*}$ such that
$V^{*}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{*}\right)=D\left(T_{1}\right)$,
one can distinguish two cases (see Figure 1). In the first case (graph on the
left) the price equilibrium is not unique, but the volume for a given price
is. In the second case (graph on the right) the price equilibrium is unique,
but the volume for a given price is not. In order to ensure uniqueness of
both, Lemma 13 requires uniqueness of volumes for a given price (i.e. forbids
that the graph is vertical), and Theorem 20 (by setting
$\mathcal{D}\tilde{\mathcal{Z}}\neq 0$) requires uniqueness of prices for a
given volume (i.e. forbids that the graph is horizontal). As we have seen in
the previous subsection, the first condition depends only on the expected
utility function, while the second condition depends on both, the expected
utility function and constraints.

$\mathbb{E}^{\mathbb{P}}\left[\Pi\right]$$Prod$$Con$$V$$\mathbb{E}^{\mathbb{P}}\left[\Pi\right]$$Prod$$Con$$V$
Figure 1: Equilibrium types for the mean maximization problems.

Even though the uniqueness of solution can not be claimed in this setting, it
is still possible to establish the following unique relation among the prices
$\Pi\left(t_{i},T_{j}\right)$, $i\in I_{j}$ for any fixed $j\in J$.

######  Proposition 26.

Electricity prices
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i},T_{j}\right)\right]$ that
constitute a CE are equal for all $i\in I_{j}$ and any fixed $j\in J$.

###### Proof.

Let there exist $j\in J$, $i^{\prime}\in I_{j}$, $i^{\prime\prime}\in I_{j}$,
$i^{\prime}\neq i^{\prime\prime}$ such that
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i^{\prime}},T_{j}\right)\right]<\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i^{\prime\prime}},T_{j}\right)\right]$.
Then for each producer $p\in P$, $V_{p}\left(t_{i^{\prime}},T_{j}\right)\geq
V_{p}\left(t_{i^{\prime\prime}},T_{j}\right)$, because otherwise they could
improve their objective functions by exchanging
$V_{p}\left(t_{i^{\prime}},T_{j}\right)$ and
$V_{p}\left(t_{i^{\prime\prime}},T_{j}\right)$. The same also holds for all
consumers $c\in C$. Note that (38) implies

$0=\sum_{p\in P}V_{p}\left(t_{i^{\prime}},T_{j}\right)+\sum_{c\in
C}V_{c}\left(t_{i^{\prime}},T_{j}\right)$

and

$0=\sum_{p\in P}V_{p}\left(t_{i^{\prime\prime}},T_{j}\right)+\sum_{c\in
C}V_{c}\left(t_{i^{\prime\prime}},T_{j}\right).$

Therefore,
$V_{p}\left(t_{i^{\prime}},T_{j}\right)=V_{p}\left(t_{i^{\prime\prime}},T_{j}\right)$
for all $p\in P$ and
$V_{c}\left(t_{i^{\prime}},T_{j}\right)=V_{c}\left(t_{i^{\prime\prime}},T_{j}\right)$
for all $c\in C$.

If $\left|V_{p}\left(t_{i^{\prime}},T_{j}\right)\right|<V_{trade}$ then this
can not be a CE, because producer $p$ could improve its objective function by
decreasing $V_{p}\left(t_{i^{\prime}},T_{j}\right)$ for some $\epsilon>0$ and
increasing $V_{p}\left(t_{i^{\prime\prime}},T_{j}\right)$ for the same
$\epsilon>0$ without changing any other decision variables or violating any
constants. The same reasoning also holds for each consumer $c\in C$. Thus,
$\left|V_{p}\left(t_{i^{\prime}},T_{j}\right)\right|=V_{trade}$ for all $p\in
P$ and $\left|V_{c}\left(t_{i^{\prime}},T_{j}\right)\right|=V_{trade}$ for all
$c\in C$. Such solution clearly does not satisfy (12). Thus,
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i^{\prime}},T_{j}\right)\right]\geq\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i^{\prime\prime}},T_{j}\right)\right]$.
One can than apply a similar reasoning to
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i^{\prime}},T_{j}\right)\right]>\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i^{\prime\prime}},T_{j}\right)\right]$
and conclude
$\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i^{\prime}},T_{j}\right)\right]=\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i^{\prime\prime}},T_{j}\right)\right]$.
∎

##  4 Conclusions

In this paper we propose a new model for modeling the electricity price and
its relation to other fuels and emission certificates. The model belongs to a
class of game theoretic equilibrium models. In this paper we rigorously show
that there exists a solution to the proposed model and develop the conditions
under which the solution is also unique. In the last part of the paper we show
why the uniqueness of solution can not be claimed for a mean maximization
problem.

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