# Supplementary material for "Tariff Versus Sanction
Under Bounded Rationality"

Fredy Pokou, Hélène Le Cadre, Marta Fochesato

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## To cite this version:

Fredy Pokou, Hélène Le Cadre, Marta Fochesato. Supplementary material for "Tariff Versus Sanction Under Bounded Rationality". 2024. hal-04606536

HAL Id: hal-04606536

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Distributed under a Creative Commons Attribution 4.0 International License
# Supplementary material for "Tariff Versus Sanction
Under Bounded Rationality"

Fredy Pokou, Hélène Le Cadre, Marta Fochesato

<figure><img src="image_1.png" /></figure>

## To cite this version:

Fredy Pokou, Hélène Le Cadre, Marta Fochesato. Supplementary material for "Tariff Versus Sanction Under Bounded Rationality". 2024. hal-04606536

HAL Id: hal-04606536

https://inria.hal.science/hal-04606536v1

Submitted on 10 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Supplementary material for "Tariff Versus Sanction Under Bounded Rationality"

Fredy POKOU *<fnref n="1" />, Marta Fochesato †<fnref n="2" />, and Hélène Le Cadre ‡<fnref n="1" />

June 10, 2024

# 1 Introduction

# 2 The Geographic Markets

# 3 Market Structures

## 3.1 Price-Taker Formulation

**Proposition 1.** There exists a NE solution of the Nash Game *G*.

*Proof.* It is easy to see that $J_n(\cdot), \forall n \in \mathcal{N}$ are concave functions by checking that their Hessian matrix is negative semi-definite. The proposition statement follows from (Rosen, 1965). □

**Proposition 2.** At equilibrium, $d_n^* = q_n^* + Q_n^*, \forall n \in \mathcal{N}$.

*Proof.* Reasoning by contradiction, we assume that at equilibrium there exists $n \in \mathcal{N}$ such that $d_n^* < q_n^* + Q_n^*$. As a result, for this $n \in \mathcal{N}$, $\lambda_n^* = 0$. But, from the KKTs and the constraint on bilateral trades price symmetry, this implies that $\lambda_n^* = 0, \forall n \in \mathcal{N}$. As a result, Russia and Asia make no profit from exports. Then, in order to maximize its profits function, while satisfying the demand constraints, the generation variables will take the values $\bar{d}_R$ and $\bar{d}_A$, respectively.

Therefore $Q_R^* = Q_A^* = 0$. This implies in turn that $q_{R,A}^* = q_{R,EU}^* = q_{A,EU}^* = 0$, and then $Q_{EU}^* = 0$. Then, from the primal feasibility constraint for EU we get that $d_{EU}^* \le 0$, which is in contradiction with the form of the EU's utility function which is quadratic in $d_{EU}$ with a maximum on $\mathbb{R}_+^*$. The proposition statement follows. □

**Proposition 3.** At equilibrium, there exist multiple NEs.

*Proof.* At equilibrium, from the stationarity conditions, we get that $p_{EU,A}^* = \lambda_{EU}^*, p_{EU,R}^* = \lambda_{EU}^*, p_{A,EU}^* = \lambda_A^*, p_{A,R}^* = \lambda_A^*, p_{R,EU}^* = \lambda_R^*, p_{R,A}^* = \lambda_R^*$.

From this set of relationships, we infer that $p_{EU,A}^* = p_{EU,R}^*, p_{A,EU}^* = p_{A,R}^*, p_{R,EU}^* = p_{R,A}^*$ and $\lambda_{EU}^* = \lambda_R^* = \lambda_A^* = \lambda^*$.

<footnote n="1">¹Inria, CNRS, Univ. of Lille, Centrale Lille, UMR 9189 - CRISTAL, F-59000 Lille, France.</footnote>

<footnote n="2">²Automatic Control Laboratory, Department of Electrical Engineering and Information Technology, ETH Zürich, Physikstrasse 38092 Zürich, Switzerland.</footnote>

<footnote>*fredy-vale-manuel.pokou@inria.fr</footnote>

<footnote>†mfochesato@ethz.ch</footnote>

<footnote>‡helene.le-cadre@inria.fr</footnote>
# Supplementary material for "Tariff Versus Sanction Under Bounded Rationality"

Fredy POKOU *<fnref n="1" />, Marta Fochesato †<fnref n="2" />, and Hélène Le Cadre ‡<fnref n="1" />

June 10, 2024

# 1 Introduction

# 2 The Geographic Markets

# 3 Market Structures

## 3.1 Price-Taker Formulation

**Proposition 1.** *There exists a NE solution of the Nash Game G.*

*Proof.* It is easy to see that $J_n(\cdot), \forall n \in \mathcal{N}$ are concave functions by checking that their Hessian matrix is negative semi-definite. The proposition statement follows from (Rosen, 1965). □

**Proposition 2.** *At equilibrium, $d_n^* = q_n^* + Q_n^*, \forall n \in \mathcal{N}$.*

*Proof.* Reasoning by contradiction, we assume that at equilibrium there exists $n \in \mathcal{N}$ such that $d_n^* < q_n^* + Q_n^*$. As a result, for this $n \in \mathcal{N}$, $\lambda_n^* = 0$. But, from the KKTs and the constraint on bilateral trades price symmetry, this implies that $\lambda_n^* = 0, \forall n \in \mathcal{N}$. As a result, Russia and Asia make no profit from exports. Then, in order to maximize its profits function, while satisfying the demand constraints, the generation variables will take the values $\bar{d}_R$ and $\bar{d}_A$, respectively.

Therefore $Q_R^* = Q_A^* = 0$. This implies in turn that $q_{R,A}^* = q_{R,EU}^* = q_{A,EU}^* = 0$, and then $Q_{EU}^* = 0$. Then, from the primal feasibility constraint for EU we get that $d_{EU}^* \le 0$, which is in contradiction with the form of the EU's utility function which is quadratic in $d_{EU}$ with a maximum on $\mathbb{R}_+$. The proposition statement follows. □

**Proposition 3.** *At equilibrium, there exist multiple NEs.*

*Proof.* At equilibrium, from the stationarity conditions, we get that $p_{EU,A}^* = \lambda_{EU}^*$, $p_{EU,R}^* = \lambda_{EU}^*$, $p_{A,EU}^* = \lambda_A^*$, $p_{A,R}^* = \lambda_A^*$, $p_{R,EU}^* = \lambda_R^*$, $p_{R,A}^* = \lambda_R^*$.

From this set of relationships, we infer that $p_{EU,A}^* = p_{EU,R}^*$, $p_{A,EU}^* = p_{A,R}^*$, $p_{R,EU}^* = p_{R,A}^*$ and $\lambda_{EU}^* = \lambda_R^* = \lambda_A^* = \lambda^*$.

<footnote n="1">¹Inria, CNRS, Univ. of Lille, Centrale Lille, UMR 9189 - CRISTAL, F-59000 Lille, France.</footnote>

<footnote n="2">²Automatic Control Laboratory, Department of Electrical Engineering and Information Technology, ETH Zürich, Physikstrasse 38092 Zürich, Switzerland.</footnote>

<footnote>*fredy-vale-manuel.pokou@inria.fr</footnote>

<footnote>†mfochesato@ethz.ch</footnote>

<footnote>‡helene.le-cadre@inria.fr</footnote>
In addition, we have the following expressions for demand and generations at NE: $d_{\text{EU}}^* = \alpha_{\text{EU}} - \beta_{\text{EU}}\lambda_{\text{EU}}^*$, $q_A^* = \frac{1}{2a_A}(\lambda_A^* - b_A)$ and $q_R^* = \frac{1}{2a_R}(\lambda_R^* - b_R)$. From Prop. 2, we have:

$$Q_{\text{EU}}^* = d_{\text{EU}}^*, Q_R^* = \bar{d}_R - q_R^*, Q_A^* = \bar{d}_A - q_A^* \quad (1)$$

Furthermore, the bilateral trading reciprocity constraint implies that $\sum_n Q_n^* = 0$. This enables us to prove the existence of an interior point NE:

$$\lambda^* = \frac{\alpha_{\text{EU}} + \left[ \bar{d}_A + \frac{b_A}{2a_A} + \bar{d}_R + \frac{b_R}{2a_R} \right]}{\beta_{\text{EU}} + \frac{b_A}{2a_A} + \frac{b_R}{2a_R}} \quad (2)$$

From Equation 1, we obtain a system of three equations with three unknown variables:

$$\left\{
\begin{array}{l}
q_{R,EU} + q_{A,EU} = d_{EU}^*, \\
-q_{R,EU} + q_{A,R} = \bar{d}_R - q_R^*, \\
-q_{A,EU} - q_{A,R} = \bar{d}_A - q_A^*
\end{array}
\right.
\quad (3)$$

It is easy to see that this resulting system is incompatible whenever $d_{\text{EU}}^* + \bar{d}_A - q_A^* + \bar{d}_R - q_R^* \neq 0$, which never holds due to the construction of the game.

Then, $d_{\text{EU}}^* + \bar{d}_A - q_A^* + \bar{d}_R - q_R^* = 0$ always and the system is linearly dependent. The interior point NE is not uniquely defined. □

**Proposition 4.** $\forall n \in \mathcal{N}, J_n(x_n^*, x_0^*) = J_n(x_n^{**}, x_0^{**}), \forall x^*, x^{**} \in \text{SOL}(\mathcal{G})$.

*Proof.* By substitution of the optimal net imports in the markets’ utility functions, at each NE we obtain the following expressions:

$$\begin{align}
J_{EU}^* &= \frac{\alpha_{EU}^2}{2\beta_{EU}} - \alpha_{EU}\lambda^* + \frac{1}{2}\beta_{EU}(\lambda^*)^2 \tag{4} \\
J_n^* &= \left[ \frac{b_n^2}{4a_n} - c_n \right] - \left[ \bar{d}_n + \frac{b_n}{2a_n} \right] \lambda^* + \frac{(\lambda^*)^2}{4a_n}, \quad \forall n \in \{A, R\} \nonumber
\end{align}$$

which depend only on $\mathcal{G}$ parameters and where $\lambda^*$ is defined in Equation (2). □

## 3.2 Price-Maker Formulation

**Proposition 5.** There exist multiple VEs solutions of $\mathcal{G}^v$.

*Proof.* At equilibrium, from the stationarity conditions, we get that $\lambda_{\text{EU,A}}^* = \lambda_{\text{EU}}^*$, $\lambda_{\text{EU,R}}^* = \lambda_{\text{EU}}^*$, $\lambda_{A,EU}^* = \lambda_A^*$, $\lambda_{A,R}^* = \lambda_A^*$, $\lambda_{R,EU}^* = \lambda_R^*$, $\lambda_{R,A}^* = \lambda_R^*$. From this set of relationships, we infer that $\lambda_{\text{EU,A}}^* = \lambda_{\text{EU,R}}^*, \lambda_{A,EU}^* = \lambda_{A,R}^*, \lambda_{R,EU}^* = \lambda_{R,A}^*$ and $\lambda_{\text{EU}}^* = \lambda_R^* = \lambda_A^* = \lambda^*$.

In addition, we have the following expressions for demand and generations at a VE: $d_{\text{EU}}^* = \alpha_{\text{EU}} - \beta_{\text{EU}}\lambda_{\text{EU}}^*$, $q_A^* = \frac{1}{2a_A}(\lambda_A^* - b_A)$ and $q_R^* = \frac{1}{2a_R}(\lambda_R^* - b_R)$.

we have: $Q_{\text{EU}}^* = d_{\text{EU}}^*, Q_R^* = \bar{d}_R - q_R^*, Q_A^* = \bar{d}_A - q_A^*$

Furthermore, the bilateral trading reciprocity constraint implies that $\sum_n Q_n^* = 0$. This enables us to prove the existence of an interior point NE in $\lambda^*$:

$$\lambda^* = \frac{\alpha_{\text{EU}} + \left[ \bar{d}_A + \frac{b_A}{2a_A} + \bar{d}_R + \frac{b_R}{2a_R} \right]}{\beta_{\text{EU}} + \frac{b_A}{2a_A} + \frac{b_R}{2a_R}}$$
In addition, we have the following expressions for demand and generations at NE: $d_{\text{EU}}^* = \alpha_{\text{EU}} - \beta_{\text{EU}}\lambda_{\text{EU}}^*$, $q_A^* = \frac{1}{2a_A}(\lambda_A^* - b_A)$ and $q_R^* = \frac{1}{2a_R}(\lambda_R^* - b_R)$. From Prop. 2, we have:

$$Q_{EU}^* = d_{EU}^*, Q_R^* = \bar{d}_R - q_R^*, Q_A^* = \bar{d}_A - q_A^*. \quad (1)$$

Furthermore, the bilateral trading reciprocity constraint implies that $\sum_n Q_n^* = 0$. This enables us to prove the existence of an interior point NE:

$$\lambda^* = \frac{\alpha_{\text{EU}} + \left[ \bar{d}_A + \frac{b_A}{2a_A} + \bar{d}_R + \frac{b_R}{2a_R} \right]}{\beta_{\text{EU}} + \frac{b_A}{2a_A} + \frac{b_R}{2a_R}} \quad (2)$$

From Equation 1, we obtain a system of three equations with three unknown variables:

$$\left\{
\begin{array}{l}
q_{R,EU} + q_{A,EU} = d_{EU}^*, \\
-q_{R,EU} + q_{A,R} = \bar{d}_R - q_R^*, \\
-q_{A,EU} - q_{A,R} = \bar{d}_A - q_A^*
\end{array}
\right.
\quad (3)$$

It is easy to see that this resulting system is incompatible whenever $d_{\text{EU}}^* + \bar{d}_A - q_A^* + \bar{d}_R - q_R^* \neq 0$, which never holds due to the construction of the game.

Then, $d_{\text{EU}}^* + \bar{d}_A - q_A^* + \bar{d}_R - q_R^* = 0$ always and the system is linearly dependent. The interior point NE is not uniquely defined. □

**Proposition 4.** $\forall n \in \mathcal{N}, J_n(x_n^*, x_0^*) = J_n(x_n^{**}, x_0^{**}), \forall x^*, x^{**} \in \text{SOL}(\mathcal{G})$.

*Proof.* By substitution of the optimal net imports in the markets’ utility functions, at each NE we obtain the following expressions:

$$\begin{align}
J_{EU}^* &= \frac{\alpha_{EU}^2}{2\beta_{EU}} - \alpha_{EU}\lambda^* + \frac{1}{2}\beta_{EU}(\lambda^*)^2 \tag{4} \\
J_n^* &= \left[ \frac{b_n^2}{4a_n} - c_n \right] - \left[ \bar{d}_n + \frac{b_n}{2a_n} \right] \lambda^* + \frac{(\lambda^*)^2}{4a_n}, \quad \forall n \in \{A, R\} \nonumber
\end{align}$$

which depend only on $\mathcal{G}$ parameters and where $\lambda^*$ is defined in Equation (2). □

## 3.2 Price-Maker Formulation

**Proposition 5.** There exist multiple VEs solutions of $\mathcal{G}^v$.

*Proof.* At equilibrium, from the stationarity conditions, we get that $\lambda_{\text{EU,A}}^* = \lambda_{\text{EU}}^*$, $\lambda_{\text{EU,R}}^* = \lambda_{\text{EU}}^*$, $\lambda_{A,EU}^* = \lambda_A^*$, $\lambda_{A,R}^* = \lambda_A^*$, $\lambda_{R,EU}^* = \lambda_R^*$, $\lambda_{R,A}^* = \lambda_R^*$. From this set of relationships, we infer that $\lambda_{\text{EU,A}}^* = \lambda_{\text{EU,R}}^*, \lambda_{A,EU}^* = \lambda_{A,R}^*, \lambda_{R,EU}^* = \lambda_{R,A}^*$ and $\lambda_{\text{EU}}^* = \lambda_R^* = \lambda_A^* = \lambda^*$.

In addition, we have the following expressions for demand and generations at a VE: $d_{\text{EU}}^* = \alpha_{\text{EU}} - \beta_{\text{EU}}\lambda_{\text{EU}}^*$, $q_A^* = \frac{1}{2a_A}(\lambda_A^* - b_A)$ and $q_R^* = \frac{1}{2a_R}(\lambda_R^* - b_R)$.

we have: $Q_{EU}^* = d_{EU}^*, Q_R^* = \bar{d}_R - q_R^*, Q_A^* = \bar{d}_A - q_A^*$

Furthermore, the bilateral trading reciprocity constraint implies that $\sum_n Q_n^* = 0$. This enables us to prove the existence of an interior point NE in $\lambda^*$:

$$\lambda^* = \frac{\alpha_{\text{EU}} + \left[ \bar{d}_A + \frac{b_A}{2a_A} + \bar{d}_R + \frac{b_R}{2a_R} \right]}{\beta_{\text{EU}} + \frac{b_A}{2a_A} + \frac{b_R}{2a_R}}$$
## 3.3 Optimum Tariff

**Proposition 6.** There exists a closed-form mapping linking the market price with no tariff $\lambda^*$ to the market price with tariff $\lambda^t(\tau) = \lambda^* + \frac{\beta_{EU}\tau}{\beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}}$. Furthermore, there exists a unique tariff $\tau^*$ solution of the Stackelberg game.

*Proof.* For fixed $\tau$, relying on the stationarity conditions of the lower level, from the trades price symmetry, we infer that $\lambda_{EU}^*(\tau) = \lambda^t(\tau) - \tau$, $\lambda_A^*(\tau) = \lambda^t(\tau)$, $\lambda_R^*(\tau) = \lambda^t(\tau)$.

In addition, we have the following expressions for demand and generations at NE of the lower level: $d_{EU}^*(\tau) = \alpha_{EU} - \beta_{EU}\lambda_{EU}^*(\tau)$, $q_A^*(\tau) = \frac{\lambda_A^*(\tau) - b_A}{2a_A}$, and $q_R^*(\tau) = \frac{\lambda_R^*(\tau) - \tau - b_R}{2a_R}$.

We have $Q_{EU}^*(\tau) = d_{EU}^*(\tau)$, $Q_R^*(\tau) = \bar{d}_R - q_R^*(\tau)$, $Q_A^*(\tau) = \bar{d}_A - q_A^*(\tau)$. Furthermore, the bilateral trading reciprocity constraint implies that $\sum_n Q_n^*(\tau) = 0$. This enables us to prove the existence of an interior point NE of the lower level:

$$ \lambda^t(\tau) = \frac{\alpha_{EU} + \bar{d}_A + \bar{d}_R + \beta_{EU}\tau + \frac{b_A}{2a_A} + \frac{b_R}{2a_R}}{\beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}} + \frac{\beta_{EU}\tau}{\beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}} \quad (5) \\ = \lambda^* + \frac{\beta_{EU}\tau}{\beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}}, $$

where $\lambda^*$ closed form has been defined in Equation (2). □

**Proposition 7.** The market price in $G_{\tau_v}^v$ can be expressed as an affine function of the market price in $G^v$, $\lambda_v^t(\tau_v) = \lambda^* - \frac{\tau_v}{\beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}} \cdot \frac{\tau_v}{2a_R}$. Furthermore, there exists a unique tariff $\tau_v^*$ solution of the Stackelberg game.

*Proof.* For fixed $\tau_v$, relying on the stationarity conditions of the lower level, from the trades price symmetry, we infer that $\lambda_{EU}^*(\tau_v) = \lambda_v^t(\tau_v) - \tau_v$, $\lambda_A^*(\tau_v) = \lambda_v^t(\tau_v)$, $\lambda_R^*(\tau_v) = \lambda_v^t(\tau_v)$.

In addition, we have the following expressions for demand and generations at NE of the lower level: $d_{EU}^*(\tau_v) = \alpha_{EU} - \beta_{EU}\lambda_{EU}^*(\tau_v)$, $q_A^*(\tau_v) = \frac{\lambda_A^*(\tau_v) - b_A}{2a_A}$, and $q_R^*(\tau_v) = \frac{\lambda_R^*(\tau_v) + \tau_v - b_R}{2a_R}$.

We have $Q_{EU}^*(\tau_v) = d_{EU}^*(\tau_v)$, $Q_R^*(\tau_v) = \bar{d}_R - q_R^*(\tau_v)$, $Q_A^*(\tau_v) = \bar{d}_A - q_A^*(\tau_v)$. Furthermore, the bilateral trading reciprocity constraint implies that $\sum_n Q_n^*(\tau_v) = 0$. This enables us to prove the existence of an interior point NE of the lower level:

$$ \lambda_v^t(\tau_v) = \frac{\alpha_{EU} + \bar{d}_A + \bar{d}_R + \frac{b_A}{2a_A} + \frac{b_R - \tau_v}{2a_R}}{\beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}}, \quad (6) \\ = \lambda^* - \frac{\frac{\tau_v}{2a_R}}{\beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}}. $$

**Theorem 1.** There exist closed-form mappings linking the geographic markets' utility functions at a SE to the markets' utility functions at a NE solution of the benchmark model: $J_n^t = A_n^t J_n^* + B_n^t$, $\forall n \in \{EU, A\}$, $J_R^t = -3A_R^t J_R^* + B_R^t$, where

* $A_{EU}^t \stackrel{\text{def}}{=} \frac{2(a_A + a_R + 2a_A a_R \beta_{EU})^2}{(2a_A + 2a_R)(a_A + a_R + 4a_A a_R \beta_{EU})}$, $B_{EU}^t \stackrel{\text{def}}{=} \lambda^* \left( \frac{2\alpha_{EU}A_2}{(2a_A + 2a_R)A_1} - \frac{\alpha_{EU}A_2}{(a_A + a_R)A_1} \right) - \frac{\alpha_{EU}^2 A_2}{\beta_{EU}(2a_A + 2a_R)A_1} + \frac{\alpha_{EU}^2 A_2}{2\beta_{EU}(a_A + a_R)A_1}$, $A_1 = a_A + a_R + 4a_A a_R \beta_{EU}$, $A_2 \stackrel{\text{def}}{=} (a_A + a_R + 2a_A a_R \beta_{EU})^2$,

* $A_A^t \stackrel{\text{def}}{=} \frac{(4a_A^2 a_R^2 \beta_{EU}^2 + 4a_A^2 a_R \beta_{EU} + a_A^2 + 4a_A a_R^2 \beta_{EU} + 2a_A a_R + a_R^2)^2}{(a_A + a_R)^2(a_A + a_R + 4a_A a_R \beta_{EU})^2}$, $B_A^t \stackrel{\text{def}}{=} B_3 - c_A - \lambda^* \left( \frac{B_2 B_1}{(a_A + a_R)B_5} - \frac{B_2 B_1^2}{B_4} + \frac{2a_A a_R^2 \alpha_{EU} \beta_{EU} B_1}{B_4} \right) + \frac{(c_A - B_3) B_1^2}{B_4} + \frac{4a_A^3 a_R^4 \alpha_{EU}^2 \beta_{EU}^2}{B_4} + \frac{4a_A^2 a_R^2 \alpha_{EU} \beta_{EU} B_2}{(a_A + a_R)B_5}$, $B_1 \stackrel{\text{def}}{=} 4a_A^2 a_R^2 \beta_{EU}^2 + 4a_A^2 a_R \beta_{EU} + a_A^2 + 4a_A a_R^2 \beta_{EU} + 2a_A a_R + a_R^2$, $B_2 \stackrel{\text{def}}{=} \bar{d}_A + \frac{b_A}{2a_A}$, $B_3 \stackrel{\text{def}}{=} \frac{b_A^2}{4a_A}$, $B_4 \stackrel{\text{def}}{=} B_4 = (a_A + a_R)^2 B_5^2$, $B_5 \stackrel{\text{def}}{=} a_A + a_R + 4a_A a_R \beta_{EU}$.
## 3.3 Optimum Tariff

**Proposition 6.** There exists a closed-form mapping linking the market price with no tariff $\lambda^*$ to the market price with tariff $\lambda^t(\tau) = \lambda^* + \frac{\beta_{EU}\tau}{\beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}}$. Furthermore, there exists a unique tariff $\tau^*$ solution of the Stackelberg game.

*Proof.* For fixed $\tau$, relying on the stationarity conditions of the lower level, from the trades price symmetry, we infer that $\lambda_{EU}^*(\tau) = \lambda^t(\tau) - \tau$, $\lambda_A^*(\tau) = \lambda^t(\tau)$, $\lambda_R^*(\tau) = \lambda^t(\tau)$.

In addition, we have the following expressions for demand and generations at NE of the lower level: $d_{EU}^*(\tau) = \alpha_{EU} - \beta_{EU}\lambda_{EU}^*(\tau)$, $q_A^*(\tau) = \frac{\lambda_A^*(\tau) - b_A}{2a_A}$, and $q_R^*(\tau) = \frac{\lambda_R^*(\tau) - \tau - b_R}{2a_R}$.

We have $Q_{EU}^*(\tau) = d_{EU}^*(\tau)$, $Q_R^*(\tau) = \bar{d}_R - q_R^*(\tau)$, $Q_A^*(\tau) = \bar{d}_A - q_A^*(\tau)$. Furthermore, the bilateral trading reciprocity constraint implies that $\sum_n Q_n^*(\tau) = 0$. This enables us to prove the existence of an interior point NE of the lower level:

$$ \lambda^t(\tau) = \frac{\alpha_{EU} + \bar{d}_A + \bar{d}_R + \beta_{EU}\tau + \frac{b_A}{2a_A} + \frac{b_R}{2a_R}}{\beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}} + \frac{\beta_{EU}\tau}{\beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}} \quad (5) \\ = \lambda^* + \frac{\beta_{EU}\tau}{\beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}}, $$

where $\lambda^*$ closed form has been defined in Equation (2). □

**Proposition 7.** The market price in $G_{\tau_v}^v$ can be expressed as an affine function of the market price in $G^v$, $\lambda_v^t(\tau_v) = \lambda^* - \frac{\tau_v}{2a_R}$. Furthermore, there exists a unique tariff $\tau_v^*$ solution of the Stackelberg game.

*Proof.* For fixed $\tau_v$, relying on the stationarity conditions of the lower level, from the trades price symmetry, we infer that $\lambda_{EU}^*(\tau_v) = \lambda_v^t(\tau_v) - \tau_v$, $\lambda_A^*(\tau_v) = \lambda_v^t(\tau_v)$, $\lambda_R^*(\tau_v) = \lambda_v^t(\tau_v)$.

In addition, we have the following expressions for demand and generations at NE of the lower level: $d_{EU}^*(\tau_v) = \alpha_{EU} - \beta_{EU}\lambda_{EU}^*(\tau_v)$, $q_A^*(\tau_v) = \frac{\lambda_A^*(\tau_v) - b_A}{2a_A}$, and $q_R^*(\tau_v) = \frac{\lambda_R^*(\tau_v) + \tau_v - b_R}{2a_R}$.

We have $Q_{EU}^*(\tau_v) = d_{EU}^*(\tau_v)$, $Q_R^*(\tau_v) = \bar{d}_R - q_R^*(\tau_v)$, $Q_A^*(\tau_v) = \bar{d}_A - q_A^*(\tau_v)$. Furthermore, the bilateral trading reciprocity constraint implies that $\sum_n Q_n^*(\tau_v) = 0$. This enables us to prove the existence of an interior point NE of the lower level:

$$ \lambda_v^t(\tau_v) = \frac{\alpha_{EU} + \bar{d}_A + \bar{d}_R + \frac{b_A}{2a_A} + \frac{b_R - \tau_v}{2a_R}}{\beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}}, \quad (6) \\ = \lambda^* - \frac{\frac{\tau_v}{2a_R}}{\beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}}. $$

**Theorem 1.** There exist closed-form mappings linking the geographic markets' utility functions at a SE to the markets' utility functions at a NE solution of the benchmark model: $J_n^t = A_n^t J_n^* + B_n^t$, $\forall n \in \{EU, A\}$, $J_R^t = -3A_R^t J_R^* + B_R^t$, where

* $A_{EU}^t \stackrel{\text{def}}{=} \frac{2(a_A+a_R+2a_A a_R \beta_{EU})^2}{(2a_A+2a_R)(a_A+a_R+4a_A a_R \beta_{EU})}$, $B_{EU}^t \stackrel{\text{def}}{=} \lambda^* \left( \frac{2\alpha_{EU}A_2}{(2a_A+2a_R)A_1} - \frac{\alpha_{EU}A_2}{(a_A+a_R)A_1} \right) - \frac{\alpha_{EU}^2 A_2}{\beta_{EU}(2a_A+2a_R)A_1} + \frac{\alpha_{EU}^2 A_2}{2\beta_{EU}(a_A+a_R)A_1}$, $A_1 = a_A + a_R + 4a_A a_R \beta_{EU}$, $A_2 \stackrel{\text{def}}{=} (a_A + a_R + 2a_A a_R \beta_{EU})^2$,

* $A_A^t \stackrel{\text{def}}{=} \frac{(4a_A^2 a_R^2 \beta_{EU}^2 + 4a_A^2 a_R \beta_{EU} + a_A^2 + 4a_A a_R^2 \beta_{EU} + 2a_A a_R + a_R^2)^2}{(a_A+a_R)^2(a_A+a_R+4a_A a_R \beta_{EU})^2}$, $B_A^t \stackrel{\text{def}}{=} B_3 - c_A - \lambda^* \left( \frac{B_2 B_1}{(a_A+a_R)B_5} - \frac{B_2 B_1^2}{B_4} + \frac{2a_A a_R^2 \alpha_{EU} \beta_{EU} B_1}{B_4} \right) + \frac{(c_A - B_3) B_1^2}{B_4} + \frac{4a_A^3 a_R^4 \alpha_{EU}^2 \beta_{EU}^2}{B_4} + \frac{4a_A^2 a_R^2 \alpha_{EU} \beta_{EU} B_2}{(a_A+a_R)B_5}$, $B_1 \stackrel{\text{def}}{=} 4a_A^2 a_R^2 \beta_{EU}^2 + 4a_A^2 a_R \beta_{EU} + a_A^2 + 4a_A a_R^2 \beta_{EU} + 2a_A a_R + a_R^2$, $B_2 \stackrel{\text{def}}{=} \bar{d}_A + \frac{b_A}{2a_A}$, $B_3 \stackrel{\text{def}}{=} \frac{b_A^2}{4a_A}$, $B_4 \stackrel{\text{def}}{=} B_4 = (a_A + a_R)^2 B_5^2$, $B_5 \stackrel{\text{def}}{=} a_A + a_R + 4a_A a_R \beta_{EU}$.
$$
\bullet \quad A_R^t \stackrel{\text{def}}{=} \frac{(4 a_A^2 a_R^2 \beta_{EU}^2 + 4 a_A^2 a_R \beta_{EU} + a_A^2 + 4 a_A a_R^2 \beta_{EU} + 2 a_A a_R + a_R^2)^2}{(a_A + a_R)^2 (a_A + a_R + 4 a_A a_R \beta_{EU})^2}, \quad B_R^t \stackrel{\text{def}}{=} \lambda^* \left( \frac{C_3 C_1}{2 a_R (a_A + a_R) C_5} - \frac{3 (\bar{d}_R + \frac{b_R}{2 a_R}) C_1^2}{C_4} + \frac{6 a_A^2 a_R \alpha_{EU} \beta_{EU} C_1}{C_4} \right) - c_R + C_2 - \frac{3 (c_R - C_2) C_1^2}{C_4} - \frac{12 a_A^4 a_R^3 \alpha_{EU}^2 \beta_{EU}^2}{C_4} - \frac{2 a_A^2 a_R \alpha_{EU} \beta_{EU} C_3}{(a_A + a_R) C_5}, \\
C_1 \stackrel{\text{def}}{=} 4 a_A^2 a_R^2 \beta_{EU}^2 + 4 a_A^2 a_R \beta_{EU} + a_A^2 + 4 a_A a_R^2 \beta_{EU} + 2 a_A a_R + a_A^2, \quad C_2 \stackrel{\text{def}}{=} \frac{b_R^2}{4 a_R}, \quad C_3 \stackrel{\text{def}}{=} b_R - 2 a_R \bar{d}_R, \\
C_4 \stackrel{\text{def}}{=} (a_A + a_R)^2 C_5^2, \quad C_5 \stackrel{\text{def}}{=} a_A + a_R + 4 a_A a_R \beta_{EU}
$$

Proof. By proceeding with a polynomial Euclidean division, we can easily obtain this theorem's results. □

**Theorem 2.** The markets' utility at a SE solution of $\mathcal{G}_{\tau^v}$ can be expressed as affine functions in the markets' utility at a NE solution of $\mathcal{G}^v: J_n^v = A_n^v J_n^v + \tilde{B}_n^v, \forall n \in \{EU, A\}, J_R^t = A_R^t J_R^t + \tilde{B}_R^t$, where $A_n^v$ is a constant, $\tilde{B}_n^v$ depends linearly on $\lambda^*$, and $J_n^*$ is quadratic in $\lambda^*, \forall n$.

*Proof.* We can easily obtain these results by proceeding in a similar way to Theorem 1. □

3.4 Optimum Sanction

**Theorem 3.** There exists a closed-form mapping linking $\tau^s$ in the optimum sanction model to $\tau^*$ in the optimum tariff model : $\tau^s = \tilde{A}_s \cdot \tau^* + \tilde{B}_s$ where $\tilde{A}_s$ and $\tilde{B}_s$ are constants defined in Supplementary Material. A special case is when $\rho=0$, leading to $\tau^s = \tau^*$.

*Proof.* Assuming $y_1 = a_1 x + b_1$ and $y_2 = a_2 x + b_2$, when we do $\frac{y_2}{y_1}$, we can write $y_2 = C_1 y_1 + C_2$. Thus, $C_1 = \frac{a_2}{a_1}$ and $C_2 = b_2 - \frac{a_2 b_1}{a_1}$.

By proceeding in this way, we obtain coefficients $\tilde{A}_s$ and $\tilde{B}_s$. When $\rho = 0$, we can easily see that
$\tilde{A}_s = 1$ and $\tilde{B}_s = 0$ then $\tau^s = \tau^*$.

We have :

$$
\tilde{A}_s = \frac{\left( \frac{\beta_{EU}^2}{\tilde{A}_1} - \frac{3 \beta_{EU} \rho}{2 a_R \tilde{A}_1} \right) \left( \beta_{EU} - \frac{\beta_{EU}^3}{\tilde{A}_1^2} \right) \tilde{A}_1}{\beta_{EU}^2 \left( \beta_{EU} - \frac{\beta_{EU}^3}{\tilde{A}_1^2} + \frac{3 \beta_{EU}^2 \rho}{2 a_R \tilde{A}_1^2} \right)} \\
\tilde{B}_s = \frac{\alpha_{EU} \left( \frac{\beta_{EU}^2}{\tilde{A}_1} - \frac{3 \beta_{EU} \rho}{2 a_R \tilde{A}_1} \right)}{\beta_{EU} \tilde{A}_2} - \frac{\rho \left( \frac{\beta_{EU} \tilde{d}_R}{\tilde{A}_1} - \frac{b_R \beta_{EU}}{2 a_R \tilde{A}_1} \right) + \frac{\alpha_{EU} \beta_{EU}}{\tilde{A}_1}}{\tilde{A}_2}
$$

$$
\tilde{A}_1 \stackrel{\text{def}}{=} \beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}, \quad \tilde{A}_2 \stackrel{\text{def}}{=} \beta_{EU} - \frac{\beta_{EU}^3}{\tilde{A}_1^2} + \frac{3\beta_{EU}^2\rho}{2a_R\tilde{A}_1^2}
$$

**Theorem 4.** There exist closed-form mappings linking objective functions in the optimum sanction model to objective functions in the optimum tariff model: $J_n^s = \tilde{A}_n^s J_n^t + \tilde{B}_n^s, \forall n \in \mathcal{N}$.

*Proof.* We can easily obtain these results by proceeding in a similar way to Theorem 1. □

Theorems 3, 4 can easily be extended to the variational setting of the price-maker model.

**Theorem 3 [Price-maker]** There exists an affine relationship between $\tau_v^s$ in the optimum sanction model and $\tau_v^*$ in the optimum tariff model: $\tau^s = \tilde{A}_s^v \tau_v^* + \tilde{B}_s^v$, where $\tilde{A}_s^v$ and $\tilde{B}_s^v$ are:

$$
\bullet \quad \tilde{A}_s^v \stackrel{\text{def}}{=} \frac{\rho \lambda_{PT}^* \left( \frac{1}{4 a_R^2 \left( \beta_{EU} + \frac{1}{2 a_A} + \frac{1}{2 a_R} \right)} - \frac{1}{2 a_R} \right)}{4 a_R^2 A_1 \left( \alpha_{EU} + \bar{d}_A + \bar{d}_R + \frac{b_A}{2 a_A} + \frac{b_R}{2 a_R} \right)}, \\
A_1 \stackrel{\text{def}}{=} 2 \rho \left( \frac{1}{16 a_R^3 \left( \beta_{EU} + \frac{1}{2 a_A} + \frac{1}{2 a_R} \right)^2} - \frac{1}{4 a_R^2 \left( \beta_{EU} + \frac{1}{2 a_A} + \frac{1}{2 a_R} \right)} + \frac{1}{4 a_R} \right) + \frac{\beta_{EU}}{4 a_R^2 \left( \beta_{EU} + \frac{1}{2 a_A} + \frac{1}{2 a_R} \right)^2}
$$
• $A_R^t \stackrel{\text{def}}{=} \frac{(4a_A^2 a_R^2 \beta_{EU}^2 + 4a_A^2 a_R \beta_{EU} + a_A^2 + 4a_A a_R^2 \beta_{EU} + 2a_A a_R + a_R^2)^2}{(a_A + a_R)^2 (a_A + a_R + 4a_A a_R \beta_{EU})^2}, B_R^t \stackrel{\text{def}}{=} \lambda^* \left( \frac{C_3 C_1}{2a_R (a_A + a_R) C_5} - \frac{3(\bar{d}_R + \frac{b_R}{2a_R}) C_1^2}{C_4} + \frac{6a_A^2 a_R \alpha_{EU} \beta_{EU} C_1}{C_4} \right) - c_R + C_2 - \frac{3(c_R - C_2) C_1^2}{C_4} - \frac{12a_A^4 a_R^3 \alpha_{EU}^2 \beta_{EU}^2}{C_4} - \frac{2a_A^2 a_R \alpha_{EU} \beta_{EU} C_3}{(a_A + a_R) C_5}, \\
C_1 \stackrel{\text{def}}{=} 4a_A^2 a_R^2 \beta_{EU}^2 + 4a_A^2 a_R \beta_{EU} + a_A^2 + 4a_A a_R^2 \beta_{EU} + 2a_A a_R + a_A^2, C_2 \stackrel{\text{def}}{=} \frac{b_R^2}{4a_R}, C_3 \stackrel{\text{def}}{=} b_R - 2a_R \bar{d}_R, \\
C_4 \stackrel{\text{def}}{=} (a_A + a_R)^2 C_5^2, C_5 \stackrel{\text{def}}{=} a_A + a_R + 4a_A a_R \beta_{EU}$

*Proof.* By proceeding with a polynomial Euclidean division, we can easily obtain this theorem's results. □

**Theorem 2.** The markets' utility at a SE solution of $G_{\tau^v}^v$ can be expressed as affine functions in the markets' utility at a NE solution of $G^v$: $J_n^{vt} = A_n^{vt} J_n^{vt} + B_n^{vt}, \forall n \in \{EU, A\}, J_R^t = A_R^{vt} J_R^* + B_R^{vt}$, where $A_n^{vt}$ is a constant, $B_n^{vt}$ depends linearly on $\lambda^*$, and $J_n^*$ is quadratic in $\lambda^*, \forall n$.

*Proof.* We can easily obtain these results by proceeding in a similar way to Theorem 1. □

## 3.4 Optimum Sanction

**Theorem 3.** There exists a closed-form mapping linking $\tau^s$ in the optimum sanction model to $\tau^*$ in the optimum tariff model: $\tau^s = \tilde{A}_s \cdot \tau^* + \tilde{B}_s$ where $\tilde{A}_s$ and $\tilde{B}_s$ are constants defined in Supplementary Material. A special case is when $\rho=0$, leading to $\tau^s = \tau^*$.

*Proof.* Assuming $y_1 = a_1 x + b_1$ and $y_2 = a_2 x + b_2$, when we do $\frac{y_2}{y_1}$, we can write $y_2 = C_1 y_1 + C_2$. Thus, $C_1 = \frac{a_2}{a_1}$ and $C_2 = b_2 - \frac{a_2 b_1}{a_1}$.

By proceeding in this way, we obtain coefficients $\tilde{A}_s$ and $\tilde{B}_s$. When $\rho = 0$, we can easily see that $\tilde{A}_s = 1$ and $\tilde{B}_s = 0$ then $\tau^s = \tau^*$.

We have:

$$ \tilde{A}_s = \frac{\left( \frac{\beta_{EU}^2}{A_1} - \frac{3\beta_{EU}\rho}{2a_R \tilde{A}_1} \right) \left( \beta_{EU} - \frac{\beta_{EU}^3}{\tilde{A}_1^2} \right) \tilde{A}_1}{\beta_{EU}^2 \left( \beta_{EU} - \frac{\beta_{EU}^3}{\tilde{A}_1^2} + \frac{3\beta_{EU}^2\rho}{2a_R \tilde{A}_1^2} \right)} \\ \tilde{B}_s = \frac{\alpha_{EU} \left( \frac{\beta_{EU}^2}{A_1} - \frac{3\beta_{EU}\rho}{2a_R \tilde{A}_1} \right)}{\beta_{EU} \tilde{A}_2} - \frac{\rho \left( \frac{\beta_{EU} \bar{d}_R}{A_1} - \frac{b_R \beta_{EU}}{2a_R \tilde{A}_1} \right) + \frac{\alpha_{EU} \beta_{EU}}{A_1}}{\tilde{A}_2} $$

$$ \tilde{A}_1 \stackrel{\text{def}}{=} \beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R}, \quad \tilde{A}_2 \stackrel{\text{def}}{=} \beta_{EU} - \frac{\beta_{EU}^3}{\tilde{A}_1^2} + \frac{3\beta_{EU}^2\rho}{2a_R \tilde{A}_1^2} $$

**Theorem 4.** There exist closed-form mappings linking objective functions in the optimum sanction model to objective functions in the optimum tariff model: $J_n^s = \tilde{A}_n^s J_n^t + \tilde{B}_n^s, \forall n \in \mathcal{N}$.

*Proof.* We can easily obtain these results by proceeding in a similar way to Theorem 1. □

Theorems 3, 4 can easily be extended to the variational setting of the price-maker model.

**Theorem 3 [Price-maker]** There exists an affine relationship between $\tau^s$ in the optimum sanction model and $\tau^*_v$ in the optimum tariff model: $\tau^s = \tilde{A}_s^v \tau^*_v + \tilde{B}_s^v$, where $\tilde{A}_s^v$ and $\tilde{B}_s^v$ are:

• $\tilde{A}_s^v \stackrel{\text{def}}{=} \frac{\rho \lambda_{PT}^* \left( \frac{1}{4a_R^2 \left( \beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R} \right)} - \frac{1}{2a_R} \right)}{4a_R^2 A_1 \left( \alpha_{EU} + \bar{d}_A + \bar{d}_R + \frac{b_A}{2a_A} + \frac{b_R}{2a_R} \right)}, \\
A_1 \stackrel{\text{def}}{=} 2 \rho \left( \frac{1}{16 a_R^3 \left( \beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R} \right)^2} - \frac{1}{4a_R^2 \left( \beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R} \right)} + \frac{1}{4a_R} \right) + \frac{\beta_{EU}}{4a_R^2 \left( \beta_{EU} + \frac{1}{2a_A} + \frac{1}{2a_R} \right)^2}$
$$ \bullet \tilde{B}_s^v \stackrel{def}{=} 0 $$

A special case is when $\rho=0$, leading to $\tau_v^s = \tau_v^*$.

*Proof.* We can easily obtain these results by proceeding in a similar way to Theorem 3 □

**Theorem 4 [Price-maker]** There exists an affine relationship between the markets' utility in the optimum sanction model and the markets' utility in the optimum tariff model: $J_n^{v^s} = \ddot{A}_n^{v^s} J_n^{v^s} + \ddot{B}_n^{v^s}$, $\forall n \in \mathcal{N}$, where $\ddot{A}_n^s$ and $\ddot{B}_n^s$ are some constants.

*Proof.* We can easily obtain these results by proceeding in a similar way to Theorem 4. □

# 4 Dealing with Bounded Rationality

## 4.1 Prospect Theory-Based Game Formulation

### Price-Taker Formulation

More formally, let $(\Xi, \mathcal{A}, \mathbb{P})$ denote a complete measure space and let $\xi \in \Xi$ be the stochastic EU's gas demand. For a subset $A$ of the real, we define the indicator function as $1_A(y) = 1$ if $y \in A$; 0 otherwise. In line with Gazioğlu and Çalışkan (2011), we model the value function $v: \mathbb{R} \to \mathbb{R}$ describing the (behavioral) value of gains or losses (i.e., the loss aversion) as a piecewise quadratic function:

$$ v(y) = (y - \zeta y^2)1_{\{y \ge 0\}} + \kappa(y + \zeta y^2)1_{\{y < 0\}}, $$

where the parameters $\kappa \ge 1$ and a $\zeta \le 1$ define the loss aversion tendency in gain and risk-seeking tendency in loss of a decision-maker. Here, $y$ denotes a measurable function of the uncertain demand $\xi$, hence it is itself a random variable.

CPT enables us to assess the way in which individuals assign a subjective value to the occurrence probabilities of an event when making decisions under uncertainty. Specifically, the expectation of (7) is no longer taken with respect to the true event probabilities $p = \mathbb{P}(y \ge 0)$ and $1-p = \mathbb{P}(y < 0)$), but with respect to a distorted probability that typically exacerbates the probability of unfavorable events. Following Tversky and Kahneman (1992), we model the probability distortion function as

$$ w(p) = w_{+}(p)1_{\{x \ge 0\}} + w_{-}(p)1_{\{x < 0\}}, $$

where $w_{+}(p) = \frac{p^{\gamma}}{(p^{\gamma}+(1-p)^{\gamma})^{1/\gamma}}$, $w_{-}(p) = \frac{p^{\delta}}{(p^{\delta}+(1-p)^{\delta})^{1/\delta}}$ and $\gamma \in (0, 1]$, $\delta \in (0, 1]$ and $w(\cdot)$ is continuous.

Let $S(\xi) = \frac{1}{\beta_{EU}}(\alpha_{EU}\xi - \frac{\xi^2}{2})$ be the consumer surplus and $R(\xi) = \xi - Q_{EU}$ a stochastic variable modeling the occurrence of a supply shortage. Then, the value function of interest is as in (7) with $y \equiv y(\xi) = S(\xi) - R(\xi)$. Note that $y \ge 0$ (i.e., $S(\xi) \ge R(\xi)$) corresponds to a non-disruptive scenario, while $y < 0$ (i.e., $S(\xi) < R(\xi)$) to a supply shortage.

Let's consider respective utility functions for the EU, Asia and Russia:

$$ J_{EU}^{\text{PT}}(\mathbf{x}_{EU}, \mathbf{x}_0) = \mathbb{E}[y] - p_{EU,R}q_{R,EU} - p_{EU,A}q_{A,EU} $$

$$ J_A^{\text{PT}}(\mathbf{x}_A, \mathbf{x}_0) = p_{A,EU}q_{EU,A} - p_{A,R}q_{R,A} - (a_A q_A^2 + b_A q_A + c_A) $$

$$ J_R^{\text{PT}}(\mathbf{x}_R, \mathbf{x}_0) = p_{R,EU}q_{EU,R} - p_{R,A}q_{A,R} - (a_R q_R^2 + b_R q_R + c_R) $$

where

$$ \mathbb{E}[y] = \left( (S(\xi) - R(\xi)) - \zeta (S(\xi) - R(\xi))^2 \right) \times w_{+}(p) + \kappa \left( (S(\xi) - R(\xi)) + \zeta (S(\xi) - R(\xi))^2 \right) w_{-}(p) $$

The perfect competition setting can be formulated as a partial equilibrium model (Arrow and Debreu, 1954; Höschle et al., 2018). In this setting, the GMO acts as price setter. Let $\mathcal{G}^{\text{PT}} = (\mathcal{N} \sqcup \{\text{GMO}\}, \mathcal{X} \times \mathcal{X}_0, (J_n^{\text{PT}})_n)$. The geographic markets solve the following non-cooperative game to determine their optimal strategy:
$$ \bullet \tilde{B}_s^v \stackrel{def}{=} 0 $$

A special case is when $\rho=0$, leading to $\tau_v^s = \tau_v^*$.

*Proof.* We can easily obtain these results by proceeding in a similar way to Theorem 3. $\square$

**Theorem 4 [Price-maker]** There exists an affine relationship between the markets' utility in the optimum sanction model and the markets' utility in the optimum tariff model: $J_n^{v^s} = \ddot{A}_n^{v^s} J_n^{v^s} + \ddot{B}_n^{v^s}$, $\forall n \in \mathcal{N}$, where $\ddot{A}_n^s$ and $\ddot{B}_n^s$ are some constants.

*Proof.* We can easily obtain these results by proceeding in a similar way to Theorem 4. $\square$

# 4 Dealing with Bounded Rationality

## 4.1 Prospect Theory-Based Game Formulation

### Price-Taker Formulation

More formally, let $(\Xi, \mathcal{A}, \mathbb{P})$ denote a complete measure space and let $\xi \in \Xi$ be the stochastic EU's gas demand. For a subset $A$ of the real, we define the indicator function as $1_A(y) = 1$ if $y \in A$; 0 otherwise. In line with Gazioğlu and Çalışkan (2011), we model the value function $v: \mathbb{R} \to \mathbb{R}$ describing the (behavioral) value of gains or losses (i.e., the loss aversion) as a piecewise quadratic function:

$$ v(y) = (y - \zeta y^2)1_{\{y \ge 0\}} + \kappa(y + \zeta y^2)1_{\{y < 0\}}, $$

where the parameters $\kappa \ge 1$ and a $\zeta \le 1$ define the loss aversion tendency in gain and risk-seeking tendency in loss of a decision-maker. Here, $y$ denotes a measurable function of the uncertain demand $\xi$, hence it is itself a random variable.

CPT enables us to assess the way in which individuals assign a subjective value to the occurrence probabilities of an event when making decisions under uncertainty. Specifically, the expectation of (7) is no longer taken with respect to the true event probabilities $p = \mathbb{P}(y \ge 0)$ and $1-p = \mathbb{P}(y < 0)$), but with respect to a distorted probability that typically exacerbates the probability of unfavorable events. Following Tversky and Kahneman (1992), we model the probability distortion function as

$$ w(p) = w_{+}(p)1_{\{x \ge 0\}} + w_{-}(p)1_{\{x < 0\}}, $$

where $w_{+}(p) = \frac{p^{\gamma}}{(p^{\gamma}+(1-p)^{\gamma})^{1/\gamma}}$, $w_{-}(p) = \frac{p^{\delta}}{(p^{\delta}+(1-p)^{\delta})^{1/\delta}}$ and $\gamma \in (0, 1]$, $\delta \in (0, 1]$ and $w(\cdot)$ is continuous.

Let $S(\xi) = \frac{1}{\beta_{EU}}(\alpha_{EU}\xi - \frac{\xi^2}{2})$ be the consumer surplus and $R(\xi) = \xi - Q_{EU}$ a stochastic variable modeling the occurrence of a supply shortage. Then, the value function of interest is as in (7) with $y \equiv y(\xi) = S(\xi) - R(\xi)$. Note that $y \ge 0$ (i.e., $S(\xi) \ge R(\xi)$) corresponds to a non-disruptive scenario, while $y < 0$ (i.e., $S(\xi) < R(\xi)$) to a supply shortage.

Let's consider respective utility functions for the EU, Asia and Russia:

$$ J_{EU}^{\text{PT}}(\mathbf{x}_{EU}, \mathbf{x}_0) = \mathbb{E}[y] - p_{EU,R}q_{R,EU} - p_{EU,A}q_{A,EU} $$

$$ J_A^{\text{PT}}(\mathbf{x}_A, \mathbf{x}_0) = p_{A,EU}q_{EU,A} - p_{A,R}q_{R,A} - (a_A q_A^2 + b_A q_A + c_A) $$

$$ J_R^{\text{PT}}(\mathbf{x}_R, \mathbf{x}_0) = p_{R,EU}q_{EU,R} - p_{R,A}q_{A,R} - (a_R q_R^2 + b_R q_R + c_R) $$

where

$$ \mathbb{E}[y] = \left( (S(\xi) - R(\xi)) - \zeta (S(\xi) - R(\xi))^2 \right) \times w_{+}(p) + \kappa \left( (S(\xi) - R(\xi)) + \zeta (S(\xi) - R(\xi))^2 \right) w_{-}(p) $$

The perfect competition setting can be formulated as a partial equilibrium model (Arrow and Debreu, 1954; Höschle et al., 2018). In this setting, the GMO acts as price setter. Let $G^{\text{PT}} = (\mathcal{N} \sqcup \{\text{GMO}\}, \mathcal{X} \times \mathcal{X}_0, (J_n^{\text{PT}})_n)$. The geographic markets solve the following non-cooperative game to determine their optimal strategy:
$$ \forall n \in \mathcal{N}, \max_{\mathbf{x}_n} J_n^{\mathrm{PT}}(\mathbf{x}_n, \mathbf{x}_0), \qquad \max_{\mathbf{x}_0} J_0(\mathbf{x}, \mathbf{x}_0), \qquad (7a) $$

$$ \text{s.t.} \quad \mathbf{x}_n \in \mathcal{X}_n, \qquad \text{s.t.} \quad q_{m,n} = -q_{m,n}, \forall n \in \mathcal{N}, \forall m \in \mathcal{N}_n, \qquad (7b) $$

$$ p_{n,m} = p_{m,n}, \forall n \in \mathcal{N}, \forall m \in \mathcal{N}_n. \qquad (7c) $$

**Proposition 3 [Price-Taker with bounded rationality]** There exist multiple NEs solutions of $\mathcal{G}^{\mathrm{PT}}$.

*Proof.* We obtain the following expressions at a NE: $p_{R,A}^* = p_{A,R}^*$, $p_{A,EU}^* = p_{R,A}^* = \lambda_{\mathrm{PT}}^*$. In addition, $q_A^* = \frac{1}{2a_A}(\lambda_A^* - b_A)$, $q_R^* = \frac{1}{2a_R}(\lambda_R^* - b_R)$. From the trade price symmetry constraint $\lambda_{ij} = \lambda_{ji}, \forall i, j$, we get that $\lambda_R^* = \lambda_A^* = \lambda_{\mathrm{PT}}$. From the bilateral trading reciprocity constraints, we have $Q_{EU}^* = \frac{\lambda_{\mathrm{PT}} - b_A}{2a_A} + \frac{\lambda_{\mathrm{PT}} - b_R}{2a_R} - \bar{d}_A - \bar{d}_R$, $Q_R^* = \bar{d}_R - q_R^*$, $Q_A^* = \bar{d}_A - q_A^*$ and we can get that $\sum_n Q_n = 0$. This enables us to prove the existence of an interior point NE with closed form expression defined through

$$ \lambda_{\mathrm{PT}}^* = \frac{w_+(p)(A_1 + 1) - \kappa w_-(p)(A_1 - 1)}{\zeta w_+(p)\left(\frac{1}{a_A} + \frac{1}{a_R}\right) - \kappa \zeta w_-(p)\left(\frac{1}{a_A} + \frac{1}{a_R}\right) + 1} \qquad (8) $$

$$ \text{where } A_1 \stackrel{\mathrm{def}}{=} 2\zeta \left( \xi + \bar{d}_A + \bar{d}_R + \frac{b_A}{2a_A} + \frac{b_R}{2a_R} - \frac{(\alpha_{\mathrm{EU}} \xi - \frac{\xi^2}{2})}{\beta_{\mathrm{EU}}} \right). \quad \square $$

**Proposition 4 [Price-Taker with bounded rationality]** $\forall n \in \mathcal{N}, J_n^{\mathrm{PT}}(\mathbf{x}_n^*, \mathbf{x}_0^*) = J_n^{\mathrm{PT}}(\mathbf{x}_n^{**}, \mathbf{x}_0^{**}), \forall \mathbf{x}^*, \mathbf{x}^{**} \in \mathrm{SOL}(\mathcal{G}^{\mathrm{PT}})$.

*Proof.* By substitution of the optimal net imports in the markets' utility functions, at each NE we obtain the following expressions: $J_{\mathrm{EU}}^{\mathrm{PT}^*} = a(B_1, B_2, B_3)\lambda_{\mathrm{PT}}^{*2} + b(B_1, B_2, B_3)\lambda_{\mathrm{PT}}^* + c(B_1, B_2, B_3)$, where $B_1 \stackrel{\mathrm{def}}{=} \xi^2 \kappa \zeta w_-(p), B_2 \stackrel{\mathrm{def}}{=} \xi^2 \zeta w_+(p), B_3 \stackrel{\mathrm{def}}{=} \alpha_{\mathrm{EU}} \xi \kappa \zeta w_-(p), J_n^{\mathrm{PT}^*} = \frac{1}{4a_n} \lambda_{\mathrm{PT}}^{*2} - \left(\bar{d}_n + \frac{b_n}{2a_n}\right) \lambda_{\mathrm{PT}}^* + \frac{b_n^2}{4a_n} - c_n, \forall n \in \{A, R\} \text{ which depend only on } \mathcal{G}^{\mathrm{PT}} \text{ parameters. } a(B_1, B_2, B_3), b(B_1, B_2, B_3) \text{ and } c(B_1, B_2, B_3) \text{ are constants. } \square$

## Price-Maker Formulation

In this setting, the geographic markets' utility functions are modified as follows: $J_{\mathrm{EU}}^{\mathrm{PT}^v}(\mathbf{x}_{\mathrm{EU}}) = \mathbb{E}[y]$ and $J_n^{\mathrm{PT}^v}(\mathbf{x}_n) = -(a_n q_n^2 + b_n q_n + c_n), \forall n \in \{A, R\}$ and the feasibility sets become coupled, i.e., $\mathcal{X}_n(\mathbf{x}_{-n}) = \{\mathbf{x}_n \in \mathcal{X}_n | q_{n,m} = -q_{m,n}, p_{n,m} = p_{m,n}, \forall m \in \mathcal{N}_n\}, \forall n$. Thus, each agent $n \in \mathcal{N}$ solves the following optimization problem:

$$ \max_{\mathbf{x}_n} J_n^{\mathrm{PT}^v}(\mathbf{x}_n), \qquad (9) \\ \text{s.t.} \quad \mathbf{x}_n \in \mathcal{X}_n(\mathbf{x}_{-n}). $$

The Variational Equilibria (VEs) solutions of the resulting game can be interpreted as NEs of the game $\mathcal{G}^{\mathrm{PT}^v} = (\mathcal{N}, (\mathcal{X}_n(\mathbf{x}_{-n}))_n, (J_n^{\mathrm{PT}^v})_n)$.

**Proposition 5 [Price-Maker with bounded rationality]** There exist multiple VEs solutions of $\mathcal{G}^{\mathrm{PT}^v}$.

*Proof.* Proceeding similarly to proposition 5, it is enables us to prove the existence of an interior point NE in $\lambda_{\mathrm{PT}}^*$ defined in (8). $\square$
$$ \forall n \in \mathcal{N}, \max_{\mathbf{x}_n} J_n^{\text{PT}}(\mathbf{x}_n, \mathbf{x}_0), \qquad \max_{\mathbf{x}_0} J_0(\mathbf{x}, \mathbf{x}_0), \qquad (7a) $$

$$ \text{s.t.} \quad \mathbf{x}_n \in \mathcal{X}_n, \qquad \text{s.t.} \quad q_{m,n} = -q_{m,n}, \forall n \in \mathcal{N}, \forall m \in \mathcal{N}_n, \qquad (7b) $$

$$ p_{n,m} = p_{m,n}, \forall n \in \mathcal{N}, \forall m \in \mathcal{N}_n. \qquad (7c) $$

**Proposition 3 [Price-Taker with bounded rationality]** There exist multiple NEs solutions of $G^{\text{PT}}$.

*Proof.* We obtain the following expressions at a NE: $p_{R,A}^* = p_{A,R}^*$, $p_{A,EU}^* = p_{R,A}^* = \lambda_{\text{PT}}^*$. In addition, $q_A^* = \frac{1}{2a_A}(\lambda_A^* - b_A)$, $q_R^* = \frac{1}{2a_R}(\lambda_R^* - b_R)$. From the trade price symmetry constraint $\lambda_{ij} = \lambda_{ji}, \forall i, j$, we get that $\lambda_R^* = \lambda_A^* = \lambda_{\text{PT}}$. From the bilateral trading reciprocity constraints, we have $Q_{\text{EU}}^* = \frac{\lambda_{\text{PT}} - b_A}{2a_A} + \frac{\lambda_{\text{PT}} - b_R}{2a_R} - \bar{d}_A - \bar{d}_R$, $Q_R^* = \bar{d}_R - q_R^*$, $Q_A^* = \bar{d}_A - q_A^*$ and we can get that $\sum_n Q_n = 0$. This enables us to prove the existence of an interior point NE with closed form expression defined through

$$ \lambda_{\text{PT}}^* = \frac{w_+(p)(A_1 + 1) - \kappa w_-(p)(A_1 - 1)}{\zeta w_+(p)\left(\frac{1}{a_A} + \frac{1}{a_R}\right) - \kappa \zeta w_-(p)\left(\frac{1}{a_A} + \frac{1}{a_R}\right) + 1} \qquad (8) $$

$$ \text{where } A_1 \stackrel{\text{def}}{=} 2\zeta \left( \xi + \bar{d}_A + \bar{d}_R + \frac{b_A}{2a_A} + \frac{b_R}{2a_R} - \frac{(\alpha_{\text{EU}} \xi - \frac{\xi^2}{2})}{\beta_{\text{EU}}} \right). \quad \square $$

**Proposition 4 [Price-Taker with bounded rationality]** $\forall n \in \mathcal{N}, J_n^{\text{PT}}(\mathbf{x}_n^*, \mathbf{x}_0^*) = J_n^{\text{PT}}(\mathbf{x}_n^{**}, \mathbf{x}_0^{**}), \forall \mathbf{x}^*, \mathbf{x}^{**} \in \text{SOL}(G^{\text{PT}})$.

*Proof.* By substitution of the optimal net imports in the markets' utility functions, at each NE we obtain the following expressions: $J_{\text{EU}}^{\text{PT}^*} = a(B_1, B_2, B_3)\lambda_{\text{PT}}^{*2} + b(B_1, B_2, B_3)\lambda_{\text{PT}}^* + c(B_1, B_2, B_3)$, where $B_1 \stackrel{\text{def}}{=} \xi^2 \kappa \zeta w_-(p), B_2 \stackrel{\text{def}}{=} \xi^2 \zeta w_+(p), B_3 \stackrel{\text{def}}{=} \alpha_{\text{EU}} \xi \kappa \zeta w_-(p)$, $J_n^{\text{PT}^*} = \frac{1}{4a_n} \lambda_{\text{PT}}^{*2} - (\bar{d}_n + \frac{b_n}{2a_n}) \lambda_{\text{PT}}^* + \frac{b_n^2}{4a_n} - c_n, \forall n \in \{A, R\}$ which depend only on $G^{\text{PT}}$ parameters. $a(B_1, B_2, B_3), b(B_1, B_2, B_3)$ and $c(B_1, B_2, B_3)$ are constants. $\square$

## Price-Maker Formulation

In this setting, the geographic markets' utility functions are modified as follows: $J_{\text{EU}}^{\text{PT}^v}(\mathbf{x}_{\text{EU}}) = \mathbb{E}[y]$ and $J_n^{\text{PT}^v}(\mathbf{x}_n) = -(a_n q_n^2 + b_n q_n + c_n), \forall n \in \{A, R\}$ and the feasibility sets become coupled, i.e., $\mathcal{X}_n(\mathbf{x}_{-n}) = \{\mathbf{x}_n \in \mathcal{X}_n | q_{n,m} = -q_{m,n}, p_{n,m} = p_{m,n}, \forall m \in \mathcal{N}_n\}, \forall n$. Thus, each agent $n \in \mathcal{N}$ solves the following optimization problem:

$$ \max_{\mathbf{x}_n} J_n^{\text{PT}^v}(\mathbf{x}_n), \qquad (9) \\ \text{s.t.} \quad \mathbf{x}_n \in \mathcal{X}_n(\mathbf{x}_{-n}). $$

The Variational Equilibria (VEs) solutions of the resulting game can be interpreted as NEs of the game $G^{\text{PT}^v} = (\mathcal{N}, (\mathcal{X}_n(\mathbf{x}_{-n}))_n, (J_n^{\text{PT}^v})_n)$.

**Proposition 5 [Price-Maker with bounded rationality]** There exist multiple VEs solutions of $G^{\text{PT}^v}$.

*Proof.* Proceeding similarly to proposition 5, it is enables us to prove the existence of an interior point NE in $\lambda_{\text{PT}}^*$ defined in (8). $\square$
## 4.2 Optimum Tariff

In the price-taker model defined in Sec. 4.1, a tariff ($\tau_{PT}$) is imposed by the GMO on imports from Russia, leading to modified utilities for the geographic markets:

$$ 
\begin{align*}
J_{\text{EU}}^{\text{PT}}(\mathbf{x}_{\text{EU}}, \mathbf{x}_0, \tau_{\text{PT}}) &= \mathbb{E}[y] - (p_{\text{EU,R}} - \tau_{\text{PT}})q_{\text{R,EU}} - p_{\text{EU,A}}q_{\text{A,EU}}, \\
J_{\text{A}}^{\text{PT}}(\mathbf{x}_{\text{A}}, \mathbf{x}_0, \tau_{\text{PT}}) &= p_{\text{A,R}}q_{\text{A,R}} + p_{\text{A,EU}}q_{\text{A,EU}} - C_{\text{A}}(q_{\text{A}}), \\
J_{\text{R}}^{\text{PT}}(\mathbf{x}_{\text{R}}, \mathbf{x}_0, \tau_{\text{PT}}) &= p_{\text{R,A}}q_{\text{R,A}} + (p_{\text{R,EU}} - \tau_{\text{PT}})q_{\text{R,EU}} + p_{\text{A,R}}q_{\text{A,R}} - C_{\text{R}}(q_{\text{R}}).
\end{align*}
$$

The tariff $\tau_{PT}$ being fixed, let $\mathcal{G}_{\tau}^{PT} = (\mathcal{N} \sqcup \mathcal{N}_0, \mathcal{X} \times \mathcal{X}_0, (J_n^{PT}(\cdot, \tau_{PT}))_n)$ and SOL($\mathcal{G}_{\tau}^{PT}$) denote the set of solutions of $\mathcal{G}_{\tau}^{PT}$. The optimum tariff problem can be formulated as a single leader-multiple followers Stackelberg game (Başar and Olsder, 1998; Von Stackelberg, 1934):

$$ 
\begin{array}{ll}
\max_{\tau_{\text{PT}}, \mathbf{x}} & J_{\text{EU}}^{\text{PT}}(\mathbf{x}_{\text{EU}}, \tau_{\text{PT}}), \\
\text{s.t.} & \forall n \in \mathcal{N}, \mathbf{x}_n \in \text{SOL}(\mathcal{G}_{\tau}^{\text{PT}}).
\end{array}
\qquad (10)
$$

**Proposition 6 [Price-taker with bounded rationality]** The market price in $\mathcal{G}_{\tau}^{PT}$ can be expressed as an affine function of the market price in $\mathcal{G}^{PT}$, $\lambda_{PT}^* + \frac{\tau_{PT} \left( \frac{\zeta w_+(p)}{a_R} - \frac{\kappa \zeta w_-(p)}{a_R} \right)}{\zeta w_+(p) \left( \frac{1}{a_A} + \frac{1}{a_R} \right) - \kappa \zeta w_-(p) \left( \frac{1}{a_A} + \frac{1}{a_R} \right) + 1}$. Furthermore, there exists a unique tariff $\tau_{PT}^*$ solution of the Stackelberg game (10).

*Proof.* Proceeding in a similar way to proposition 6, we easily obtain these results. □

**Proposition 7 [Price-Maker with bounded rationality]** Proceeding in a similar way to proposition 6, we easily obtain these results.

**Theorem 1 [Price-Taker with bounded rationality]** Proceeding in a similar way to theorem 1, we easily obtain these results.

**Theorem 2 [Price-Maker with bounded rationality]** Proceeding in a similar way to theorem 3, we easily obtain these results.

**Theorem 3 [Price-Taker with bounded rationality]** Proceeding in a similar way to theorem 4, we easily obtain these results.

**Theorem 4 [Price-Maker with bounded rationality]** Proceeding in a similar way to theorem 4, we easily obtain these results.

## 4.3 Price of Irrationality

## 4.4 Numerical Results

## References

K. J. Arrow and G. Debreu. Existence of an equilibrium for a competitive economy. *Econometrica: Journal of the Econometric Society*, pages 265–290, 1954.

T. Başar and G. J. Olsder. *Dynamic noncooperative game theory*. SIAM, 1998.

Ş. Gazioğlu and N. Çalışkan. Cumulative prospect theory challenges traditional expected utility theory. *Applied Financial Economics*, 21(21):1581–1586, 2011.

H. Höschle, H. Le Cadre, Y. Smeers, A. Papavasiliou, and R. Belmans. An admm-based method for computing risk-averse equilibrium in capacity markets. *IEEE Transactions on Power Systems*, 33 (5):4819–4830, 2018.
## 4.2 Optimum Tariff

In the price-taker model defined in Sec. 4.1, a tariff ($\tau_{PT}$) is imposed by the GMO on imports from Russia, leading to modified utilities for the geographic markets:

$$ 
\begin{aligned}
J_{\text{EU}}^{\text{PT}}(\mathbf{x}_{\text{EU}}, \mathbf{x}_0, \tau_{\text{PT}}) &= \mathbb{E}[y] - (p_{\text{EU,R}} - \tau_{\text{PT}})q_{\text{R,EU}} - p_{\text{EU,A}}q_{\text{A,EU}}, \\
J_{\text{A}}^{\text{PT}}(\mathbf{x}_{\text{A}}, \mathbf{x}_0, \tau_{\text{PT}}) &= p_{\text{A,R}}q_{\text{A,R}} + p_{\text{A,EU}}q_{\text{A,EU}} - C_{\text{A}}(q_{\text{A}}), \\
J_{\text{R}}^{\text{PT}}(\mathbf{x}_{\text{R}}, \mathbf{x}_0, \tau_{\text{PT}}) &= p_{\text{R,A}}q_{\text{R,A}} + (p_{\text{R,EU}} - \tau_{\text{PT}})q_{\text{R,EU}} + p_{\text{A,R}}q_{\text{A,R}} - C_{\text{R}}(q_{\text{R}}).
\end{aligned}
$$

The tariff $\tau_{PT}$ being fixed, let $\mathcal{G}_{\tau}^{PT} = (\mathcal{N} \sqcup \mathcal{N}_0, \mathcal{X} \times \mathcal{X}_0, (J_n^{PT}(\cdot, \tau_{PT}))_n)$ and SOL($\mathcal{G}_{\tau}^{PT}$) denote the set of solutions of $\mathcal{G}_{\tau}^{PT}$. The optimum tariff problem can be formulated as a single leader-multiple followers Stackelberg game (Başar and Olsder, 1998; Von Stackelberg, 1934):

$$ 
\begin{array}{ll}
\max_{\tau_{PT}, \mathbf{x}} & J_{\text{EU}}^{\text{PT}}(\mathbf{x}_{\text{EU}}, \tau_{PT}), \\
\text{s.t.} & \forall n \in \mathcal{N}, \mathbf{x}_n \in \text{SOL}(\mathcal{G}_{\tau}^{\text{PT}}).
\end{array}
\qquad (10)
$$

**Proposition 6 [Price-taker with bounded rationality]** The market price in $\mathcal{G}_{\tau}^{PT}$ can be expressed as an affine function of the market price in $\mathcal{G}^{PT}$, $\lambda_{PT}^* + \frac{\tau_{PT} \left( \frac{\zeta w_+(p)}{a_R} - \frac{\kappa \zeta w_-(p)}{a_R} \right)}{\zeta w_+(p) \left( \frac{1}{a_A} + \frac{1}{a_R} \right) - \kappa \zeta w_-(p) \left( \frac{1}{a_A} + \frac{1}{a_R} \right) + 1}$. Furthermore, there exists a unique tariff $\tau_{PT}^*$ solution of the Stackelberg game (10).

*Proof.* Proceeding in a similar way to proposition 6, we easily obtain these results. □

**Proposition 7 [Price-Maker with bounded rationality]** Proceeding in a similar way to proposition 6, we easily obtain these results.

**Theorem 1 [Price-Taker with bounded rationality]** Proceeding in a similar way to theorem 1, we easily obtain these results.

**Theorem 2 [Price-Maker with bounded rationality]** Proceeding in a similar way to theorem 3, we easily obtain these results.

**Theorem 3 [Price-Taker with bounded rationality]** Proceeding in a similar way to theorem 4, we easily obtain these results.

**Theorem 4 [Price-Maker with bounded rationality]** Proceeding in a similar way to theorem 4, we easily obtain these results.

## 4.3 Price of Irrationality

## 4.4 Numerical Results

## References

K. J. Arrow and G. Debreu. Existence of an equilibrium for a competitive economy. *Econometrica: Journal of the Econometric Society*, pages 265–290, 1954.

T. Başar and G. J. Olsder. *Dynamic noncooperative game theory*. SIAM, 1998.

Ş. Gazioğlu and N. Çalışkan. Cumulative prospect theory challenges traditional expected utility theory. *Applied Financial Economics*, 21(21):1581–1586, 2011.

H. Höschle, H. Le Cadre, Y. Smeers, A. Papavasiliou, and R. Belmans. An admm-based method for computing risk-averse equilibrium in capacity markets. *IEEE Transactions on Power Systems*, 33 (5):4819–4830, 2018.
<table><caption>Table 1: Variables used</caption><thead><tr><th>Variables</th><th>values</th></tr></thead><tbody><tr><td>a<sub>A</sub></td><td>0.30</td></tr><tr><td>b<sub>A</sub></td><td>2.61</td></tr><tr><td>c<sub>A</sub></td><td>4.00</td></tr><tr><td>d&#773;<sub>A</sub></td><td>28.00</td></tr><tr><td>a<sub>R</sub></td><td>0.18</td></tr><tr><td>b<sub>R</sub></td><td>2.61</td></tr><tr><td>c<sub>R</sub></td><td>4.00</td></tr><tr><td>d&#773;<sub>R</sub></td><td>19.00</td></tr><tr><td>&alpha;<sub>EU</sub></td><td>15.30</td></tr><tr><td>&beta;<sub>EU</sub></td><td>3.20</td></tr><tr><td>&gamma;</td><td>0.60</td></tr><tr><td>&delta;</td><td>0.30</td></tr><tr><td>&kappa;</td><td>2.59</td></tr><tr><td>&zeta;</td><td>0.60</td></tr><tr><td>&mu; (d<sub>EU</sub>)</td><td>7</td></tr></tbody></table>

J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. *Econometrica*, pages 520–534, 1965.

A. Tversky and D. Kahneman. Advances in prospect theory: Cumulative representation of uncertainty. *Journal of Risk and uncertainty*, 5:297–323, 1992.

H. Von Stackelberg. *Marktform und Gleichgewicht*. Oxford University Press, 1934.
<table><caption>Table 1: Variables used</caption><thead><tr><th>Variables</th><th>values</th></tr></thead><tbody><tr><td>a<sub>A</sub></td><td>0.30</td></tr><tr><td>b<sub>A</sub></td><td>2.61</td></tr><tr><td>c<sub>A</sub></td><td>4.00</td></tr><tr><td>d&#773;<sub>A</sub></td><td>28.00</td></tr><tr><td>a<sub>R</sub></td><td>0.18</td></tr><tr><td>b<sub>R</sub></td><td>2.61</td></tr><tr><td>c<sub>R</sub></td><td>4.00</td></tr><tr><td>d&#773;<sub>R</sub></td><td>19.00</td></tr><tr><td>&alpha;<sub>EU</sub></td><td>15.30</td></tr><tr><td>&beta;<sub>EU</sub></td><td>3.20</td></tr><tr><td>&gamma;</td><td>0.60</td></tr><tr><td>&delta;</td><td>0.30</td></tr><tr><td>&kappa;</td><td>2.59</td></tr><tr><td>&varsigma;</td><td>0.60</td></tr><tr><td>&mu; (d<sub>EU</sub>)</td><td>7</td></tr></tbody></table>

J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. *Econometrica*, pages 520–534, 1965.

A. Tversky and D. Kahneman. Advances in prospect theory: Cumulative representation of uncertainty. *Journal of Risk and uncertainty*, 5:297–323, 1992.

H. Von Stackelberg. *Marktform und Gleichgewicht*. Oxford University Press, 1934.