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# Supplementary Material to "Learning Market Equilibria Preserving Statistical Privacy Using Performative Prediction"

Hélène Le Cadre, Mandar Datar, Mathis Guckert, Eitan Altman

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Hélène Le Cadre, Mandar Datar, Mathis Guckert, Eitan Altman. Supplementary Material to "Learning Market Equilibria Preserving Statistical Privacy Using Performative Prediction". 2025. hal-05052889

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Distributed under a Creative Commons Attribution 4.0 International License
# Supplementary Material to “Learning Market Equilibria Preserving Statistical Privacy Using Performative Prediction”

Hélène Le Cadre, Mandar Datar, Mathis Guckert, and Eitan Altman

## I. NOMENCLATURE TABLE

We provide a nomenclature table in Tab. I.

<table><caption>TABLE I: Nomenclature used throughout the paper.</caption><tr><td>EU</td><td>End User</td></tr><tr><td>DA</td><td>Data Aggregator</td></tr><tr><td>PSE</td><td>Performatively Stable Equilibrium</td></tr><tr><td>NE</td><td>Nash Equilibrium</td></tr><tr><td>SE</td><td>Stackelberg Equilibrium</td></tr><tr><td>RSGM</td><td>Repeated Stochastic Gradient Method</td></tr><tr><td>GDM</td><td>Gradient Descent Method</td></tr><tr><td>GNEs</td><td>Generalized Nash Equilibria</td></tr><tr><td>VE</td><td>Variational Equilibrium</td></tr><tr><td>Dist-RSGM</td><td>Distributed Repeated Stochastic Gradient Method</td></tr><tr><td>Seq-RSGM</td><td>Sequential Repeated Stochastic Gradient Method</td></tr><tr><td>TTSM</td><td>Two-Timescale Stochastic Method</td></tr></table>

## II. SYMBOLS AND NOTATION TABLE

Symbols and notation are listed in Tab. II.

<table><caption>TABLE II: Main notations used throughout the paper.</caption><tr><td>N</td><td>Set of N EUs</td></tr><tr><td>Γ<sub>n</sub></td><td>Set of neighbors of EU n</td></tr><tr><td>D<sub>n</sub></td><td>EU n demand</td></tr><tr><td>G<sup>c</sup><sub>n</sub></td><td>EU n controllable generation</td></tr><tr><td>G<sup>nc</sup><sub>n</sub></td><td>EU n non-controllable generation</td></tr><tr><td>(q<sub>mn</sub>)<sub>m∈Γ<sub>n</sub></sub></td><td>EU n trades with neighbors</td></tr><tr><td>Q<sub>n</sub></td><td>EU n's net import</td></tr><tr><td>x<sub>n</sub> ∈ X<sub>n</sub> ⊆ R<sup>m<sub>n</sub></sup></td><td>EU n's decision variable</td></tr><tr><td>j<sub>n</sub>(·)</td><td>EU n's objective function</td></tr><tr><td>U<sub>n</sub>(·)</td><td>EU n's usage benefit</td></tr><tr><td>θ<sub>n</sub> ∈ Θ<sub>n</sub> ⊆ R</td><td>EU n's nominal demand</td></tr><tr><td>p<sup>a</sup><sub>n</sub>(·)</td><td>EU n's contract payment</td></tr><tr><td>C<sup>G</sup><sub>n</sub>(·)</td><td>EU n's controllable generation cost</td></tr><tr><td>c<sub>nm</sub> = 1</td><td>Network charges or taxes</td></tr><tr><td>SC(·)</td><td>Social cost</td></tr><tr><td>(ζ<sub>nm</sub>)<sub>n,m</sub></td><td>Duals of the trading reciprocity eq.</td></tr><tr><td>F<sup>0</sup><sub>x</sub>(·)</td><td>Pseudo gradient of (j<sub>n</sub>(·))<sub>n</sub></td></tr><tr><td>A<sub>n</sub>, B<sub>n</sub>, C<sub>n</sub></td><td>Parameters of x<sub>n</sub></td></tr><tr><td>θ̃<sub>n</sub></td><td>EU n's reading</td></tr><tr><td>ε<sub>n</sub></td><td>EU n's reading noise</td></tr><tr><td>θ̃<sub>n</sub></td><td>EU n's reading noise bias</td></tr></table>

<table><tr><td>V<sub>n</sub></td><td>EU n's reading noise variance</td></tr><tr><td>x<sup>a</sup><sub>n</sub></td><td>DA's estimate of EU n's decisions</td></tr><tr><td>A<sup>a</sup><sub>n</sub>, B<sup>a</sup><sub>n</sub>, C<sup>a</sup><sub>n</sub></td><td>Parameters of x<sup>a</sup><sub>n</sub></td></tr><tr><td>γ<sub>n</sub>, β<sub>n</sub></td><td>EU n's contract payment parameters</td></tr><tr><td>N(·)</td><td>Gaussian density function</td></tr><tr><td>I<sub>n</sub>(·)</td><td>EU n's Kullback-Leibler divergence</td></tr><tr><td>c<sup>l</sup><sub>n</sub></td><td>EU n's information unit cost</td></tr><tr><td>J<sub>n</sub>(·)</td><td>EU n's extended penalized utility</td></tr><tr><td>ξ</td><td>penalty coefficient</td></tr><tr><td>α<sub>n</sub></td><td>EU n's adjacency level</td></tr><tr><td>V̄<sub>n</sub>, V̄<sub>n</sub></td><td>bounds on EU n's variance</td></tr><tr><td>v<sub>n</sub> = (θ̂<sub>n</sub>, V<sub>n</sub>) ∈ U<sub>n</sub></td><td>EU n's hidden ability parameters</td></tr><tr><td>y<sub>n</sub> = (x<sub>n</sub>, ̂θ<sub>n</sub>, V<sub>n</sub>) ∈ Y<sub>n</sub></td><td>EU n's vector of decisions</td></tr><tr><td>F<sub>z<sub>n</sub></sub>(·)</td><td>Partial derivative of E<sub>ε</sub>[J<sub>n</sub>(·)] in z<sub>n</sub></td></tr><tr><td>H<sub>z<sub>n</sub></sub>(·)</td><td>Partial Hessian of E<sub>ε</sub>[J<sub>n</sub>(·)] in z<sub>n</sub></td></tr><tr><td>Ĥ<sub>y<sub>n</sub></sub>(·)</td><td>Pseudo Hessian of (E<sub>ε</sub>[J<sub>n</sub>(·)])<sub>n</sub></td></tr><tr><td>K<sub>n</sub></td><td>(A<sup>a</sup><sub>n</sub>)<sup>2</sup>(1,n) + (A<sup>a</sup><sub>n</sub>)<sup>2</sup>(2,n)</td></tr><tr><td>J̃<sub>n</sub>(·)</td><td>EU n's extended utility (without penalty)</td></tr><tr><td>L<sup>a</sup>(·)</td><td>DA's objective function</td></tr><tr><td>π</td><td>DA's conversion weight</td></tr><tr><td>l(·)</td><td>DA's loss function</td></tr><tr><td>p<sup>r</sup></td><td>Conventional retailer's unit price</td></tr><tr><td>Ψ<sub>n</sub>(·)</td><td>Implicitly defined map</td></tr><tr><td>β̄, β̄</td><td>Bounds on β</td></tr><tr><td>K<sub>n</sub></td><td>(A<sup>a</sup><sub>n</sub>)<sup>2</sup>(1,n) + (A<sup>a</sup><sub>n</sub>)<sup>2</sup>(2,n)</td></tr><tr><td>φ</td><td>Dual variable of (∑<sub>n</sub> Q<sub>n</sub>)<sup>2</sup> = 0</td></tr><tr><td>E<sup>a</sup><sub>n</sub></td><td>C<sup>a</sup><sub>n</sub> - B<sup>a</sup><sub>n</sub>(1,·)G<sup>nc</sup></td></tr><tr><td>F<sup>a</sup><sub>n</sub></td><td>C<sup>a</sup><sub>n</sub> - B<sup>a</sup><sub>n</sub>(2,·)G<sup>nc</sup></td></tr><tr><td>η</td><td>Dist-RSGM stepsize</td></tr><tr><td>η̄</td><td>Seq-RSGM stepsize</td></tr><tr><td>η<sup>t</sup><sub>1</sub>, η<sup>t</sup><sub>2</sub></td><td>TTSM stepsize at t</td></tr><tr><td>ω</td><td>Strong monotonicity coefficient of F<sub>y<sub>n</sub></sub>(·)</td></tr><tr><td>τ<sub>i</sub></td><td>Upper-bound on the Wasserstein distance</td></tr><tr><td>τ</td><td>max<sub>i</sub> τ<sub>i</sub></td></tr><tr><td>ρ</td><td>√(∑<sub>i=1</sub><sup>N</sup> (κ<sub>i</sub>τ<sub>i</sub>)<sup>2</sup>)</td></tr><tr><td>κ<sub>n</sub></td><td>Lipschitz continuity parameter of ∇y<sub>n</sub> J<sub>n</sub>(y, ε)</td></tr><tr><td>σ<sup>2</sup></td><td>Upper-bound on the variance of ∇y<sub>n</sub> J<sub>n</sub>(·)</td></tr><tr><td>S<sub>n</sub></td><td>Scalar quadratic in the elements of A<sup>a</sup><sub>n</sub></td></tr><tr><td>P</td><td>Upper-bound coef. on the bias of the gradient estimator</td></tr><tr><td>δ</td><td>Accuracy of the algorithm</td></tr></table>
### III. DETAILED COMPUTATION FOR $\hat{H}_y$

From the definitions:

$$ \hat{H}_y = \begin{bmatrix} \begin{bmatrix} 2\frac{c_1}{V_1} + 4\beta_1 \left( \left( \frac{A_1^a}{V_1^{(1,1)}} + \left( \frac{A_1^a}{V_1^{(2,1)}} \right)^2 \right) \frac{-2c_1(\hat{\theta}_n - \theta_1)}{V_1^2} - 4\beta_1 A_1^a_{(1,1)} - 4\beta_1 A_1^a_{(2,1)} \right. \\ \left. -c_1(\hat{\theta}_n - \theta_1) \frac{2c_1(\hat{\theta}_n - \theta_1)^2}{V_1^2} \right) & 0 & 0 & 0 \\ -4\beta_1 A_1^a_{(1,1)} & 0 & 4\tilde{a}_1 + 4\beta_1 + 4\xi & -4\xi \\ -4\beta_1 A_1^a_{(2,1)} & 0 & -4\xi & 2a_1 + 4\beta_1 + 4\xi \end{bmatrix} & \begin{bmatrix} 2\beta_1 \sum_{l=1,2} A_1^a_{(1,1)} A_1^a_{(2,1)} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ -2\beta_1 A_1^a_{(1,2)} & 0 & 2\xi - 2\xi & 0 \\ -2\beta_1 A_1^a_{(2,2)} & 0 & 2\xi - 2\xi & 0 \end{bmatrix} + \begin{bmatrix} 2\beta_1 \sum_{l=1,2} A_1^a_{(1,1)} A_1^a_{(2,1)} & 0 & 2\xi - 2\xi & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 2\xi & 2\xi \\ 0 & 0 & -2\xi & -2\xi \end{bmatrix} + \begin{bmatrix} 2\frac{c_1}{V_1} + 4\beta_1 \left( \left( \frac{A_1^a}{V_1^{(1,1)}} + \left( \frac{A_1^a}{V_1^{(2,1)}} \right)^2 \right) \frac{-2c_1(\hat{\theta}_n - \theta_1)}{V_1^2} - 4\beta_1 A_1^a_{(1,1)} - 4\beta_1 A_1^a_{(2,1)} \right. \\ \left. -c_1(\hat{\theta}_n - \theta_1) \frac{2c_1(\hat{\theta}_n - \theta_1)^2}{V_1^2} \right) & 0 & 0 & 0 \\ -4\beta_1 A_1^a_{(1,1)} & 0 & 4\tilde{a}_1 + 4\beta_1 + 4\xi & -4\xi \\ -4\beta_1 A_1^a_{(2,1)} & 0 & -4\xi & 2a_1 + 4\beta_1 + 4\xi \end{bmatrix} \end{bmatrix} $$

which can be rewritten as:

$$ \hat{H}_y = \begin{bmatrix} \begin{bmatrix} 2\frac{c_1}{V_1} + 4\beta_1 \left( \left( \frac{A_1^a}{V_1^{(1,1)}} + \left( \frac{A_1^a}{V_1^{(2,1)}} \right)^2 \right) \frac{-2c_1(\hat{\theta}_n - \theta_1)}{V_1^2} - 4\beta_1 A_1^a_{(1,1)} - 4\beta_1 A_1^a_{(2,1)} \right. \\ \left. -c_1(\hat{\theta}_n - \theta_1) \frac{2c_1(\hat{\theta}_n - \theta_1)^2}{V_1^2} \right) & 0 & 0 & 0 \\ -4\beta_1 A_1^a_{(1,1)} & 0 & 4\tilde{a}_1 + 4\beta_1 + 4\xi & -4\xi \\ -4\beta_1 A_1^a_{(2,1)} & 0 & -4\xi & 2a_1 + 4\beta_1 + 4\xi \end{bmatrix} & \begin{bmatrix} 2\sum_{n=1,2} \sum_{m,m \neq n} \beta_n \sum_{l=1,2} A_n^a_{(l,n)} A_n^a_{(l,m)} & 0 & -2\beta_2 A_2^a_{(1,1)} & -2\beta_2 A_2^a_{(2,1)} \\ 0 & 0 & 0 & 0 \\ -2\beta_1 A_1^a_{(1,2)} & 0 & 4\xi & -4\xi \\ -2\beta_1 A_1^a_{(2,2)} & 0 & 4\xi & -4\xi \end{bmatrix} + \begin{bmatrix} 2\frac{c_1}{V_2} + 4\beta_2 \left( \left( \frac{A_2^a}{V_2^{(1,2)}} + \left( \frac{A_2^a}{V_2^{(2,2)}} \right)^2 \right) \frac{-2c_1(\hat{\theta}_n - \theta_2)}{V_2^2} - 4\beta_2 A_2^a_{(1,2)} & -4\beta_2 A_2^a_{(2,2)} \\ \frac{-c_1(\hat{\theta}_n - \theta_2)}{V_2^2} \frac{2c_1(\hat{\theta}_n - \theta_2)^2}{V_2^2} & 0 \\ -4\beta_2 A_2^a_{(1,2)} & 0 \\ -4\beta_2 A_2^a_{(2,2)} & 0 \end{bmatrix} \end{bmatrix} $$

### IV. DERIVATION ON BOUNDS ON VARIANCE

In this section, we detail how bounds on the End Users' variances can be derived related to (8e) and (8f), and allowing the treatment of problem (8) as an unconstrained problem.

By computing the gradient of $\mathbb{E}_{\epsilon}[J_n(\cdot)]$ w.r.t. $v_n$, $\forall n$, we obtain the following expression at equilibrium in $\hat{\theta}_n, V_n$:

$$ \forall n \in \mathcal{N}, \quad c_n^I (\hat{\theta}_n - \theta_n) + 2\beta_n \left( (A_n^a)_{(1,n)} (D_n^a(\hat{\theta}) - D_n) + (A_n^a)_{(2,n)} (G_n^c a(\hat{\theta}) - G_n^c) \right) = 0, \quad (1a) $$

$$ -c_n^I (\hat{\theta}_n - \theta_n)^2 + \beta_n K_n = 0. \quad (1b) $$

From (1b), we get $(\hat{\theta}_n - \theta_n)^2 = \frac{2\beta_n K_n}{V_n}$.

Now, assuming that the constraint (8e) is strict, we get:

$$ \begin{align*} (\hat{\theta}_n - \theta_n)^2 &< \alpha_n, \\ \Leftrightarrow \frac{2\beta_n K_n}{c_n^I} &< \frac{\alpha_n}{V_n^2} \quad \text{from (1b)}, \\ \Leftrightarrow V_n &< \sqrt{\frac{c_n^I \alpha_n}{2\beta_n K_n}}. \end{align*} $$

To get that (8f) is strict, i.e., on the one side, $V_n < \bar{V}_n$, it suffices to choose $\sqrt{\frac{c_n^I \alpha_n}{2\beta_n K_n}} \le \bar{V}_n$.

On the other side

$$ 0 < \underline{V}_n < V_n, \\ \Leftrightarrow \sqrt{\frac{c_n^I \alpha_n}{2\beta_n K_n}} > \underline{V}_n. $$

We conclude that if $\underline{V}_n < \sqrt{\frac{c_n^I \alpha_n}{2\beta_n K_n}} \le \bar{V}_n$, then (8e), (8f) are non binding.

### V. EXTENDED PROOF OF PROPOSITION 6

The Incentive Compatibility (IC) definition assumes that the induced equilibrium between the End Users (EUs) is a Nash Equilibrium (NE). It can be extended considering that the EUs reach a Performatively Stable Equilibrium (PSE). In that case, IC is enforced by the Data Aggregator (DA) designing $p_n^a(\cdot)$ such that:

$$ \mathbf{y}_n^{\text{PSE}}(\beta_n) = \underset{\mathbf{y}_n \in \mathcal{Y}_n}{\operatorname{argmin}} \mathbb{E}_{\epsilon \sim \mathcal{N}(0, \nu^{\text{PSE}})} \left[ \tilde{J}_n \left( \mathbf{p}_n^a (\gamma_n, \beta_n, \mathbf{y}_n, \mathbf{y}_{-n}^{\text{PSE}}), \mathbf{y}_n \right) \right]. \quad (2) $$

Define $\Psi_n: \mathbb{R}_+ \to \mathcal{Y}_n$ to be the implicitly defined map such that $\Psi_n: \beta_n \mapsto \mathbf{y}_n^{\text{PSE}}$ returns the solution of (2). Furthermore, from Prop. 3, we know that the lower level of the DA's optimization problem admits a unique PSE, which is characterized
by the set of conditions:

$$ \hat{\theta}_{n}^{\text{PSE}} = \theta_{n}, $$

$$ D_{n}^{\text{PSE}}(\beta_{n}) = \frac{\tilde{a}_{n}\theta_{n} + \beta_{n}(\mathbf{A}_{n}^{a}_{(1,:)}\boldsymbol{\theta} - E_{n}^{a})}{\tilde{a}_{n} + \beta_{n}} - \frac{1}{2(\tilde{a}_{n} + \beta_{n})}, $$

$$ G_{n}^{c \text{PSE}}(\beta_{n}) = \frac{2\beta_{n}(\mathbf{A}_{n}^{a}_{(2,:)}\boldsymbol{\theta} - F_{n}^{a})}{a_{n} + 2\beta_{n}} - \frac{1}{a_{n} + 2\beta_{n}}, $$

$$ V_{n}^{\text{PSE}} = \bar{V}_{n}. $$

We elucidate the transformations that occur in the EUs' extended utility functions giving rise to the following bilevel optimization problem:

$$ \min_{\gamma, \beta} \sum_{n} \left( \frac{\gamma_n}{\beta_n} + \left(\pi - \frac{1}{\beta_n}\right) \left[ j_n \left( \mathbf{x}_n^{\text{PSE}}(\beta_n), \theta_n \right) - p^r D_n^{\text{PSE}}(\beta_n) \right] \right), $$

$$ \text{s.t.} \quad \hat{\theta}_{n}^{\text{PSE}} = \theta_{n}, \forall n, $$

$$ D_{n}^{\text{PSE}}(\beta_{n}) = \frac{\tilde{a}_{n}\theta_{n} + \beta_{n}(\mathbf{A}_{n}^{a}_{(1,:)}\boldsymbol{\theta} - E_{n}^{a})}{\tilde{a}_{n} + \beta_{n}} - \frac{1}{2(\tilde{a}_{n} + \beta_{n})}, \forall n, $$

$$ G_{n}^{c \text{PSE}}(\beta_{n}) = \frac{2\beta_{n}(\mathbf{A}_{n}^{a}_{(2,:)}\boldsymbol{\theta} - F_{n}^{a})}{a_{n} + 2\beta_{n}} - \frac{1}{a_{n} + 2\beta_{n}}, \forall n, $$

$$ V_{n}^{\text{PSE}} = \bar{V}_{n}, \forall n, $$

$$ j_n(\mathbf{x}_n^{\text{PSE}}(\beta_n), \theta_n) - p^r D_n^{\text{PSE}}(\beta_n) \geq 0, \forall n \in \mathcal{N}, $$

$$ \sum_{n} (D_{n}^{\text{PSE}}(\beta_{n}) - G_{n}^{c \text{PSE}}(\beta_{n}) - G_{n}^{nc}) = 0, $$

$$ \Psi_{n}(\beta_{n}) \in \mathcal{Y}_{n}, \gamma_{n} \geq 0, \forall n \in \mathcal{N}. $$

By differentiating $D_n^{\text{PSE}}(\beta_n)$ and $G_n^{c \text{PSE}}(\beta_n)$ w.r.t. $\beta_n$, we obtain that $\frac{\partial D_n^{\text{PSE}}(\beta_n)}{\partial \beta_n} > 0$ and $\frac{\partial^2 D_n^{\text{PSE}}(\beta_n)}{\partial \beta_n^2} > 0$ if and only if $\mathbf{A}_n^a_{(1,:)}\boldsymbol{\theta} + \mathbf{B}_n^a_{(1,:)}\mathbf{G}_n^{c \text{PSE}}(\boldsymbol{\beta}) > C_n^a_{(1,1)}$ and $\frac{\partial G_n^{c \text{PSE}}}{\partial \beta_n} > 0$ and $\frac{\partial^2 G_n^{c \text{PSE}}}{\partial \beta_n^2} > 0$ if and only if $\mathbf{A}_n^a_{(2,:)}\boldsymbol{\theta} + \mathbf{B}_n^a_{(2,:)}\mathbf{G}_n^{c \text{PSE}}(\boldsymbol{\beta}) > C_n^a_{(2,1)}$.

Under these assumptions, $D_n^{\text{PSE}}(\cdot)$ and $G_n^{c \text{PSE}}(\cdot)$ are strictly increasing concave in $\beta_n$, therefore $\Psi_n(\cdot)$ is bijective over $B_n \stackrel{\text{def}}{=} [\underline{\beta}_n, \bar{\beta}_n]$, where $\Psi_n(\underline{\beta}_n) = \underline{y}_n$ and $\Psi_n(\bar{\beta}_n) = \bar{y}_n$. Finally, we check that the bilevel optimization problem's objective function is strongly convex in $\mathbf{x}$, thus it admits a unique solution.