Learning Market Equilibria Preserving Statistical Privacy Using Performative Prediction

We consider a peer-to-peer electricity marketmodeled as a private network game, where End Users (EUs)minimize their cost by computing their demand and controllablegeneration. Their nominal demand constitutes sensitiveinformation, that they might want to keep private. We provethat the private network game admits a unique VariationalEquilibrium, which depends on the private information of allthe EUs. Thus, to update their strategy, EUs rely on readingsfrom the other EUs. However, to preserve their privacy, theEUs might report randomized readings. A Data Aggregator(DA) is introduced, which aims to learn the EUs’ privateinformation, while remunerating them depending on the qualityof their readings. Using performative prediction, we define adecision-dependent game Gstoch, to explicitly take into accountthe distribution shift caused by the EUs’ hidden ability. Wefocus on three solution concepts: (⋆) Nash Equilibrium (NE), (⋆)Performatively Stable Equilibrium (PSE), and (⋆) StackelbergEquilibrium (SE). Relying on the strong monotonicity of thegame, we prove that there exist unique NE and PSE solutionsof Gstoch. Both PSE and SE account for the feedback-loop effectof the EUs’ hidden ability. The related market robustness can beevaluated by comparing the market efficiency in the PSE andSE with respect to the social optimum. We show that undermild assumptions, the PSE can be found by distributed andsequential variants of the repeated (stochastic) gradient methodwhile we propose a two-timescale stochastic approximationmethod to learn SE. Finally, we formulate the DA’s optimalcontract design as a bilevel optimization problem that we castas a more tractable nonlinear nonconvex optimization problemwhich can be solved using simulated annealing. Simulations onsmall and large scale problem instances illustrate the results.