<p>In most electricity markets, producers submit supply functions to a procurement uniform-price auction under uncertainty before demand has been realized. In the Supply Function Equilibrium (SFE), every producer commits to the supply function that maximises his expected profit given the bids of competitors. </p><p>The presence of multiple equilibria is a basic weakness of the SFE framework. Essay I shows that with (i) symmetric producers, (ii) perfectly inelastic demand, (iii) a reservation price (price cap), and (iv) capacity constraints that bind with a positive probability, a unique symmetric SFE exists. The equilibrium price reaches the price cap exactly when capacity constraints bind.</p><p>Another weakness is difficulty finding a valid asymmetric SFE with non-decreasing supply functions. Essay II shows that for firms with asymmetric capacity constraints but identical constant marginal costs there exists a unique and valid SFE. Equilibrium supply functions exhibit kinks as well as vertical and horizontal segments. The price at which the capacity constraint of a firm binds is increasing in the firm’s share of market capacity. The capacity constraint of the second largest firm binds when the market price reaches the price cap. Thereafter, the largest firm supplies its remaining capacity with a perfectly elastic segment at the price cap. Essay III presents a numerical algorithm that calculates a similar SFE for asymmetric firms with increasing marginal costs. </p><p>Essay IV derives the SFE of a pay-as-bid auction such as the balancing market for electric power in Britain. A unique SFE always exists if the demand’s hazard rate is monotonically decreasing, as for a Pareto distribution of the second kind. Assuming this probability distribution, the pay-as-bid procurement auction is compared to the SFE of a uniform-price procurement auction. Two theorems in Essay V prove that the demand-weighted average price is (weakly) lower in the pay-as-bid procurement auction. </p>

In most electricity markets, producers submit supply functions to a procurement uniform-price auction under uncertainty before demand has been realized. In the Supply Function Equilibrium (SFE), every producer commits to the supply function that maximises his expected profit given the bids of competitors. The presence of multiple equilibria is a basic weakness of the SFE framework. Essay I shows that with (i) symmetric producers, (ii) perfectly inelastic demand, (iii) a reservation price (price cap), and (iv) capacity constraints that bind with a positive probability, a unique symmetric SFE exists. The equilibrium price reaches the price cap exactly when capacity constraints bind. Another weakness is difficulty finding a valid asymmetric SFE with non-decreasing supply functions. Essay II shows that for firms with asymmetric capacity constraints but identical constant marginal costs there exists a unique and valid SFE. Equilibrium supply functions exhibit kinks as well as vertical and horizontal segments. The price at which the capacity constraint of a firm binds is increasing in the firm’s share of market capacity. The capacity constraint of the second largest firm binds when the market price reaches the price cap. Thereafter, the largest firm supplies its remaining capacity with a perfectly elastic segment at the price cap. Essay III presents a numerical algorithm that calculates a similar SFE for asymmetric firms with increasing marginal costs. Essay IV derives the SFE of a pay-as-bid auction such as the balancing market for electric power in Britain. A unique SFE always exists if the demand’s hazard rate is monotonically decreasing, as for a Pareto distribution of the second kind. Assuming this probability distribution, the pay-as-bid procurement auction is compared to the SFE of a uniform-price procurement auction. Two theorems in Essay V prove that the demand-weighted average price is (weakly) lower in the pay-as-bid procurement auction.

Dans cette thèse, notre but est de donner un cadre numérique général pour approcher les solutions des problèmes du transport optimal (TO). L’idée générale est d’introduire une régularisation entropique du problème initial. Le problème régularisé correspond à minimiser une entropie relative par rapport à une mesure de référence donnée. En effet, cela équivaut à trouver la projection d’un couplage par rapport à la divergence de Kullback-Leibler. Cela nous permet d’utiliser l’algorithme de Bregman/Dykstra et de résoudre plusieurs problèmes variationnels liés au TO. Nous nous intéressons particulièrement à la résolution des problèmes du transport optimal multi-marges (TOMM) qui apparaissent dans le cadre de la dynamique des fluides (équations d’Euler incompressible à la Brenier) et de la physique quantique (la théorie de fonctionnelle de la densité ). Dans ces cas, nous montrons que la régularisation entropique joue un rôle plus important que de la simple stabilisation numérique. De plus, nous donnons des résultats concernant l’existence des transports optimaux (par exemple des transports fractals) pour le problème TOMM. / In this thesis we aim at giving a general numerical framework to approximate solutions to optimal transport (OT) problems. The general idea is to introduce an entropic regularization of the initialproblems. The regularized problem corresponds to the minimization of a relative entropy with respect a given reference measure. Indeed, this is equivalent to find the projection of the joint coupling with respect the Kullback-Leibler divergence. This allows us to make use the Bregman/Dykstra’s algorithm and solve several variational problems related to OT. We are especially interested in solving multi-marginal optimal transport problems (MMOT) arising in Physics such as in Fluid Dynamics (e.g. incompressible Euler equations à la Brenier) and in Quantum Physics (e.g. Density Functional Theory). In these cases we show that the entropic regularization plays a more important role than a simple numerical stabilization. Moreover, we also give some important results concerning existence and characterization of optimal transport maps (e.g. fractal maps) for MMOT .