Mean-Field Optimal Control

We introduce the concept of {\it mean-field optimal control} which is the rigorous limit process connecting finite dimensional optimal control problems with ODE constraints modeling multi-agent interactions to an infinite dimensional optimal control problem with a constraint given by a PDE of Vlasov-type, governing the dynamics of the probability distribution of interacting agents. While in the classical mean-field theory one studies the behavior of a large number of small individuals {\it freely interacting} with each other, by simplifying the effect of all the other individuals on any given individual by a single averaged effect, we address the situation where the individuals are actually influenced also by an external {\it policy maker}, and we propagate its effect for the number $N$ of individuals going to infinity. On the one hand, from a modeling point of view, we take into account also that the policy maker is constrained to act according to optimal strategies promoting its most parsimonious interaction with the group of individuals. This will be realized by considering cost functionals including $L^1$-norm terms penalizing a broadly distributed control of the group, while promoting its sparsity. On the other hand, from the analysis point of view, and for the sake of generality, we consider broader classes of convex control penalizations. In order to develop this new concept of limit rigorously, we need to carefully combine the classical concept of mean-field limit, connecting the finite dimensional system of ODE describing the dynamics of each individual of the group to the PDE describing the dynamics of the respective probability distribution, with the well-known concept of $\Gamma$-convergence to show that optimal strategies for the finite dimensional problems converge to optimal strategies of the infinite dimensional problem.Comment: 31 page

On the presence of ultra-fast outflows in the WAX sample of Seyfert galaxies

The study of winds in active galactic nuclei (AGN) is of utmost importance as they may provide the long sought-after link between the central black hole and the host galaxy, establishing the AGN feedback. Recently, Laha et al. (2014) reported the X-ray analysis of a sample of 26 Seyferts observed with XMM-Newton, which are part of the so-called warm absorbers in X-rays (WAX) sample. They claim the non-detection of Fe K absorbers indicative of ultra-fast outflows (UFOs) in four observations previously analyzed by Tombesi et al. (2010). They mainly impute the Tombesi et al. detections to an improper modeling of the underlying continuum in the E=4-10 keV band. We therefore re-address here the robustness of these detections and we find that the main reason for the claimed non-detections is likely due to their use of single events only spectra, which reduces the total counts by 40%. Performing a re-analysis of the data in the whole E=0.3-10 keV energy band using their models and spectra including also double events, we find that the blue-shifted Fe K absorption lines are indeed detected at >99%. This work demonstrates the robustness of these detections in XMM-Newton even including complex model components such as reflection, relativistic lines and warm absorbers.Comment: 5 pages, 1 figure, accepted for publication in MNRA

Mean-Field Pontryagin Maximum Principle

International audienceWe derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ordinary differential equations and a partial differential equation of Vlasov-type with smooth interaction kernel. Such problems arise naturally as Gamma-limits of optimal control problems constrained by ordinary differential equations, modeling, for instance, external interventions on crowd dynamics by means of leaders. We obtain these first-order optimality conditions in the form of Hamiltonian flows in the Wasserstein space of probability measures with forward-backward boundary conditions with respect to the first and second marginals, respectively. In particular, we recover the equations and their solutions by means of a constructive procedure, which can be seen as the mean-field limit of the Pontryagin Maximum Principle applied to the optimal control problem for the discretized density, under a suitable scaling of the adjoint variables

A dynamic entry and price game with capacity indivisibility

Strategic market interaction is here modelled as a two-stage game in which potential entrants choose capacities and active firms compete in prices. Due to capital indivisibility, the capacity choice is made from a finite grid and there are substantial economies of scale. In the simplest version of the model assuming a single production technique, the equilibrium of the game is shown to depend on the market size - namely, on total demand at a price equal to the minimum average cost -relative to the firm minimum efficient scale: if the market is sufficiently large, then the competitive price (the minimum of average cost) emerges at a subgame-perfect equilibrium of the game; if the market is not that large, then the firms randomize in prices on the equilibrium path of the game. The role of the market size for the competitive outcome is even more important for the case of two production techniquesBertrand-Edgeworth, oligopoly, price game, mixed strategy equilibrium, capacity indivisibility

Pricing and matching under duopoly with imperfect buyer mobility

Recent contributions have explored how lack of buyer mobility affects pricing. For example, Burdett, Shi, and Wright (2001) envisage a two-stage game where, once prices are set by the firms, the buyers play a static game by choosing independently which firm to visit. We incorporate imperfect mobility in a duopolistic pricing game where the buyers are involved into a multi-stage game. The firms are shown to have an incentive to give service priority to loyal customers. Under this rationing rule, equilibrium prices converge to their value under perfect buyer mobility as the number of stages of the buyer game increasesBertrand competition, matching, imperfect mobility, sequential equilibrium, buyerloyalty

On a property of mixed strategy equilibria of the pricing game

Before solving a two-stage capacity and pricing game for oligopoly, Boccard and Wauthy (2000) argue that, as under duopoly, at a mixed strategy equilibrium of the pricing game the largest firm's expected profit is the profit accruing to it as a Stackelberg follower when the rivals supply their entire capacity. We point to a serious mistake in their argument and then we see how this important property can be satisfactorily established.Bertrand competition

Bad politicians

We present a simple theory of the quality of elected officials. Quality has (at least) two dimensions: competence and honesty. Voters prefer competent and honest policymakers, so high-quality citizens have a greater chance of being elected to office. But low-quality citizens have a “comparative advantage” in pursuing elective office, because their market wages are lower than the market wages of high-quality citizens (competence), and/or because they reap higher returns from holding office (honesty). In the political equilibrium, the average quality of the elected body depends on the structure of rewards from holding public office. Under the assumption that the rewards from office are increasing in the average quality of office holders there can be multiple equilibria in quality. Under the assumption that incumbent policymakers set the rewards for future policymakers there can be path dependence in quality.Corruption