Cheap Non-standard Analysis and Computability

Non standard Analysis is an area of Mathematics dealing with notions of infinitesimal and infinitely large numbers, in which many statements from classical Analysis can be expressed very naturally. Cheap non-standard analysis introduced by Terence Tao in 2012 is based on the idea that considering that a property holds eventually is sufficient to give the essence of many of its statements. This provides constructivity but at some (acceptable) price. We consider computability in cheap non-standard analysis. We prove that many concepts from computable analysis as well as several concepts from computability can be very elegantly and alternatively presented in this framework. It provides a dual view and dual proofs to several statements already known in these fields.

Learning Equilibria in Games by Stochastic Distributed Algorithms

We consider a class of fully stochastic and fully distributed algorithms, that we prove to learn equilibria in games. Indeed, we consider a family of stochastic distributed dynamics that we prove to converge weakly (in the sense of weak convergence for probabilistic processes) towards their mean-field limit, i.e an ordinary differential equation (ODE) in the general case. We focus then on a class of stochastic dynamics where this ODE turns out to be related to multipopulation replicator dynamics. Using facts known about convergence of this ODE, we discuss the convergence of the initial stochastic dynamics: For general games, there might be non-convergence, but when convergence of the ODE holds, considered stochastic algorithms converge towards Nash equilibria. For games admitting Lyapunov functions, that we call Lyapunov games, the stochastic dynamics converge. We prove that any ordinal potential game, and hence any potential game is a Lyapunov game, with a multiaffine Lyapunov function. For Lyapunov games with a multiaffine Lyapunov function, we prove that this Lyapunov function is a super-martingale over the stochastic dynamics. This leads a way to provide bounds on their time of convergence by martingale arguments. This applies in particular for many classes of games that have been considered in literature, including several load balancing game scenarios and congestion games.