Rida Laraki

A Theory of Measuring, Electing and Ranking

The impossibility theorems that abound in the theory of social choice show that there can be no satisfactory method for electing and ranking in the context of the traditional, 700-year-old model. A more realistic model, whose antecedents may be traced to Laplace and Galton, leads to a new theory that avoids all impossibilities with a simple and eminently practical method, “the majority judgement.” It has already been tested.

The Value of Zero-Sum Stopping Games in Continuous Time

We study two-player zero-sum stopping games in continuous time and infinite horizon. We prove that the value in randomized stopping times exists as soon as the payoff processes are right-continuous. In particular, as opposed to existing literature, we do not assume any conditions on the relations between the payoff processes.

Continuous-time games of timing

We address the question of existence of equilibrium in general timing games with complete information. Under weak assumptions, any two-player timing game has a Markov subgame perfect var epsilon-equilibrium, for each var epsilon>0. This result is tight. For some classes of games (symmetric games, games with cumulative payoffs), stronger existence results are established.

Election by Majority Judgment: Experimental Evidence

The majority judgement is a method of election. It is the consequence of a new theory of social choice where voters judge candidates instead of ranking them. The theory is explained elsewhere [2, 4]. This article describes and analyzes electoral experiments conducted in parallel with the last two French presidential elections to: (1) show that the majority judgement is a practical method, (2) describe it and its salient properties, (3) establish that it escapes the classical paradoxes, (4) illustrate how in practice the well known electoral mechanisms all fail to meet important criteria. The demonstrations introduce new concepts and methods.

A Continuous Time Approach for the Asymptotic Value in Two-Person Zero-Sum Repeated Games

We consider the asymptotic value of two person zero sum repeated games with general evaluations of the stream of stage payoffs. We show existence for incomplete information games, splitting games and absorbing games. The technique of proof consists in embedding the discrete repeated game into a continuous time one and to use viscosity solution tools.

Equilibrium in two-player non-zero-sum Dynkin games in continuous time

We prove that every two-player non-zero-sum Dynkin game in continuous time admits an ϵ-equilibrium in randomized stopping times. We provide a condition that ensures the existence of an ϵ-equilibrium in non-randomized stopping times.

The splitting game and applications

Abstract. First we define the splitting operator, which is related to the Shapley operator of the splitting game introduced by Sorin (2002). It depends on two compact convex sets C and D and associates to a function defined on C×D a saddle function, extending the usual convexification or concavification operators. We first prove general properties on its domain and its range. Then we give conditions on C and D allowing to preserve continuity or Lipschitz properties, extending the results in Laraki (2001a) obtained for the convexification operator. These results are finally used, through the analysis of the asymptotic behavior of the splitting game, to prove the existence of a continuous solution for the Mertens-Zamir system of functional equations (Mertens and Zamir (1971–72) and (1977)) in a quite general framework.

Higher order game dynamics

Continuous-time game dynamics are typically first order systems where payoffs determine the growth rate of the playersʼ strategy shares. In this paper, we investigate what happens beyond first order by viewing payoffs as higher order forces of change, specifying e.g. the acceleration of the playersʼ evolution instead of its velocity (a viewpoint which emerges naturally when it comes to aggregating empirical data of past instances of play). To that end, we derive a wide class of higher order game dynamics, generalizing first order imitative dynamics, and, in particular, the replicator dynamics. We show that strictly dominated strategies become extinct in n-th order payoff-monotonic dynamics n orders as fast as in the corresponding first order dynamics; furthermore, in stark contrast to first order, weakly dominated strategies also become extinct for n⩾2. All in all, higher order payoff-monotonic dynamics lead to the elimination of weakly dominated strategies, followed by the iterated deletion of strictly dominated strategies, thus providing a dynamic justification of the well-known epistemic rationalizability process of Dekel and Fudenberg [7]. Finally, we also establish a higher order analogue of the folk theorem of evolutionary game theory, and we show that convergence to strict equilibria in n-th order dynamics is n orders as fast as in first order.

Semidefinite programming for min–max problems and games

We consider two min–max problems (1) minimizing the supremum of finitely many rational functions over a compact basic semi-algebraic set and (2) solving a 2-player zero-sum polynomial game in randomized strategies with compact basic semi-algebraic sets of pure strategies. In both problems the optimal value can be approximated by solving a hierarchy of semidefinite relaxations, in the spirit of the moment approach developed in Lasserre (SIAM J Optim 11:796–817, 2001; Math Program B 112:65–92, 2008). This provides a unified approach and a class of algorithms to compute Nash equilibria and min–max strategies of several static and dynamic games. Each semidefinite relaxation can be solved in time which is polynomial in its input size and practice on a sample of experiments reveals that few relaxations are needed for a good approximation (and sometimes even for finite convergence), a behavior similar to what was observed in polynomial optimization.

On the Existence of Approximate Equilibria and Sharing Rule Solutions in Discontinuous Games

This paper studies the existence of equilibrium solution concepts in a large class of economic models with discontinuous payoff functions. The issue is well understood for Nash equilibria, thanks to Renys better-reply security condition (Reny (1999)), and its recent improvements (Barelli and Meneghel (2013); McLennan et al. (2011); Reny (2009, 2011)). We propose new approaches, related to Renys work, and obtain tight conditions for the existence of approximate equilibria and of sharing rule solutions in pure and mixed strategies (Simon and Zame (1990)). As byproducts, we prove that many auction games with correlated types admit an approximate equilibrium, and that many competition models have a sharing rule solution.