86 research outputs found
On an unified framework for approachability in games with or without signals
We unify standard frameworks for approachability both in full or partial
monitoring by defining a new abstract game, called the "purely informative
game", where the outcome at each stage is the maximal information players can
obtain, represented as some probability measure. Objectives of players can be
rewritten as the convergence (to some given set) of sequences of averages of
these probability measures. We obtain new results extending the approachability
theory developed by Blackwell moreover this new abstract framework enables us
to characterize approachable sets with, as usual, a remarkably simple and clear
reformulation for convex sets. Translated into the original games, those
results become the first necessary and sufficient condition under which an
arbitrary set is approachable and they cover and extend previous known results
for convex sets. We also investigate a specific class of games where, thanks to
some unusual definition of averages and convexity, we again obtain a complete
characterization of approachable sets along with rates of convergence
Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition
In the present work, we consider 2-person zero-sum stochastic differential
games with a nonlinear pay-off functional which is defined through a backward
stochastic differential equation. Our main objective is to study for such a
game the problem of the existence of a value without Isaacs condition. Not
surprising, this requires a suitable concept of mixed strategies which, to the
authors' best knowledge, was not known in the context of stochastic
differential games. For this, we consider nonanticipative strategies with a
delay defined through a partition of the time interval . The
underlying stochastic controls for the both players are randomized along
by a hazard which is independent of the governing Brownian motion, and knowing
the information available at the left time point of the subintervals
generated by , the controls of Players 1 and 2 are conditionally
independent over . It is shown that the associated lower and
upper value functions and converge uniformly on compacts to
a function , the so-called value in mixed strategies, as the mesh of
tends to zero. This function is characterized as the unique viscosity
solution of the associated Hamilton-Jacobi-Bellman-Isaacs equation.Comment: Published in at http://dx.doi.org/10.1214/13-AOP849 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Limit value for optimal control with general means
We consider optimal control problem with an integral cost which is a mean of
a given function. As a particular case, the cost concerned is the Ces\`aro
average. The limit of the value with Ces\`aro mean when the horizon tends to
infinity is widely studied in the literature. We address the more general
question of the existence of a limit when the averaging parameter converges,
for values defined with means of general types.
We consider a given function and a family of costs defined as the mean of the
function with respect to a family of probability measures -- the evaluations --
on R_+. We provide conditions on the evaluations in order to obtain the uniform
convergence of the associated value function (when the parameter of the family
converges).
Our main result gives a necessary and sufficient condition in term of the
total variation of the family of probability measures on R_+. As a byproduct,
we obtain the existence of a limit value (for general means) for control
systems having a compact invariant set and satisfying suitable nonexpansive
property.Comment: 21 pages, 2 figure
Playable differential games
AbstractPlayability conditions of differential games are studied by using Viability Theory. First, the results on payability of time independent differential games are extended to time dependent games. In fact, time is introduced in the dynamics of the game, in the state dependent constraints bearing on controls, and in state constraints. Second, some examples of pursuit games are studied. Necessary and sufficient conditions of playability of the game are provided. Here, pursuit games are directly considered as “games of kind” (in Isaacs's sense) and are not considered as “games of degree.” The viability condition does not always provide the “optimal strategy” to be as close as possible to a certain goal, but it supplies strategies allowing the system to reach a given goal
Differential games with asymmetric information and without Isaacs condition
We investigate a two-player zero-sum differential game with asymmetric
information on the payoff and without Isaacs condition. The dynamics is an
ordinary differential equation parametrised by two controls chosen by the
players. Each player has a private information on the payoff of the game, while
his opponent knows only the probability distribution on the information of the
other player. We show that a suitable definition of random strategies allows to
prove the existence of a value in mixed strategies. Moreover, the value
function can be characterised in term of the unique viscosity solution in some
dual sense of a Hamilton-Jacobi-Isaacs equation. Here we do not suppose the
Isaacs condition which is usually assumed in differential games
Differential games through viability theory : old and recent results.
This article is devoted to a survey of results for differential games obtained through Viability Theory. We recall the basic theory for differential games (obtained in the 1990s), but we also give an overview of recent advances in the following areas : games with hard constraints, stochastic differential games, and hybrid differential games. We also discuss several applications.Game theory; Differential game; viability algorithm;
Control Systems with Constraints and Uncertain Initial Conditions
AMS subject classification: 49N55, 93B52, 93C15, 93C10, 26E25.We study the problem of finding a control such that all solutions of a control systems, starting from a given set of initial conditions, satisfy a given constraint. This problem is an extension of the well-known Viability Problem when the initial condition is a set. The present paper is mainly a survey of results recently obtained by the authors, but some new results with proofs are also included
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