46 research outputs found
The asymptotic value in finite stochastic games
We provide a direct, elementary proof for the existence of , where is the value of a -discounted
finite two-person zero-sum stochastic game
Differential games with asymmetric and correlated information
Differential games with asymmetric information were introduced by
Cardaliaguet (2007). As in repeated games with lack of information on both
sides (Aumann and Maschler (1995)), each player receives a private signal (his
type) before the game starts and has a prior belief about his opponent's type.
Then, a differential game is played in which the dynamic and the payoff
function depend on both types: each player is thus partially informed about the
differential game that is played. The existence of the value function and some
characterizations have been obtained under the assumption that the signals are
drawn independently. In this paper, we drop this assumption and extend these
two results to the general case of correlated types. This result is then
applied to repeated games with incomplete information: the characterization of
the asymptotic value obtained by Rosenberg and Sorin (2001) and Laraki (2001)
for the independent case is extended to the general case.Comment: 22 page
A uniform Tauberian theorem in optimal control
In an optimal control framework, we consider the value of the
problem starting from state with finite horizon , as well as the value
of the -discounted problem starting from . We prove
that uniform convergence (on the set of states) of the values as
tends to infinity is equivalent to uniform convergence of the values
as tends to 0, and that the limits are identical.
An example is also provided to show that the result does not hold for pointwise
convergence. This work is an extension, using similar techniques, of a related
result in a discrete-time framework \cite{LehSys}.Comment: 14 page
A formula for the value of a stochastic game
In 1953, Lloyd Shapley defined the model of stochastic games, which were the
first general dynamic model of a game to be defined, and proved that
competitive stochastic games have a discounted value. In 1982,
Jean-Fran\c{c}ois Mertens and Abraham Neyman proved that competitive stochastic
games admit a robust solution concept, the value, which is equal to the limit
of the discounted values as the discount rate goes to 0. Both contributions
were published in PNAS. In the present paper, we provide a tractable formula
for the value of competitive stochastic games
Absorbing games with irrational values
Can an absorbing game with rational data have an irrational limit value? Yes:
In this note we provide the simplest examples where this phenomenon arises.
That is, the following absorbing game and a sequence of absorbing games whose limit
values are , for all integer . Finally, we conjecture that any
algebraic number can be represented as the limit value of an absorbing game
Constant payoff in zero-sum stochastic games
In a zero-sum stochastic game, at each stage, two adversary players take
decisions and receive a stage payoff determined by them and by a random
variable representing the state of nature. The total payoff is the discounted
sum of the stage payoffs. Assume that the players are very patient and use
optimal strategies. We then prove that, at any point in the game, players get
essentially the same expected payoff: the payoff is constant. This solves a
conjecture by Sorin, Venel and Vigeral (2010). The proof relies on the
semi-algebraic approach for discounted stochastic games introduced by Bewley
and Kohlberg (1976), on the theory of Markov chains with rare transitions,
initiated by Friedlin and Wentzell (1984), and on some variational inequalities
for value functions inspired by the recent work of Davini, Fathi, Iturriaga and
Zavidovique (2016