Appl Math Optim 50:87–118 (2004)
2004 Springer-Verlag New York, LLC
Some Classes of Imperfect Information Finite State-Space
Stochastic Games with Finite-Dimensional Solutions
William M. McEneaney
Departments of Mathematics and Mechanical/Aerospace Engineering,
University of California at San Diego,
La Jolla, CA 92093-0112, USA
Abstract. Stochastic games under imperfect information are typically computa-
tionally intractable even in the discrete-time/discrete-state case considered here.
We consider a problem where one player has perfect information. A function of
a conditional probability distribution is proposed as an information state. In the
problem form here, the payoff is only a function of the terminal state of the system,
and the initial information state is either linear or a sum of max-plus delta func-
tions. When the initial information state belongs to these classes, its propagation
is ﬁnite-dimensional. The state feedback value function is also ﬁnite-dimensional,
and obtained via dynamic programming, but has a nonstandard form due to the
necessity of an expanded state variable. Under a saddle point assumption, Certainty
Equivalence is obtained and the proposed function is indeed an information state.
Key Words. Dynamic games, Stochastic games, Imperfect information, Certainty
equivalence, Markov chains.
AMS Classiﬁcation. 91A15, 93C41, 91A50, 90C39, 49K35.
A class of discrete stochastic games where one player has imperfect information is
considered. Background material on such games can be found in ,  and . The
focus here is on minimax-type values , , , . We ﬁrst put the problem difﬁculty
This reasearch was partially supported by AFRL and DARPA (Contract F30602-01-C-0201) and NSF