56
N. VIEILLE
Isr. J. Math.
actions selected. The game never ends. We assume that the sets S, A and B are
finite.
This model was introduced by Shapley [17], who proved that, when payoffs
are zero-sum, and the infinite stream of payoffs (gn)n_>l is evaluated according
to a geometric average )~ )-]~ 1 (1 -
n-1
=
A)
gn, the game has a value v~, and, for
both players, stationary optimal strategies do exist. This was extended to the
non zero-sum case by Fink [8]. Many results followed, relaxing the finiteness
assumptions on S, A and B; see Mertens and Parthasarathy [13] for general
conditions.
Unlike the one-player case (Blackwell [5]), the optimal strategies
vary with )~, even when A is arbitrarily close to 0. This dependency has been
investigated by Bewley and Kohlberg [3, 2, 4]. Using the algebraic structure of
the graph of the equilibrium correspondence, they proved that vx has a Puiseux
expansion in a neighborhood of 0, and that similar properties hold for optimal
strategies.
Shortly after Shapley, the undiscounted evaluation was introduced by Gillette
[9], in the zero-sum case. The type of requirement introduced by Gillette has
been strengthened by Aumann and Maschler [1] in the framework of games with
incomplete information. Assume the game is stopped after the n-th stage, and
each player wishes to maximize the arithmetic average of the payoffs he received
up to that stage. This defines a finite game F~, which has a value v~. Does limv~
exist ? If so, does there exist a
single
strategy which is optimal (or e-optimal) in
every
F,~, for n large enough ? If the answer is positive, then v = lim,~-~o~ v,~ is
called the value of the game. In the same volume, Milnor and Shapley [15] and