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ISRAEL JOURNAL OF MATHEMATICS 119 (2000), 93126
TWOPLAYER STOCHASTIC GAMES II:
THE CASE OF RECURSIVE GAMES
BY
NICOLAS VIEILLE
Laboratoire d'Economdtrie, Ecole Polytechnique
1 rue Descartes, 75 005 Paris, France
and
Grape, Universitd Montesquieu Bordeaux
av. Duguit, 33608 Pessac, France
email: [email protected]
ABSTRACT
This paper contains the second step in the proof of existence of equilib
rium payoffs for twoplayer stochastic games. It deals with the case of
positive absorbing recursive games
This paper* complements [12]. We prove here the existence of equilibrium
payoffs in twoplayer, absorbing positive recursive games. Recursive games are
stochastic games in which the players receive a payoff equal to zero until an
absorbing state is reached. Positive recursive games are recursive games in which
the payoff to one of the players is positive in each absorbing state. Such a game is
absorbing if the other player cannot prevent the play from reaching an absorbing
state in finite time.
Zerosum recursive games were first introduced by Everett [2], who proved the
existence of stationary eoptimal strategies. Flesch, Thuijsman and Vrieze [3]
exhibited a twoplayer recursive game with no stationary eequilibrium profile.
Independently of our work, Solan [9] proved the existence of equilibrium payoffs
* This is a thoroughly revised version of a discussion paper which circulated under
a slightly different title [10]. I wish to thank Eilon Solan, Sylvain Sorin and an
anonymous referee for many helpful comments.
Received January 28, 1998
93
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N. VIEILLE
Isr. J. Math.
in twoplayer, positive recursive games with two nonabsorbing states that have
the absorbing property.
The paper is organized as follows. Section 1 contains definitions and the state
ment of the main result. Sections 2 and 3 are devoted to examples. The first
example is a variation on the example in [3]: it is a twoplayer positive absorbing
recursive game with no stationary eequilibrium profile. The second example is
used to present the main features of the eequilibrium profiles that we obtain.
Section 4 provides a sufficient condition for the existence of equilibrium payoffs.
In Section 6, we define a family of constrained games, indexed by c > 0, and
analyze the asymptotics of this family, as 6 goes to zero.
1. Definitions and main result
A twoplayer
recursive game
is given by (i) a finite set of states S partitioned
into S* and S\S*; (ii) finite sets A and B of available actions; (iii) a transition
function p:
S\S* x A x B + A(S),
where A(S) is the space of all probability
distributions over S, and (iv) a payoff function g = (gl, g2): S* + R 2.
The game is played as follows. As long as S* has not been reached, the players
choose actions, and the state changes from stage to stage according to p. As soon
as a state s* C S* is reached, the game stops and the players receive the payoff
g(s*).
The elements of S* are called
absorbing states.
It is convenient to formalize this as follows. The set of stages is the set N*
of positive integers. The initial state sl is given. At stage n, the current state
sn is announced to the players. Player 1 and player 2 choose an action an and
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.
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