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# Weighted-average stochastic games with constant payoff

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## Abstract

In a zero-sum stochastic game, at each stage, two opponents make decisions which determine a stage reward and the law of the state of nature at the next stage, and the aim of the players is to maximize the weighted-average of the stage rewards. In this paper we solve the constant-payoff conjecture formulated by Sorin, Venel and Vigeral in 2010 for two classes of stochastic games with weighted-average rewards: (1) absorbing games, a well-known class of stochastic games where the state changes at most once during the game, and (2) smooth stochastic games, a newly introduced class of stochastic games where the state evolves smoothly under optimal play.
Vol.:(0123456789)
Operational Research (2022) 22:1675–1696
https://doi.org/10.1007/s12351-021-00625-6
1 3
REVIEW
Weighted‑average stochastic games withconstant payoﬀ
MiquelOliu‑Barton1
Received: 13 July 2020 / Revised: 18 December 2020 / Accepted: 1 February 2021 /
Published online: 2 March 2021
© The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature 2021
Abstract
In a zero-sum stochastic game, at each stage, two opponents make decisions which
determine a stage reward and the law of the state of nature at the next stage, and the
aim of the players is to maximize the weighted-average of the stage rewards. In this
paper we solve the constant-payoﬀ conjecture formulated by Sorin, Venel and Vig-
eral in 2010 for two classes of stochastic games with weighted-average rewards: (1)
absorbing games, a well-known class of stochastic games where the state changes
at most once during the game, and (2) smooth stochastic games, a newly introduced
class of stochastic games where the state evolves smoothly under optimal play.
Keywords Stochastic game· Value· Markov chain· Constant payoﬀ
1 Introduction
Model Stochastic games were introduced by  Shapley (1953) in order to model a
repeated interaction between two opponent players in a changing environment. The
game proceeds in stages. At each stage
m
of the game, players play a zero-
sum game that depends on a state variable. Formally, knowing the current state
km
, Player1 chooses an action
im
and Player2 chooses an action
jm
. Their choices
occur independently and have two consequences: ﬁrst, they produce a stage reward
g(km,im,jm)
which is observed by the players and, second, they determine the law
q(km,im,jm)
of the next period’s state
km+1
. Thus, the sequence of states follows a
Markov chain controlled by the actions of both players. To any sequence of non-
negative weights
𝜃=(𝜃m)
and any initial state k corresponds the
-weighted aver-
age stochastic game which is one where Player 1 maximizes the expectation of
m1𝜃m
g(k
m
,i
m
,j
m)
, given that
k1=k
, while Player2 minimizes the same amount.
A crucial aspect in this model is that the current state is commonly observed by the
players at every stage. Another one is stationarity: the transition and stage reward
functions do not change over time.
* Miquel Oliu-Barton
miquel.oliu.barton@normalesup.org
1 Université Paris-Dauphine, Paris, France
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