On stochastic games

Maitra, A.; Parthasarathy, T.

doi:10.1007/bf00927915

Cited by 90 publications

(36 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Such results have also been extended to non-zero-sum stochastic games (Fink, 1964, Takahashi, 1964, Sobel, 1971) and also to stochastic games with infinite state space and infinite action space (Maitra and Parthasarathy, 1970). Maitra and Sudderth (1996) proposed an alternative proof of the existence of value in the finite, undiscounted, zero-sum case which extends to the case when the state space is uncountable as well.…”

Section: Introductionmentioning

confidence: 95%

Orderfield property of mixtures of stochastic games

Krishnamurthy

Parthasarathy

Ravindran

2010

Sankhya

Self Cite

View full text Add to dashboard Cite

We consider certain mixtures, Γ, of classes of stochastic games and provide sufficient conditions for these mixtures to possess the orderfield property. For 2-player zero-sum and non-zero sum stochastic games, we prove that if we mix a set of states S1 where the transitions are controlled by one player with a set of states S2 constituting a sub-game having the orderfield property (where S1 ∩ S2 = ∅), the resulting mixture Γ with states S = S1 ∪ S2 has the orderfield property if there are no transitions from S2 to S1. This is true for discounted as well as undiscounted games. This condition on the transitions is sufficient when S1 is perfect information or SC (Switching Control) or ARAT (Additive Reward Additive Transition). In the zero-sum case, S1 can be a mixture of SC and ARAT as well. On the other hand,when S1 is SER-SIT (Separable Reward -State Independent Transition), we provide a counter example to show that this condition is not sufficient for the mixture Γ to possess the orderfield property. In addition to the condition that there are no transitions from S2 to S1, if the sum of all transition probabilities from S1 to S2 is independent of the actions of the players, then Γ has the orderfield property even when S1 is SER-SIT. When S1 and S2 are both SER-SIT, their mixture Γ has the orderfield property even if we allow transitions from S2 to S1. We also extend these results to some multi-player games namely, mixtures with one player control Polystochastic games. In all the above cases, we can inductively mix many such games and continue to retain the orderfield property.

show abstract

Section: Introductionmentioning

confidence: 95%

Orderfield property of mixtures of stochastic games

Krishnamurthy

Parthasarathy

Ravindran

2010

Sankhya

Self Cite

View full text Add to dashboard Cite

show abstract

“…K w is continuous in (s, 9, 3') (reasoning almost as in Maitra/Parthasarathy [1970], Lemma 2.1 ; use our lemma 2.2). For fixed s E S we may apply the minimax theorem of Sion [ 1958, Theorem 3.4] on K w (s,.…”

Section: Then For Allfef G E G We Have (L ~ G) W) (S) = K W (S F(s)mentioning

confidence: 94%

“…In the literature several sets of conditions have been given for the existence of "stationary" optimal strategies. We mention results of Vrieze [1976] and lCessels [1977] in case of a countable state space, and of Maitra/Parthasarathy [1970] and T. Parthasarathy [ 1973] when S is a compact metric space.…”

Section: Introductionmentioning

confidence: 98%

Stochastic games with metric state space

Couwenbergh

1980

Int J Game Theory

View full text Add to dashboard Cite

“…are jointly continuous, then Assumptions 2 and 3 hold (see [11, Proposition VII.1.4, p.394]). Maitra and Parthasarathy [8] provide even weaker conditions under which Assumptions 2 and 3 hold in the n stage-case and the λ-discounted case, which generalize immediately to the θ-weighted case.…”

Section: Zero-sum Stochastic Gamesmentioning

confidence: 99%

Tauberian theorems for general iterations of operators: Applications to zero-sum stochastic games

Ziliotto

2018

Games and Economic Behavior

View full text Add to dashboard Cite

This paper proves several Tauberian theorems for general iterations of operators, and provides two applications to zero-sum stochastic games where the total payoff is a weighted sum of the stage payoffs. The first application is to provide conditions under which the existence of the asymptotic value implies the convergence of the values of the weighted game, as players get more and more patient. The second application concerns stochastic games with finite state space and action sets. This paper builds a simple class of asymptotically optimal strategies in the weighted game, that at each stage play optimally in a discounted game with a discount factor corresponding to the relative weight of the current stage. models (see for instance the recent surveys [5], [2] and [18]). A natural question is to ask whether there is a link between the convergence of (v n ), (v λ ) and (v θ ) * . Ziliotto [22] has proved that in a very general stochastic game model, with possibly infinitely many states and actions, (v n ) converges uniformly if and only if (v λ ) converges uniformly † . This paper aims at generalizing such a result to a more general family of values (v θ ). Note that for recursive games (stochastic games in which the stage payoff is 0 in nonabsorbing states), Li and Venel [7] have proved that if (v n ) or (v λ ) converges uniformly, then (v θ ) converges uniformly when θ is decreasing and θ 1 goes to 0. In a dynamic programming framework (one player), Monderer and Sorin [9] have proved the convergence property, but for a more restricted class of decreasing evaluations. The contribution of this paper is twofold. In the same stochastic game model as in [22], it provides conditions under which the uniform convergence of (v n ) or (v λ ) implies the uniform convergence of (v θ ). Second, in the case where the state space and the action sets are finite, it proves that the following discounted strategy is asymptotically optimal in the θ-weighted game, as θ 1 goes to 0: at each stage m ≥ 1, play optimally in the discounted game with discount factor θ m /( m ′ ≥m θ m ′ ) ‡ . Such a discount factor corresponds to the weight of stage m relative to the weight of future stages. This result is new even when θ is a n-stage evaluation. Finally, this paper provides an example that illustrates the sharpness of the first result.The proof of the results rely on the operator approach, introduced by Rosenberg and Sorin [16]. This approach builds on the fact that the value of the θ-weighted game satisfies a functional equation, called the Shapley equation (see [17]). The properties of the associated operator can be exploited to infer convergence properties of (v θ ) (see [16]). This paper first proves several Tauberian theorems in an operator setting, and apply them to stochastic games to get the first result. Surprisingly, the proof of the second result also follows from a Tauberian theorem for operators. It is due to the fact that the payoff w θ guaranteed by a discounted strategy in the θ-weighted game satisfies a functional equatio...

show abstract

On stochastic games

Cited by 90 publications

References 6 publications

Orderfield property of mixtures of stochastic games

Orderfield property of mixtures of stochastic games

Stochastic games with metric state space

Tauberian theorems for general iterations of operators: Applications to zero-sum stochastic games

Contact Info

Product

Resources

About