Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates

2003 ◽  
Vol 40 (2) ◽  
pp. 327-345 ◽  
Author(s):  
Xianping Guo ◽  
Onésimo Hernández-Lerma
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Transition Rates ◽  
Zero Sum Games ◽  
Stationary Strategies ◽  
Continuous Time Markov Chains ◽  
Average Payoff ◽  
Optimal Stationary Strategies ◽  
Zero Sum

This paper is a first study of two-person zero-sum games for denumerable continuous-time Markov chains determined by given transition rates, with an average payoff criterion. The transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. In the spirit of the ‘drift and monotonicity’ conditions for continuous-time Markov processes, we give conditions on the controlled system's primitive data under which the existence of the value of the game and a pair of strong optimal stationary strategies is ensured by using the Shapley equations. Also, we present a ‘martingale characterization’ of a pair of strong optimal stationary strategies. Our results are illustrated with a birth-and-death game.

Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates

2003 ◽  
Vol 40 (02) ◽  
pp. 327-345 ◽  
Author(s):  
Xianping Guo ◽  
Onésimo Hernández-Lerma
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Transition Rates ◽  
Zero Sum Games ◽  
Stationary Strategies ◽  
Continuous Time Markov Chains ◽  
Average Payoff ◽  
Optimal Stationary Strategies ◽  
Zero Sum

This paper is a first study of two-person zero-sum games for denumerable continuous-time Markov chains determined by given transition rates, with an average payoff criterion. The transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. In the spirit of the ‘drift and monotonicity’ conditions for continuous-time Markov processes, we give conditions on the controlled system's primitive data under which the existence of the value of the game and a pair of strong optimal stationary strategies is ensured by using the Shapley equations. Also, we present a ‘martingale characterization’ of a pair of strong optimal stationary strategies. Our results are illustrated with a birth-and-death game.

scholarly journals Nonzero-sum games for continuous-time Markov chains with unbounded discounted payoffs

2005 ◽  
Vol 42 (2) ◽  
pp. 303-320 ◽  
Author(s):  
Xianping Guo ◽  
Onésimo Hernández-Lerma
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Queueing System ◽  
Transition Rates ◽  
Markov Game ◽  
Continuous Time Markov Chains ◽  
Payoff Functions ◽  
Zero Sum ◽  
Action Spaces ◽  
Uniformly Bounded

In this paper, we study two-person nonzero-sum games for continuous-time Markov chains with discounted payoff criteria and Borel action spaces. The transition rates are possibly unbounded, and the payoff functions might have neither upper nor lower bounds. We give conditions that ensure the existence of Nash equilibria in stationary strategies. For the zero-sum case, we prove the existence of the value of the game, and also provide arecursiveway to compute it, or at least to approximate it. Our results are applied to a controlled queueing system. We also show that if the transition rates areuniformly bounded, then a continuous-time game is equivalent, in a suitable sense, to a discrete-time Markov game.

scholarly journals Zero-sum games for continuous-time jump Markov processes in Polish spaces: discounted payoffs

2007 ◽  
Vol 39 (03) ◽  
pp. 645-668 ◽  
Author(s):  
Xianping Guo ◽  
Onésimo Hernández-Lerma
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Infinitesimal Operator ◽  
Polish Spaces ◽  
Zero Sum Games ◽  
Stationary Strategies ◽  
Value Of The Game ◽  
Optimal Stationary Strategies ◽  
Zero Sum ◽  
Jump Markov Processes

This paper is devoted to the study of two-person zero-sum games for continuous-time jump Markov processes with a discounted payoff criterion. The state and action spaces are all Polish spaces, the transition rates are allowed to beunbounded, and the payoff rates may haveneither upper nor lower bounds. We give conditions on the game'sprimitive dataunder which the existence of a solution to the Shapley equation is ensured. Then, from the Shapley equation, we obtain the existence of the value of the game and of a pair of optimal stationary strategies using theextended infinitesimal operatorassociated with the transition function of a possibly nonhomogeneous continuous-time jump Markov process. We also provide arecursiveway of computing (or at least approximating) the value of the game. Moreover, we present a ‘martingale characterization’ of a pair of optimal stationary strategies. Finally, we apply our results to a controlled birth and death system and a Schlögl first model, and then we use controlled Potlach processes to illustrate our conditions.

scholarly journals Nonzero-sum games for continuous-time Markov chains with unbounded discounted payoffs

2005 ◽  
Vol 42 (02) ◽  
pp. 303-320 ◽  
Author(s):  
Xianping Guo ◽  
Onésimo Hernández-Lerma
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Queueing System ◽  
Transition Rates ◽  
Markov Game ◽  
Continuous Time Markov Chains ◽  
Payoff Functions ◽  
Zero Sum ◽  
Action Spaces ◽  
Uniformly Bounded

In this paper, we study two-person nonzero-sum games for continuous-time Markov chains with discounted payoff criteria and Borel action spaces. The transition rates are possibly unbounded, and the payoff functions might have neither upper nor lower bounds. We give conditions that ensure the existence of Nash equilibria in stationary strategies. For the zero-sum case, we prove the existence of the value of the game, and also provide arecursiveway to compute it, or at least to approximate it. Our results are applied to a controlled queueing system. We also show that if the transition rates areuniformly bounded, then a continuous-time game is equivalent, in a suitable sense, to a discrete-time Markov game.

scholarly journals Zero-sum games for continuous-time jump Markov processes in Polish spaces: discounted payoffs

2007 ◽  
Vol 39 (3) ◽  
pp. 645-668 ◽  
Author(s):  
Xianping Guo ◽  
Onésimo Hernández-Lerma
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Infinitesimal Operator ◽  
Polish Spaces ◽  
Zero Sum Games ◽  
Stationary Strategies ◽  
Value Of The Game ◽  
Optimal Stationary Strategies ◽  
Zero Sum ◽  
Jump Markov Processes

This paper is devoted to the study of two-person zero-sum games for continuous-time jump Markov processes with a discounted payoff criterion. The state and action spaces are all Polish spaces, the transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. We give conditions on the game's primitive data under which the existence of a solution to the Shapley equation is ensured. Then, from the Shapley equation, we obtain the existence of the value of the game and of a pair of optimal stationary strategies using the extended infinitesimal operator associated with the transition function of a possibly nonhomogeneous continuous-time jump Markov process. We also provide a recursive way of computing (or at least approximating) the value of the game. Moreover, we present a ‘martingale characterization’ of a pair of optimal stationary strategies. Finally, we apply our results to a controlled birth and death system and a Schlögl first model, and then we use controlled Potlach processes to illustrate our conditions.

Markovian couplings staying in arbitrary subsets of the state space

2002 ◽  
Vol 39 (01) ◽  
pp. 197-212 ◽  
Author(s):  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Transition Rates ◽  
Arbitrary Subset ◽  
State Spaces ◽  
Continuous Time Markov Chains ◽  
Countable State ◽  
Necessary And Sufficient

Let (X t ) and (Y t ) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (X t ) and (Y t ) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

On the Problem of Establishing the Existence of Stationary Distributions for Continuous-Time Markov Chains

1993 ◽  
Vol 7 (4) ◽  
pp. 529-543 ◽  
Author(s):  
P. K. Pollett ◽  
P. G. Taylor
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Transition Rates ◽  
Necessary And Sufficient Condition ◽  
Stationary Distributions ◽  
Continuous Time Markov Chains ◽  
Necessary And Sufficient ◽  
Networks Of Queues

We consider the problem of establishing the existence of stationary distributions for continuous-time Markov chains directly from the transition rates Q. Given an invariant probability distribution m for Q, we show that a necessary and sufficient condition for m to be a stationary distribution for the minimal process is that Q be regular. We provide sufficient conditions for the regularity of Q that are simple to verify in practice, thus allowing one to easily identify stationary distributions for a variety of models. To illustrate our results, we shall consider three classes of multidimensional Markov chains, namely, networks of queues with batch movements, semireversible queues, and partially balanced Markov processes.

Markovian couplings staying in arbitrary subsets of the state space

2002 ◽  
Vol 39 (1) ◽  
pp. 197-212 ◽  
Author(s):  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Transition Rates ◽  
Arbitrary Subset ◽  
State Spaces ◽  
Continuous Time Markov Chains ◽  
Countable State ◽  
Necessary And Sufficient

Let (Xt) and (Yt) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (Xt) and (Yt) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

Sign in / Sign up

Export Citation Format

Share Document