Risk-sensitive Average Cost Criterion Research Articles

Zero-sum stochastic games in continuous-time with risk-sensitive average cost criterion on a countable state space

We consider zero-sum stochastic games in continuous time with controlled Markov chains and with risk-sensitive average cost criterion. Here the transition and the cost rates may be unbounded. We prove the existence of the value of the game and a saddle-point equilibrium in the class of all stationary strategies under a Lyapunov stability condition. This is accomplished by establishing the existence of a principal eigenpair for the corresponding Hamilton-Jacobi-Isaacs (HJI) equation. This, in turn, is established by using a nonlinear version of Krein-Rutman theorem. We then obtain a characterization of the saddle-point equilibrium in terms of the corresponding HJI equation. Finally, we use a controlled population system to illustrate our results.

Discrete-time zero-sum games for Markov chains with risk-sensitive average cost criterion

We study zero-sum stochastic games for controlled discrete time Markov chains with risk-sensitive average cost criterion with countable/compact state space and Borel action spaces. The payoff function is nonnegative and possibly unbounded for countable state space case and for compact state space case it is a real-valued and bounded function. For countable state space case, under a certain Lyapunov type stability assumption on the dynamics we establish the existence of the value and a saddle point equilibrium. For compact state space case we establish these results without any Lyapunov type stability assumptions. Using the stochastic representation of the principal eigenfunction of the associated optimality equation, we completely characterize all possible saddle point strategies in the class of stationary Markov strategies. Also, we present and analyze an illustrative example.

Nonzero-sum Risk-Sensitive Average Stochastic Games: The Case of Unbounded Costs

In this paper, we study discrete-time nonzero-sum stochastic games under the risk-sensitive average cost criterion. The state space is a denumerable set, the action spaces of players are Borel spaces, and the cost functions are unbounded. Under suitable conditions, we first introduce the risk-sensitive first passage payoff functions and obtain their properties. Then, we establish the existence of a solution to the risk-sensitive average cost optimality equation of each player for the case of unbounded cost functions and show the existence of a randomized stationary Nash equilibrium in the class of randomized history-dependent strategies. Finally, we use a controlled population system to illustrate the main results.

Risk-sensitive average continuous-time Markov decision processes with unbounded rates

In this paper we study the risk-sensitive average cost criterion for continuous-time Markov decision processes in the class of all randomized Markov policies. The state space is a denumerable set, and the cost and transition rates are allowed to be unbounded. Under the suitable conditions, we establish the optimality equation of the auxiliary risk-sensitive first passage optimization problem and obtain the properties of the corresponding optimal value function. Then by a technique of constructing the appropriate approximating sequences of the cost and transition rates and employing the results on the auxiliary optimization problem, we show the existence of a solution to the risk-sensitive average optimality inequality and develop a new approach called the risk-sensitive average optimality inequality approach to prove the existence of an optimal deterministic stationary policy. Furthermore, we give some sufficient conditions for the verification of the simultaneous Doeblin condition, use a controlled birth and death system to illustrate our conditions and provide an example for which the risk-sensitive average optimality strict inequality occurs.

Characterization of the Optimal Risk-Sensitive Average Cost in Denumerable Markov Decision Chains

This work is concerned with Markov decision chains on a denumerable state space. The controller has a positive risk-sensitivity coefficient, and the performance of a control policy is measured by a risk-sensitive average cost criterion. Besides standard continuity-compactness conditions, it is assumed that the state process is communicating under any stationary policy, and that the simultaneous Doeblin condition holds. In this context, it is shown that if the cost function is bounded from below, and the superior limit average index is finite at some point, then (i) the optimal superior and inferior limit average value functions coincide and are constant, (ii) the optimal average cost is characterized via an extended version of the Collatz-Wielandt formula in the theory of positive matrices, and (iii) an optimality inequality is established, from which a stationary optimal policy is obtained. Moreover, an explicit example is given to show that, even if the cost function is bounded, the strict inequality may occur in the optimality relation.

Average cost criterion induced by the regular utility function for continuous-time Markov decision processes

In this paper we study the average cost criterion induced by the regular utility function (U-average cost criterion) for continuous-time Markov decision processes. This criterion is a generalization of the risk-sensitive average cost and expected average cost criteria. We first introduce an auxiliary risk-sensitive first passage optimization problem and obtain the properties of the corresponding optimal value function under the slight conditions. Then we show that the pair of the optimal value functions of the risk-sensitive average cost criterion and the risk-sensitive first passage criterion is a solution to the optimality equation of the risk-sensitive average cost criterion allowing the risk-sensitivity parameter to take any nonzero value. Moreover, we have that the optimal value function of the risk-sensitive average cost criterion is continuous with respect to the risk-sensitivity parameter. Finally, we give the connections between the U-average cost criterion and the average cost criteria induced by the identity function and the exponential utility function, and prove the existence of a U-average optimal deterministic stationary policy in the class of all randomized Markov policies.

Local Poisson Equations Associated with Discrete-Time Markov Control Processes

This paper provides a characterization of the optimal average cost function, when the long-run (risk-sensitive) average cost criterion is used. The Markov control model has a denumerable state space with finite set of actions, and the characterization presented is given in terms of a system of local Poisson equations, which gives as a by-product the existence of an optimal stationary policy.

Continuous-time Markov decision processes under the risk-sensitive average cost criterion

This paper studies continuous-time Markov decision processes under the risk-sensitive average cost criterion. The state space is a finite set, the action space is a Borel space, the cost and transition rates are bounded, and the risk-sensitivity coefficient can take any positive real number. Under the mild conditions, we develop a new approach to establish the existence of a solution to the risk-sensitive average cost optimality equation and obtain the existence of an optimal deterministic stationary policy.

Controlled Semi-Markov Chains with Risk-Sensitive Average Cost Criterion

This work concerns with semi-Markov decision chains on a finite state space. Assuming that the controller has a constant and positive risk-sensitive coefficient, an optimality equation for the corresponding (long-run) risk-sensitive average cost index is formulated and, under suitable continuity-compactness conditions, it is shown that a solution of such an equation determines the optimal average cost, as well as an optimal stationary policy. Additionally, if the underlying Markov chain is communicating, then it is proved that the optimality equation has a solution.

A poisson equation for the risk-sensitive average cost in semi-markov chains

This work concerns semi-Markov chains evolving on a finite state space. The system development generates a cost when a transition is announced, as well as a holding cost which is incurred continuously during each sojourn time. It is assumed that these costs are paid by an observer with positive and constant risk-sensitivity, and the overall performance of the system is measured by the corresponding (long-run) risk-sensitive average cost criterion. In this framework, conditions are provided under which the average index does not depend on the initial state and is characterized in terms of a single Poisson equation.

Discounted Approximations for Risk-Sensitive Average Criteria in Markov Decision Chains with Finite State Space

This work concerns Markov decision processes with finite state space and compact action set. The performance of a control policy is measured by a risk-sensitive average cost criterion and, under standard continuity-compactness conditions, it is shown that the discounted approximations converge to the optimal value function, and that the superior and inferior limit average criteria have the same optimal value function. These conclusions are obtained for every nonnull risk-sensitivity coefficient, and regardless of the communication structure induced by the transition law.

A note on risk-sensitive control of invariant models

According to Assaf, a dynamic programming problem is called invariant if its transition mechanism depends only on the chosen action. This paper studies properties of risk-sensitive invariant problems with a general state space. The main result establishes the optimality equation for the risk-sensitive average cost criterion without any restrictions on the risk factor. Moreover, a practical algorithm is provided for solving the optimality equation in case of a finite action space.

Average optimality for risk-sensitive control with general state space

This paper deals with discrete-time Markov control processes on a general state space. A long-run risk-sensitive average cost criterion is used as a performance measure. The one-step cost function is nonnegative and possibly unbounded. Using the vanishing discount factor approach, the optimality inequality and an optimal stationary strategy for the decision maker are established.

Nonstationary value iteration in controlled Markov chains with risk-sensitive average criterion

This work concerns Markov decision chains with finite state spaces and compact action sets. The performance index is the long-run risk-sensitive average cost criterion, and it is assumed that, under each stationary policy, the state space is a communicating class and that the cost function and the transition law depend continuously on the action. These latter data are not directly available to the decision-maker, but convergent approximations are known or are more easily computed. In this context, the nonstationary value iteration algorithm is used to approximate the solution of the optimality equation, and to obtain a nearly optimal stationary policy.

Nonstationary value iteration in controlled Markov chains with risk-sensitive average criterion

This work concerns Markov decision chains with finite state spaces and compact action sets. The performance index is the long-run risk-sensitive average cost criterion, and it is assumed that, under each stationary policy, the state space is a communicating class and that the cost function and the transition law depend continuously on the action. These latter data are not directly available to the decision-maker, but convergent approximations are known or are more easily computed. In this context, the nonstationary value iteration algorithm is used to approximate the solution of the optimality equation, and to obtain a nearly optimal stationary policy.

A characterization of the optimal risk-sensitive average cost in finite controlled Markov chains

This work concerns controlled Markov chains with finite state and action spaces. The transition law satisfies the simultaneous Doeblin condition, and the performance of a control policy is measured by the (long-run) risk-sensitive average cost criterion associated to a positive, but otherwise arbitrary, risk sensitivity coefficient. Within this context, the optimal risk-sensitive average cost is characterized via a minimization problem in a finite-dimensional Euclidean space.

The Value Iteration Algorithm in Risk-Sensitive Average Markov Decision Chains with Finite State Space

This work concerns discrete-time Markov decision chains with finite state space and bounded costs. The controller has constant risk sensitivity λ, and the performance of a control policy is measured by the corresponding risk-sensitive average cost criterion. Assuming that the optimality equation has a solution, it is shown that the value iteration scheme can be implemented to obtain, in a finite number of steps, (1) an approximation to the optimal λ-sensitive average cost with an error less than a given tolerance, and (2) a stationary policy whose performance index is arbitrarily close to the optimal value. The argument used to establish these results is based on a modification of the original model, which is an extension of a transformation introduced by Schweitzer (1971) to analyze the the risk-neutral case.

Solution to the risk-sensitive average cost optimality equation in a class of Markov decision processes with finite state space

This work concerns discrete-time Markov decision processes with finite state space and bounded costs per stage. The decision maker ranks random costs via the expectation of the utility function associated to a constant risk sensitivity coefficient, and the performance of a control policy is measured by the corresponding (long-run) risk-sensitive average cost criterion. The main structural restriction on the system is the following communication assumption: For every pair of states x and y, there exists a policy π, possibly depending on x and y, such that when the system evolves under π starting at x, the probability of reaching y is positive. Within this framework, the paper establishes the existence of solutions to the optimality equation whenever the constant risk sensitivity coefficient does not exceed certain positive value.

Solution to the risk-sensitive average optimality equation in communicating Markov decision chains with finite state space: An alternative approach

This note concerns Markov decision chains with finite state and action sets. The decision maker is assumed to be risk-averse with constant risk sensitive coefficient λ, and the performance of a control policy is measured by the risk-sensitive average cost criterion. In their seminal paper Howard and Matheson established that, when the whole state space is a communicating class under the action of each stationary policy, then there exists a solution to the optimality equation for every λ>0. This paper presents an alternative proof of this fundamental result, which explicitly highlights the essential role of the communication properties in the analysis of the risk-sensitive average cost criterion.

Controlled Markov chains with risk-sensitive criteria: Average cost, optimality equations, and optimal solutions

We study controlled Markov chains with denumerable state space and bounded costs per stage. A (long-run) risk-sensitive average cost criterion, associated to an exponential utility function with a constant risk sensitivity coefficient, is used as a performance measure. The main assumption on the probabilistic structure of the model is that the transition law satisfies a simultaneous Doeblin condition. Working within this framework, the main results obtained can be summarized as follows: If the constant risk-sensitivity coefficient is small enough, then an associated optimality equation has a bounded solution with a constant value for the optimal risk-sensitive average cost; in addition, under further standard continuity-compactness assumptions, optimal stationary policies are obtained. However, it is also shown that the above conclusions fail to hold, in general, for large enough values of the risk-sensitivity coefficient. Our results therefore disprove previous claims on this topic. Also of importance is the fact that our developments are very much self-contained and employ only basic probabilistic and analysis principles.