We study infinite horizon discounted-cost and ergodic-cost risk-sensitive zero-sum stochastic games for controlled continuous time Markov chains on a countable state space. For the discounted-cost game we prove the existence of value and saddle-point equilibrium in the class of Markov strategies under nominal conditions. For the ergodic-cost game we prove the existence of values and saddle point equilibrium by studying the corresponding Hamilton-Jacobi-Isaacs equation under a certain Lyapunov condition.
In the paper we consider the controlled continuous-time Markov chain describing the interacting particles system with the finite number of types. The system is controlled by two players with the opposite purposes. The limiting game as the number of particles tends to infinity is a zero-sum differential game. Krasovskii--Subbotin extremal shift provides the optimal strategy in the limiting game. The main result of the paper is the near optimality of the Krasovskii--Subbotin extremal shift rule for the original Markov game.
In this paper, we investigate a partially observable zero sum games where the state process is a discrete time Markov chain. We consider a general utility function in the optimization criterion. We show the existence of value for both finite and infinite horizon games and also establish the existence of optimal polices. The main step involves converting the partially observable game into a completely observable game which also keeps track of the total discounted accumulated reward/cost.
This paper considers a two-person zero-sum continuous-time Markov pure jump game in Borel state and action spaces over a fixed finite horizon. The main assumption on the model is the existence of a drift function, which bounds the reward rate. Under some regularity conditions, we show that the game has a value, and both of the players have their optimal policies.
In this article we consider zero and non-zero sum risk-sensitive average criterion games for semi-Markov processes with a finite state space. For the zero-sum case, under suitable assumptions we show that the game has a value. We also establish the existence of a stationary saddle point equilibrium. For the non-zero sum case, under suitable assumptions we establish the existence of a stationary Nash equilibrium.
In this paper, we study a nonzero-sum stochastic differential game in Markovian framework. We show the existence of the Nash equilibrium point which is discontinuous and of bang-bang type under natural conditions. The main tool is the notion of backward stochastic differential equations which, in our case, are multidimensional with discontinuous generators with respect to z component.
In this paper, we present an optimal control problem for stochastic differential games under Markov regime-switching forward-backward stochastic differential equations with jumps and partial information. First, we prove a sufficient maximum principle for non zero-sum stochastic differential game problems and obtain equilibrium point for such games. Second, we prove an equivalent maximum principle for non zero-sum stochastic differential games. The zero-sum stochastic differential games equivalent maximum principle is then obtained as a corollary. We apply the obtained results to study a proble…
We study the optimal use of information in Markov games with incomplete information on one side and two states. We provide a finite-stage algorithm for calculating the limit value as the gap between stages goes to 0, and an optimal strategy for the informed player in the limiting game in continuous time. This limiting strategy induces an-optimal strategy for the informed player, provided the gap between stages is small. Our results demonstrate when the informed player should use his information and how.
We study zero-sum stochastic games for controlled discrete time Markov chains with risk-sensitive average cost criterion with countable state space and Borel action spaces. The payoff function is nonnegative and possibly unbounded. Under a certain Lyapunov stability assumption on the dynamics, we establish the existence of a value and saddle point equilibrium. Further we completely characterize all possible saddle point strategies in the class of stationary Markov strategies. Finally, we present and analyze an illustrative example.
We consider time-homogeneous uniformly nondegenerate stochastic differential games in domains and propose constructing $\varepsilon$-optimal strategies and policies by using adjoint Markov strategies and adjoint Markov policies which are actually time-homogeneous Markov, however, relative not to the original process but to a couple of processes governed by a system consisting of the main original equation and of an adjoint stochastic equations of the same type as the main one. We show how to find $\varepsilon$-optimal strategies and policies in these classes by using the solvability in Sobolev…
We consider a nonzero-sum Markov game on an abstract measurable state space with compact metric action spaces. The goal of each player is to maximize his respective discounted payoff function under the condition that some constraints on a discounted payoff are satisfied. We are interested in the existence of a Nash or noncooperative equilibrium. Under suitable conditions, which include absolute continuity of the transitions with respect to some reference probability measure, additivity of the payoffs and the transition probabilities (ARAT condition), and continuity in action of the payoff func…
We introduce a modification of Perron's method, where semi-solutions are considered in a carefully defined asymptotic sense. With this definition, we can show, in a rather elementary way, that in a zero-sum game or a control problem (with or without model uncertainty), the value function over all strategies coincides with the value function over Markov strategies discretized in time. Therefore, there are always discretized Markov $\varepsilon$-optimal strategies, (uniform with respect to the bounded initial condition). With a minor modification, the method produces a value and approximate sadd…
We propose and investigate a discrete-time mean field game model involving risk-averse agents. The model under study is a coupled system of dynamic programming equations with a Kolmogorov equation. The agents' risk aversion is modeled by composite risk measures. The existence of a solution to the coupled system is obtained with a fixed point approach. The corresponding feedback control allows to construct an approximate Nash equilibrium for a related dynamic game with finitely many players.
In this paper, we analyze mean-field game modulated by finite states markov chains. We first develop a sufficient stochastic maximum principle for the optimal control of a Markov-modulated stochastic differential equation (SDE) of mean-field type whose coefficients depend on the state of the process, some functional of its law as well as variation of time and sample. As coefficients are perturbed by a Markov chain and thus random, to study such SDEs, we analyze existence and uniqueness of solutions of a class of mean-field type SDEs whose coefficients are random Lipschitz as well as the proper…
We study a two-player, zero-sum, stochastic game with incomplete information on one side in which the players are allowed to play more and more frequently. The informed player observes the realization of a Markov chain on which the payoffs depend, while the non-informed player only observes his opponent's actions. We show the existence of a limit value as the time span between two consecutive stages vanishes; this value is characterized through an auxiliary optimization problem and as the solution of an Hamilton-Jacobi equation.
This article is related to risk-sensitive nonzero-sum stochastic differential games in the Markovian framework. This game takes into account the attitudes of the players toward risk and the utility is of exponential form. We show the existence of a Nash equilibrium point for the game when the drift is no longer bounded and only satisfies a linear growth condition. The main tool is the notion of backward stochastic differential equation, which in our case, is multidimensional with continuous generator involving both a quadratic term of Z and a stochastic linear growth component with respect to …
We use Markov risk measures to formulate a risk-averse version of the undiscounted total cost problem for a transient controlled Markov process. We derive risk-averse dynamic programming equations and we show that a randomized policy may be strictly better than deterministic policies, when risk measures are employed. We illustrate the results on an optimal stopping problem and an organ transplant problem.
We study two person nonzero-sum stochastic differential games with risk-sensitive discounted and ergodic cost criteria. Under certain conditions we establish a Nash equilibrium in Markov strategies for the discounted cost criterion and a Nash equilibrium in stationary strategies for the ergodic cost criterion. We achieve our results by studying the relevant systems of coupled HJB equations.
We consider zero-sum repeated games with incomplete information on both sides, where the states privately observed by each player follow independent Markov chains. It generalizes the model, introduced by Aumann and Maschler in the sixties and solved by Mertens and Zamir in the seventies, where the private states of the players were fixed. It also includes the model introduced in Renault \cite{R2006}, of Markov chain repeated games with lack of information on one side, where only one player privately observes the sequence of states. We prove here that the limit value exists, and we obtain a cha…
We study nonzero-sum stochastic games for continuous time Markov chains on a denumerable state space with risk sensitive discounted and ergodic cost criteria. For the discounted cost criterion we first show that the corresponding system of coupled HJB equations has an appropriate solution. Then under an additional additive structure on the transition rate matrix and payoff functions, we establish the existence of a Nash equilibrium in Markov strategies. For the ergodic cost criterion we assume a Lyapunov type stability assumption and a small cost condition. Under these assumptions we show that…
We develop an approach to time-consistent risk evaluation of continuous-time processes in Markov systems. Our analysis is based on dual representation of coherent risk measures, differentiability concepts for multivalued mappings, and a refined concept of time consistency. We prove that the risk measures are defined by a family of risk evaluation functionals (transition risk mappings), which depend on state, time, and the transition function. Their dual representations are risk multikernels of the Markov system. We introduce the concept of a semi-derivative of a risk multikernel and use it to …
We consider zero-sum stochastic games for continuous time Markov decision processes with risk-sensitive average cost criterion. Here the transition and 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 the nonlinear version of Krein-Rutman theorem. We then obtain a characterization of th…
The infinite horizon risk-sensitive discounted-cost and ergodic-cost nonzero-sum stochastic games for controlled Markov chains with countably many states are analyzed. For the discounted-cost game, we prove the existence of Nash equilibrium strategies in the class of Markov strategies under fairly general conditions. Under an additional geometric ergodicity condition and a small cost criterion, the existence of Nash equilibrium strategies in the class of stationary Markov strategies is proved for the ergodic-cost game.
We study nonzero-sum stochastic differential games with risk-sensitive ergodic cost criterion. Under certain conditions, using multi-parameter eigenvalue approach, we establish the existence of a Nash equilibrium in the space of stationary Markov strategies. We achieve our results by studying the relevant systems of coupled HJB equations. Exploiting the stochastic representation of the principal eigenfunctions we completely characterize Nash equilibrium points in the space of stationary Markov strategies.
In this paper, we consider risk-sensitive discounted control problem for continuous-time jump Markov processes taking values in general state space. The transition rates of underlying continuous-time jump Markov processes and the cost rates are allowed to be unbounded. Under certain Lyapunov condition, we establish the existence and uniqueness of the solution to the Hamilton-Jacobi-Bellman (HJB) equation. Also we prove the existence of optimal risk-sensitive control in the class of Markov control.
In this paper we study a class of risk-sensitive Markovian control problems in discrete time subject to model uncertainty. We consider a risk-sensitive discounted cost criterion with finite time horizon. The used methodology is the one of adaptive robust control combined with machine learning.
In this paper we consider two-person zero-sum risk-sensitive stochastic dynamic games with Borel state and action spaces and bounded reward. The term risk-sensitive refers to the fact that instead of the usual risk neutral optimization criterion we consider the exponential certainty equivalent. The discounted reward case on a finite and an infinite time horizon is considered, as well as the ergodic reward case. Under continuity and compactness conditions we prove that the value of the game exists and solves the Shapley equation and we show the existence of optimal (non-stationary) strategies. …
In this article we consider risk-sensitive control of semi-Markov processes with a discrete state space. We consider general utility functions and discounted cost in the optimization criteria. We consider random finite horizon and infinite horizon problems. Using a state augmentation technique we characterise the value functions and also prescribe optimal controls.
We study a two-player zero-sum stochastic differential game with asymmetric information where the payoff depends on a controlled continuous-time Markov chain X with finite state space which is only observed by player 1. This model was already studied in Cardaliaguet et al. (2015) through an approximating sequence of discrete-time games. Our first contribution is the proof of the existence of the value in the continuous-time model based on duality techniques. This value is shown to be the unique solution of the same Hamilton-Jacobi equation with convexity constraints which characterized the lim…
In this paper, we consider a large class of constrained non-cooperative stochastic Markov games with countable state spaces and discounted cost criteria. In one-player case, i.e., constrained discounted Markov decision models, it is possible to formulate a static optimisation problem whose solution determines a stationary optimal strategy (alias control or policy) in the dynamical infinite horizon model. This solution lies in the compact convex set of all occupation measures induced by strategies, defined on the set of state-action pairs. In case of n-person discounted games the occupation mea…
To analyze strategic interactions arising in the cyber-security context, we develop a new reputation game model in which an attacker can pretend to be a normal user and a defender may have to announce attack detection at a certain point of time without knowing whether he has been attacked. We show the existence and uniqueness of sequential equilibrium in Markov strategies, and explicitly characterize the players' equilibrium strategies. Using our model, we suggest empirical and theoretical ways of calibrating the attack probability, which is an important element of cyber-risks.
Recently, Sidford, Wang, Wu and Ye (2018) developed an algorithm combining variance reduction techniques with value iteration to solve discounted Markov decision processes. This algorithm has a sublinear complexity when the discount factor is fixed. Here, we extend this approach to mean-payoff problems, including both Markov decision processes and perfect information zero-sum stochastic games. We obtain sublinear complexity bounds, assuming there is a distinguished state which is accessible from all initial states and for all policies. Our method is based on a reduction from the mean payoff pr…
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 arbitrary positive real numbers. 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.
The existence of stationary Markov perfect equilibria in stochastic games is shown under a general condition called "(decomposable) coarser transition kernels". This result covers various earlier existence results on correlated equilibria, noisy stochastic games, stochastic games with finite actions and state-independent transitions, and stochastic games with mixtures of constant transition kernels as special cases. A remarkably simple proof is provided via establishing a new connection between stochastic games and conditional expectations of correspondences. New applications of stochastic gam…
This work establishes sufficient conditions for existence of saddle points in discrete Markov games. The result reveals the relation between dynamic games and static games using dynamic programming equations. This result enables us to prove existence of saddle points of non-separable stochastic differential games of regime-switching diffusions under appropriate conditions.
We consider mean field games with ergodic cost in the framework of a general discrete time controlled Markov processes. The state space of the processes is given by a general $σ$-compact Polish space. Under certain conditions, we show the existence of a mean field game equilibrium. We also study the $N$-person game where the players interacts with each other via their empirical measure. We show that the $N$-person game has Nash equilibrium and as $N$ tends to infinity the equilibria converge to a mean field game solution.
This paper is related to nonzero-sum stochastic differential games in the Markovian framework. We show existence of a Nash equilibrium point for the game when the drift is no longer bounded and only satisfies a linear growth condition. The main tool is the notion of backward stochastic differential equations which, in our case, are multidimensional with continuous coefficient and stochastic linear growth.
Semi-Markov model is one of the most general models for stochastic dynamic systems. This paper deals with a two-person zero-sum game for semi-Markov processes. We focus on the expected discounted payoff criterion with state-action-dependent discount factors. The state and action spaces are both Polish spaces, and the payoff function is $ω$-bounded. We first construct a fairly general model of semi-Markov games under a given semi-Markov kernel and a pair of strategies. Next, based on the standard regularity condition and the continuity-compactness condition for semi-Markov games, we derive a "d…
We establish existence of controlled Markov chain of mean-field type with unbounded jump intensities by means of a fixed point argument using the Wasserstein distance. Using a Markov chain entropic backward SDE approach, we further suggest conditions for existence of an optimal control and a saddle-point for respectively a control problem and a zero-sum differential game associated with risk sensitive payoff functionals of mean-field type.
Markov decision process (MDP) congestion game is an extension of classic congestion games, where a continuous population of selfish agents solves Markov decision processes with congestion: the payoff of a strategy decreases as more population uses it. We draw parallels between key concepts from capacitated congestion games and MDP. In particular, we show that population mass constraints in MDP congestion games are equivalent to imposing tolls/incentives on the reward function, which can be utilized by social planners to achieve auxiliary objectives. We demonstrate such methods in a simulated S…
The paper solves constrained Dynkin games with risk-sensitive criteria, where two players are allowed to stop at two independent Poisson random intervention times, via the theory of backward stochastic differential equations. This generalizes the previous work of [Liang and Sun, Dynkin games with Poisson random intervention times, SIAM Journal on Control and Optimization, 2019] from the risk-neutral criteria and common signal times for both players to the risk-sensitive criteria and two heterogenous signal times. Furthermore, the paper establishes a connection of such constrained risk-sensitiv…
A zero-sum two-person Perfect Information Semi-Markov game (PISMG) under limiting ratio average payoff has a value and both the maximiser and the minimiser have optimal pure semi-stationary strategies. We arrive at the result by first fixing an arbitrary initial state and forming the matrix of undiscounted payoffs corresponding to each pair of pure stationary strategies of the two players and proving that this matrix has a pure saddle point.
Mean field games are studied in the framework of controlled martingale problems, and general existence theorems are proven in which the equilibrium control is Markovian. The framework is flexible enough to include degenerate volatility, which may depend on both the control and the mean field. The objectives need not be strictly convex, and the mean field interactions considered are nonlocal and Wasserstein-continuous. When the volatility is nondegenerate, continuity assumptions may be weakened considerably. The proofs first use relaxed controls to establish existence. Then, using a convexity a…
Multi-agent reinforcement learning (MARL) is often modeled using the framework of Markov games (also called stochastic games or dynamic games). Most of the existing literature on MARL concentrates on zero-sum Markov games but is not applicable to general-sum Markov games. It is known that the best-response dynamics in general-sum Markov games are not a contraction. Therefore, different equilibria in general-sum Markov games can have different values. Moreover, the Q-function is not sufficient to completely characterize the equilibrium. Given these challenges, model based learning is an attract…
The paper is concerned with the deterministic limit of mean field games with the nonlocal coupling. It is assumed that the dynamics of mean field games are given by nonlinear Markov processes. This type of games includes stochastic mean field games as well as mean field games with finite state space. We consider the limiting deterministic mean field game within the framework of minimax approach. The concept of minimax solutions is close to the probabilistic formulation. In this case the Hamilton--Jacobi equation is considered in the minimax/viscosity sense, whereas the flow of probabilities is…
The paper is concerned with a variant of the continuous-time finite state Markov game of control and stopping where both players can affect transition rates, while only one player can choose a stopping time. We use the dynamic programming principle and reduce this problem to a system of ODEs with unilateral constraints. This system plays the role of the Bellman equation. We show that its solution provides the optimal strategies of the players. Additionally, we prove the existence and uniqueness theorem for the deduced system of ODEs with unilateral constraints.
This paper presents a theoretical framework for probably approximately correct (PAC) multi-agent reinforcement learning (MARL) algorithms for Markov games. The paper offers an extension to the well-known Nash Q-learning algorithm, using the idea of delayed Q-learning, in order to build a new PAC MARL algorithm for general-sum Markov games. In addition to guiding the design of a provably PAC MARL algorithm, the framework enables checking whether an arbitrary MARL algorithm is PAC. Comparative numerical results demonstrate performance and robustness.
