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.
Saddle point with a given Morse index on a potential energy surface is an important object related to energy landscape in physics and chemistry. Efficient numerical methods based on iterative minimization formulation have been proposed in the forms of the sequence of minimization subproblems or the continuous dynamics. We here present a differential game interpretation of this formulation and theoretically investigate the Nash equilibrium of the proposed game and the original saddle point on potential energy surface. To define this differential game, a new proximal function growing faster than…
The purpose of this paper is to study 2-person zero-sum stochastic differential games, in which one player is a major one and the other player is a group of $N$ minor agents which are collectively playing, statistically identical and have the same cost-functional. The game is studied in a weak formulation; this means in particular, we can study it as a game of the type "feedback control against feedback control". The payoff/cost functional is defined through a controlled backward stochastic differential equation, for which driving coefficient is assumed to satisfy strict concavity-convexity wi…
In this paper, we consider a linear quadratic stochastic two-person zero-sum differential game. The controls for both players are allowed to appear in both drift and diffusion of the state equation. The weighting matrices in the performance functional are not assumed to be definite/non-singular. A necessary and sufficient condition for the existence of a closed-loop saddle point is established in terms of the solvability of a Riccati differential equation with certain regularity. It is possible that the closed-loop saddle point fails to exist, and at the same time, the corresponding Riccati eq…
The paper is concerned with two-person games with saddle point. We investigate the limits of value functions for long-time-average payoff, discounted average payoff, and the payoff that follows a probability density. Most of our assumptions restrict the dynamics of games. In particular, we assume the closedness of strategies under concatenation. It is also necessary for the value function to satisfy Bellman's optimality principle, even if in a weakened, asymptotic sense. We provide two results. The first one is a uniform Tauber result for games: if the value functions for long-time-average pay…
We consider the problem of computing optimal policies in average-reward Markov decision processes. This classical problem can be formulated as a linear program directly amenable to saddle-point optimization methods, albeit with a number of variables that is linear in the number of states. To address this issue, recent work has considered a linearly relaxed version of the resulting saddle-point problem. Our work aims at achieving a better understanding of this relaxed optimization problem by characterizing the conditions necessary for convergence to the optimal policy, and designing an optimiza…
The paper studies the open-loop saddle point and the open-loop lower and upper values, as well as their relationship for two-person zero-sum stochastic linear-quadratic (LQ, for short) differential games with deterministic coefficients. It derives a necessary condition for the finiteness of the open-loop lower and upper values and a sufficient condition for the existence of an open-loop saddle point. It turns out that under the sufficient condition, a strongly regular solution to the associated Riccati equation uniquely exists, in terms of which a closed-loop representation is further establis…
The paper is concerned with two-person zero-sum mean-field linear-quadratic stochastic differential games over finite horizons. By a Hilbert space method, a necessary condition and a sufficient condition are derived for the existence of an open-loop saddle point. It is shown that under the sufficient condition, the associated two Riccati equations admit unique strongly regular solutions, in terms of which the open-loop saddle point can be represented as a linear feedback of the current state. When the game only satisfies the necessary condition, an approximate sequence is constructed by solvin…
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.
This paper is concerned with a linear quadratic stochastic two-person zero-sum differential game with constant coefficients in an infinite time horizon. Open-loop and closed-loop saddle points are introduced. The existence of closed-loop saddle points is characterized by the solvability of an algebraic Riccati equation with a certain stabilizing condition. A crucial result makes our approach work is the unique solvability of a class of linear backward stochastic differential equations in an infinite horizon.
In this paper, we consider a linear quadratic stochastic two-person nonzero-sum differential game. Open-loop and closed-loop Nash equilibria are introduced. The existence of the former is characterized by the solvability of a system of forward-backward stochastic differential equations, and that of the latter is characterized by the solvability of a system of coupled symmetric Riccati differential equations. Sometimes, open-loop Nash equilibria admit a closed-loop representation, via the solution to a system of non-symmetric Riccati equations, which is different from the outcome of the closed-…
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 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.
We establish existence of nearly-optimal controls, conditions for existence of an optimal control and a saddle-point for respectively a control problem and zero-sum differential game associated with payoff functionals of mean-field type, under dynamics driven by weak solutions of stochastic differential equations of mean-field type.
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.
Many problems in machine learning and game theory can be formulated as saddle-point problems, for which various first-order methods have been developed and proven efficient in practice. Under the general convex-concave assumption, most first-order methods only guarantee an ergodic convergence rate, that is, the uniform averages of the iterates converge at a $O(1/T)$ rate in terms of the saddle-point residual. However, numerically, the iterates themselves can often converge much faster than the uniform averages. This observation motivates increasing averaging schemes that put more weight on lat…
Zero sum games with risk-sensitive cost criterion are considered with underlying dynamics being given by controlled stochastic differential equations. Under the assumption of geometric stability on the dynamics , we completely characterize all possible saddle point strategies in the class of stationary Markov controls. In addition, we also establish existence-uniqueness result for the value function of the Hamilton-Jacobi-Isaacs equation.
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…
We consider a zero-sum stochastic game for continuous-time Markov chain with countable state space and unbounded transition and pay-off rates. The additional feature of the game is that the controllers together with taking actions are also allowed to stop the process. Under suitable hypothesis we show that the game has a value and it is the unique solution of certain dynamic programming inequalities with bilateral constraints. In the process we also prescribe a saddle point equilibrium.
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 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 work considers two-player zero-sum semi-Markov games with incomplete information on one side and perfect observation. At the beginning, the system selects a game type according to a given probability distribution and informs to Player 1 only. After each stage, the actions chosen are observed by both players before proceeding to the next stage. Firstly, we show the existence of the value function under the expected discount criterion and the optimality equation. Secondly, the existence and iterative algorithm of the optimal policy for Player 1 are introduced through the optimality equation…
We provide a simple and generic adaptive restart scheme for convex optimization that is able to achieve worst-case bounds matching (up to constant multiplicative factors) optimal restart schemes that require knowledge of problem specific constants. The scheme triggers restarts whenever there is sufficient reduction of a distance-based potential function. This potential function is always computable. We apply the scheme to obtain the first adaptive restart algorithm for saddle-point algorithms including primal-dual hybrid gradient (PDHG) and extragradient. The method improves the worst-case bou…
Brookhaven National Laboratory（布鲁克海文国家实验室）；Rutgers University（罗格斯大学）
We present a new, stochastic variant of the projective splitting (PS) family of algorithms for monotone inclusion problems. It can solve min-max and noncooperative game formulations arising in applications such as robust ML without the convergence issues associated with gradient descent-ascent, the current de facto standard approach in such situations. Our proposal is the first version of PS able to use stochastic (as opposed to deterministic) gradient oracles. It is also the first stochastic method that can solve min-max games while easily handling multiple constraints and nonsmooth regulariz…
Bilevel Optimization over Saddle Points of Zero-Sum Markov Games
Shun Hing Institute of Advanced Engineering, The Chinese University of Hong Kong（香港中文大学先进工程学院）；Department of Computer Science and Engineering, The Chinese University of Hong Kong（香港中文大学计算机科学与工程系）
We establish existence of Markov chains of mean-field type with unbounded jump intensities by means of a fixed point argument using the Total Variation distance. We further show existence of nearly-optimal controls and, using a Markov chain backward SDE approach, we 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 payoff functionals of mean-field type, under dynamics driven by such Markov chains of mean-field type.
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.
This paper studies the mixed zero-sum stochastic differential game problem. We allow the functionals and dynamics to be of polynomial growth. The problem is formulated as an extended doubly reflected BSDEs with a specific generator. We show the existence of solution for this doubly reflected BSDEs and we prove the existence of a saddle-point of the game. Moreover, in the Markovian framework we prove that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation.
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.
A conflict between rational and autonomous agents is considered. The paper addresses a differential game of protecting a target in the 3-D space. This problem highlights the strong correlation between the highly dynamic scenario, the uncertainty on the behavior of the adversary, and the online and robust computation of state-feedback strategies which guarantee the required level of performance of each player. This work significantly expands previous results around this problem by providing the players' state-feedback saddle-point strategies. Additionally, the continuously differentiable Value …
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.
School of Computer Science（计算机科学学院）；University of Waterloo（滑铁卢大学）；Vector Institute（向量研究所）
Convergence to a saddle point for convex-concave functions has been studied for decades, while recent years has seen a surge of interest in non-convex (zero-sum) smooth games, motivated by their recent wide applications. It remains an intriguing research challenge how local optimal points are defined and which algorithm can converge to such points. An interesting concept is known as the local minimax point, which strongly correlates with the widely-known gradient descent ascent algorithm. This paper aims to provide a comprehensive analysis of local minimax points, such as their relation with o…
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…
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.
Saddle-point problems appear in various settings including machine learning, zero-sum stochastic games, and regression problems. We consider decomposable saddle-point problems and study an extension of the alternating direction method of multipliers to such saddle-point problems. Instead of solving the original saddle-point problem directly, this algorithm solves smaller saddle-point problems by exploiting the decomposable structure. We show the convergence of this algorithm for convex-concave saddle-point problems under a mild assumption. We also provide a sufficient condition for which the a…
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…
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…
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 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…
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 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…
