Pierre Cardaliaguet, Catherine Rainer
We investigate a two-player zero-sum stochastic differential game in which the players have an asymmetric information on the random payoff. We prove that the game has a value and characterize this value in terms of dual solutions of some second order Hamilton-Jacobi equation.
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We introduce a new notion of pathwise strategies for stochastic differential games. This allows us to give a correct meaning to some statement asserted in [Cardaliaguet-Rainer 2009].
Linear-Quadratic Non-zero Sum Differential Game with Asymmetric Delayed Information
本文研究具有非对称延迟信息的线性二次非零和微分博弈，通过随机最大值原理和离散化方法，建立了状态估计反馈纳什均衡。
This paper focuses on a kind of linear quadratic non-zero sum differential game driven by backward stochastic differential equation with asymmetric information, which is a natural continuation of Wang and Yu [IEEE TAC (2010) 55: 1742-1747, Automatica (2012) 48: 342-352]. Different from Wang and Yu [IEEE TAC (2010) 55: 1742-1747, Automatica (2012) 48: 342-352], novel motivations for studying this kind of game are provided. Some feedback Nash equilibrium points are uniquely obtained by forward-backward stochastic differential equations, their filters and the corresponding Riccati equations with …
This paper is concerned with a leader-follower stochastic differential game with asymmetric information, where the information available to the follower is based on some sub-$σ$-algebra of that available to the leader. Such kind of game problem has wide applications in finance, economics and management engineering such as newsvendor problems, cooperative advertising and pricing problems. Stochastic maximum principles and verification theorems with partial information are obtained, to represent the Stackelberg equilibrium. As applications, a linear-quadratic leader-follower stochastic different…
Differential games with asymmetric information were introduced by Cardaliaguet (2007). As in repeated games with lack of information on both sides (Aumann and Maschler (1995)), each player receives a private signal (his type) before the game starts and has a prior belief about his opponent's type. Then, a differential game is played in which the dynamic and the payoff function depend on both types: each player is thus partially informed about the differential game that is played. The existence of the value function and some characterizations have been obtained under the assumption that the sig…
We investigate a two-player zero-sum differential game with asymmetric information on the payoff and without Isaacs condition. The dynamics is an ordinary differential equation parametrised by two controls chosen by the players. Each player has a private information on the payoff of the game, while his opponent knows only the probability distribution on the information of the other player. We show that a suitable definition of random strategies allows to prove the existence of a value in mixed strategies. Moreover, the value function can be characterised in term of the unique viscosity solutio…
This paper is concerned with a Stackelberg stochastic differential game with asymmetric noisy observation, with one follower and one leader. In our model, the follower cannot observe the state process directly, but could observe a noisy observation process, while the leader can completely observe the state process. Open-loop Stackelberg equilibrium is considered. The follower first solve an stochastic optimal control problem with partial observation, the maximum principle and verification theorem are obtained. Then the leader turns to solve an optimal control problem for a conditional mean-fie…
This paper is concerned with a three-level stochastic linear-quadratic Stackelberg differential game with asymmetric information, in which three players participate credited as Player 1, Player 2 and Player 3. Player 3 acts as the leader of Player 2 and Player 1, Player 2 acts as the leader of Player 1 and Player 1 acts as the follower. The asymmetric information considered is: the information available to Player 1 is based on the sub-$σ$-algebra of the information available to Player 2, and the information available to Player 2 is based on the sub-$σ$-algebra of the information available to P…
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…
Stochastic Linear-Quadratic Stackelberg Differential Game with Asymmetric Informational Uncertainties: Robust Optimization Approach
本文研究具有非对称信息不确定性的随机线性二次Stackelberg微分博弈，采用鲁棒优化方法，通过求解跟随者的min-max问题和领导者的max-min问题，并利用Riccati方程给出鲁棒Stackelberg均衡的显式状态反馈表示。
Asymmetric Feedback Learning in Online Convex Games
本文放宽智能体对称性假设，针对部分智能体获一阶梯度信息、部分获零阶预言机的非对称场景，提出结合两类信息机制的学习算法，分析其遗憾值与纳什均衡收敛性，数值实验验证了算法性能。
Noncooperative Game in Multi-controller System under Delayed and Asymmetric Information
本文研究了在存在延迟和不对称信息的多控制器系统中非合作博弈问题，提出了一种公共-私人信息分解方法，通过解耦控制输入和状态估计，得到闭式纳什均衡，并证明信息集丰富度对玩家成本的影响。
We consider a stochastic game with partial, asymmetric and non-classical information, where the agents are trying to acquire as many available opportunities/locks as possible. Agents have access only to local information, the information updates are asynchronous and our aim is to obtain relevant equilibrium policies. Our approach is to consider optimal open-loop control until the information update, which allows managing the belief updates in a structured manner. The agents continuously control the rates of their Poisson search clocks to acquire the locks, and they get rewards at every success…
Asymmetric Perturbation in Solving Bilinear Saddle-Point Optimization
提出非对称扰动方法，仅扰动一个玩家的收益函数，以线性收敛速率求解双线性鞍点优化问题，并提供了无参数变体。
Best Response Dynamics for Zero-Sum Dynamic Games with Partial-Asymmetric Information
针对部分不对称信息下的零和随机线性二次动态博弈，本文提出基于最佳响应动力学的方法，推导纯线性动态输出反馈控制策略的最佳响应显式表达式，实验表明博弈值迭代几次后收敛，还分析了相关因素对线性二次追逃博弈值的影响。
Forward-Backward Dynamic Programming for LQG Dynamic Games with Partial and Asymmetric Information
本文研究了具有部分和不对称信息的两人零和随机动态博弈，提出了一种新的向前-向后算法，用于计算有限时间 horizon 问题中的信念状态和均衡策略，以及平均成本无限时间 horizon 问题中的平稳信念状态和均衡策略。
We study a discrete-time finite-horizon two-players nonzero-sum stopping game where the filtration of Player 1 is richer than the filtration of Player 2. A major difficulty which is caused by the information asymmetry is that Player 2 may not know whether Player 1 has already stopped the game or not. Furthermore, the classical backward-induction approach is not applicable in the current setup. This is because when the informed player decides not to stop, he reveals information to the uninformed player and hence the decision of the uninformed player at time $t$ may not be determined by the play…
We establish, for the first time, a connection between stochastic games and multiparameter eigenvalue problems, using the theory developed by Shapley and Snow (1950). This connection provides new results, new proofs, and new tools for studying stochastic games.
In this paper, we consider a partial information two-person zero-sum stochastic differential game problem where the system is governed by a backward stochastic differential equation driven by Teugels martingales associated with a Lévy process and an independent Brownian motion. One sufficient (a verification theorem) and one necessary conditions for the existence of optimal controls are proved. To illustrate the general results, a linear quadratic stochastic differential game problem is discussed.
We consider discrete time partially observable zero-sum stochastic game with average payoff criterion. We study the game using an equivalent completely observable game. We show that the game has a value and also we come up with a pair of optimal strategies for both the players.
Constant Payoff Property in Zero-Sum Stochastic Games with a Finite Horizon
本文研究有限零和随机博弈，证明当博弈时长足够长时存在近似最优策略使任意时刻期望平均收益接近其值，首次在该类博弈中确立此前仅适用于吸收博弈和折扣随机博弈的常收益性质。
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…
In this paper, we consider constrained discounted stochastic games with a countably generated state space and norm continuous transition probability having a density function. We prove existence of approximate stationary equilibria and stationary weak correlated equilibria. Our results imply the existence of stationary Nash equilibrium in $ARAT$ stochastic games.
We provide a direct, elementary proof for the existence of $\lim_{λ\to 0} v_λ$, where $v_λ$ is the value of a $λ$-discounted finite two-person zero-sum stochastic game.
Zero-Sum State-Blind Stochastic Games with Vanishing Stage Duration
本文研究了阶段时长趋于0时状态盲随机博弈的值的行为，证明其值收敛于某偏微分方程的唯一粘性解。
In this paper, we consider a zero-sum undiscounted stochastic game which has finite state space and finitely many pure actions. Also, we assume the transition probability of the undiscounted stochastic game is controlled by one player and all the optimal strategies of the game are strictly positive. Under all the above assumptions, we show that the $β$-discounted stochastic games with the same payoff matrices and $β$ sufficiently close to 1 are also completely mixed. We give a counterexample to show that the converse of the above result in not true. We also show that, if we have non-zero value…
We study a zero-sum stochastic differential switching game in infinite horizon. We prove the existence of the value of the game and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities with bilateral obstacles. We also obtain a verification theorem which provides an optimal strategy of the game. Finally, some numerical examples with two regimes are given.
We develop an approach for two player constraint zero-sum and nonzero-sum stochastic differential games, which are modeled by Markov regime-switching jump-diffusion processes. We provide the relations between a usual stochastic optimal control setting and a Lagrangian method. In this context, we prove corresponding theorems for two different type of constraints, which lead us to find real valued and stochastic Lagrange multipliers, respectively. Then, we illustrate our results for a nonzero-sum game problem with stochastic maximum principle technique. Our application is an example of cooperati…
We study a model of two-player, zero-sum, stopping games with asymmetric information. We assume that the payoff depends on two continuous-time Markov chains (X, Y), where X is only observed by player 1 and Y only by player 2, implying that the players have access to stopping times with respect to different filtrations. We show the existence of a value in mixed stopping times and provide a variational characterization for the value as a function of the initial distribution of the Markov chains. We also prove a verification theorem for optimal stopping rules which allows to construct optimal sto…
We study stochastic differential games of jump diffusions, where the players have access to inside information. Our approach is based on anticipative stochastic calculus, white noise, Hida-Malliavin calculus, forward integrals and the Donsker delta functional. We obtain a characterization of Nash equilibria of such games in terms of the corresponding Hamiltonians. This is used to study applications to insider games in finance, specifically optimal insider consumption and optimal insider portfolio under model uncertainty.
In this paper, an open-loop two-person non-zero sum stochastic differential game is considered for forward-backward stochastic systems. More precisely, the controlled systems are described by a fully coupled nonlinear multi- dimensional forward-backward stochastic differential equation driven by a multi-dimensional Brownian motion. one sufficient (a verification theorem) and one necessary conditions for the existence of open-loop Nash equilibrium points for the corresponding two-person non-zero sum stochastic differential game are proved. The control domain need to be convex and the admissible…
Definable zero-sum stochastic games involve a finite number of states and action sets, reward and transition functions that are definable in an o-minimal structure. Prominent examples of such games are finite, semi-algebraic or globally subanalytic stochastic games. We prove that the Shapley operator of any definable stochastic game with separable transition and reward functions is definable in the same structure. Definability in the same structure does not hold systematically: we provide a counterexample of a stochastic game with semi-algebraic data yielding a non semi-algebraic but globally …
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 consider a finite state, finite action, zero-sum stochastic games with data defining the game lying in the ordered field of algebraic numbers. In both the discounted and the limiting average versions of these games we prove that the value vector also lies in the same field of algebraic numbers. In a prescribed sense, our results settle a problem that has remained open since, at least, 1991.
This paper investigates the game theory of resource-allocation situations where the "first come, first serve" heuristic creates inequitable, asymmetric benefits to the players. Specifically, this problem is formulated as a Generalized Nash Equilibrium Model where the players are arranged sequentially along a directed line graph. The goal of the model is to reduce the asymmetric benefits among the players using a policy instrument. It serves as a more realistic, alternative approach to the line-graph models considered in the cooperative game-theoretic literature. An application-oriented formula…
We study two classes of zero-sum stochastic games with compact action sets and a finite product state space. These two classes assume a communication property on the state spaces of the players. For strongly communicating on one side games, we prove the existence of the uniform value. For weakly communicating on both sides games, we prove that the asymptotic value, and therefore the uniform value, may fail to exist.
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 …
We consider zero sum stochastic games. For every discount factor $λ$, a time normalization allows to represent the game as being played on the interval [0, 1]. We introduce the trajectories of cumulated expected payoff and of cumulated occupation measure up to time t $\in$ [0, 1], under $ε$-optimal strategies. A limit optimal trajectory is defined as an accumulation point as the discount factor tends to 0. We study existence, uniqueness and characterization of these limit optimal trajectories for absorbing games.
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.
Asymptotic Value in Zero-Sum Stochastic Games with Vanishing Stage Duration and Public Signals
研究了在阶段持续时间消失和公共信号条件下，零和随机博弈中折扣因子和阶段持续时间趋近于零时渐近值的存在性。
Linear-quadratic-singular stochastic differential games and applications
本文研究具有奇异控制的非合作N人非零和随机微分博弈，在自然假设下通过利普希茨连续策略逼近证明了开环纳什均衡的存在性，并将其应用于寡头垄断市场的产能扩张模型。
This paper obtains the maximum principle for both stochastic (global) open-loop and stochastic (global) closed-loop Stackelberg differential games. For the closed-loop case, we use the theory of controlled forward-backward stochastic differential equations to derive the maximum principle for the leader's optimal strategy. In the special case of the open-loop linear quadratic Stackelberg game, we consider the follower's Hamiltonian system as the leader's state equation, derive the related stochastic Riccati equation, and show the existence and uniqueness of the solution to the Riccati equation …
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 Stackelberg stochastic differential game on a finite horizon in feedback information pattern. A system of parabolic partial differential equations is obtained at the level of Hamiltonian to give the verification theorem of the feedback Stackelberg equilibrium. As an example, a linear quadratic Stackelberg stochastic differential game is investigated. Riccati equations are introduced to express the feedback Stackelberg equilibrium, analytical and numerical solutions to these Riccati equations are discussed in some special cases.
We consider a class of two-player zero-sum stochastic games with finite state and compact control spaces, which we call stochastic shortest path (SSP) games. They are undiscounted total cost stochastic dynamic games that have a cost-free termination state. Exploiting the close connection of these games to single-player SSP problems, we introduce novel model conditions under which we show that the SSP games have strong optimality properties, including the existence of a unique solution to the dynamic programming equation, the existence of optimal stationary policies, and the convergence of valu…
Martingale theory for Dynkin games with asymmetric information
本文提出非马尔可夫框架下非对称信息零和Dynkin博弈中随机停止时间对构成鞍点的充要条件，通过超/子鞅识别与Doob-Meyer分解刻画均衡策略。
Common Information based Markov Perfect Equilibria for Linear-Gaussian Games with Asymmetric Information
本文研究了具有不对称信息的线性高斯博弈，提出基于公共信息的马尔可夫完美均衡概念，通过求解线性方程组计算均衡，并讨论了二次成本函数下的唯一性及与其他纳什均衡的区别。
We prove that zero-sum Dynkin games in continuous time with partial and asymmetric information admit a value in randomised stopping times when the stopping payoffs of the players are general \cadlag measurable processes. As a by-product of our method of proof we also obtain existence of optimal strategies for both players. The main novelties are that we do not assume a Markovian nature of the game nor a particular structure of the information available to the players. This allows us to go beyond the variational methods (based on PDEs) developed in the literature on Dynkin games in continuous t…
The General Lotto game is a popular variant of the famous Colonel Blotto game, in which two opposing players allocate limited resources over many battlefields. In this paper, we consider incomplete and asymmetric information formulations regarding the resource budgets of the players. In particular, one of the player's resource budget is common knowledge while the other player's is private. We provide complete equilibrium characterizations in the scenario where the private resource budget is drawn from an arbitrary Bernoulli distribution. We then show that these characterizations can be used to…
We study a general class of dynamic games with asymmetric information where agents' beliefs are strategy dependent, i.e. signaling occurs. We show that the notion of sufficient information, introduced in the companion paper team, can be used to effectively compress the agents' information in a mutually consistent manner that is sufficient for decision-making purposes. We present instances of dynamic games with asymmetric information where we can characterize a time-invariant information state for each agent. Based on the notion of sufficient information, we define a class of equilibria for dyn…
We consider a partially asymmetric multi-players zero-sum game with two strategic variables. All but one players have the same payoff functions, and one player (Player $n$) does not. Two strategic variables are $t_i$'s and $s_i$'s for each player $i$. Mainly we will show the following results. 1) The equilibrium when all players choose $t_i$'s is equivalent to the equilibrium when all but one players choose $t_i$'s and Player $n$ chooses $s_n$ as their strategic variables. 2) The equilibrium when all players choose $s_i$'s is equivalent to the equilibrium when all but one players choose $s_i$'…
We study the value and the optimal strategies for a two-player zero-sum optimal stopping game with incomplete and asymmetric information. In our Bayesian set-up, the drift of the underlying diffusion process is unknown to one player (incomplete information feature), but known to the other one (asymmetric information feature). We formulate the problem and reduce it to a fully Markovian setup where the uninformed player optimises over stopping times and the informed one uses randomised stopping times in order to hide their informational advantage. Then we provide a general verification result wh…
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…
Best Response Strategies for Asymmetric Sensing in Linear-Quadratic Differential Games
本文研究线性二次微分博弈中一方仅能间歇采样状态的非对称感知问题，推导出劣势方在平稳随机化策略类中的最优传感器策略，并通过仿真验证期望代价随感知限制放宽而趋近安全水平。
We consider both discrete and continuous time finite state-action stochastic games. In discrete time stochastic games, it is known that a stationary Blackwell-Nash equilibrium (BNE) exists for a single controller additive reward (SC-AR) stochastic game which is a special case of a general stochastic game. We show that, in general, the additive reward condition is needed for the existence of a BNE. We give an example of a single controller stochastic game which does not satisfy additive reward condition. We show that this example does not have a stationary BNE. For a general discrete time disco…
In this Note, assuming that the generator is uniform Lipschitz in the unknown variables, we relate the solution of a one dimensional backward stochastic differential equation with the value process of a stochastic differential game. Under a domination condition, a filtration-consistent evaluations is also related to a stochastic differential game. This relation comes out of a min-max representation for uniform Lipschitz functions as affine functions. The extension to reflected backward stochastic differential equations is also included.
We show that every two-player stochastic game with finite state and action sets and bounded, Borel-measurable, and shift-invariant payoffs, admits an $\ep$-equilibrium for all $\varepsilon>0$.
In this paper, we investigate Nash equilibrium payoffs for nonzero-sum stochastic differential games with reflection. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations.
We establish regularity for functions satisfying a dynamic programming equation, which may arise for example from stochastic games or discretization schemes. Our results can also be utilized in obtaining regularity and existence results for the corresponding partial differential equations.
On the Global Convergence of Stochastic Fictitious Play in Stochastic Games with Turn-based Controllers
本文针对带回合式控制器的随机博弈，结合经典随机虚拟博弈与值迭代提出具备几乎必然收敛性的学习动力学，揭示了有限 horizon 长度递增与回合式控制器保证辅助博弈策略等价性的关键作用。
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