Eran Shmaya, Eilon Solan
We prove that every two-player nonzero-sum stopping game in discrete time admits an ε-equilibrium in randomized strategies for every ε>0. We use a stochastic variation of Ramsey's theorem, which enables us to reduce the problem to that of studying properties of ε-equilibria in a simple class of stochastic games with finite state space.
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A multi-player optimal stopping game based on a spider Brownian bridge
研究由蜘蛛布朗桥驱动的多人非零和最优停止博弈，先分析两人情形下纳什均衡存在及对称情形下唯一性的条件并举例说明多均衡情况，后扩展到多人情形并给出相应存在条件。
We prove that every two-player non-zero-sum Dynkin game in continuous time admits an epsilon-equilibrium in randomized stopping times. We provide a condition that ensures the existence of an epsilon-equilibrium in non-randomized stopping times.
Non-zero-sum optimal stopping game with continuous versus periodic exercise opportunities
该研究提出一种非对称行权机会的非零和最优停时博弈，在带正跳的Lévy过程下证明并构造了以尺度函数表示的纳什均衡，还通过数值示例分析策略行为与行权机会价值。
In the nonzero-sum setting, we establish a connection between Nash equilibria in games of optimal stopping (Dynkin games) and generalised Nash equilibrium problems (GNEP). In the Dynkin game this reveals novel equilibria of threshold type and of more complex types, and leads to novel uniqueness and stability results.
This paper analyses two-player nonzero-sum games of optimal stopping on a class of linear regular diffusions with not non-singular boundary behaviour (in the sense of Itô and McKean (1974), p.\ 108). We provide sufficient conditions under which Nash equilibria are realised by each player stopping the diffusion at one of the two boundary points of an interval. The boundaries of this interval solve a system of algebraic equations. We also provide conditions sufficient for the uniqueness of the equilibrium in this class.
A Dynkin Game with Independent Processes and Incomplete Information
针对两人非零和不完全信息停止型Dynkin博弈，分析独立不可观测布朗运动场景下的均衡，证明奖励函数满足增长条件时存在无穷多无穷期望收益纳什均衡，而有限期望收益均衡仅要求至少一方立即停止，结果适用于纯与混合策略。
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…
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.
A forward algorithm for a class of Markov zero-sum stopping games
本文提出了一种高效算法，用于计算有限状态空间中马尔可夫零和停止博弈的价值函数，并讨论了迭代次数及计算实现。
We study two-player zero-sum stopping games in continuous time and infinite horizon. We prove that the value in randomized stopping times exists as soon as the payoff processes are right-continuous. In particular, as opposed to existing literature, we do not assume any conditions on the relations between the payoff processes. We also show that both players have simple epsilon- optimal randomized stopping times; namely, randomized stopping times which are small perturbations of non-randomized stopping times.
A new class of multi-player competitive stochastic games in discrete-time with an affine specification of the redistribution of payoffs at exercise is proposed and examined. Our games cover as a very special case the classic two-person stochastic stopping games introduced by Dynkin (1969). We first extend to the case of a single-period deterministic affine game the results from Guo and Rutkowski (2012,2014) where a particular subclass of competitive stopping games was studied. We identify conditions under which optimal equilibria and value for a multi-player competitive game with affine redist…
In this paper we study the N-player nonzero-sum Dynkin game ($N\geq 3$) in continuous time, which is a non-cooperative game where the strategies are stopping times. We show that the game has a Nash equilibrium point for general payoff processes.
Mean-field games of optimal stopping: master equation and weak equilibria
该研究针对参与者面临最优停时问题的随机博弈，构建平均场问题并引入弱公式化表述，证明了一大类准则下纳什均衡的存在性，同时验证平均场均衡可作为N人博弈的近似纳什均衡，还推导了对应的主方程。
We study the infinite horizon discrete time N-player nonzero-sum Dynkin game ($N \geq 2$) with stopping times as strategies (or pure strategies). We prove existence of an $\varepsilon$-Nash equilibrium point for the game by presenting a constructive algorithm. One of the main features is that the payoffs of the players depend on the set of players that stop at the termination stage which is the minimal stage in which at least one player stops. The existence result is extended to the case of a nonzero-sum game with finite horizon. Finally, the algorithm is illustrated by two explicit examples i…
A Dynkin game is a zero-sum, stochastic stopping game between two players where either player can stop the game at any time for an observable payoff. Typically the payoff process of the max-player is assumed to be smaller than the payoff process of the min-player, while the payoff process for simultaneous stopping is in between the two. In this paper, we study general Dynkin games whose payoff processes are in arbitrary positions. In both discrete and continuous time settings, we provide necessary and sufficient conditions for the existence of pure strategy Nash equilibria and epsilon-optimal …
We consider a stochastic game of control and stopping specified in terms of a process $X_t=-θΛ_t+W_t$, representing the holdings of Player 1, where $W$ is a Brownian motion, $θ$ is a Bernoulli random variable indicating whether Player 2 is active or not, and $Λ$ is a non-decreasing process representing the accumulated "theft" or "fraud" performed by Player 2 (if active) against Player 1. Player 1 cannot observe $θ$ or $Λ$ directly, but can merely observe the path of the process $X$ and may choose a stopping rule $τ$ to deactivate Player 2 at a cost $M$. Player 1 thus does not know if she is th…
We study the optimal stopping problem of maximizing the variance of an unkilled linear diffusion. Especially, we demonstrate how the problem can be solved as a convex two-player zero-sum game, and reveal quite surprising application of game theory by doing so. Our main result shows that an optimal solution can, in general case, be found among stopping times that are mixtures of two hitting times. This and other revealed phenomena together with suggested solution methods could be helpful when facing more complex non-linear optimal stopping problems. The results are illustrated by a few examples…
We consider a general class of nonzero-sum $N$-player stochastic games with impulse controls, where players control the underlying dynamics with discrete interventions. We adopt a verification approach and provide sufficient conditions for the Nash equilibria (NEs) of the game. We then consider the limit situation of $N \to \infty$, that is, a suitable mean-field game (MFG) with impulse controls. We show that under appropriate technical conditions, the existence of unique NE solution to the MFG, which is an $ε$-NE approximation to the $N$-player game, with $ε=O\left(\frac{1}{\sqrt{N}}\right)$.…
The sero-sum stopping game for the stochastic sequences has been formulated in late sixties of the twenty century by Dynkin (1969). The formulation had the assumption about separability of decision moment of the players which simplified the construction of the solution. Further research by Neveu (1975) extended the model by admitting more general behaviour of the players and their pay--offs. In new formulation there is the problem with existence of the equilibrium. The proper approach to solution of the problem without restriction of former models was developed by Yasuda (1975). The results wa…
We study a two-player nonzero-sum stochastic differential game where one player controls the state variable via additive impulses while the other player can stop the game at any time. The main goal of this work is characterize Nash equilibria through a verification theorem, which identifies a new system of quasi-variational inequalities whose solution gives equilibrium payoffs with the correspondent strategies. Moreover, we apply the verification theorem to a game with a one-dimensional state variable, evolving as a scaled Brownian motion, and with linear payoff and costs for both players. Two…
On the existence of Markovian randomized equilibria in Dynkin games of war-of-attrition-type
本文针对消耗战型非零和 Dynkin 博弈，通过构建马尔可夫随机化停止时间的适当拓扑，证明了马尔可夫完美均衡的存在性，填补了随机化情形下马尔可夫停止博弈均衡理论的空白。
We first study an optimal stopping problem in which a player (an agent) uses a discrete stopping time in order to stop optimally a payoff process whose risk is evaluated by a (non-linear) $g$-expectation. We then consider a non-zero-sum game on discrete stopping times with two agents who aim at minimizing their respective risks. The payoffs of the agents are assessed by g-expectations (with possibly different drivers for the different players). By using the results of the first part, combined with some ideas of S. Hamad{è}ne and J. Zhang, we construct a Nash equilibrium point of this game by a…
Nonzero-Sum Stochastic Differential Games for Controlled Convection-Diffusion SPDEs
本文研究了受控对流-扩散SPDE主导的双玩家非零和随机微分博弈，证明了正向和反向SPDE在Hilbert空间中的解存在性，利用哈密顿方法推导最大原理，并通过复合材料示例展示界面传输条件。
In the best choice problem with random arrivals, an unknown number $n$ of rankable items arrive at times sampled from the uniform distribution. As is well known, a real-time player can ensure stopping at the overall best item with probability at least $1/e$, by waiting until time $1/e$ then selecting the first relatively best item to appear (if any). This paper discusses the issue of dominance in a wide class of stopping strategies of best choice, and argues that in fact the player faces a trade-off between success probabilities for various values of $n$. We argue that the $1/e$-strategy is no…
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…
This paper shows that penalized backward stochastic differential equation (BSDE), which is often used to approximate and solve the corresponding reflected BSDE, admits both optimal stopping representation and optimal control representation. The new feature of the optimal stopping representation is that the player is allowed to stop at exogenous Poisson arrival times. The convergence rate of the penalized BSDE then follows from the optimal stopping representation. The paper then applies to two classes of equations, namely multidimensional reflected BSDE and reflected BSDE with a constraint on t…
In this paper we study the nonzero-sum Dynkin game in continuous time which is a two player non-cooperative game on stopping times. We show that it has a Nash equilibrium point for general stochastic processes. As an application, we consider the problem of pricing American game contingent claims by the utility maximization approach.
We formulate a stochastic game of mean field type where the agents solve optimal stopping problems and interact through the proportion of players that have already stopped. Working with a continuum of agents, typical equilibria become functions of the common noise that all agents are exposed to, whereas idiosyncratic randomness can be eliminated by an Exact Law of Large Numbers. Under a structural monotonicity assumption, we can identify equilibria with solutions of a simple equation involving the distribution function of the idiosyncratic noise. Solvable examples allow us to gain insight into…
Nonzero-sum stochastic differential games with impulse controls: a verification theorem with applications
本文研究了一类非零和脉冲博弈问题，提出了一种验证定理，为两个玩家的价值函数和最优策略提供了合适的拟变分不等式系统，并应用该定理分析了一维状态变量随实值布朗运动变化的脉冲博弈，给出了纳什均衡的显式表达式，并证明了干预成本的渐进行为。
A new probabilistic approach for mean field games of optimal stopping
该研究为最优停止平均场博弈提出新概率公式，用耦合反射正倒向随机微分方程刻画均衡，证明其与均衡的等价性，通过不动点定理建立均衡存在性与唯一性，还给出序理论方法及与分析方法的联系，以及对多人博弈的作用。
We consider a symmetric $n$-player nonzero-sum stochastic differential game with controlled jumps and mean-field type interaction among the players. Each player minimizes some expected cost by affecting the drift as well as the jump part of their own private state process. We consider the corresponding limiting mean-field game and, under the assumption that the latter admits a regular Markovian solution, we prove that an approximate Nash equilibrium for the $n$-player game can be constructed for $n$ large enough, and provide the rate of convergence. This extends to a class of games with contro…
Holistic Decision-Making in Stopping Problems: Emphasizing Psychological Aspects
本文探讨了在停止游戏中综合考虑心理因素的决策方法，通过分析不同停止时间的影响，提出基于偏离理性策略的诊断方法。
This paper deals with an extension of the concept of correlated strategies to Markov stopping games. The Nash equilibrium approach to solving nonzero-sum stopping games may give multiple solutions. An arbitrator can suggest to each player the decision to be applied at each stage based on a joint distribution over the players' decisions. This is a form of equilibrium selection. Examples of correlated equilibria in nonzero-sum games related to the staff selection competition in the case of two departments are given. Utilitarian, egalitarian, republican and libertarian concepts of correlated equi…
Zero-sum Dynkin games under common and independent Poisson constraints
本文研究了在共同和独立泊松约束下零和Dynkin游戏的解等价性，通过马尔可夫框架和BSDE技术，建立了价值函数和最优停止集的条件，展示了无限时间跨度游戏的适用性。
Trapping the Ultimate Success
本研究引入一种猜测最后一次成功时刻的博弈游戏，其与有限试验序列中停在最后一次成功的序贯决策问题相关，研究者利用该关联分析了Polya-Lundberg过程驱动下停在最后一个记录值的问题。
We study the convergence of Nash equilibria in a game of optimal stopping. If the associated mean field game has a unique equilibrium, any sequence of $n$-player equilibria converges to it as $n\to\infty$. However, both the finite and infinite player versions of the game often admit multiple equilibria. We show that mean field equilibria satisfying a transversality condition are limit points of $n$-player equilibria, but we also exhibit a remarkable class of mean field equilibria that are not limits, thus questioning their interpretation as "large $n$" equilibria.
We introduce a generalized Dynkin game problem with non linear conditional expectation ${\cal E}$ induced by a Backward Stochastic Differential Equation (BSDE) with jumps. Let $ξ, ζ$ be two RCLL adapted processes with $ξ\leq ζ$. The criterium is given by \begin{equation*} {\cal J}_{τ, σ}= {\cal E}_{0, τ\wedge σ} \left(ξ_τ\textbf{1}_{\{ τ\leq σ\}}+ζ_σ\textbf{1}_{\{σ<τ\}}\right) \end{equation*} where $τ$ and $ σ$ are stopping times valued in $[0,T]$. Under Mokobodski's condition, we establish the existence of a value function for this game, i.e. $\inf_σ\sup_τ {\cal J}_{τ, σ} = \sup_τ \inf_σ {\ca…
This paper studies a 2-players zero-sum Dynkin game arising from pricing an option on an asset whose rate of return is unknown to both players. Using filtering techniques we first reduce the problem to a zero-sum Dynkin game on a bi-dimensional diffusion $(X,Y)$. Then we characterize the existence of a Nash equilibrium in pure strategies in which each player stops at the hitting time of $(X,Y)$ to a set with moving boundary. A detailed description of the stopping sets for the two players is provided along with global $C^1$ regularity of the value function.
We study a class of optimal stopping games (Dynkin games) of preemption type, with uncertainty about the existence of competitors. The set-up is well-suited to model, for example, real options in the context of investors who do not want to publicly reveal their interest in a certain business opportunity. We show that there exists a Nash equilibrium in randomized stopping times which is described explicitly in terms of the corresponding one-player game.
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 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…
This article studies a biased version of the naming game in which players located on a connected graph interact through successive conversations to bootstrap a common name for a given object. Initially, all the players use the same word B except for one bilingual individual who also uses word A. Both words are attributed a fitness, which measures how often players speak depending on the words they use and how often each word is pronounced by bilingual individuals. The limiting behavior depends on a single parameter: phi = the ratio of the fitness of word A to the fitness of word B. The main ob…
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。
Markovian randomized equilibria for general Markovian Dynkin games in discrete time
本文研究离散时间马尔可夫框架下的两人 Dynkin 博弈，提出马尔可夫随机停时策略，推导出 Nash 均衡的 Wald-Bellman 型显式刻画，并证明了零和及一般情形下均衡的存在性与构造方法。
Nonstationary nonzero-sum Markov games under a probability criterion
针对非平稳 N 人非零和 Markov 博弈，本文以首次到达目标集前累积报酬达标概率为准则，证明历史依赖策略下 Nash 均衡存在，并提出 epsilon-Nash 均衡算法。
On the Robust Optimal Stopping Problem
针对一组相互奇异概率测度，研究鲁棒最优停止问题，通过定义上Snell包络和相应的非线性期望，证明最优停止时间的存在性及零和博弈的值。
Each of two players, by turns, rolls a dice several times accumulating the successive scores until he decides to stop, or he rolls an ace. When stopping, the accumulated turn score is added to the player account and the dice is given to his opponent. If he rolls an ace, the dice is given to the opponent without adding any point. In this paper we formulate this game in the framework of competitive Markov decision processes (also known as stochastic games), show that the game has a value, provide an algorithm to compute the optimal minimax strategy, and present results of this algorithm in three…
Toral introduced so-called cooperative Parrondo games, in which there are N players (3 or more) arranged in a circle. At each turn one player is randomly chosen to play. He plays either game A or game B. Game A results in a win or loss of one unit based on the toss of a fair coin. Game B results in a win or loss of one unit based on the toss of a biased coin, with the amount of the bias depending on whether none, one, or two of the player's two nearest neighbors have won their most recent games. Game A is fair, so the games are said to exhibit the Parrondo effect if game B is losing or fair an…
Mean field games with option to buy information
研究连续时间有限时域平均场博弈，其中代表性玩家的目标函数依赖于隐藏状态，玩家可选择付费观察该状态，通过连接最优控制与自由停止模型给出解的表征和显式示例。
We study Nash equilibria for a sequence of symmetric $N$-player stochastic games of finite-fuel capacity expansion with singular controls and their mean-field game (MFG) counterpart. We construct a solution of the MFG via a simple iterative scheme that produces an optimal control in terms of a Skorokhod reflection at a (state-dependent) surface that splits the state space into action and inaction regions. We then show that a solution of the MFG of capacity expansion induces approximate Nash equilibria for the $N$-player games with approximation error $\varepsilon$ going to zero as $N$ tends to…
We adapt the Stochastic Perron's method in Bayraktar and Sirbu (ArXiv: 1103.0538) to the case of double obstacle problems associated to Dynkin games. We construct, symmetrically, a viscosity sub-solution which dominates the upper value of the game and a viscosity super-solution lying below the lower value of the game. If the double obstacle problem satisfies the viscosity comparison property, then the game has a value which is equal to the unique and continuous viscosity solution. In addition, the optimal strategies of the two players are equal to the first hitting times of the two stopping re…
In this paper we investigate Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games whose cost functionals are defined by a system of coupled backward stochastic differential equations. We obtain an existence theorem and a characterization theorem for Nash equilibrium payoffs. For this end the problem is described equivalently by a stochastic differential game with jumps. But, however, unlike the work by Buckdahn, Hu and Li [9], here the important tool of a dynamic programming principle for stopping times has to be developed. Moreover, we prove that the lower and upp…
Symmetric Equilibria in Stochastic Timing Games
本文研究了随机时机博弈中的对称均衡，分析了局部先动或后动优势下的策略差异，通过Snell envelope理论刻画耗尽战的停止率，并提出算法确定 preemptive 必要性及对称均衡。
Exit game with private information
针对竞争与不确定下的战略退出决策难题，研究了参与者不知对手退出价值的随机退出博弈，基于一般一维扩散收益流构建了含状态变量与后验信念的显式均衡策略，并证明其在对称贝叶斯均衡子类中唯一。
Dynkin games for Markov processes associated with semi-Dirichlet forms
针对与半 Dirichlet 型相关的 Markov 过程上的 Dynkin 最优停止博弈，本文证明双障碍变分不等式的解给出均衡价格，并构造得到博弈鞍点。
Dynkin ghost games with asymmetry and consolation
研究不确定竞争下两位参与者的先发制人停止博弈，引入收益不对称与安慰补偿，提出一般性验证定理以构造均衡解。
Deep Learning for Energy Market Contracts: Dynkin Game with Doubly RBSDEs
本文提出利用深度学习方法解决能源市场差价合约中的双反射反向随机微分方程问题，通过神经网络求解器实现高维下的最优停止策略计算。
We study a robust Dynkin game over a set of mutually singular probabilities. We first prove that for the conservative player of the game, her lower and upper value processes coincide (i.e. She has a value process $V $ in the game). Such a result helps people connect the robust Dynkin game with second-order doubly reflected backward stochastic differential equations. Also, we show that the value process $V$ is a submartingale under an appropriately defined nonlinear expectations up to the first time $τ_*$ when $V$ meets the lower payoff process $L$. If the probability set is weakly compact, one…
We study a nonzero-sum game of two players which is a generalization of the antagonistic noisy duel of discrete type. The game is considered from the point of view of various criterions of optimality. We prove existence of epsilon-equilibrium situations and show that the epsilon-equilibrium strategies that we have found are epsilon-maxmin. Conditions under which the equilibrium plays are Pareto-optimal are given. Keywords: noisy duel, payoff function, strategy, equilibrium situation, Pareto optimality, the value of a game.
A zero-sum stochastic differential game with impulses, precommitment, and unrestricted cost functions
本文研究带脉冲与随机控制的零和随机微分博弈，通过有限值停时的动态规划原理，在无成本递减假设下证明了博弈值的存在性及相应拟变分不等式粘性解的唯一性。
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