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Showing papers on "Stochastic game published in 1996"


BookDOI
01 Dec 1996
TL;DR: In this article, the authors present a series of courses and prerequisites for the development of stochastic games with a focus on reducing the complexity of the problem of finding the optimal solution.
Abstract: 1 Introduction.- 1.0 Background.- 1.1 Raison d'Etre and Limitations.- 1.2 A Menu of Courses and Prerequisites.- 1.3 For the Cognoscenti.- 1.4 Style and Nomenclature.- I Mathematical Programming Perspective.- 2 Markov Decision Processes: The Noncompetitive Case.- 2.0 Introduction.- 2.1 The Summable Markov Decision Processes.- 2.2 The Finite Horizon Markov Decision Process.- 2.3 Linear Programming and the Summable Markov Decision Models.- 2.4 The Irreducible Limiting Average Process.- 2.5 Application: The Hamiltonian Cycle Problem.- 2.6 Behavior and Markov Strategies.- 2.7 Policy Improvement and Newton's Method in Summable MDPs.- 2.8 Connection Between the Discounted and the Limiting Average Models.- 2.9 Linear Programming and the Multichain Limiting Average Process.- 2.10 Bibliographic Notes.- 2.11 Problems.- 3 Stochastic Games via Mathematical Programming.- 3.0 Introduction.- 3.1 The Discounted Stochastic Games.- 3.2 Linear Programming and the Discounted Stochastic Games.- 3.3 Modified Newton's Method and the Discounted Stochastic Games.- 3.4 Limiting Average Stochastic Games: The Issues.- 3.5 Zero-Sum Single-Controller Limiting Average Game.- 3.6 Application: The Travelling Inspector Model.- 3.7 Nonlinear Programming and Zero-Sum Stochastic Games.- 3.8 Nonlinear Programming and General-Sum Stochastic Games.- 3.9 Shapley's Theorem via Mathematical Programming.- 3.10 Bibliographic Notes.- 3.11 Problems.- II Existence, Structure and Applications.- 4 Summable Stochastic Games.- 4.0 Introduction.- 4.1 The Stochastic Game Model.- 4.2 Transient Stochastic Games.- 4.2.1 Stationary Strategies.- 4.2.2 Extension to Nonstationary Strategies.- 4.3 Discounted Stochastic Games.- 4.3.1 Introduction.- 4.3.2 Solutions of Discounted Stochastic Games.- 4.3.3 Structural Properties.- 4.3.4 The Limit Discount Equation.- 4.4 Positive Stochastic Games.- 4.5 Total Reward Stochastic Games.- 4.6 Nonzero-Sum Discounted Stochastic Games.- 4.6.1 Existence of Equilibrium Points.- 4.6.2 A Nonlinear Compementarity Problem.- 4.6.3 Perfect Equilibrium Points.- 4.7 Bibliographic Notes.- 4.8 Problems.- 5 Average Reward Stochastic Games.- 5.0 Introduction.- 5.1 Irreducible Stochastic Games.- 5.2 Existence of the Value.- 5.3 Stationary Strategies.- 5.4 Equilibrium Points.- 5.5 Bibliographic Notes.- 5.6 Problems.- 6 Applications and Special Classes of Stochastic Games.- 6.0 Introduction.- 6.1 Economic Competition and Stochastic Games.- 6.2 Inspection Problems and Single-Control Games.- 6.3 The Presidency Game and Switching-Control Games.- 6.4 Fishery Games and AR-AT Games.- 6.5 Applications of SER-SIT Games.- 6.6 Advertisement Models and Myopic Strategies.- 6.7 Spend and Save Games and the Weighted Reward Criterion.- 6.8 Bibliographic Notes.- 6.9 Problems.- Appendix G Matrix and Bimatrix Games and Mathematical Programming.- G.1 Introduction.- G.2 Matrix Game.- G.3 Linear Programming.- G.4 Bimatrix Games.- G.5 Mangasarian-Stone Algorithm for Bimatrix Games.- G.6 Bibliographic Notes.- Appendix H A Theorem of Hardy and Littlewood.- H.1 Introduction.- H.2 Preliminaries, Results and Examples.- H.3 Proof of the Hardy-Littlewood Theorem.- Appendix M Markov Chains.- M.1 Introduction.- M.2 Stochastic Matrix.- M.3 Invariant Distribution.- M.4 Limit Discounting.- M.5 The Fundamental Matrix.- M.6 Bibliographic Notes.- Appendix P Complex Varieties and the Limit Discount Equation.- P.1 Background.- P.2 Limit Discount Equation as a Set of Simultaneous Polynomials.- P.3 Algebraic and Analytic Varieties.- P.4 Solution of the Limit Discount Equation via Analytic Varieties.- References.

1,191 citations


Journal ArticleDOI
TL;DR: In this paper, a class of noncooperative games in which the players share a common set of strategies is described, and the payoff a player receives for playing a particular strategy depends only on the total number of players playing the game.

727 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that every ANN-person game has the fictitious play property and has identical interests if it is best response equivalent in mixed strategies to a game with identical payoff functions.

505 citations


Journal ArticleDOI
Francis Bloch1
TL;DR: In this paper, the authors analyzed a sequential game of coalition formation when the division of the coalitional surplus is fixed and the payoffs are defined relative to the whole coalition structure and showed that any core stable coalition structure can be attained as a stationary perfect equilibrium of the game.

480 citations


Journal ArticleDOI
TL;DR: In this article, the authors established existence and uniqueness results for adapted solutions of backward stochastic differential equations with two reflecting barriers, generalizing the work of El Karoui, Kapoudjian, Pardoux, Peng and Quenez.
Abstract: We establish existence and uniqueness results for adapted solutions of backward stochastic differential equations (BSDE's) with two reflecting barriers, generalizing the work of El Karoui, Kapoudjian, Pardoux, Peng and Quenez. Existence is proved first by solving a related pair of coupled optimal stopping problems, and then, under different conditions, via a penalization method. It is also shown that the solution coincides with the value of a certain Dynkin game, a stochastic game of optimal stopping. Moreover, the connection with the backward SDE enables us to provide a pathwise (deterministic) approach to the game.

326 citations


Journal ArticleDOI
TL;DR: This work considers duality relations between risk-sensitive stochastic control problems and dynamic games derived from two basic duality results, the first involving free energy and relative entropy and resulting from a Legendre-type transformation, the second involving power functions.
Abstract: We consider duality relations between risk-sensitive stochastic control problems and dynamic games. They are derived from two basic duality results, the first involving free energy and relative entropy and resulting from a Legendre-type transformation, the second involving power functions. Our approach allows us to treat, in essentially the same way, continuous- and discrete-time problems, with complete and partial state observation, and leads to a very natural formal justification of the structure of the cost functional of the dual. It also allows us to obtain the solution of a stochastic game problem by solving a risk-sensitive control problem.

190 citations


Book
14 Mar 1996
TL;DR: In this article, the authors present an algorithm for the value of a non-leavable game and the optimality equation for a two-player, zero-sum game.
Abstract: 1 Introduction.- 1.1 Preview.- 1.2 Prerequisites.- 1.3 Numbering.- 2 Gambling Houses and the Conservation of Fairness.- 2.1 Introduction.- 2.2 Gambles, Gambling Houses, and Strategies.- 2.3 Stopping Times and Stop Rules.- 2.4 An Optional Sampling Theorem.- 2.5 Martingale Convergence Theorems.- 2.6 The Ordinals and Transfinite Induction.- 2.7 Uncountable State Spaces and Continuous-Time.- 2.8 Problems for Chapter 2.- 3 Leavable Gambling Problems.- 3.1 The Fundamental Theorem.- 3.2 The One-Day Operator and the Optimality Equation.- 3.3 The Utility of a Strategy.- 3.4 Some Examples.- 3.5 Optimal Strategies.- 3.6 Backward Induction: An Algorithm for U.- 3.7 Problems for Chapter 3.- 4 Nonleavable Gambling Problems.- 4.1 Introduction.- 4.2 Understanding u(?).- 4.3 A Characterization of V.- 4.4 The Optimality Equation for V.- 4.5 Proving Optimality.- 4.6 Some Examples.- 4.7 Optimal Strategies.- 4.8 Another Characterization of V.- 4.9 An Algorithm for V.- 4.10 Problems for Chapter 4.- 5 Stationary Families of Strategies.- 5.1 Introduction.- 5.2 Comparing Strategies.- 5.3 Finite Gambling Problems.- 5.4 Nonnegative Stop-or-Go Problems.- 5.5 Leavable Houses.- 5.6 An Example of Blackwell and Ramakrishnan.- 5.7 Markov Families of Strategies.- 5.8 Stationary Plans in Dynamic Programming.- 5.9 Problems for Chapter 5.- 6 Approximation Theorems.- 6.1 Introduction.- 6.2 Analytic Sets.- 6.3 Optimality Equations.- 6.4 Special Cases of Theorem 1.2.- 6.5 The Going-Up Property of $$ \overline M $$.- 6.6 Dynamic Capacities and the Proof of Theorem 1.2.- 6.7 Approximating Functions.- 6.8 Composition Closure and Saturated House.- 6.9 Problems for Chapter 6.- 7 Stochastic Games.- 7.1 Introduction.- 7.2 Two-Person, Zero-Sum Games.- 7.3 The Dynamics of Stochastic Games.- 7.4 Stochastic Games with lim sup Payoff.- 7.5 Other Payoff Functions.- 7.6 The One-Day Operator.- 7.7 Leavable Games.- 7.8 Families of Optimal Strategies for Leavable Games.- 7.9 Examples of Leavable Games.- 7.10 A Modification of Leavable Games and the Operator T.- 7.11 An Algorithm for the Value of a Nonleavable Game.- 7.12 The Optimality Equation for V.- 7.13 Good Strategies in Nonleavable Games.- 7.14 Win, Lose, or Draw.- 7.15 Recursive Matrix Games.- 7.16 Games of Survival.- 7.17 The Big Match.- 7.18 Problems for Chapter 7.- References.- Symbol Index.

170 citations


Journal ArticleDOI
TL;DR: This work examines decision making in two- person extensive form game trees using nine treatments that vary matching protocol, payoffs, and payoff information to establish replicable principles of cooperative versus noncooperative behavior.
Abstract: We examine decision making in two-person extensive form game trees using nine treatments that vary matching protocol, payoffs, and payoff information. Our objective is to establish replicable principles of cooperative versus noncooperative behavior that involve the use of signaling, reciprocity, and backward induction strategies, depending on the availability of dominated direct punishing strategies and the probability of repeated interaction with the same partner. Contrary to the predictions of game theory, we find substantial support for cooperation under complete information even in various single-play treatments.

154 citations


03 Oct 1996
TL;DR: It is shown that for hybrid automata with rectangular inclusions, the reachability question can be answered in a finite number of steps and that an $\omega$-automata game with the chain acceptance condition can be solved as a mean payoff game.
Abstract: A continuous system has a continuous state space and an evolution law given by a differential or a difference equation. A discrete event system is modeled by an automaton which changes state in response to events. A hybrid system contains both continuous and discrete event sub-systems. In this thesis we study some theoretical problems in the design and analysis of hybrid systems and discrete event systems. We first consider the reachability question for a hybrid system--is a target state reachable from an initial state? We show that for hybrid automata with rectangular inclusions, the reachability question can be answered in a finite number of steps. Hybrid systems with more general dynamics can be reduced to hybrid systems with rectangular inclusions using abstractions. We next consider an Automated Vehicle Highway System (AVHS) design. We consider the safety question: can there be a collision between two vehicles on the AVHS? We show that the AVHS is safe provided the controllers in the vehicles satisfy a set of constraints. The constraints require the reach set $Reach\sb{f}(X\sb0,t$)--the set of states reached after time t starting from an initial set $X\sb0$ for a differential inclusion $\dot x\ \in\ f(x$)--to satisfy a simple criterion. We show that this problem is equivalent to solving an optimal control problem. We then consider some computational questions for differential inclusions. For a Lipschitz differential inclusion $\dot x\ \in\ f(x$), we give a method to compute an arbitrary close approximation of $Reach\sb{f}(X\sb0,t$). For a differential inclusion $\dot x\ \in\ f(x$), and any $\epsilon>$ 0, we define a finite sample graph $A\sp{\epsilon}$. Using graph $A\sp{\epsilon}$, we can compute the $\epsilon$-invariant sets of the differential inclusion--the sets that remain invariant under $\epsilon$-perturbations in f. We also consider some dynamical games played on graphs. The synthesis and the control problem for $\omega$-automata can be formulated as a game between two players. We discuss games on $\omega$-automata and the payoff games. We show that $\omega$-automata games do not necessarily have a value when restricted to positional strategies. We exhibit a bound on the amount of memory required to play these games. We then consider the discounted and mean payoff games. We present the successive approximation and the policy iteration algorithm for solving payoff games. We then show that an $\omega$-automata game with the chain acceptance condition can be solved as a mean payoff game. Solving a chain game is equivalent to solving the model checking problem for propositional $\mu$-calculus. Hence, the policy iteration method can be used to model check $\mu$-calculus formula. This is at present the most efficient algorithm for model checking propositional $\mu$-calculus.

151 citations


Journal ArticleDOI
TL;DR: In this article, a discounted stochastic game of common-property capital accumulation with nonsymmetric players, bounded one-period extraction capacities, and a transition law satisfying a general strong convexity condition is considered.

140 citations


Journal ArticleDOI
Itai Sened1
TL;DR: In this paper, a new core solution concept, the IVCORE, is introduced, which allows the analysis of the trade-off between ideological, policy payoffs, and office-related sidepayments, in the bargaining process over future coalitions.
Abstract: Coalition formation is modeled as a cooperative game. Each party enters the game endowed with a proportion (weight) of votes that it obtained in the election, and a preferred policy position. The payoffs to any party that joins a coalition are a function of the distance between the party's and the government's respective policy positions, and the office related payoff that the party receives as a member of the coalition. A new core solution concept, the IVCORE, is introduced. It allows the analysis of the trade-off between ideological, policy payoffs, and office-related sidepayments, in the bargaining process over future coalitions. It turns out that policy concerns of parties "induce" a core in a typical transferable payoffs game that would, otherwise, have a generically empty core. At the same time, the "budget constraint" on the office-related sidepayments, determines the composition of the coalition. The process of coalition formation in Israel, after the 1992 election, is used to illustrate the empir...

Journal ArticleDOI
TL;DR: In this article, the authors considered infinite horizon risk-sensitive control of Markov processes with discrete time and denumerable state space, and proved that there exists a bounded solution to the dynamic programming equation.

Journal ArticleDOI
TL;DR: In this paper, the authors assume that the players who participate in such a game are part of some permission structure, which means there are players who need permission from one or more other players before they can act or cooperate.

Journal ArticleDOI
TL;DR: In this article, the authors considered repeated games where each player can be observed by only a subset of the other players, and where players can make public announcements about the behavior of the players they observed.

Journal ArticleDOI
TL;DR: In this article, the authors explore some solution concepts resulting from a payoff vector selection based on the excess vector but by means of an assessment of their relative fairness different from that given by the lexicographical order.
Abstract: The nucleolus and the prenucleolus are solution concepts for TU games based on the excess vector that can be associated to any payoff vector. Here we explore some solution concepts resulting from a payoff vector selection based also on the excess vector but by means of an assessment of their relative fairness different from that given by the lexicographical order. We take the departure consisting of choosing the payoff vector which minimizes the variance of the resulting excesses of the coalitions. This procedure yields two interesting solution concepts, both a prenucleolus-like and a nucleolus-like notion, depending on which set is chosen to set up the minimizing problem: the set of efficient payoff vectors or the set of inputations. These solution concepts, which, paralleling the prenucleolus and the nucleolus, we call least square prenucleolus and least square nucleolus, are easy to calculate and exhibit nice properties. Different axiomatic characterizations of the former are established, some of them by means of consistency for a reasonable reduced game concept.

Journal ArticleDOI
TL;DR: In this article, three measures of task complexity-cardinality of choice space, level of iterative knowledge of rationality, and level of knowledge of strategy-were manipulated and tested.
Abstract: We consider several coordination games with multiple equilibria each of which is a different division of a fixed pie. Laboratory experiments are conducted to address whether "task complexity" affects the selection of equilibrium by subjects. Three measures of task complexity-cardinality of choice space, level of iterative knowledge of rationality, and level of iterative knowledge of strategy-are manipulated and tested. Results suggest the three measures can predict choice behavior. Since strategically equivalent games can have different task complexity measures, our results imply that subjects are sensitive to game form presentation. We also fit data using three adaptive learning models: 1 Cournot, 2 Fictitious Play, and 3 Payoff Reinforcement, in increasing order of required cognitive effort. The Fictitious Play model, which tracks only cumulative frequencies of opponents' past behaviors fits the data best.

Journal ArticleDOI
TL;DR: The existence of a Pareto best element in the set of strategies that survive iterated elimination of dominated strategies implies the existence of coalition-proof correlated equilibrium for any specification of coalitional communication possibilities that always permits individual deviations as mentioned in this paper.

Journal ArticleDOI
TL;DR: In this article, the authors prove optimality inequalities of dynamic programming for viscosity sub-and supersolutions of the associated Bellman-Isaacs equations for stochastic differential games.

Patent
25 Apr 1996
TL;DR: A non-banking gambling method for use with conventional gambling games such as card games was proposed in this article, where each player risks only the amount such player wagers and wherein each player plays against other players and wins or loses according the player's collection of cards or other gambling devices regardless of the amount wagered.
Abstract: A non-banking gambling method for use with conventional gambling games such as card games wherein each player risks only the amount such player wagers and wherein each player plays against other players and wins or loses according the player's collection of cards or other gambling devices regardless of the amount wagered.

Journal ArticleDOI
TL;DR: In this article, a stochastic model is developed to describe behavioral changes by imitative pair interactions of individuals, which leads to the derivation of covariance equations, a measure of the reliability of game dynamical descriptions.
Abstract: A stochastic model is developed to describe behavioral changes by imitative pair interactions of individuals. ‘Microscopic’ assumptions on the specific form of the imitative processes lead to a stochastic version of the game dynamical equations, which means that the approximate mean value equations of these equations are the game dynamical equations of evolutionary game theory. The stochastic version of the game dynamical equations allows the derivation of covariance equations. These should always be solved along with the ordinary game dynamical equations. On the one hand, the average behavior is affected by the covariances so that the game dynamical equations must be corrected for increasing covariances; otherwise they may become invalid in the course of time. On the other hand, the covariances are a measure of the reliability of game dynamical descriptions. An increase of the covariances beyond a critical value indicates a phase transition, i.e. a sudden change in the properties of the social system under consideration. The applicability and use of the equations introduced are illustrated by computational results for the social self-organization of behavioral conventions.

Journal ArticleDOI
In-Koo Cho1
TL;DR: In this paper, the authors examined an infinitely repeated principal agent game without discounting, where the agent may engage in multiple projects and select an action according to a single threshold rule.
Abstract: We examine an infinitely repeated principal agent game without discounting (Radner [1985]), in which the agent may engage in multiple projects. We focus on “linear” strategies that summarize each history into a linear function of public outcomes, and select an action according to a single threshold rule. We claim that linear strategies significantly simplify the computation needed to make strategic decisions following each history. Despite the simplicity of linear strategies, we can virtually recover the folk theorem. For any individually rational payoff vector in the interior of the set of feasible expected payoff vectors, there exists a pair of linear strategies that form a Nash equilibrium supporting the target payoff. The equilibrium strategies and the equilibrium payoff vectors form a globally stable solution (Smale [1980]).

Journal ArticleDOI
TL;DR: In this article, it was shown that equilibrium selection is incompatible with One Person Rationality, Consistency and (restricted) Non-Emptiness for mixed extensions of finite games and games with compact and convex strategy spaces and continuous-concave payoff functions.

Journal ArticleDOI
TL;DR: In this article, a dynamic model of duopoly is considered in which a fixed fraction of the customer population changes loyalties in each period from the current high-price setter to the current low price setter, and it is shown to have a Markov perfect equilibrium with distinctive economic features.

Journal ArticleDOI
TL;DR: In this paper, a stochastic differential game model of a common-property commercial fishery is presented and a feedback Nash equilibrium of the game is determined, where closed-form expressions for the value functions, the equilibrium harvesting strategies, and stationary distributions of the fish stock are derived.
Abstract: The paper presents a stochastic differential game model of a common-property commercial fishery and determines a feedback Nash equilibrium of the game. Closed-form expressions for the value functions, the equilibrium harvesting strategies, and stationary distributions of the fish stock are derived. Sensitivity analyses with respect tot he model parameters are carried out. The paper also considers equilibrium outcomes under joint maximization and surplus maximization. In the latter case, an optimal market size (i.e., number of firms) is identified.

Journal ArticleDOI
TL;DR: This paper investigates situations of non-cooperative dynamic control of queueing systems by two agents, having different objectives, and establishes the equilibrium of a policy pair for which the router uses the intuitive “Join the shortest queue” policy.
Abstract: The purpose of this paper is to investigate situations of non-cooperative dynamic control of queueing systems by two agents, having different objectives. The main part of the paper is devoted to analyzing a problem of an admission and a service (vacation) control. The admission controller has to decide whether to allow arrivals to occur. Once the queue empties, the server goes on vacation, and controls the vacations duration (according to the state and past history of the queue). The immediate costs for each controller are increasing in the number of customers, but no convexity assumptions are made. The controllers are shown to have a stationary equilibrium policy pair, for which each controller uses a stationary threshold type policy with randomization in at most one state. We then investigate a problem of a non-zero sum stochastic game between a router into several queues, and a second controller that allocates some extra service capacity to one of the queues. We establish the equilibrium of a policy pair for which the router uses the intuitive “Join the shortest queue” policy.

Journal ArticleDOI
TL;DR: In this article, the authors study a repeated game in which a long-run player without discounting faces another long run player with strict discounting, and the game is perturbed so that there is uncertainty in the patient player's strategy choice: she may be a rational player, but she may also be one of many irrational types who are committed to various repeated game strategies.

Journal ArticleDOI
TL;DR: In this article, a new approach based on occupation measures is introduced for studying stochastic differential games, and the existence of values and optimal strategies for both players is established for various payoff criteria.
Abstract: A new approach based on occupation measures is introduced for studying stochastic differential games. For two-person zero-sum games, the existence of values and optimal strategies for both players is established for various payoff criteria. ForN-person games, the existence of equilibria in Markov strategies is established for various cases.

Journal ArticleDOI
TL;DR: In this article, reputation effects in perturbed repeated games with discounting are analyzed and a lower bound for the equilibrium payoff of a long-run player facing a sequence of short-run opponents is provided.

Journal ArticleDOI
TL;DR: In this article, a simple and basic signaling game is studied in an experimental environment and the experimental results are related to the predictions of two competing behavioral models: a game model, in which subjects are assumed to behave in line with (refined) sequential equilibrium theory, and a non-strategic decision makers.
Abstract: In this paper a simple and basic signaling game is studied in an experimental environment. First, we check whether we can replicate some of the findings in the literature concerning equilibrium selection and the use and impact of costly signals. Second, and foremost, the comparative statics implications of the game are studied. The experimental results are related to the predictions of two competing behavioral models: a game model, in which subjects are assumed to behave in line with (refined) sequential equilibrium theory, and a decision model, in which subjects are assumed to behave as non-strategic decision makers. The experimental outcomes replicate the finding in the literature that costly messages are sent more frequently by ‘higher’ sender types (whose information is such that persuasion is also profitable to the responder), and that such messages have an impact on the behavior of the responder. These results are consistent with (versions of) both the game model and the decision model. The comparative statics results, however, clearly point in the direction of the decision model. Play is most strongly affected by ‘own’ payoff parameters, as predicted by the decision model, and less so by opponent's payoff parameters, as predicted by the mixed strategies of the refined sequential equilibrium. Particularly, a decision model in which players are assumed to adapt beliefs about opponents' choice probabilities in response to experience in previous play, appears to succeed best in organizing the data.

Journal ArticleDOI
TL;DR: The main result is that the existence of a solution ψ* to a partial differential equation with appropriate boundary conditions and regularity properties implies the uniform convergence of ψn to the Fenchel conjugate of ω*.
Abstract: Let vnp denote the value of the n-times repeated zero-sum game with incomplete information on one side and full monitoring and let up be the value of the average game Gp. The error term enp = vnp-cavup is then converging to zero at least as rapidly as 1/√n. In this paper, we analyze the convergence of ψnp = √nenp in the games with square payoff matrices such that the optimal strategy of the informed player in the average game Gp is unique, is completely mixed and does not depend on p. Our main result is that the existence of a solution ψ* to a partial differential equation with appropriate boundary conditions and regularity properties implies the uniform convergence of ψn to the Fenchel conjugate of ψ*. In particular cases, the P.D.E. problem is linear and its solution ψ* is then related to the multidimensional normal distribution.