the value of repeated games with an informed controller we consider the general model of zero sum repeated games or stochastic games with signals and assume that one of the players is fully informed and controls the transitions of the state variable we prove the existence of the uniform value generalizing several results of the literature a preliminary existence result is obtained for a particular class of stochastic games played with pure strategies

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

the value of markov chain games with lack of information on one side we consider a two player zero sum game given by a markov chain over a finite set of states and a family of matrix games indexed by states the sequence of states follows the markov chain at the beginning of each stage only player 1 is informed of the current state then the corresponding matrix game is played and the actions chosen are observed by both players before proceeding to the next stage we call such a game a markov chain game with lack of information on one side this model generalizes the model of aumann and maschler of zero sum repeated games with lack of information on one side which corresponds to the case where the transition matrix of the markov chain is the identity matrix we generalize the proof of aumann and maschler and from the definition and the study of appropriate nonrevealing auxiliary games with infinitely many stages show the existence of the uniform value an important difference with aumann and maschler s model is that here the notions for player 1 of using the information and revealing a relevant information are distinct

long term values in markov decision processes and repeated games and a new distance for probability spaces we study long term markov decision processes mdps and gambling houses with applications to any partial observation mdps with finitely many states and zero sum repeated games with an informed controller we consider a decision maker who is maximizing the weighted sum t 1 x1d703 trt where rt is the expected reward of the t th stage we prove the existence of a very strong notion of long term value called general uniform value representing the fact that the decision maker can play well independently of the evaluations x1d703 t t 1 over stages provided the total variation or impatience t 1 x1d703 t 1 x1d703 t is small enough this result generalizes previous results of the literature that focus on arithmetic means and discounted evaluations moreover we give a variational characterization of the general uniform value via the introduction of appropriate invariant measures for the decision problems generalizing the fundamental theorem of gambling or the aumann maschler cav u formula for repeated games with incomple

searching for an agent who may or may not want to be found there is an extensive theory regarding optimal continuous path search for a mobile or immobile target the traditional theory assumes that the target is one of three types i an object with a known distribution of paths ii a mobile or immobile hider who wants to avoid or delay capture or iii a rendezvouser who wants to find the searcher this paper introduces a new type of search problem by assuming that aims of the target are not known to the searcher the target may be either a type iii cooperator with a known cooperation probabilityc or a type ii evader this formulation models search problems like that for a lost teenager who may be a runaway or a lost intelligence agent who may be a defector in any given search context it produces a continuum of search problems g c 0 c 1 linking a zero sum searchgame withc 0 to a rendezvous problem withc 1 these models thus provide a theoretical bridge between two previously distinct parts of search theory namely searchgames and rendezvous search

the value of markov chain games with incomplete information on both sides we consider zero sum repeated games with incomplete information on both sides where the states privately observed by each player follow independent markov chains it generalizes the model introduced by aumann and maschler in the sixties and solved by mertens and zamir in the seventies where the private states of the players were fixed it also includes the model introduced in renault 19 renault j 2006 the value of markov chain games with lack of information on one side math oper res 31 3 490 512 of markov chain repeated games with lack of information on one side where only one player privately observes the sequence of states we prove here that the limit value exists and we obtain a characterization via the mertens zamir system where the nonrevealing value function plugged in the system is now defined as the limit value of an auxiliary nonrevealing dynamic game this nonrevealing game is defined by restricting the players not to reveal any information on the limit behavior of their

multiplicity of nontrivial solutions for quasilinear elliptic equation two nontrivial solutions are obtained by the reduction method for the nonautonomous semilinear elliptic dirichlet boundary value problem some well known multiplicity results are generalized

the global artificial boundary conditions for numerical simulations of the flow around a submerged body we consider the numerical approximations of the two dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface one vertical segment is introduced as the upstream artificial boundary and two vertical segments are introduced as the downstream artificial boundaries on the artificial boundaries a sequence of high order global artificial boundary conditions are given then the original problem is reduced to a problem defined on a finite computational domain which is equivalent to a variational problem after solving the variational problem by the finite element method we obtain the numerical approximation of the original problem the numerical examples show that the artificial boundary conditions given in this paper are very effective

existence of solutions for 2mth order periodic boundary value problems abstract in this paper the existence results of solutions are obtained for the 2mth order differential equation periodic boundary value problem 1 m u 2 m t i 1 m 1 m i a i u 2 m i t f t u t for all t 0 1 with u i 0 u i 1 i 0 1 2m 1 where f c 1 0 1 r 1 r 1 a i r 1 i 1 2 m by using the morse theory we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution two nontrivial solutions and infinitely many solutions separately

a fast iterative method to compute the flow around a submerged body the authors develop an efficient iterative method for computing steady linearized potential flow around a submerged body moving in a liquid of finite constant depth in this paper they restrict the presentation to the two dimensional problem but the method is readily generalizable to the three dimensional case i e the flow in a canal the problem is indefinite which makes the convergence of most iterative methods unstable to circumvent this difficulty the authors decompose the problem into two more easily solvable subproblems and form a schwarz type iteration to solve the original problem the first subproblem is definite and can therefore be solved by standard iterative methods the second subproblem is indefinite but has no body it is therefore easily and efficiently solvable by separation of variables the authors prove that the iteration converges for sufficiently small froude numbers in addition they present numerical results for a second order accurate discretization of the problem they demonstrate that the iterative method converges rapidly and that the convergences rate improves when the froude number decreases they also verify numerically that the convergence rate is essentially independent of the grid size 20 refs 6 figs 10 tabs

existence of a nontrivial solution for a class of hemivariational inequality problems at double resonance abstract in this paper we discuss a class of semilinear elliptic hemivariational inequality problems by using the nonsmooth minimax principle for locally lipschitz functions we establish the existence of a nontrivial solution for the semilinear elliptic hemivariational inequality problem where incomplete double resonance occurs at infinity between two distinct consecutive eigenvalues

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

computing the flow around a submerged body using composite grids the subject of this paper is an accurate numerical method for solving the linear two dimensional steady potential flow around a body which moves in a liquid of finite constant depth at constant speed and distance below the free surface the differential equation is discretized by a second order accurate finite difference scheme on a composite grid the composite grid consists of two overlapping component grids one curvilinear grid close to the body and one cartesian grid which covers the surrounding liquid to solve the problem numerically the infinite domain is truncated to finite length the inflow and outflow boundary conditions are formed by making an eigenfunction expansion of the solution ahead of and behind the body each eigenfunction is required to be bounded and satisfy the upstream condition at infinity this is imposed by functional relations between the solution and its normal derivative at the inflow and outflow boundaries the method is carefully validated and the computed solutions are found to be in very good agreement with existing results

on a markov game with one sided information we apply the average cost optimality equation to zero sum markov games by considering a simple game with one sided incomplete information that generalizes an example of aumann and maschler aumann r j m b maschler 1995 repeated games with incomplete information mit press cambridge ma we determine the value and identify the optimal strategies for a range of parameters

the artificial boundary conditions for computing the flow around a submerged body in this paper we consider the numerical simulations of the steady linearized potential flow around a body which moves in a liquid of finite constant depth at constant speed and distance below a free surface two vertical segments are introduced as the artificial boundary to limit the unbounded physical domain to a bounded computational domain a series of approximate artificial boundary conditions at the given artificial boundary are designed by imposing the continuity of the solution and its normal derivative then the original problem is reduced to a problem defined in a bounded computational domain which is solved numerically by the finite element method the numerical examples show that our artificial boundary conditions are very effective

the rendezvous search problem the author considers the problem faced by two people who are placed randomly in a known search region and move about at unit speed to find each other in the least expected time this time is called the rendezvous value of the region it is shown how symmetries in the search region may hinder the process by preventing coordination based on concepts such as north or clockwise a general formulation of the rendezvous search problem is given for a compact metric space endowed with a group of isometrics which represents the spatial uncertainties of the players these concepts are illustrated by considering upper bounds for various rendezvous values for the circle and an arbitrary metric network the discrete rendezvous problem on a cycle graph for players restricted to symmetric markovian strategies is then solved finally the author considers the problem faced by two people on an infinite line who each know the distribution of the distance but not the direction to each other

dynamical analysis of a repeated game with incomplete information we study a two player repeated zero sum game with asymmetric information introduced by renault in which the underlying state of the game undergoes markov evolution parameterized by a transition probability p in the range 12 to 1 horner rosenberg solan and vieille identified an optimal strategy for the informed player for p in the range 12 23 we extend the range on which is proved to be optimal to about 12 0 719 and prove that it fails to be optimal at a value around 0 7328 our techniques make use of tools from dynamical systems specifically the notion of pressure introduced by d ruelle

rendezvous search on the line with distinguishable players two players are placed on the real line at a distance d with a distribution f known to both neither knows the direction of the other nor do they have a common notion of a positive direction on the line we seek the least expected rendezvous time r r left f right in which they can meet given maximum speeds of one we consider the cases where f is a bounded point discrete or finite mean distribution we obtain upper bounds or exact values for r and in one case an optimality condition for search strategies a connection with beck s linear search problem is established

a numerical method to calculate the two dimensional flow around an underwater obstacle the linear two dimensional steady potential flow around a body travelling in a liquid of finite constant depth at constant speed and distance below a free surface is considered the analytical solutions ahead of and behind the body are used to form functional relations between the potential and its normal derivative these relations are imposed as boundary conditions for the problem close to the body to satisfy the radiation condition one more relation is imposed ahead of the body than behind it a second order accurate finite difference discretization of the problem is also analyzed an analogous theory is developed which results in matrix relations as boundary conditions for the problem close to the body

rendezvous evasion search in two boxes an agent who may or may not want to be found is located in one of two boxes at time 0 suppose that he is in box b with probability p he wishes to be found in which case he has been asked to stay in box b with probability 1 p he tries to evade the searcher in which case he may move between boxes a and b the searcher looks into one of the boxes at times 1 2 3 between each search the agent may change boxes if he wants the searcher is trying to minimise the expected time to discovery the agent is trying to minimise this time if he wants to be found but trying to maximise it otherwise this paper finds a solution to this game in a sense defined in the paper associated strategies for the searcher and each type of agent and a continuous value function v p giving the expected time for the agent to be discovered the solution method which is to solve an associated zero sum game would apply generally to this type of game of incomplete information

multiplicity results for semilinear elliptic problems at resonance and with jumping nonlinearities abstract in the present paper we develop a method to deal with multiple existence of solutions of nonlinear elliptic problems and give some multiple existence results for elliptic boundary value problems at resonance and problems with jumping nonlinearities

markov games with frequent actions and incomplete information the limit case we study the asymptotics of a class of two player zero sum stochastic game with incomplete information on one side when the time span between two consecutive stages vanishes the informed player observes the realization of a markov chain on which the payoffs depend whereas the noninformed player only observes his opponent s actions we show the existence of a limit value this value is characterized through an auxiliary optimization problem and as the solution of a hamilton jacobi equation

multiplicity results for semilinear resonant elliptic problems with discontinuous nonlinearities in the present paper some generalized critical points theorems for locally lipschitz functions are given as applications some multiplicity results for zero boundary value elliptic resonant problems are obtained two classical important theorems are improved

rendezvous search on the line we present two new results for the asymmetric rendezvous problem on the line we first show that it is never optimal for one player to be stationary during the entire search period in the two player rendezvous then we consider the meeting time of n players in the worst case and show that it has an asymptotic behavior of n 2 o log n

a mixed strategy minimax theorem without compactness minimax theorems for infinite games generally require that both players choose their pure strategies from compact sets and have semicontinuity requirements in both variables this paper proves the following theorem let x be a compact hausdorfl space and let y a be a measurable space let f x times y be a measurable function which is bounded below and lower semicontinuous on x for all fixed y in y let m be any convex set of probability measures mixed strategies on y a then mathop min limits mu in b x mathop sup limits rho in m iint f x y d mu d rho mathop sup limits rho in m mathop min limits mu in b x iint f x y d mu d rho where b x denotes the regular borel probability measures on x