Catálogo de publicaciones - libros

This chapter provides an introductory exposition of stochastic games with imperfect monitoring. These are stochastic games in which the players imperfectly observe the play. We discuss at length a few basic issues, and describe selected contributions.

In this chapter one considers a linear antagonistic differential game with fixed terminal time , geometric constraints on the players’ controls, and continuous quasi-convex payoff function depending on two components , of the phase vector . Let be a level set (a Lebesgue set) of the payoff function. One says that the function possesses the level sweeping property if for any pair of constants < the relation holds. Here, the symbols + and mean algebraic sum (Minkowski sum) and geometric difference (Minkowski difference). Let be a level set of the value function . The main result of this work is the proof of the fact that if the payoff function possesses the level sweeping property, then for any ∈ [, ] the function also has the property: . Such an inheritance of the level sweeping property by the value function is specific to the case where the payoff function depends on two components of the phase vector. If it depends on three or more components of the vector , the statement, generally speaking, is wrong. This is shown by a counterexample.

We consider the problem of generalized shortest path. The task is to transit optimally from the origin through a system , of intermediate sets in ℝ to a fixed destination point (or set), under conditions that only one node in can be chosen for passing. Any returns to the sets that have already been passed, are prohibited. The (combinatorial) cost function to minimize is either additive or bottleneck. The visiting nodes , are either governed by an antagonistic nature or by a rational antagonist. For this multistage game problem both open-loop and feedback settings are suggested. The feedback problem is posed in the class of feedback strategies which can change route during motion, depending on the current moves of the opponent. They provide, in general, a strictly better value of the problem, with respect to the open-loop minimax setting. The optimal feedback minimax strategy is constructed, and some (polynomial)heuristics are given.

This chapter presents a new approach to improve the homing performance of a pursuer with delayed information on the evader’s lateral acceleration. This approach reduces the of the pursuer, created due to the estimation delay, by considering not only the current (pure feedback) measurements but also the available measurement history during the period of the estimation delay. The reduced is computed by solving two auxiliary optimization problems. By using the center of the new convex hull as a new state variable, the original game is transformed to a nonlinear delayed dynamics game with perfect information for both players. The solution of this new game is obtained in pure strategies for the pursuer and mixed ones for the evader. The of this game (the guaranteed miss distance) is substantially less than the one obtained in previous works by using only the current measurements.

Conflict-controlled processes for systems with Riemann-Liouville derivatives of arbitrary order are studied here. A solution of such a system is presented in the form of a Cauchy formula analog. Using the resolving functions method, sufficient conditions for termination of the game are obtained. These conditions are based on the modified Pontryagin condition, expressed in terms of the generalized matrix functions of Mittag-Leffler. To find the latter, the interpolating polynomial of Lagrange-Sylvester is used. An illustrative example is given.

Necessary conditions are obtained for the capture of several evaders in a group pursuit problem, where all evaders use the same control. Necessary conditions for capture in such a group pursuit problem are also obtained for the special case of “soft” capture.

A cooperative stochastic -person game on a finite graph tree is considered. The subtree of cooperative trajectories maximizing the sum of expected players’ payoffs is defined, and the solution of the game along the paths of this tree is investigated. The new notion of cooperative payoff distribution procedure (CPDP) is defined, and the time-consistent Shapley value is constructed.

In this paper, we examine the uniqueness of a reduced game in an axiomatic characterization of the core of transferable utility (TU) games in terms of consistency. Tadenuma [] establishes that the core is the only solution satisfying non-emptiness, individual rationality, and consistency with respect to a natural reduced game due to Moulin []. However, the core satisfies consistency with respect to many other reduced games, including unnatural ones. Then we ask whether there are other reduced games that can be used to characterize the core based on the same three axioms. The answer is no: the Moulin reduced game is the only reduced game such that the core is characterized by the three axioms, since for any other reduced game, there is a solution that satisfies the three axioms, but it differs from the core. Many other unnatural reduced games cannot be used to characterize the core based on the three axioms. Funaki [] provides another axiomatization of the core: the core is the only solution satisfying non-emptiness, Pareto optimality, sub-grand rationality, and consistency with respect to a simple reduced game similar to a so-called subgame. We show that the simple reduced game is the only reduced game that can be used to characterize the core by the four axioms.

This chapter develops a framework for analyzing the interaction between individual players (actors) and collective players (coalitions) who mutually adapt the allocation of investment to their values and each other’s decisions. The dynamic process of coalition formation can be described by a coupled evolutionary game of allocation controls. Potential fields of applications are outlined, and an example analyzing the management of energy and carbon emissions is discussed in more detail.

This chapter surveys recent developments on some basic solution concepts, like stable sets, the core, the nucleolus and the modiclus for a very special class of cooperative games, namely assignment games with transferable utility. The existence of a stable set for assignment games is still an open problem.