William H. Ruckle

Ambushing Random Walks I: Finite Models

We consider a two-person competitive problem in which one player blue wishes to move across a rectangular lattice in such a way as to avoid being ambushed by his opponent red, who has placed obstacles in blues path. Under some conditions, optimal strategies for blue and red are obtained and the value of the game is computed.

Initial Point Search on Weighted Trees

An initial point search game on a weighted graph involves a searcher who wants to minimize search and travel costs seeking a hider who wants to maximize these costs. The searcher starts from a specified vertex 0 and searches each vertex in some order. The hider chooses a nonzero vertex and remains there. We solve the game in which the graph is a simple tree, and use this solution to solve a search game on a tree in which each branch is itself a weighted graph with a certain property, and the searcher is obliged to search the entire branch before departing.

Ambushing Random Walks II: Continuous Models

We consider a two person competitive problem in which a traveler (BLUE) wishes to cross a rectangle in such a way to avoid being ambushed by his adversary (RED) who has placed obstacles within the rectangle. Optimal strategies for BLUE and RED are derived in several cases, and applications of these results are suggested.

Factoring Absolutely Convergent Series

It is a well known fact that if p > l and 1 / p + l / q = l , every sequence (x.) in 11 (absolutely convergent series) can be factored in the form (x,,)=(u,,v,,) where (u.) is ha 1 p (p-absolutely convergent series) and (v.) is in t q. A related fact is the familiar exercise in advanced calculus that every sequence (x.) in 11 can be factored in the form (x.)= (u.v.) where (u.) is in Co (sequences convergent to 0) and (v.) is again in 11. Our purpose in this paper is to prove a natural generalization of this fact in the setting of K6the sequence spaces [6]. Our main result states essentially that 11 factors through every balanced Banach sequence space and its K6the dual. The proof is not easy and constitutes a nice exercise in non-linear functional analysis. Some consequences of this result concerning the structure of Banach spaces will appear in [9]. To prove the main result (the theorem in Section 2) we first establish a corresponding statement in the finite dimensional case and proceed to the infinite dimensional case via a compactness argument. Throughout this paper we assume our vector spaces to be over the field of real numbers. The complex case of the main theorem, however, follows immediately from the real case.

The upper risk of an inspection agreement

An inspection agreement, contained within a treaty proposal, determines a two-person, zero sum game which we call the implicit game. The value of the implicit game, called the upper risk, is an important parameter of the agreement. The upper risk and other parameters in the solution of the implicit game are useful for evaluating the proposal and comparing it to other proposals. The purpose of this paper is to define the implicit game which arises from an inspection arrangement, define the upper risk and other parameters, and then to illustrate the theory with examples, several of which originated in the analysis of actual inspection proposals.

Ambushing Random Walks III: More Continuous Models

We continue our study of the following game theory problem: a traveler (BLUE) wishes to choose a path across a rectangle from left to right in such a way to avoid certain obstacles in the rectangle (called ambushes) which are chosen by his antagonist (RED). The game theoretic payoff might depend upon whether BLUEs path and the ambushes have a common point, or the payoff may be a function of how far the path proceeds before meeting the first ambush. These ambush games have analogous search games.

Pursuit on a cyclic graph — The symmetric stochastic case

In the cyclic pursuit game the evader, BLUE, and the pursuer, RED, choose one of the vertices of ann point cyclic graph at discrete time 1. If they initially choose the same vertex BLUE receives payoff one. At each subsequent time BLUE may remain where he is or move to an adjacent vertex. RED has the same capability. At no time do RED or BLUE know the others location. The game ends when RED and BLUE arrive at the same vertex. BLUE then receives a payoff equal to the time of this arrival, i.e. the amount of time for which he eludes RED. In this paper we solve this game under the assumption that both RED and BLUE are restricted to stochastic strategies for which each moves right or left with equal probility.

Schauder Decompositions, Approximations and Control Problems

A Schauder decomposition for a Banach space

A Discrete Search Game

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An extension of the Aumann-shapley value concept to functions on arbitrary banach spaces

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