A tribute to Anatole Beck (1930-2014)

A little over a year since his passing, Adam Ostaszewki, Professor of Mathematics at LSE, remembers Anatole Beck, our friend and colleague, with input from Steve Alpern and Kenneth Binmore. They have also put together a bibliography of Anatole’s work

Optimal trade-off between speed and acuity when searching for a small object

A Searcher seeks to find a stationary Hider located at some point H (not necessarily a node) on a given network Q. The Searcher can move along the network from a given starting point at unit speed, but to actually find the Hider she must pass it while moving at a fixed slower speed (which may depend on the arc). In this “bimodal search game,” the payoff is the first time the Searcher passes the Hider while moving at her slow speed. This game models the search for a small or well hidden object (e.g., a contact lens, improvised explosive device, predator search for camouflaged prey). We define a bimodal Chinese postman tour as a tour of minimum time δ which traverses every point of every arc at least once in the slow mode. For trees and weakly Eulerian networks (networks containing a number of disjoint Eulerian cycles connected in a tree-like fashion) the value of the bimodal search game is δ/2. For trees, the optimal Hider strategy has full support on the network. This differs from traditional search games, where it is optimal for him to hide only at leaf nodes. We then consider the notion of a lucky Searcher who can also detect the Hider with a positive probability q even when passing him at her fast speed. This paper has particular importance for demining problems

Patrolling a pipeline

A pipeline network can potentially be attacked at any point and at any time, but such an attack takes a known length of time. To counter this, a Patroller moves around the network at unit speed, hoping to intercept the attack while it is being carried out. This is a zero sum game between the mobile Patroller and the Attacker, which we analyze and solve in certain case

Extensional and Intensional Strategies

This paper is a contribution to the theoretical foundations of strategies. We first present a general definition of abstract strategies which is extensional in the sense that a strategy is defined explicitly as a set of derivations of an abstract reduction system. We then move to a more intensional definition supporting the abstract view but more operational in the sense that it describes a means for determining such a set. We characterize the class of extensional strategies that can be defined intensionally. We also give some hints towards a logical characterization of intensional strategies and propose a few challenging perspectives

Optimizing randomized patrols

A key operational problem for those charged with the security of vulnerable facilities (such as airports or art galleries) is the scheduling and deployment of patrols. Motivated by the problem of optimizing randomized, and thus unpredictable, patrols, we present a class of patrolling games on graphs. The facility can be thought of as a graph Q of interconnected nodes (e.g. rooms, terminals) and the Attacker can choose to attack any node of Q within a given time T: He requires m consecutive periods there, uninterrupted by the Patroller, to commit his nefarious act (and win). The Patroller can follow any path on the graph. Thus the patrolling game is a win-lose game, where the Value is the probability that the Patroller successfully intercepts an attack, given best play on both sides. We determine analytically optimal (minimax) patrolling strategies for various classes of graphs, and discuss how our results could support decisions about hardening facilities or changing the topology of the terrain to be patrolled

Patrolling games

A key operational problem for those charged with the security of vulnerable facilities (such as airports or art galleries) is the scheduling and deployment of patrols. Motivated by the problem of optimizing randomized, and thus unpredictable, patrols, we present a class of patrolling games. The facility to be patrolled can be thought of as a network or graph Q of interconnected nodes (e.g., rooms, terminals), and the Attacker can choose to attack any node of Q within a given time T . He requires m consecutive periods there, uninterrupted by the Patroller, to commit his nefarious act (and win). The Patroller can follow any path on the graph. Thus, the patrolling game is a win-lose game, where the Value is the probability that the Patroller successfully intercepts an attack, given best play on both sides. We determine analytically either the Value of the game, or bounds on the Value, for various classes of graphs, and we discuss possible extensions and generalizations. Subject classifications: games; noncooperative; military, search/surveillance; decision analysis; risk; networks/graphs. Area of review: Military and Homeland Security. History: Received November 2009; revisions received March 2010, September 2010; accepted November 201

Bilateral street searching in Manhattan (line-of-sight rendezvous on a planar lattice)

We consider the rendezvous problem faced by two mobile agents, initially placed according to a known distribution on intersections in Manhatten (nodes of the integer lattice Z2): We assume they can distinguish streets from avenues (the two axes) but have no common notion of North or East (positive directions along axes). How should they move, from node to adjacent node, so as to minimize the expected time required to �see�each other, to be on a common street or avenue. This problem can be viewed either as a bilateral form (with two players) of the street searching problems of computer science, or a �line-of-sight�version of the rendezvous problem studied in operations research. We show how this problem can be reduced to a Double Alternating Search (DAS) problem in which a single searcher minimizes the time required to nd one of two objects hidden according to known distributions in distinct regions (e.g. a datum held on multiple disks), and we develop a theory for solving the latter problem. The DAS problem generalizes a related one introduced earlier by the author and J. V. Howard. We solve the original rendezvous problem in the case that the searchers are initially no more than four streets or avenues apart

Search games on trees with asymmetric travel times

A point H is hidden in a rooted tree Q which is endowed with asymmetric distances (travel times) between nodes. We determine the randomized search strategy, starting from the root, which minimizes the expected time to reach H, in the worst case. This is equivalent to a zero-sum search game Γ(Q), with minimizing Searcher, maximizing Hider, and payoff equal to the capture time. The worst Hiding distribution (over the leaves) from the Searcher's viewpoint is one where at every node i the probability of each branch is proportional to the minimum time required to tour it from i. The optimal randomized search is a mixture over depth-first searches. We also consider briefly some other networks and the possibility of a mobile Hider. Our formulation with asymmetric travel times generalizes that of Gal [SIAM J. Control Optim., 17 (1979), pp. 99-122] for symmetric travel times and also the search games of Kikuta [J. Oper. Res., 38 (1995), pp. 70-88] and Kikuta and Ruckle [Naval Res. Logist., 41 (1994), pp. 821-831], who posited search costs Ci at each node i which were added to the travel time to obtain the payoff. We also briefly consider what happens if we allow the Searcher (Hider) to start (hide) at any leaf node. We determine when properties found by Dagan and Gal [Networks, 52 (2008), pp. 156-161] for the symmetric version of such games hold in our asymmetric context

Line-of-sight rendezvous

We consider the rendezvous problem faced by two mobile agents, initially placed according to a known distribution on intersections in Manhattan (nodes of the integer lattice Z2). We assume they can distinguish streets from avenues (the two axes) and move along a common axis in each period (both to an adjacent street or both to an adjacent avenue). However they have no common notion of North or East (positive directions along axes). How should they move, from node to adjacent node, so as to minimize the expected time required to ‘see’ each other, to be on a common street or avenue. This is called ‘line-of-sight’ rendezvous. It is equivalent to a rendezvous problem where two rendezvousers attempt to find each other via two means of communication. We show how this problem can be reduced to a double alternating search (DAS) problem in which a single searcher minimizes the time required to find one of two objects hidden according to known distributions in distinct regions (e.g. a datum held on multiple disks), and we develop a theory for solving the latter problem. The DAS problem generalizes a related search problem introduced earlier by the author and J.V. Howard. We solve the original rendezvous problem in the case that the searchers are initially no more than four streets or avenues apart