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ABSTRACT: Scatter hoarders are animals (e.g. squirrels) who cache food (nuts) over a number of sites for later collection. A certain minimum amount of food must be recovered, possibly after pilfering by another animal, in order to survive the winter. An optimal caching strategy is one that maximizes the survival probability, given worst case behaviour of the pilferer. We modify certain 'accumulation games' studied by Kikuta & Ruckle (2000 J. Optim. Theory Appl.) and Kikuta & Ruckle (2001 Naval Res. Logist.), which modelled the problem of optimal diversification of resources against catastrophic loss, to include the depth at which the food is hidden at each caching site. Optimal caching strategies can then be determined as equilibria in a new 'caching game'. We show how the distribution of food over sites and the sitedepths of the optimal caching varies with the animal's survival requirements and the amount of pilfering. We show that in some cases, 'decoy nuts' are required to be placed above other nuts that are buried further down at the same site. Methods from the field of search games are used. Some empirically observed behaviour can be shown to be optimal in our model.Journal of The Royal Society Interface 10/2011; 9(70):86979. DOI:10.1098/rsif.2011.0581  [Show abstract] [Hide abstract]
ABSTRACT: In data analysis problems where the data are represented by vectors of real numbers, it is often the case that some of the datapoints will have “missing values”, meaning that one or more of the entries of the vector that describes the datapoint is not observed. In this paper, we propose a new approach to the imputation of missing binary values. The technique we introduce employs a “similarity measure” introduced by Anthony and Hammer (2006) [1]. We compare experimentally the performance of our technique with ones based on the usual Hamming distance measure and multiple imputation.Discrete Applied Mathematics 06/2011; 159(10159):10401047. DOI:10.1016/j.dam.2011.01.024  [Show abstract] [Hide abstract]
ABSTRACT: We advance and apply the mathematical theory of search games to model the problem faced by a predator searching for prey. Two search modes are available: ambush and cruising search. Some species can adopt either mode, with their choice at a given time traditionally explained in terms of varying habitat and physiological conditions. We present an additional explanation of the observed predator alternation between these search modes, which is based on the dynamical nature of the search game they are playing: the possibility of ambush decreases the propensity of the prey to frequently change locations and thereby renders it more susceptible to the systematic cruising search portion of the strategy. This heuristic explanation is supported by showing that in a new idealized search game where the predator is allowed to ambush or search at any time, and the prey can change locations at intermittent times, optimal predator play requires an alternation (or mixture) over time of ambush and cruise search. Thus, our game is an extension of the wellstudied 'Princess and Monster' search game. Search games are zero sum games, where the payoff is the capture time and neither the Searcher nor the Hider knows the location of the other. We are able to determine the optimal mixture of the search modes when the predator uses a mixture which is constant over time, and also to determine how the mode mixture changes over time when dynamic strategies are allowed (the ambush probability increases over time). In particular, we establish the 'square root law of search predation': the optimal proportion of active search equals the square root of the fraction of the region that has not yet been explored.Journal of The Royal Society Interface 05/2011; 8(64):166572. DOI:10.1098/rsif.2011.0154 
Chapter: Rendezvous Search Games
Wiley Encyclopedia of Operations Research and Management Science, 02/2011; , ISBN: 9780470400531  [Show abstract] [Hide abstract]
ABSTRACT: Motivated by recent work, we establish the Baire Theorem in the broad context afforded by weak forms of completeness implied by analyticity and Kanalyticity, thereby adding to the ‘Baire space recognition literature’ (cf. Aarts and Lutzer (1974) [1], Haworth and McCoy (1977) [43]). We extend a metric result of van Mill, obtaining a generalization of Oxtoby's weak αfavourability conditions (and therefrom variants of the Baire Theorem), in a form in which the principal role is played by Kanalytic (in particular analytic) sets that are ‘heavy’ (everywhere large in the sense of some σideal). From this perspective finetopology versions are derived, allowing a unified view of the Baire Theorem which embraces classical as well as generalized Gandy–Harrington topologies (including the Ellentuck topology), and also various separation theorems. A multipletarget form of the Choquet Banach–Mazur game is a primary tool, the key to which is a restatement of the Cantor Theorem, again in Kanalytic form.Topology and its Applications 02/2011; 158(3158):253275. DOI:10.1016/j.topol.2010.11.001 
Article: Regular variation without limits
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ABSTRACT: Karamata theory (N.H. Bingham et al. (1987) [8, Ch. 1]) explores functions f for which the limit function g(λ):=f(λx)/f(x) exists (as x→∞) and for which g(λ)=λρ subject to mild regularity assumptions on f. Further Karamata theory (N.H. Bingham et al. (1987) [8, Ch. 2]) explores functions f for which the upper limit , as x→∞, remains bounded. Here the usual regularity assumptions invoke boundedness of f* on a Baire nonmeagre/measurable nonnull set, with f Baire/measurable, and the conclusions assert uniformity over compact λsets (implying upper bounds of the form f(λx)/f(x)⩽Kλρ for all large λ, x). We give unifying combinatorial conditions which include the two classical cases, deriving them from a combinatorial semigroup theorem. We examine character degradation in the passage from f to f* (using some standard descriptive set theory) and thus identify natural classes in which the theory may be established.Journal of Mathematical Analysis and Applications 10/2010; 370(2370):322338. DOI:10.1016/j.jmaa.2010.04.013  [Show abstract] [Hide abstract]
ABSTRACT: This paper extends the topological theory of regular variation of the slowly varying case of Bingham and Ostaszewski (2010) [5] to the regularly varying functions between metric groups, viewed as normed groups (see also Bingham and Ostaszewski (2010) [6]). This employs the language of topological dynamics, especially flows and cocycles. In particular we show that regularly varying functions obey the chain rule and in the noncommutative context we characterize pairs of regularly varying functions whose product is regularly varying. The latter requires the use of a ‘differential modulus’ akin to the modulus of Haar integration.Topology and its Applications 08/2010; 157(13157):20242037. DOI:10.1016/j.topol.2010.04.002  [Show abstract] [Hide abstract]
ABSTRACT: Motivated by the Category Embedding Theorem, as applied to convergent automorphisms (Bingham and Ostaszewski (in press) [11]), we unify and extend the multivariate regular variation literature by a reformulation in the language of topological dynamics. Here the natural setting are metric groups, seen as normed groups (mimicking normed vector spaces). We briefly study their properties as a preliminary to establishing that the Uniform Convergence Theorem (UCT) for Baire, groupvalued slowlyvarying functions has two natural metric generalizations linked by the natural duality between a homogenous space and its group of homeomorphisms. Each is derivable from the other by duality. One of these explicitly extends the (topological) group version of UCT due to Bajšanski and Karamata (1969) [4] from groups to flows on a group. A multiplicative representation of the flow derived in Ostaszewski (2010) [45] demonstrates equivalence of the flow with the earlier group formulation. In companion papers we extend the theory to regularly varying functions: we establish the calculus of regular variation in Bingham and Ostaszewski (2010) [13] and we extend to locally compact, σcompact groups the fundamental theorems on characterization and representation (Bingham and Ostaszewski (2010) [14]). In Bingham and Ostaszewski (2009) [15], working with topological flows on homogeneous spaces, we identify an index of regular variation, which in a normedvector space context may be specified using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure.Topology and its Applications 08/2010; 157(13157):19992013. DOI:10.1016/j.topol.2010.04.001  [Show abstract] [Hide abstract]
ABSTRACT: This paper investigates fundamental theorems of regular variation (Uniform Convergence, Representation, and Characterization Theorems) some of which, in the classical setting of regular variation in R, rely in an essential way on the additive semigroup of natural numbers N (e.g. de Bruijn's Representation Theorem for regularly varying functions). Other such results include Goldie's direct proof of the Uniform Convergence Theorem and Seneta's version of Kendall's theorem connecting sequential definitions of regular variation with their continuous counterparts (for which see Bingham and Ostaszewski (2010) [13]). We show how to interpret these in the topological group setting established in Bingham and Ostaszewski (2010) [12] as connecting Nflow and Rflow versions of regular variation, and in so doing generalize these theorems to Rd. We also prove a flow version of the classical Characterization Theorem of regular variation.Topology and its Applications 08/2010; 157(13157):20142023. DOI:10.1016/j.topol.2010.04.003 
Article: π options
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ABSTRACT: We consider a discretionary stopping problem that arises in the context of pricing a class of perpetual Americantype call options, which include the perpetual American, Russian and lookbackAmerican call options as special cases. We solve this genuinely twodimensional optimal stopping problem by means of an explicit construction of its value function. In particular, we fully characterise the freeboundary that provides the optimal strategy, and which involves the analysis of a highly nonlinear ordinary differential equation (ODE). In accordance with other optimal stopping problems involving a running maximum process that have been studied in the literature, it turns out that the associated variational inequality has an uncountable set of solutions that satisfy the socalled principle of smooth fit.Stochastic Processes and their Applications 07/2010; 8(7120):10331059. DOI:10.1016/j.spa.2010.02.008
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