[Show abstract][Hide abstract] 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 site-depths 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):869-79.
[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 well-studied 'Princess and Monster' search game. Search games are zero sum games, where the pay-off 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):1665-72.
[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 K-analyticity, thereby adding to the ‘Baire space recognition literature’ (cf. Aarts and Lutzer (1974) , Haworth and McCoy (1977) ). We extend a metric result of van Mill, obtaining a generalization of Oxtoby's weak α-favourability conditions (and therefrom variants of the Baire Theorem), in a form in which the principal role is played by K-analytic (in particular analytic) sets that are ‘heavy’ (everywhere large in the sense of some σ-ideal). From this perspective fine-topology 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 multiple-target form of the Choquet Banach–Mazur game is a primary tool, the key to which is a restatement of the Cantor Theorem, again in K-analytic form.
[Show abstract][Hide abstract] ABSTRACT: We consider a discretionary stopping problem that arises in the context of pricing a class of perpetual American-type call options, which include the perpetual American, Russian and lookback-American call options as special cases. We solve this genuinely two-dimensional optimal stopping problem by means of an explicit construction of its value function. In particular, we fully characterise the free-boundary 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 so-called principle of smooth fit.
Stochastic Processes and their Applications. 01/2010;
[Show abstract][Hide abstract] ABSTRACT: Problems of matching have long been studied in the operations research literature (assignment problem, secretary problem, stable marriage problem). All of these consider a centralized mechanism whereby a single decision maker chooses a complete matching which optimizes some criterion. This paper analyzes a more realistic scenario in which members of the two groups (buyers–sellers, employers–workers, males–females) randomly meet each other in pairs (interviews, dates) over time and form couples if there is mutual agreement to do so. We assume members of each group have common preferences over members of the other group. Generalizing an earlier model of Alpern and Reyniers [Alpern, S., Reyniers, D.J., 2005. Strategic mating with common preferences. J. Theor. Biol. 237, 337–354], we assume that one group (called males) is r times larger than the other, r⩾1. Thus all females, but only 1/r of the males, end up matched. Unmatched males have negative utility -c. We analyze equilibria of this matching game, depending on the parameters r and c. In a region of (r,c) space with multiple equilibria, we compare these, and analyze their ‘efficiency’ in several respects. This analysis should prove useful for designers of matching mechanisms who have some control over the sex ratio (e.g. by capping numbers of males at a ‘singles event’or by having ‘ladies free’ nights) or the nonmating cost c (e.g. tax benefits to married couples).
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: We present a closed form solution to be considered in Kramkov and Mordecki [Kramkov, D.O., Mordecki, E., 1994. Integral option. Theory of Probability and its Applications 39 (1), 201-211] optimal stopping problem for the case of geometric compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problem to an integro-differential free-boundary problem and solving the latter by using continuous and smooth fit. The result can be interpreted as pricing perpetual integral options in a model with jumps.
Statistics [?] Probability Letters 01/2008; 78(16):2623-2631.
[Show abstract][Hide abstract] ABSTRACT: Strategies in a stochastic game are δ-perfect if the induced one-stage games have certain δ-equilibrium properties. In special cases the existence of δ-perfect strategies for all positive δ implies the existence of ϵ-equilibria for every positive ϵ. Using this approach we prove the existence of ϵ-equilibria for every positive ϵ for a special class of quitting games. The proof reveals that more general proofs for the existence of ϵ-equilibria in stochastic games must involve the topological structure of how the equilibria of one-stage games are related to changes in the payoffs.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we use reinforcement learning (RL) as a tool to study price dynamics in an electronic retail market consisting of two competing sellers, and price sensitive and lead time sensitive customers. Sellers, offering identical products, compete on price to satisfy stochastically arriving demands (customers), and follow standard inventory control and replenishment policies to manage their inventories. In such a generalized setting, RL techniques have not previously been applied. We consider two representative cases: 1) no information case, were none of the sellers has any information about customer queue levels, inventory levels, or prices at the competitors; and 2) partial information case, where every seller has information about the customer queue levels and inventory levels of the competitors. Sellers employ automated pricing agents, or pricebots, which use RL-based pricing algorithms to reset the prices at random intervals based on factors such as number of back orders, inventory levels, and replenishment lead times, with the objective of maximizing discounted cumulative profit. In the no information case, we show that a seller who uses Q-learning outperforms a seller who uses derivative following (DF). In the partial information case, we model the problem as a Markovian game and use actor-critic based RL to learn dynamic prices. We believe our approach to solving these problems is a new and promising way of setting dynamic prices in multiseller environments with stochastic demands, price sensitive customers, and inventory replenishments
IEEE Transactions on Systems Man and Cybernetics Part C (Applications and Reviews) 02/2006;
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