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Optimal Search in Discrete Locations: Extensions and New Findings

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Optimal Search in Discrete Locations: Extensions and New Findings Optimal Search in Discrete Locations: Extensions and New Findings Jake Clarkson, M.Sci.(Hons.), M.Res Submitted for the degree of Doctor of Philosophy at Lancaster University. July 2020 Abstract A hidden target needs to be found by a searcher in many real-life situations, some of which involve large costs and significant consequences with failure. Therefore, efficient search methods are paramount. In our search model, the target lies in one of several discrete locations according to some hiding distribution, and the searcher's goal is to discover the target in minimum expected time by making successive searches of individual locations. In Part I of the thesis, the searcher knows the hiding distribution. Here, if there is only one way to search each location, the solution to the search problem, discovered in the 1960s, is simple; search next any location with a maximal probability per unit time of detecting the target. An equivalent solution is derived by viewing the search problem as a multi-armed bandit and following a Gittins index policy. Motivated by modern search technology, we introduce two modes|fast and slow| to search each location. The fast mode takes less time, but the slow mode is more likely to find the target. An optimal policy is difficult to obtain in general, because it requires an optimal sequence of search modes for each location, in addition to a set of sequence-dependent Gittins indices for choosing between locations. For each mode, we identify a sufficient condition for a location to use only that search mode in an optimal policy. For locations meeting neither sufficient condition, an optimal choice of search mode is extremely complicated, depending both on the hiding distribution and the search parameters of the other locations. We propose several heuristic policies motivated by our analysis, and demonstrate their near-optimal performance in an I Abstract II extensive numerical study. In Part II of the thesis, the searcher has only one search mode per location, but does not know the hiding distribution, which is chosen by an intelligent hider who aims to maximise the expected time until the target is discovered. Such a search game, modelled via two-person, zero-sum game theory, is relevant if the target is a bomb, intruder, or, of increasing importance due to advances in technology, a computer hacker. By Part I, if the hiding distribution is known, an optimal counter strategy for the searcher is any corresponding Gittins index policy. To develop an optimal search strategy in the search game, the searcher must account for the hider's motivation to choose an optimal hiding distribution, and consider the set of corresponding Gittins index policies. However, the searcher must choose carefully from this set of Gittins index policies to ensure the same expected time to discover the target regardless of where it is hidden by the hider. As a result, finding an optimal search strategy, or even proving one exists, is difficult. We extend several results for special cases from the literature to the fully-general search game; in particular, we show an optimal search strategy exists and may take a simple form. Using a novel test, we investigate the frequency of the optimality of a particular hiding strategy that gives the searcher no preference over any location at the beginning of the search. Acknowledgements I'll begin by thanking my supervisors, without whom this PhD would not have been possible. First, Kevin. It seems a long time since I mistakenly thought you were called Keith (coincidently, the name of your twin) after my STOR-i welcome day! Since then we've spent many hours discussing challenging problems in your office, but enough about the owners of our respective football clubs; I think we made more progress talking about search problems. Thanks for your invaluable guidance, encouragement and wisdom regarding not only the PhD but also my future career. You were also an excellent tour guide in the Monterey Peninsula, teaching me a lot about, inter alia, wildlife (sorry no whale watching) and Motown. I also encountered some fantastic new fish-based dishes there; who couldn't get swept up in your love of cioppino and, in particular, clam chowder? Monterey brings me on to Kyle, who enriched each NPS visit by being a most-welcoming host. Yet, your influence extended far beyond Monterey trips, thank you for your continual help and meticulous feedback throughout the three years - your input was crucial. In particular, you've helped my writing; now I tend to find that I use less unnecessary words and phrases and, therefore, can be more `to the point', so to speak, without losing any information that I wish to convey to the reader I'm more succinct. I hope to work more with both of you in the future. Trips to NPS and the PhD itself would not have been possible without the financial support of ESPRC, of which I am grateful, and, of course, STOR-i itself, a wonderfully supportive, friendly and vibrant atmosphere in which to do research. I'd like to thank everyone involved at STOR-i, staff and students alike, but in particular Jon Tawn and III Acknowledgements IV the admin team, Kim Wilson, Jen Bull and Wendy Shimmin, who make everything run so smoothly whilst all being so friendly and approachable. I'd especially like to thank them for dealing with my requests regarding lumbar support and ever-changing thesis hand in date. I've had so many enjoyable experiences at STOR-i: dressing up for the Christmas meal, dancing until cramp hits at the STOR-i ball, away days (the good and the `interesting'). I've made lots of great friends who I'd like to thank for enduring my endless tales of buses, bargains and dreams, humouring my latest fad, and supporting me through tougher times. Since I celebrated it recently, in reverse-chronological order of the date of their golden birthday (Paperless Post, 2020), I'd like, in particular, to thank: Graham: no-one else takes such an interest in my dreams (or my kitchen uten- sils...); I raise my eyebrows at others! Sean: at his peak, a fantastic food critic and user of satire, and otherwise still pretty funny. Lucy: our humour never failed to brighten my day, most of it Terribly Silly, some slightly crude, laughing until we're both crazy like a fool! Sam: the most superb, astounding, majestic good sir, someone Hoo's always great for a Chuckle no matter Watt the situation; you think of everything! Thank you also for introducing me to the concept of a `golden birthday'; I'm sure your unbelievable memory for dates will correct me if the order of this list is wrong. Emily: always there for support whenever I needed it, and always up for a good old natter! Never lose your Twinkling Sparkle! Kathryn: a regularly-supportive ear over peppermint tea, always helping put the pieces into place ... Luke: a fount of wise words and stimulating conversation, not least about who should open and how many spinners to play in the next test. Euan: a natural quipster, who kept me from going nuts by turning the seemingly Acknowledgements V mundane into a zesty mix of trpamolining dogs, fetching emails and much more. Next come the only four people I've met who come close to making me look sane. Jack: the man with the best-smelling arms in Devon. Bike-safety gear aside, a fellow seeker of good value; I got the best value with your friendship. David: a bin-diving, graffiti-dishing, ‘fit’-assessing, Spaniard-in-Belgium-spotting, sign-slapping, `vegetarian' Tasmanian devil. Whilst your antics never failed to en- lighten my day, in some tough times you've been a knight in shining armour. Harry: a frequent partner in crime, always egging eachother on in our latest craze. A refreshingly-free spirit with relentless enthusiasm and tr`esBon life advice when required. Er ist alles in Butter (as long as it's grass fed). Harjit: I've mentioned the best Baker, now time for the worst baker, the confident, insightful, whimsical and practical Harjit Hullait. Thanks for all the moments that can only be described as `CH' (and C doesn't stand for conventional); social events were always better when you hadn't food in the oven. And that just leaves . Rob, my LIC1 for 8 (eight) glorious years, where do I start? From the Brummie hipster who I Met on an unquestionably-hot day in October 2011, you've changed a fair bit. You've learnt about the washing machine, how to use pots and pans (as opposed to baking trays and more baking trays), and that AM doesn't end when you go to bed. On a serious note, your excellent sense of humour, willingness to play `guessing games' and continuing interest in my Cricket Captain saves has significantly enriched my time at Lancaster; we've laughed uncontrollably an uncountable number of times, mainly at things an outsider would find most baffling. Away from STOR-i, I'd like to thank my home friends (who've been bored my tales far more than anyone else) Aidan, Imogen, James and Robyn for many jokes, mainly directed (amicably) at eachother. I'd also like to thank Andrew for his Terrific Sass. Finally, I'd like to thank my family, especially my parents, grandparents and sister. 1live-in-chum Acknowledgements VI I don't think they were expecting me to have 23 years of education, but they've been a pillar of strength throughout all of it. I consider myself extremely lucky to have them; they've picked me up when I'm down, and we've regularly laughed with (and often at) eachother in better times.
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