In a stopping rule problem, a real-time player decides to stop or continue at stage n based on the observations up to that stage, but in a k-step look-ahead stopping rule problem, we suppose the player knows k steps ahead. The aim of this Ph.D. dissertation is to study this type of prophet problems for simple random walk, determine the optimal stopping rule and calculate the expected return for them. The optimal one-step look-ahead stopping rule for a finite simple random walk is determined in this work. We also study two infinite horizon stopping rule problems, sum with negative drift problems and discounted sum problems. The optimal one, two and three-step look-ahead stopping rules are introduced for the sum with negative drift problem for simple random walk. We also compare the maximum expected returns and calculate the upper bound for the advantage of the prophet over the decision maker. The last chapter of this dissertation concentrates on the discounted sum problem for simple random walk. Optimal one-step look-ahead stopping rule is defined and lastly we compare the optimal expected return for one-step look-ahead prophet with a real-time decision maker.

This thesis studies two discrete optimization problems: ordering problems in optimal stopping theory and popular matchings. The main goal of this thesis is to find the boundary between NP-hardness and tractability for these problems, and whenever possible, designs polynomial-time algorithms for the easy cases and approximation schemes or prophet inequalities for the hard cases. In the first part of the thesis, we study ordering problems in optimal stopping theory. In the optimal stopping problem, a player is presented with 𝓃 random variables 𝑋₁, . . . , 𝑋n, whose distributions are known to the player, but not their realizations. After observing the realization of 𝑋ᵢ, the player can choose to stop and earn reward 𝑋ᵢ, or reject 𝑋ᵢ and probe the next variable 𝑋ᵢ₊₁. If 𝑋ᵢ is rejected, it cannot be accepted in the future. The goal of the player is to maximize the expected reward at stopping time. If the order of observation is fixed, the player can find the optimal stopping criteria using a dynamic program. In this thesis, we investigate the variant in which the player is able to choose the order of observation. What is the best ordering and what benefits does ordering bring? Chapter 2 introduces the optimal ordering problem in optimal stopping theory. We prove that the problem of finding an optimal ordering is NP-hard even in very restricted cases where the support of each distribution has support on at most three points. Next, we prove an FPTAS for the hardness case and provide a tractable algorithm and a prophet inequality for two-point distributions. Chapter 3 studies the optimal ordering problem when the player can choose 𝑘 > 1 rewards before stopping. We show that finding an optimal static ordering is NP-hard even for very simple two-point distributions. Next, we prove an FPTAS for the hardness case and give prophet inequalities under static and dynamic policies for two-point distributions. In the second part of the thesis, we study popular matchings. Suppose we are given a bipartite graph with independent sets 𝑨 and 𝐵. Each vertex in 𝑨 has a ranked order of preferences on the vertices in 𝐵, and vice versa. A matching 𝑴 is popular if for any other matching 𝑴′, the number of vertices that prefer 𝑴 is at least as much as the number of vertices that prefer 𝑴′. Chapter 4 studies popular matchings. In the first part, we provide a general reduction which, through minor adjustments, proves NP-Hardness for a variety of different questions, including that of finding a max-weight popular matching. In the second part, we restrict our attention to graphs of bounded treewidth and provide a tractable algorithm for finding a max-weight popular matching.

京都大学 / 0048 / 新制・課程博士 / 博士(経済学) / 甲第20145号 / 経博第543号 / 新制||経||280(附属図書館) / 京都大学大学院経済学研究科経済学専攻 / (主査)教授 江上 雅彦, 教授 原 千秋, 准教授 砂川 伸幸 / 学位規則第4条第1項該当 / Doctor of Economics / Kyoto University / DGAM

京都大学 / 0048 / 新制・課程博士 / 博士(経済学) / 甲第21530号 / 経博第598号 / 新制||経||289(附属図書館) / 京都大学大学院経済学研究科経済学専攻 / (主査)教授 江上 雅彦, 教授 西山 慶彦, 准教授 砂川 伸幸 / 学位規則第4条第1項該当 / Doctor of Economics / Kyoto University / DGAM

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This study addresses the application and the effectiveness of radar obstacle sensors for forklift trucks during reverse travel. Two different discriminating radar obstacle sensors with different outputs are evaluated. This study reviews the safety of human exposure to emissions from these radar sensors; documents the field of view obtained from experiments with the two systems; gives the results from experiments with sensors on lift trucks. The influence of obstacle reflectivity, composition and area on the size and shape of the radar detection zone are discussed. An experimental setup for measuring position and velocity of the obstacle crossing the truck path is described. The combination of obstacle sensors required for full coverage of the back of the lift trucks and the mounting height and angle are discussed.

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