arXiv subjects

Thomas Lidbetter

Publications and source records attributed to Thomas Lidbetter.

At least 19 recordsLinked to original sources

arXiv2026

Quiz Show Games: Searching with Bimodal Hiding

We consider a quiz show game in which a contestant is presented with a sequence of questions. Each time the contestant answers a question correctly, she receives a prize and proceeds to the next question; the probability of answering each question correctly is given. If the contestant answers a question incorrectly, she receives a consolation prize and the game ends. The contestant's problem of determining the optimal order in which to answer questions, for known fixed parameters, is a classic one studied in Kadane (1969), and admits a simple index-based solution. We consider game-theoretic versions of this problem in which a game show host can choose how to allocate a fixed prize budget. Our models are motivated by operational search problems in national security involving reconnaissance missions and inspecting for evidence of nuclear enrichment, as well as certain scheduling problems. We study three variants of the game, corresponding to different ways in which the host can distribute the prize money. For the first variant, we provide complete closed-form solutions, including equilibrium strategies and the value of the game. We reduce the second variant to a game solved in the literature. For the third variant, we obtain partial results by analyzing a more general game with a geometric structure.

cs.GT↗

arXiv2025

Alignment Games

This paper introduces alignment games, a new class of zero-sum games modeling strategic interventions where effectiveness depends on alignment with an underlying hidden state. Motivated by operational problems in medical diagnostics, economic sanctions, and resource allocation, this framework features two players, a Hider and a Searcher, who choose subsets of a given space. Payoffs are determined by their misalignment (symmetric difference), explicitly modeling the trade-off between commission errors (unnecessary action) and omission errors (missed targets), given by a cost function and a penalty function, respectively. We provide a comprehensive theoretical analysis, deriving closed-form equilibrium solutions that contain interesting mathematical properties based on the game's payoff structure. When cost and penalty functions are unequal, optimal strategies are consistently governed by cost-penalty ratios. On the unit circle, optimal arc lengths are direct functions of this ratio, and in discrete games, optimal choice probabilities are proportional to element-specific ratios. When costs are equal, the solutions exhibit rich structural properties and sharp threshold behaviors. On the unit interval, this manifests as a geometric pattern of minimal covering versus maximal non-overlapping strategies. In discrete games with cardinality constraints, play concentrates on the highest-cost locations, with solutions changing discontinuously as parameters cross critical values. Our framework extends the theory of geometric and search games and is general enough that classical models, such as Matching Pennies, emerge as special cases. These results provide a new theoretical foundation for analyzing the strategic tension between comprehensive coverage and precise targeting under uncertainty.

math.OC↗

arXiv2025

A review of minimum cost box searching games

We consider a class of zero-sum search games in which a Hider hides one or more target among a set of $n$ boxes. The boxes may require differing amount of time to search, and detection may be imperfect, so that there is a certain probability that a target may not be found when a box is searched, even when it is there. A Searcher must choose how to search the boxes sequentially, and wishes to minimize the expected time to find the target(s), whereas the Hider wishes to maximize this payoff. We review some known solutions to different cases of this game.

math.OC↗

arXiv2024

A Booby Trap Game

This paper presents a booby trap game played between a defender and an attacker on a search space, which may be a compact subset of Euclidean space or a network. The defender has several booby traps and chooses where to plant them. The attacker, aware of the presence of these booby traps but not their locations, chooses a subset of the space and collects a reward equal to the measure of the subset. If the attacker does not encounter any booby traps, then the attacker keeps the reward; otherwise, the attacker gets nothing. The attacker's objective is to maximize the expected reward, while the defender's objective is to minimize it. We solve this game in the case that the search space is a compact subset of Euclidean space, and then turn our attention to the case where the search space is a network in which the attacker must choose a connected subset of the network. We solve the game when the network is a circle or a line. For the case of one booby trap, we solve the game for 2-connected networks, and when the network is a tree we present an upper bound and a lower bound for the value of the game whose ratio is at most 27/25. We also present an optimal solution for each player in a few cases where the tree is a star network.

math.OC↗

arXiv2024

Search Games with Predictions

We introduce the study of search games between a mobile Searcher and an immobile Hider in a new setting in which the Searcher has some potentially erroneous information, i.e., a prediction on the Hider's position. The objective is to establish tight tradeoffs between the consistency of a search strategy (i.e., its worst case expected payoff assuming the prediction is correct) and its robustness (i.e., the worst case expected payoff with no assumptions on the quality of the prediction). Our study is the first to address the full power of mixed (randomized) strategies; previous work focused only on deterministic strategies, or relied on stochastic assumptions that do not guarantee worst-case robustness in adversarial situations. We give Pareto-optimal strategies for three fundamental problems, namely searching in discrete locations, searching with stochastic overlook, and searching in the infinite line. As part of our contribution, we provide a novel framework for proving optimal tradeoffs in search games which is applicable, more broadly, to any two-person zero-sum games in learning-augmented settings.

cs.GT↗

arXiv2023

A Local Search Algorithm for the Min-Sum Submodular Cover Problem

We consider the problem of solving the Min-Sum Submodular Cover problem using local search. The Min-Sum Submodular Cover problem generalizes the NP-complete Min-Sum Set Cover problem, replacing the input set cover instance with a monotone submodular set function. A simple greedy algorithm achieves an approximation factor of 4, which is tight unless P=NP [Streeter and Golovin, NeurIPS, 2008]. We complement the greedy algorithm with analysis of a local search algorithm. Building on work of Munagala et al. [ICDT, 2005], we show that, using simple initialization, a straightforward local search algorithm achieves a $(4+ε)$-approximate solution in time $O(n^3\log(n/ε))$, provided that the monotone submodular set function is also second-order supermodular. Second-order supermodularity has been shown to hold for a number of submodular functions of practical interest, including functions associated with set cover, matching, and facility location. We present experiments on two special cases of Min-Sum Submodular Cover and find that the local search algorithm can outperform the greedy algorithm on small data sets.

cs.DS↗

arXiv2023

Optimal Pure Strategies for a Discrete Search Game

Consider a two-person zero-sum search game between a Hider and a Searcher. The Hider chooses to hide in one of $n$ discrete locations (or "boxes") and the Searcher chooses a search sequence specifying which order to look in these boxes until finding the Hider. A search at box $i$ takes $t_i$ time units and finds the Hider - if hidden there - independently with probability $q_i$, for $i=1,\ldots,n$. The Searcher wants to minimize the expected total time needed to find the Hider, while the Hider wants to maximize it. It is shown in the literature that the Searcher has an optimal search strategy that mixes up to $n$ distinct search sequences with appropriate probabilities. This paper investigates the existence of optimal pure strategies for the Searcher - a single deterministic search sequence that achieves the optimal expected total search time regardless of where the Hider hides. We identify several cases in which the Searcher has an optimal pure strategy, and several cases in which such optimal pure strategy does not exist. An optimal pure search strategy has significant practical value because the Searcher does not need to randomize their actions and will avoid second guessing themselves if the chosen search sequence from an optimal mixed strategy does not turn out well.

cs.DM↗

arXiv2023

The Search and Rescue Game on a Cycle

We consider a search and rescue game introduced recently by the first author. An immobile target or targets (for example, injured hikers) are hidden on a graph. The terrain is assumed to dangerous, so that when any given vertex of the graph is searched, there is a certain probability that the search will come to an end, otherwise with the complementary {\em success probability} the search can continue. A Searcher searches the graph with the aim of finding all the targets with maximum probability. Here, we focus on the game in the case that the graph is a cycle. In the case that there is only one target, we solve the game for equal success probabilities, and for a class of games with unequal success probabilities. For multiple targets and equal success probabilities, we give a solution for an adaptive Searcher and a solution in a special case for a non-adaptive Searcher. We also consider a continuous version of the model, giving a full solution for an adaptive Searcher and approximately optimal solutions in the non-adaptive case.

cs.DS↗

arXiv2023

Search for an Immobile Hider on a Binary Tree with Unreliable Locational Information

Adversarial search of a network for an immobile Hider (or target) was introduced and solved for rooted trees by Gal (1979). In this zero-sum game, a Hider picks a point to hide on the tree and a Searcher picks a unit speed trajectory starting at the root. The payoff (to the Hider) is the search time. In Gal's model (and many subsequent investigations), the Searcher receives no additional information after the Hider chooses his location. In reality, the Searcher will often receive such locational information. For homeland security, mobile sensors on vehicles have been used to locate radioactive material stashed in an urban environment. In a military setting, mobile sensors can detect chemical signatures from land mines. In predator-prey search, the predator often has specially attuned senses (hearing for wolves, vision for eagles, smell for dogs, sonar for bats, pressure sensors for sharks) that may help it locate the prey. How can such noisy locational information be used by the Searcher to modify her route? We model such information as signals which indicate which of two branches of a binary tree should be searched first, where the signal has a known accuracy p<1. Our solution calculates which branch (at every branch node) is favored, meaning it should always be searched first when the signal is in that direction. When the signal is in the other direction, we calculate the probability the signal should be followed. Compared to the optimal Hider strategy in the classic search game of Gal, the Hider's optimal distribution for this model is more skewed towards leaf nodes that are further from the root.

cs.DM↗

arXiv2023

Optimal Patrolling Strategies for Trees and Complete Networks

We present solutions to a continuous patrolling game played on network. In this zero-sum game, an Attacker chooses a time and place to attack a network for a fixed amount of time. A Patroller patrols the network with the aim of intercepting the attack with maximum probability. Our main result is the proof of a recent conjecture on the optimal patrolling strategy for trees. The conjecture asserts that a particular patrolling strategy called the E-patrolling strategy is optimal for all tree networks. The conjecture was previously known to be true in a limited class of special cases. The E-patrolling strategy has the advantage of being straightforward to calculate and implement. We prove the conjecture by presenting $\varepsilon$-optimal strategies for the Attacker which provide upper bounds for the value of the game that come arbitrarily close to the lower bound provided by the E-patrolling strategy. We also solve the patrolling game in some cases for complete networks.

cs.GT↗

arXiv2022

Continuous Patrolling Games

We study a patrolling game played on a network $Q$, considered as a metric space. The Attacker chooses a point of $Q$ (not necessarily a node) to attack during a chosen time interval of fixed duration. The Patroller chooses a unit speed path on $Q$ and intercepts the attack (and wins) if she visits the attacked point during the attack time interval. This zero-sum game models the problem of protecting roads or pipelines from an adversarial attack. The payoff to the maximizing Patroller is the probability that the attack is intercepted. Our results include the following: (i) a solution to the game for any network $Q$, as long as the time required to carry out the attack is sufficiently short, (ii) a solution to the game for all tree networks that satisfy a certain condition on their extremities, and (iii) a solution to the game for any attack duration for stars with one long arc and the remaining arcs equal in length. We present a conjecture on the solution of the game for arbitrary trees and establish it in certain cases.

cs.DM↗

arXiv2021

A Game Theoretic Approach to a Problem in Polymatroid Maximization

We consider the problem of maximizing the minimum (weighted) value of all components of a vector over a polymatroid. This is a special case of the lexicographically optimal base problem introduced and solved by Fujishige. We give an alternative formulation of the problem as a zero-sum game between a maximizing player whose mixed strategy set is the base of the polymatroid and a minimizing player whose mixed strategy set is a simplex. We show that this game and three variations of it unify several problems in search, sequential testing and queuing. We give a new, short derivation of optimal strategies for both players and an expression for the value of the game. Furthermore, we give a characterization of the set of optimal strategies for the minimizing player and we consider special cases for which optimal strategies can be found particularly easily.

math.OC↗

arXiv2021

A General Framework for Approximating Min Sum Ordering Problems

We consider a large family of problems in which an ordering (or, more precisely, a chain of subsets) of a finite set must be chosen to minimize some weighted sum of costs. This family includes variations of Min Sum Set Cover (MSSC), several scheduling and search problems, and problems in Boolean function evaluation. We define a new problem, called the Min Sum Ordering Problem (MSOP) which generalizes all these problems using a cost and a weight function defined on subsets of a finite set. Assuming a polynomial time $α$-approximation algorithm for the problem of finding a subset whose ratio of weight to cost is maximal, we show that under very minimal assumptions, there is a polynomial time $4 α$-approximation algorithm for MSOP. This approximation result generalizes a proof technique used for several distinct problems in the literature. We apply this to obtain a number of new approximation results.

cs.DS↗

arXiv2020

The Largest-Z-ratio-First algorithm is 0.8531-approximate for scheduling unreliable jobs on m parallel machines

In this paper we analyze the worst-case performance of a greedy algorithm called Largest-Z-ratio-First for the problem of scheduling unreliable jobs on m parallel machines. Each job is characterized by a success probability and a reward earned in the case of success. In the case of failure, the jobs subsequently sequenced on that machine cannot be performed. The objective is to maximize the expected reward. We show the algorithm provides an approximation ratio of approximately 0.853196, and that the bound is tight.

math.OC↗

arXiv2020

Search and Delivery Man Problems: When Are Depth-First Paths Optimal?

Let h be a probability measure on the nodes and arcs of a network Q, viewed either as the location of a hidden object to be found or as the continuous distribution of customers receiving packages. We wish to find a trajectory starting from a specified root, or depot O that minimizes the expected search or delivery time. We call such a trajectory optimal. When Q is a tree, we ask for which h there is an optimal trajectory that is depth-first, and we find sufficient conditions and in some cases necessary and sufficient conditions on h. A consequence of our analysis is a determination of the optimal depot location in the Delivery Man Problem, correcting an error in the literature. We concentrate mainly on the search problem, with the Delivery Man Problem arising as a special case.

math.OC↗

arXiv2020

A Search Game on a Hypergraph with Booby Traps

A set of n boxes, located on the vertices of a hypergraph G, contain known but different rewards. A Searcher opens all the boxes in some hyperedge of G with the objective of collecting the maximum possible total reward. Some of the boxes, however, are booby trapped. If the Searcher opens a booby trapped box, the search ends and she loses all her collected rewards. We assume the number k of booby traps is known, and we model the problem as a zero-sum game between the maximizing Searcher and a minimizing Hider, where the Hider chooses k boxes to booby trap and the Searcher opens all the boxes in some hyperedge. The payoff is the total reward collected by the Searcher. This model could reflect a military operation in which a drone gathers intelligence from guarded locations, and a booby trapped box being opened corresponds to the drone being destroyed or incapacitated. It could also model a machine scheduling problem, in which rewards are obtained from successfully processing jobs but the machine may crash. We solve the game when G is a 1-uniform hypergraph (the hyperedges are all singletons), so the Searcher can open just 1 box. When G is the complete hypergraph (containing all possible hyperedges), we solve the game in a few cases: (1) same reward in each box, (2) k=1, and (3) n=4 and k=2. The solutions to these few cases indicate that a general simple, closed form solution to the game appears unlikely.

math.OC↗

arXiv2019

Search and Rescue in the Face of Uncertain Threats

We consider a search problem in which one or more targets must be rescued by a search party, or Searcher. The targets may be survivors of some natural disaster, or prisoners held by an adversary. The targets are hidden among a finite set of locations, but when a location is searched, there is a known probability that the search will come to an end, perhaps because the Searcher becomes trapped herself, or is captured by the adversary. If this happens before all the targets have been recovered, then the rescue attempt is deemed a failure. The objective is to find the search that maximizes the probability of recovering all the targets. We present and solve a game theoretic model for this problem, by placing it in a more general framework that encompasses another game previously introduced by the author. We also consider an extension to the game in which the targets are hidden on the vertices of a graph. In the case that there is only one target, we give a solution of the game played on a tree.

cs.DS↗

arXiv2019

Competitive Search in a Network

We study the classic problem in which a Searcher must locate a hidden point, also called the Hider in a network, starting from a root point. The network may be either bounded or unbounded, thus generalizing well-known settings such as linear and star search. We distinguish between pathwise search, in which the Searcher follows a continuous unit-speed path until the Hider is reached, and expanding search, in which, at any point in time, the Searcher may restart from any previously reached point. The former has been the usual paradigm for studying search games, whereas the latter is a more recent paradigm that can model real-life settings such as hunting for a fugitive, demining a field, or search-and-rescue operations. We seek both deterministic and randomized search strategies that minimize the competitive ratio, namely the worst-case ratio of the Hider's discovery time, divided by the shortest path to it from the root. Concerning expanding search, we show that a simple search strategy that applies a "waterfilling" principle has optimal deterministic competitive ratio; in contrast, we show that the optimal randomized competitive ratio is attained by fairly complex strategies even in a very simple network of three arcs. Motivated by this observation, we present and analyze an expanding search strategy that is a 5/4 approximation of the randomized competitive ratio. Our approach is also applicable to pathwise search, for which we give a strategy that is a 5 approximation of the randomized competitive ratio, and which improves upon strategies derived from previous work.

math.OC↗