Efficiency and fairness of resource utilisation under uncertainty

Abstract

The efficient use of resources is a crucial problem of our time. Besides the constraints of efficient usage of scarce resources, in real-world problems the ubiquitous constraint of uncertainty further affects the use or distribution of most resources. Solution approaches are problem-dependent and have various benefits and difficulties. In this work we examine these benefits and difficulties in two different settings of uncertainty, both with their own benefits and difficulties. Moreover, we address the two problems using different techniques applicable to other settings. In the first problem the uncertainty is with respect to the resource itself. In the well-studied problem of fair multi-agent resource allocation it is generally assumed that the quantity of each resource is known a priori. However, in many real-world problems, such as the production of renewable energy which is typically weather-dependent, the exact amount of each resource may not be known at the time of decision making. This work investigates the fair division of a homogeneous, divisible resource where the available amount is given by a probability distribution. Specifically, the notion of ex-ante envy-freeness, where, in expectation, agents weakly prefer their allocation over every other agent’s allocation is considered. This work shows how uncertainty changes the relationship of fairness and efficiency, how the solution space is affected, how difficult the problem becomes, and gives algorithms for the case of two agents with utility function that are linear up to a maximal value. This is achieved by showing how in expectation a higher efficiency can be attained; the worst case might still affect the results; and that the problem is strongly NP-hard. Additionally, we provide two variants of a greedy algorithm for the case of two agents. One variant is optimal for the case of uniform probability distributions over the events. For the case of arbitrary probability distributions, we show that this problems is also NP-hard. Accordingly, we address the possibility of approximation. We show that one variant is not able to approximate all instances. Nevertheless, we show empirically that for realistic instances both variants of the algorithm can approximate the optimal solution. Hence the work lays the foundation for further research into homogeneous resources under uncertainty. In the second problem, the uncertainty comes from the behaviour of an agent and this behaviour is countered by random strategies. In full-knowledge multi-robot adversarial patrolling, a group of robots have to detect an adversary who knows the robots’ strategy. The adversary can easily take advantage of any deterministic patrolling strategy, which necessitates the employment of a randomised strategy. Previous algorithms have to be repeated to calculate the solution for different instances and lack insight into the strategy space. In comparison, this work shows how enumerative combinatorics can be used to provide the closed formulae of the probabilities of detecting the adversary. Hence, it facilitates characterising optimal random defence strategies in comparison to formerly used iterative black-box models. We provide the probability functions for four cases based on open and closed polylines using two different robot movement patterns. Moreover, we show how analysing the structure of the strategy space can further reduce the runtime. Hence, the work introduces a new technique into adversarial patrolling that can be used to improve runtime and foster further research. In conclusion, the work provides progress in two established research areas, and highlights the potential and importance of the consideration of the effects of uncertainty. Foremost, including uncertainty opens up research which is more attuned to real-world problems. Additionally, addressing these problems, including with novel approaches, allows finding (computationally) more efficient solutions

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ORCID for Jan Buermann: orcid.org/0000-0002-4981-6137

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