Lexicographic Maximin Research Articles

A lexicographic maximin approach to the selective assessment routing problem

Max-min approaches have been widely applied to address equity as an essential consideration in humanitarian operations. These approaches, however, have a significant drawback of being neutral when it comes to solutions with the same minimum values. These equivalent solutions, from a max-min point of view, might be significantly different. We address this problem using the lexicographic maximin approach, a refinement of the classic max-min approach. We apply this approach in the rapid needs assessment process, which is carried out immediately after the onset of a disaster, to investigate the disaster’s impact on the affected community groups through field visits. We construct routes for an assessment plan to cover community groups, each carrying a distinct characteristic, such that the vector of coverage ratios are maximized. We define the leximin selective assessment problem, which considers the bi-objective optimization of total assessment time and coverage ratio vector maximization. We solve the bi-objective problem by a heuristic approach based on the multi-directional local search framework.

Institutions and their ethical evaluation

An institution is viewed as an arrangement of the society through which various social states emerge as outcomes from the actions and interactions of individuals belonging to the society. Following an earlier contribution by Pattanaik and Suzumura (Oxford Economic Papers 48:194–212, 1996), we subscribe to the view that the ethical desirability of an institution is based on the goodness of the outcomes emerging from individuals’ actions and interactions via the institution as well as the intrinsic attractiveness of the institution itself. In this paper, we explore the structure of an individual’s ethical evaluations of institutions. For this purpose, we discuss how social choices are made through institutions given profiles of individual preferences over social states, propose several a priori properties of institutions and examine their plausibility, and introduce and discuss a lexicographic maximin rule for evaluating institutions ethically.

Egalitarianism in the queueing problem

We use the Lorenz criterion to select new allocation rules for the queueing problem. We first show that the Lorenz criterion selects a unique rule from four distinct subsets of Pareto efficient allocation rules satisfying some fairness concept. We then compare these four rules whenever possible. When we cannot compare, we use the lexicographic maximin and the lexicographic minimax criteria to make a comparison. However, a Lorenz optimal rule does not exist on the set of Pareto efficient and strategy-proof rules.

Fair profit distribution in multi-echelon supply chains via transfer prices

The total profit maximisation of a supply chain may result in an uneven and impractical profit distribution among the members. This work addresses the fair profit distribution within a multi-echelon supply chain using transfer prices. A mixed integer linear programming (MILP) model framework is proposed for the optimal production, distribution and capacity planning of a supply chain of an active ingredient (AI), consisting of AI plants, formulation plants and markets. The transfer prices of the AI from AI plants to formulation plants, and those of products from formulation plants to markets are to be optimised. The proportional and max–min fairness criteria are adopted to define fair profit distributions. Considering bargaining powers of supply chain members, game theoretic solution approaches are developed for fair solutions using Nash bargaining and lexicographic maximin principles. Especially, a hierarchical approach is developed to obtain an approximate optimal fair solution efficiently. The applicability and efficiency of the proposed approaches are demonstrated by two examples, including a real world agrochemical supply chain.

Fair Optimization and Networks: A Survey

Optimization models related to designing and operating complex systems are mainly focused on some efficiency metrics such as response time, queue length, throughput, and cost. However, in systems which serve many entities there is also a need for respecting fairness: each system entity ought to be provided with an adequate share of the system’s services. Still, due to system operations-dependant constraints, fair treatment of the entities does not directly imply that each of them is assigned equal amount of the services. That leads to concepts of fair optimization expressed by the equitable models that represent inequality averse optimization rather than strict inequality minimization; a particular widely applied example of that concept is the so-called lexicographic maximin optimization (max-min fairness). The fair optimization methodology delivers a variety of techniques to generate fair and efficient solutions. This paper reviews fair optimization models and methods applied to systems that are based on some kind of network of connections and dependencies, especially, fair optimization methods for the location problems and for the resource allocation problems in communication networks.

A distributed algorithm for equitable bandwidth allocation for content distribution in a tree network

We present a distributed algorithm for bandwidth allocation for content distribution in tree networks, where the intensive computations are executed independently at the nodes while some information is exchanged among the nodes. The root node has a server that stores and broadcasts multiple programs requested at the nodes through links with limited capacity. The bandwidth allocated for a specific program can be decreased from one link to the next in a path from the root node to an end-node, but it cannot be increased. The objective is to determine equitable bandwidth allocations among all programs across all links, which leads to a formulation of a lexicographic maximin optimization model. The performance function at each of the nodes for a specific program represents customers’ satisfaction as a function of the incoming bandwidth into the node. The distributed algorithm that determines the equitable bandwidth allocations extends the algorithm described in Luss (2010).

Transplant quality and patientsʼ preferences in paired kidney exchange

Paired Kidney Exchange (PKE) programs solve incompatibility problems of donor–patient pairs in living donor kidney transplantation by arranging exchanges of donors among several pairs. Further efficiency gains may emerge if the programs consider the quality of the matches between patients and donors. Limitations on the number of simultaneous required operations imply that every efficient PKE program introduces incentives for the patients to misreport how they rank the option of remaining in dialysis with respect to the available kidneys. Truthfully revealing their preferences is however, the unique protective (lexicographic maximin) strategy for patients under pairwise exchange maximizing PKE programs.

Equitable bandwidth allocation in content distribution networks

AbstractApplications for content distribution over networks, such as Video‐on‐Demand (VOD), are expected to grow significantly over time. Effective bandwidth allocation schemes that can be repeatedly executed must be deployed since new programs are often installed at various servers while other are deleted. We present a model for bandwidth allocation in a content distribution network that consists of multiple trees, where the root of each tree has a server that broadcasts multiple programs throughout the tree. Each network link has limited capacity and may be used by one or more of these trees. The model is formulated as an equitable resource allocation problem with a lexicographic maximin objective function that attempts to provide equitable service performance for all requested programs at the various nodes. The constraints include link capacity constraints and tree‐like ordering constraints imposed on each of the programs. We present an algorithm that provides an equitable solution in polynomial time for certain performance functions. At each iteration, the algorithm solves single‐link maximin optimization problems while relaxing the ordering constraints. The algorithm selects a bottleneck link, fixes various variables at their lexicographic optimal solution while enforcing the ordering constraints, and proceeds with the next iteration. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010

Lexicographic maximin optimisation for fair bandwidth allocation in computer networks

This paper addresses the problem of bandwidth allocation in multi-application computer network environments. Allocations are determined from the solution of a multiple objective optimisation problem under network constraints, where the lexicographic maximin criterion is applied to solve the problem and guarantees fairness and efficiency properties to the solution. An algorithm based on a series of maximum concurrent multicommodity flow subproblems is proposed. Numerical results show the advantage of the approach compared to other traditional bandwidth allocation solutions.

Lexicographic use of Sugeno integrals and monotonicity conditions

The Sugeno integral is very useful but its sensitivity is low. In order to overcome this disadvantage, the author proposes the lexicographic order induced by Sugeno integrals, which is called the LS order. It is an extension of useful orderings such as the ordinary lexicographic order, the leximax (the lexicographic maximax rule), and the leximin (the lexicographic maximin rule). Concerning the sensitivity of orderings, the author discusses three monotonicity conditions: the ordinary monotonicity condition (M), the weak Pareto principle (WP), and the strong Pareto principle (SP). As well known, the Sugeno integral always satisfies (M). The author shows that it satisfies (WP) under a special condition, and that it never satisfies (SP). In addition, he shows necessary and sufficient conditions for an LS order to satisfy (WP) and (SP). By using these conditions one can design LS orders satisfying desired monotonicity conditions.

An extended maximin approach for decision analysis with uncontrollable factors

In this paper an extension to the maximin approach to decision analysis in the presence of uncontrollable factors is proposed. This extension is based on the assumption that probabilities of consequences are known. Using the language of stochastic dominance, one decision alternative is preferred to another if the cumulative distribution function of the first alternative dominates that of the second in some area of low value consequences. This approach is an extension of a standard lexicographic maximin procedure to a case in which decision alternatives are characterised by arbitrary, including continuous, sets of consequences. Applications of the suggested approach to an ‘attack–defence’ type game and to the problems of location of public facilities are discussed.

A characterization of the leximin solution of the bargaining problem

We show that the lexicographic maximin solution is the only solution that satisfies the axioms of Pareto optimality, comprehensive hull axiom and duplicated individual monotonicity.

The Theory of Social Choice since 1951 / Die Theorie kollektiver Entscheidungen seit 1951

Summary Starting point for our analysis is Arrow’s famous impossibility result from 1951. Arrow had proved that within an ordinal framework no non-dictatorial social welfare function exists that satisfies the condition of unrestricted domain, the weak Pareto principle and the requirement of independence of irrelevant alternatives. In the first few sections, we discuss what kinds of positive results can be obtained once we relax one or the other of his axioms. The simple majority rule, for example, is an Arrow social welfare function under a domain restriction that is more general than single-peakedness. The Borda rule violates Arrow’s independence axiom but satisfies a weak independence requirement together with some other rather attractive properties. We next look at preference domains that admit the existence of Arrovian social welfare functions, i. e. aggregation rules fulfilling both the Pareto and the independence condition and being nondictatorial. We also characterize domains for non-manipulable voting procedures. The subsequent section analyzes social aggregation functions that allow for an interpersonal comparison of both utility levels and utility gains. Utilitarian rules and the Rawlsian lexicographic maximin principle are discussed in particular. We finally examine the exercise of individual rights within the social choice context, which means that non-utility information is considered as well.

Efficiency and monotonicity of bargaining solutions

A continuity axiom for bargaining solutions is introduced, which is satisfied by all Pareto optimal and continuous (in the Hausdorff metric) solutions. It is shown by two examples how this axiom can be used to characterize solutions having certain kind of monotonicity properties. One of the solutions is the lexicographic maximin solution. The other is the lexicographic extension of the Kalai-Smorodinsky solution. The former is an efficient (Pareto optimal) extension of the symmetric proportional solution. The latter is an efficient extension of the Kalai-Smorodinsky solution.

A note on the redundancy of an axiom in the Pattanaik-Peleg characterization of the lexicographic maximin extension

P. K. Pattanaik and B. Peleg have provided an axiomatic characterization of the lexicographic maximin extension of an ordering over a finite set to an ordering over the set of all non-empty subsets of that set. This note shows that one of their axioms, namely the Union axiom, is redundant, thus obtaining a more economical characterization composed of the remaining four axioms (which are independent by virtue of the examples given by Pattanaik and Peleg).

An axiomatic characterization of the lexicographic maximin extension of an ordering over a set to the power set

The lexicographic maximin extension of an ordering is an important and widely used tool in social choice theory. We provide an axiomatization of it by means of five axioms. When the basic ordering is linear the following four (independent) axioms are sufficient: (1) Gardenfors principle; (2) Neutrality; (3) Strong Fishburn monotonicity; and (4) Extension. Our result may also have applications in the theory of individual choice under uncertainty.

Fondation axiomatique et comparaison de trois régles d'agrégation de la théorie cardinale des décisions collectives

As a contribution to the present discussion between utilitarians and ‘Rawlsians’, the social choice rule of maximizing the sum of individual utilities, on one hand, and the simple as well as the lexicographic maximin rule, on the other hand, are defined by means of distinct sets of characterizing axioms.

A Theorem on Utilitarianism

In a recent paper [1], d'Aspremont and Gevers establish that if a social welfare functional (SWFL) satisfies several reasonable properties, and if interpersonal comparisons of absolute welfare levels are prohibited, although unit interpersonal comparisons of welfare are not (for a discussion of unit and level welfare comparisons, see Sen [9]), the SWFL must be the principle of utilitarianism. In this paper we derive a similar result when full comparability of welfare is permitted, so that both interpersonal evels and units are significant. By characterizing utilitarianism under full comparability, we avoid the somewhat arbitrary informational basis from which the d'Aspremont-Gevers esult is derived (although we are still, of course, open to criticism of interpersonal comparisons in general). Our result also gives a new slant to the contrast between utilitarianism and the lexicographic maximin principle (Rawls' generalized difference principle [6]), which Hammond [4] has characterized under full comparability. In another paper [5], we point out that the theorem is actually an immediate consequence of a proposition from the theory of decision-making under ignorance. Rather than exploiting here the isomorphism between social choice and individual decision theory, however, we shall use results, as well as terminology, from the d'Aspremont-Gevers paper. Let X be a set of social alternatives containing at least three elements. Let