This paper proposes incremental preference elicitation methods for multicriteria decision making with a Choquet integral. The Choquet integral is an evaluation function that performs a weighted aggregation of criterion values using a capacity function assigning a weight to any coalition of criteria, thus enabling positive and/or negative interactions among them and covering an important range of possible decision behaviors. However, the specification of the capacity involves many parameters which raises challenging questions, both in terms of elicitation burden and guarantee on the quality of the final recommendation. In this paper, we investigate the incremental elicitation of the capacity through a sequence of preference queries (questions) selected one-by-one using a minimax regret strategy so as to progressively reduce the set of possible capacities until the regret (the worst-case "loss" due to reasoning with only partially specified capacities) is low enough. We propose a new approach designed to efficiently compute minimax regret for the Choquet model and we show how this approach can be used in different settings: 1) the problem of recommending a single alternative, 2) the problem of ranking alternatives from best to worst, and 3) sorting several alternatives into ordered categories. Numerical experiments are provided to demonstrate the practical efficiency of our approach for each of these situations.

In this paper, we study the problem of matching a set of items to a set of agents partitioned into types so as to balance fairness towards the types against overall utility/efficiency. We extend multiple desirable properties of indivisible goods allocation to our model and investigate the possibility and hardness of achieving combinations of these properties, e.g. we prove that maximizing utilitarian social welfare under constraints of typewise envy-freeness up to one item (TEF1) is computationally intractable. We also define a new concept of waste for this setting, show experimentally that augmenting an existing algorithm with a marginal utility maximization heuristic can produce a TEF1 solution with reduced waste, and also provide a polynomial-time algorithm for computing a non-wasteful TEF1 allocation for binary agent-item utilities.

In this paper, we present new results on the fair and efficient allocation of indivisible goods to agents whose preferences correspond to matroid rank functions. This is a versatile valuation class, with several desirable properties (monotonicity, submodularity) which naturally models several real-world domains. We use these properties to our advantage; first, we show that when agent valuations are matroid rank functions, a socially optimal (i.e. utilitarian social welfare-maximizing) allocation that achieves envy-freeness up to one item (EF1) exists and is computationally tractable. We also prove that the Nash welfare-maximizing and the leximin allocations both exhibit this fairness/efficiency combination, by showing that they can be achieved by minimizing any symmetric strictly convex function over utilitarian optimal outcomes. Moreover, for a subclass of these valuation functions based on maximum (unweighted) bipartite matching, we show that a leximin allocation can be computed in polynomial time.

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In this article, we present new results on the fair and efficient allocation of indivisible goods to agents whose preferences correspond to matroid rank functions . This is a versatile valuation class with several desirable properties (such as monotonicity and submodularity), which naturally lends itself to a number of real-world domains. We use these properties to our advantage; first, we show that when agent valuations are matroid rank functions, a socially optimal (i.e., utilitarian social welfare-maximizing) allocation that achieves envy-freeness up to one item (EF1) exists and is computationally tractable. We also prove that the Nash welfare-maximizing and the leximin allocations both exhibit this fairness/efficiency combination by showing that they can be achieved by minimizing any symmetric strictly convex function over utilitarian optimal outcomes. To the best of our knowledge, this is the first valuation function class not subsumed by additive valuations for which it has been established that an allocation maximizing Nash welfare is EF1. Moreover, for a subclass of these valuation functions based on maximum (unweighted) bipartite matching, we show that a leximin allocation can be computed in polynomial time. Additionally, we explore possible extensions of our results to fairness criteria other than EF1 as well as to generalizations of the above valuation classes.

This paper aims to introduce an adaptive preference elicitation method for interactive decision support in sequential decision problems. The Decision Maker's preferences are assumed to be representable by an additive utility, initially unknown or imperfectly known. We first study the determination of possibly optimal policies when admissible utilities are imprecisely defined by some linear constraints derived from observed preferences. Then, we introduce a new approach interleaving elicitation of utilities and backward induction to incrementally determine a near-optimal policy. We propose an interactive algorithm with performance guarantees and describe numerical tests demonstrating the practical efficiency of our approach.

This paper proposes incremental preference elicitation methods for multiobjective state space search. Our approach consists in integrating weight elicitation and search to determine, in a vector-valued state-space graph, a solution path that best fits the Decision Maker's preferences. We first assume that the objective weights are imprecisely known and propose a state space search procedure to determine the set of possibly optimal solutions. Then, we introduce incremental elicitation strategies during the search that use queries to progressively reduce the set of admissible weights until a nearly-optimal path can be identified. The validity of our algorithms is established and numerical tests are provided to test their efficiency both in terms of number of queries and solution times.

We propose an introduction to the use of incremental preference elicitation methods in the field of multiobjective combinatorial optimization. We consider three different optimization problems in vector-valued graphs, namely the shortest path problem, the minimum spanning tree problem and the assignment problem. In each case, the preferences of the decision maker over cost vectors are assumed to be representable by a weighted sum but the weights of criteria are initially unknown. We then explain how to interweave preference elicitation and search in order to quickly determine a near-optimal solution with a limited number of preference queries. This leads us to successively introduce an interactive version of dynamic programming, greedy search, and branch and bound to solve the problems under consideration. We then present numerical tests showing the practical efficiency of these algorithms that achieve a good compromise between the number of queries asked and the solution times.