Multi-Objective Decision Making. Diederik M. Roijers

via Gaussian processes.

      In the unknown weights scenario, a priori scalarization is impossible, because it would shift the burden of computation toward a point in time where it is not available. The scalarization f is known, and the weights w will become available in the selection phase, where a single policy is selected for execution.

      By contrast, in the decision support scenario (Figure 1.1b), w and f are never made explicit. In fact, scalarization is infeasible throughout the entire decision-making process because of the difficulty of specifying w and/or f. For example, when a community is considering the construction of a new metro line, economists may not be able to accurately compute the economic benefit of reduced commuting times. The users may also have “fuzzy” preferences that defy meaningful quantification. For example, if construction of the new metro line could be made more efficient by building it in such a way that it obstructs a beautiful view, then a human designer may not be able to quantify the loss of beauty. The difficulty of specifying the exact scalarization is especially apparent when the designer is not a single person but a committee or legislative body whose members have different preferences and agendas, such as the politicians and interest groups involved in constructing the metro line. In such a system, the multi-objective planning method is used to calculate a coverage set with respect to the constraints that can safely be imposed on f and w. For example, we can safely assume that gaining value in one objective, without reducing the value in any of the others cannot reduce the utility to the user (i.e., the scalarized value).

      As shown in Figure 1.1b, the decision support scenario proceeds similarly to the unknown weights scenario in the planning phase. In the selection phase, however, the user or users directly select a policy from the coverage set according to their idiosyncratic preferences, rather than explicitly computing a numerical utility by applying the scalarization function to each value vector.

      In the decision support scenario, one could still argue that scalarization before planning (or learning) is possible in principle. For example, the loss of beauty can be quantified by measuring the resulting drop in housing prices in neighborhoods that previously enjoyed an unobstructed view. However, the difficulty with explicit scalarization is not only that doing so may be impractical but, more importantly, that it forces the users to express their preferences in a way that may be inconvenient and unnatural. This is because selecting w requires weighing hypothetical trade-offs, which can be much harder than choosing from a set of actual alternatives. This is a well understood phenomenon in the field of decision analysis [Clemen, 1997], where the standard workflow involves presenting alternatives before soliciting preferences. In the same way, algorithms for multi-objective decision problems can provide critical decision support; rather than forcing the users to specify f and w in advance, these algorithms just prune policies that would not be optimal for any f and w that fit the known constraints on the preferences of the users, and produce a coverage set. Because this coverage set contains optimal solutions for all f and w that fit the known constraints—rather than just all w for a specified f, as in the unknown weights scenario—it offers a range of alternatives from which the users can select, according to preferences whose relative importance is not easily quantified.

      Finally, we present one more scenario, which we call the known weights scenario, that requires explicit multi-objective methods. In this scenario, we assume that w is known at the time of planning and thus scalarization is possible. However, it may be undesirable because of the difficulty of the second step in the conversion. In particular, if f is nonlinear, then the resulting single-objective problem may be much more complex than the original multi-objective problem. As a result, finding the optimal policy may be intractable while the original multi-objective problem is tractable. For example, in multi-objective Markov decision processes (MOMDPs²), which we formally define in Chapter 2, nonlinear scalarization destroys the additivity property on which single-objective solution methods rely (see Section 3.2.3).

      Figure 1.1c depicts the known weights scenario. In contrast to the other scenarios, in the known weights scenario, the multi-objective method only produces one policy, which is then executed, i.e., there is no separate selection phase.

      The scenarios we have presented here require explicit multi-objective methods because a priori scalarization of the multi-objective decision problems, and subsequent solving with standard single-objective methods, does not apply. In this book, we focus on the two multi-policy scenarios, i.e., the unknown weights and decision support scenarios, in which the goal of a multi-objective planning method is to produce a coverage set. From this coverage set, the policy that maximizes user utility is selected in the selection phase.

      Of course, computing a coverage is in general more difficult than finding a single policy, and thus multi-objective methods are typically more expensive than their single-objective counterparts. It is natural then to wonder whether the complications of a multi-objective approach are merited. After all, many single-objective problems are already intractable. In this book, we take a pragmatic perspective: multi-objective methods are not a panacea and are not always the best option, even if the problem is naturally modeled with multiple objectives. If the scalarization weights are known (or can be reasonably estimated) before planning begins, and a priori scalarization does not yield an intractable problem, then converting a multi-objective problem to a single-objective one may be the best option. However, in any of the many cases where such a conversion is not possible or practical, then the multi-objective methods discussed in this book may prove indispensable.

1.2 UTILITY-BASED APPROACH

      The goal of solving all—including multi-objective—decision problems is to maximize user utility. However, in the unknown weights and decision support scenarios, we cannot optimize user utility directly because, at the time when planning or learning takes place, the parameters w of the scalarization function f, which maps the multi-objective values to a scalar utility, are unknown. Therefore, we must compute a coverage set: a set of policies such that, for every possible scalarization, a maximizing policy is in the set (see Definition 3.5).

      In this book, we argue that which policies should be included in the coverage set should be derived from what we know about f. We call this the utility-based approach, in contrast to the axiomatic approach that axiomatically assumes the coverage set is the Pareto front, which we define formally in Chapter 3. In short, the Pareto front is the set of all policies for which there is no other policy that has at least equal value in all objectives and has a higher value in at least one objective. Indeed, the Pareto front contains at least one optimal policy for most, if not all, scalarization functions that occur in practice. However, we argue that, while the Pareto front is sufficient, it is often not necessary. In fact, we show in Chapter 3 that the full Pareto front is required only in the special case where the scalarization function is nonlinear and policies must be deterministic. A utility-based approach thus often results in a much smaller coverage set, which is typically less expensive to compute and easier for the user to examine.

      Another advantage of the utility-based approach is that it is possible to derive how much utility is maximally lost when an exact coverage set cannot be computed [Zintgraf et al., 2015]. Such bounds are often crucial for a meaningful interpretation of the quality of approximate methods for decision-theoretic planning or learning, especially when comparing algorithms [Oliehoek et al., 2015]. Furthermore, the bounds provide insight into the quality and reliability of the selected final policy.

      ¹We provide a formal definition of the term coverage set in Chapter