to solve it. On the contrary, if the decision problem can be scalarized, i.e., the vector-valued reward function can be converted to a scalar reward function, the problem may be solvable with existing single-objective methods. Such a conversion involves two steps [Roijers et al., 2013a]. The first step is to specify a scalarization function that expresses the utility of the user for different trade-offs between the objectives.
Definition 1.1 A scalarization function f is a function that maps a multi-objective value of a policy π of a decision problem, Vπ, to a scalar value
where w is a weight vector that parameterizes f.
The second step of the conversion is to define a single-objective version of the decision problem such that the utility of each policy π equals the scalarized value of the original multi-objective decision problem
Though it is rarely stated explicitly, all research on automated multi-objective decision making rests on the premise that there are decision problems for which performing one or both of these conversion steps is impossible, infeasible, or undesirable at the moment at which planning or learning must be performed. Here, we discuss three scenarios, illustrated in Figure 1.1, where this is the case, and specialized multi-objective methods are therefore preferable. In these scenarios, we assume a planning setting. However, in Chapter 6, we briefly address the learning setting.
Figure 1.1a depicts the unknown weights scenario. In this scenario, w is unknown at the moment when planning must occur: the planning phase. For example, consider a company that mines different resources from different mines spread out through a mountain range. The people who work for the company live in villages at the foot of the mountains. In Figure 1.2, we depict the problem this company faces: in the morning, one van per village needs to transport workers from that village to a nearby mine, where various resources can be mined. Different mines yield different quantities of resource per worker. The market prices per unit of resource vary through a stochastic process and every price change can affect the optimal assignment of vans. Furthermore, the expected price variation increases with time. It is therefore critical to act based on the latest possible price information in order to maximize performance.
Figure 1.1: The three motivating scenarios for multi-objective decision-theoretic planning: (a) the unknown weights scenario, (b) the decision support scenario, and (c) the known weights scenario.
Figure 1.2: Mining company example.
Because computing the optimal van assignment takes time, computing the optimal assignment with each price change is impossible. Therefore, we need a multi-objective method that computes a set containing an optimal solution for every possible value of the prices, w. We call such a set a coverage set, as it “covers” all possible preferences of the user (i.e., the possible prices) with respect to the objectives (as specified by f).1 Although computing a coverage set is computationally more expensive than computing a single optimal policy for a given price, it needs to be done only once. Furthermore, the planning phase can take place in advance, when more computational resources are available.
In the selection phase, when the prices (w) are revealed and we want to use as little computation time as possible, we can use the coverage set to determine the best policy by simple maximization. Finally, in the execution phase, the selected policy is employed.
In this book, we focus on algorithms for the planning/learning phase. For the selection phase, specialized preference elicitation algorithms for selecting the policy with the optimal utility from the coverage set may be necessary when the coverage set is large. For example, Chu and Ghahramani [2005], propose an approach to preference elicitation 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
