Multi-Objective Decision Making. Diederik M. Roijers
agents have enormous potential to improve our productivity and quality of life.
In order to successfully complete tasks, autonomous agents require the capacity to reason about their environment and the consequences of their actions, as well as the desirability of those consequences. The field of decision theory uses probabilistic models of the environment, called decision problems, to formalize the tasks about which such agents reason. Decision problems can include the states the environment can be in, the possible actions that agents can perform in each state, and how the state is affected by these actions. Furthermore, the desirability of actions and their effects are modeled as numerical feedback signals. These feedback signals are typically referred to as reward, utility, payoff, or cost functions. Solving a decision problem consists of finding a policy, i.e., rules for how to behave in each state, that is optimal in some sense with respect to these feedback signals.
In most research on planning and learning in decision problems, the desirability of actions and their effects are codified in a scalar reward function [Busoniu et al., 2008, Oliehoek, 2010, Thiébaux et al., 2006, Wiering and Van Otterlo, 2012]. In such scenarios, agents aim to maximize the expected (cumulative) reward over time.
However, many real-world decision problems have multiple objectives. For example, for a computer network we may want to maximize performance while minimizing power consumption [Tesauro et al., 2007]. Similarly, for traffic control, we may want to maximize throughput, minimize latency, maximize fairness to drivers, and minimize noise and pollution. In response to a query, we may want a search engine to provide a balanced list of documents that maximizes both the relevance to the query and the readability of the documents [Van Doorn et al., 2016]. In probabilistic planning, e.g., path planning for robots, we may want to maximize the probability of reaching a goal, while minimizing the expected cost of executing the plan [Bryce, 2008, Bryce et al., 2007]. Countless other real-world scenarios are naturally characterized by multiple objectives.
In all the cases mentioned above, the problem is more naturally expressed using a vector-valued reward function. When the reward function is vector-valued, the value of a policy is also vector-valued. Typically, there is no single policy that maximizes the value for all objectives simultaneously. For example, in a computer network, we can often achieve higher performance by using more power. If we do not know the exact preferences of the user with respect to these objectives, or indeed if these preferences may change over time, it can be crucial to produce a set of policies that offer different trade-offs between the objectives, rather than a single optimal policy.
The field of multi-objective decision making addresses how to formalize and solve decision problems with multiple objectives. This book provides an introductory overview of this field from the perspective of artificial intelligence. In addition to describing multi-objective decision problems and algorithms for solving them, we aim to make explicit the key assumptions that underly work in this area. Such assumptions are often left implicit in the multi-objective literature, which can be a source of confusion, especially for readers new to the topic. We also aim to synthesize these assumptions and offer a coherent, holistic view of the field.
We start by explicitly formulating the motivation for developing algorithms that are specific to multi-objective decision problems.
1.1 MOTIVATION
The existence of multiple objectives in a decision problem does not automatically imply that we require specialized multi-objective methods 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