text: In this paper, we present a link between preference-based and multiobjective sequential decision-making. While transforming a multiobjective problem to a preference-based one is quite natural, the other direction is a bit less obvious. We present how this transformation (from preference-based to multiobjective) can be done under the classic condition that preferences over histories can be represented by additively decomposable utilities and that the decision criterion to evaluate policies in a state is based on expectation. This link yields a new source of multiobjective sequential decision-making problems (i.e., when reward values are unknown) and justifies the use of solving methods developed in one setting in the other one.
  @cite_31: A Markov decision process (MDP) is a general model for solving planning problems under uncertainty. It has been extended to multiobjective MDP to address multicriteria or multiagent problems in which the value of a decision must be evaluated according to several viewpoints, sometimes conflicting. Although most of the studies concentrate on the determination of the set of Pareto-optimal policies, we focus here on a more specialized problem that concerns the direct determination of policies achieving well-balanced tradeoffs. To this end, we introduce a reference point method based on the optimization of a weighted ordered weighted average (WOWA) of individual disachievements. We show that the resulting notion of optimal policy does not satisfy the Bellman principle and depends on the initial state. To overcome these difficulties, we propose a solution method based on a linear programming (LP) reformulation of the problem. Finally, we illustrate the feasibility of the proposed method on two types of planning problems under uncertainty arising in navigation of an autonomous agent and in inventory management.
  @cite_29: In this paper, we propose an exact solution method to generate fair policies in Multiobjective Markov Decision Processes (MMDPs). MMDPs consider n immediate reward functions, representing either individual payoffs in a multiagent problem or rewards with respect to different objectives. In this context, we focus on the determination of a policy that fairly shares regrets among agents or objectives, the regret being defined on each dimension as the opportunity loss with respect to optimal expected rewards. To this end, we propose to minimize the ordered weighted average of regrets (OWR). The OWR criterion indeed extends the minimax regret, relaxing egalitarianism for a milder notion of fairness. After showing that OWR-optimality is state-dependent and that the Bellman principle does not hold for OWR-optimal policies, we propose a linear programming reformulation of the problem. We also provide experimental results showing the efficiency of our approach.
  @cite_32: A Markov Decision Process (MDP) is a general model for solving planning problems under uncertainty. It has been extended to multiobjective MDP to address multicriteria or multiagent problems in which the value of a decision must be evaluated according to several viewpoints, sometimes conflicting. Although most of the studies concentrate on the determination of the set of Pareto-optimal policies, we focus here on a more specialized problem that concerns the direct determination of policies achieving well-balanced tradeoffs. We first explain why this problem cannot simply be solved by optimizing a linear combination of criteria. This leads us to use an alternative optimality concept which formalizes the notion of best compromise solution, i.e. a policy yielding an expected-utility vector as close as possible (w.r.t. Tchebycheff norm) to a reference point. We show that this notion of optimality depends on the initial state. Moreover, it appears that the best compromise policy cannot be found by a direct adaptation of value iteration. In addition, we observe that in some (if not most) situations, the optimal solution can only be obtained with a randomized policy. To overcome all these problems, we propose a solution method based on linear programming and give some experimental results.
  @cite_23: We describe an algorithm for learning in the presence of multiple criteria. Our technique generalizes previous approaches in that it can learn optimal policies for all linear preference assignments over the multiple reward criteria at once. The algorithm can be viewed as an extension to standard reinforcement learning for MDPs where instead of repeatedly backing up maximal expected rewards, we back up the set of expected rewards that are maximal for some set of linear preferences (given by a weight vector, w). We present the algorithm along with a proof of correctness showing that our solution gives the optimal policy for any linear preference function. The solution reduces to the standard value iteration algorithm for a specific weight vector, w.
summary: Still another approach to solve multiobjective problems is to assume the existence of a scalarizing function @math , which, given a vector @math , returns a scalar valuation. Two cases can be considered: @math can be either linear @cite_23 or nonlinear @cite_32 @cite_31 @cite_29 .