GUBS: a Utility-Based Semantic for Goal-Directed Markov Decision Processes

A key question in stochastic planning is how to evaluate policies when a goal cannot be reached with probability one. Usually, in the Goal-Directed Markov Decision Process (GD-MDP) formalization, the outcome of a policy is summarized on two criteria: probability of reaching a goal state and expected cost to reach a goal state. The dual criterion solution considers a lexicography preference, by prioritizing probability of reaching the goal state and then minimizing the expected cost to goal. Some other solutions, consider only cost by using some math trick to guaranteed that every policy has a finite expected cost. In this paper we show that the lexicography solution does not allow a smooth trade-off between goal and cost, while the expected cost solution does not define a goal-semantic. We propose GUBS (Goals with Utility-Based Semantic), a new model to evaluate policies based on the expected utility theory; this model defines a trade-off between cost and goal by proposing an axiomatization for goal semantics in GD-MDPs. We show that our model can be solved by any continuous state MDP solver and propose an algorithm to solve a special class of GD-MDPs.