Reachability and Safety Objectives in Markov Decision Processes on Long but Finite Horizons

Abstract

We consider discrete-time Markov decision processes in which the decision maker is interested in long but finite horizons. First we consider reachability objective: the decision maker’s goal is to reach a specific target state with the highest possible probability. A strategy is said to overtake another strategy, if it gives a strictly higher probability of reaching the target state on all sufficiently large but finite horizons. We prove that there exists a pure stationary strategy that is not overtaken by any pure strategy nor by any stationary strategy, under some condition on the transition structure and respectively under genericity. A strategy that is not overtaken by any other strategy, called an overtaking optimal strategy, does not always exist. We provide sufficient conditions for its existence. Next we consider safety objective: the decision maker’s goal is to avoid a specific state with the highest possible probability. We argue that the results proven for reachability objective extend to this model.

Strategy Complexity of Point Payoff, Mean Payoff and Total Payoff Objectives in Countable MDPs

“…Markov decision processes (MDPs) are a standard model for dynamic systems that exhibit both stochastic and controlled behavior [17]. Applications include control theory [6,1], operations research and finance [2,4,19], artificial intelligence and machine learning [22,20], and formal verification [10,3].…”

“…directly. (2) The mean payoff considers the lim inf of the sequence 1 (3) The total payoff considers the lim inf of the sequence n−1 i=0 r i n∈N , i.e., the sum of all rewards seen so far. For each of the three cases above, the lim inf threshold objective is to maximize the probability that the lim inf of the respective type of sequence is ≥ 0.…”