On Satisficing in Quantitative Games

Abstract

Several problems in planning and reactive synthesis can be reduced to the analysis of two-player quantitative graph games. Optimization is one form of analysis. We argue that in many cases it may be better to replace the optimization problem with the satisficing problem, where instead of searching for optimal solutions, the goal is to search for solutions that adhere to a given threshold bound.This work defines and investigates the satisficing problem on a two-player graph game with the discounted-sum cost model. We show that while the satisficing problem can be solved using numerical methods just like the optimization problem, this approach does not render compelling benefits over optimization. When the discount factor is, however, an integer, we present another approach to satisficing, which is purely based on automata methods. We show that this approach is algorithmically more performant – both theoretically and empirically – and demonstrates the broader applicability of satisficing over optimization.

Synthesis from Satisficing and Temporal Goals

“…with discount factor d > 1 is given by DS (W, d) = ∞ i=0 wi d i . The comparator automata for the discounted-sum has been shown to be a safety or co-safety automata when the discount factor d > 1 is an integer, for all values of R, µ and v. It is further known to not form a Büchi automata for non-integer discount factors d > 1, for all values of R, µ and v (Bansal and Vardi 2019;Bansal, Chatterjee, and Vardi 2021).…”

“…the threshold value. The advantage of this approach is that when the discount factor is an integer, an existing synthesis algorithm is both sound and complete (Bansal, Chatterjee, and Vardi 2021). The method builds on novel automata-based technique for quantitative reasoning called comparator automata (Bansal, Chaudhuri, and Vardi 2018a,b).…”