Concurrent reachability games

Abstract

We consider concurrent two-player games with reachability objectives. In such games, at each round, player 1 and player 2 independently and simultaneously choose moves, and the two choices determine the next state of the game. The objective of player 1 is to reach a set of target states; the objective of player 2 is to prevent this. These are zero-sum games, and the reachability objective is one of the most basic objectives: determining the set of states from which player 1 can win the game is a fundamental problem in control theory and system verification. There are three types of winning states, according to the degree of certainty with which player 1 can reach the target. From type-1 states, player 1 has a deterministic strategy to always reach the target. From type-2 states, player 1 has a randomized strategy to reach the target with probability 1. From type-3 states, player 1 has for every real ε > 0 a randomized strategy to reach the target with probability greater than 1 − ε. We show that for finite state spaces, all three sets of winning states can be computed in polynomial time: type-1 states in linear time, and type-2 and type-3 states in quadratic time. The algorithms to compute the three sets of winning states also enable the construction of the winning and spoiling strategies.

Algorithms for Omega-Regular Games with Imperfect Information

“…To our knowledge, for these games no algorithms (symbolic or not) are present in the literature. Following [7], we refer to winning with probability 1 as almost-sure winning (almost winning, for short), in contrast to sure winning with deterministic strategies. We provide a symbolic Exptime algorithm to compute the set of almost-winning states for games of imperfect information with Büchi objectives (reachability objectives can be obtained as a special case, and for safety objectives almost winning and sure winning coincide).…”

Multi‐Agent Planning in the Presence of Multiple Goals

“…Second, as for the strong cyclic algorithm, we build up a strong cyclic adversarial plan incrementally from the goal states. Alternatively, the algorithm could iteratively prune a largest possible plan as the algorithm suggested in [4]. However, this approach seems less efficient and has to our knowledge never been implemented and experimentally evaluated.…”

“…Such problems have been considered in the AI literature on multiagent systems, in particular, concerning co-operation and negotiation (e.g., [2,19,11,7,5,9]). They have also been studied in game theory and formal verification under various forms (e.g., [15,4]). …”

The Winning Ways of Concurrent Games

“…Melliès et al [1], [9], [10] have done extensive work on concurrent games on asynchronous transition systems; however, determinacy issues were not addressed. Concurrent games on graphs [3], [4] have also been studied in order to solve verification problems for open systems; such games are undetermined in the general case and as a consequence stochastic strategies are used. Of the several treatments of winning conditions in game semantics ours is close to Hyland's [6], which it can be seen as generalizing directly.…”