Abstract

We study two-player reachability games on finite graphs. At each state the interaction between the players is concurrent and there is a stochastic Nature. Players also play stochastically. The literature tells us that 1) Player 𝖡, who wants to avoid the target state, has a positional strategy that maximizes the probability to win (uniformly from every state) and 2) from every state, for every ε > 0, Player 𝖠 has a strategy that maximizes up to ε the probability to win. Our work is two-fold.
First, we present a double-fixed-point procedure that says from which state Player 𝖠 has a strategy that maximizes (exactly) the probability to win. This is computable if Nature’s probability distributions are rational. We call these states maximizable. Moreover, we show that for every ε > 0, Player 𝖠 has a positional strategy that maximizes the probability to win, exactly from maximizable states and up to ε from sub-maximizable states.
Second, we consider three-state games with one main state, one target, and one bin. We characterize the local interactions at the main state that guarantee the existence of an optimal Player 𝖠 strategy. In this case there is a positional one. It turns out that in many-state games, these local interactions also guarantee the existence of a uniform optimal Player 𝖠 strategy. In a way, these games are well-behaved by design of their elementary bricks, the local interactions. It is decidable whether a local interaction has this desirable property.

@InProceedings{bordais_et_al:LIPIcs.CSL.2022.7,
  author =	{Bordais, Benjamin and Bouyer, Patricia and Le Roux, St\'{e}phane},
  title =	{{Optimal Strategies in Concurrent Reachability Games}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15727},
  URN =		{urn:nbn:de:0030-drops-157278},
  doi =		{10.4230/LIPIcs.CSL.2022.7},
  annote =	{Keywords: Concurrent reachability games, Game forms, Optimal strategies}
}