Network-Formation Games with Regular Objectives

Classical network-formation games are played on a directed graph. Players have reachability objectives, and each player has to select a path satisfying his objective. Edges are associated with costs, and when several players use the same edge, they evenly share its cost. The theoretical and practical aspects of network-formation games have been extensively studied and are well understood. We introduce and study network-formation games with regular objectives. In our setting, the edges are labeled by alphabet letters and the objective of each player is a regular language over the alphabet of labels, given by means of an automaton or a temporal-logic formula. Thus, beyond reachability properties, a player may restrict attention to paths that satisfy certain properties, referring, for example, to the providers of the traversed edges, the actions associated with them, their quality of service, security, etc. Unlike the case of network-formation games with reachability objectives, here the paths selected by the players need not be simple, thus a player may traverse some transitions several times. Edge costs are shared by the players with the share being proportional to the number of times the transition is traversed. We study the existence of a pure Nash equilibrium (NE), convergence of best-response-dynamics, the complexity of finding the social optimum, and the inefficiency of a NE compared to a social-optimum solution. We examine several classes of networks (for example, networks with uniform edge costs, or alphabet of size 1) and several classes of regular objectives. We show that many properties of classical networkformation games are no longer valid in our game. In particular, a pure NE might not exist and the Price of Stability equals the number of players (as opposed to logarithmic in the number of players in the classic setting, where a pure NE always exists). In light of these results, we also present special cases for which the resulting game is more stable.