Network-Formation Games with Regular Objectives

Guy Avni, Orna Kupferman, Tami Tamir

Abstract:

Classical network-formation games are played on a directed graph. Players have reach-
ability 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 al-
phabet 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, be-
yond 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 exis-
tence 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, net-
works with uniform edge costs, or alphabet of size 1) and several classes of regular

objectives. We show that many properties of classical network-formation games are no longer valid in our game.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.