Game theory on graphs is a basic tool in computer science. In this paper, we propose a new game-theoretic framework for studying the privacy protection of a user who interactively uses a software service. Our framework is based on the idea that an objective of a user using software services should not be known to an adversary because the objective is often closely related to personal information of the user. We propose two new notions, O-indistinguishable strategy (O-IS) and objective-indistinguishability equilibrium (OIE). For a given game and a subset O of winning objectives (or objectives in short), a strategy of a player is O-indistinguishable if an adversary cannot shrink O by excluding any objective from O as an impossible objective. A strategy profile, which is a tuple of strategies of all players, is an OIE if the profile is locally maximal in the sense that no player can expand her set of objectives indistinguishable from her real objective from the viewpoint of an adversary. We show that for a given multiplayer game with Muller objectives, both of the existence of an O-IS and that of OIE are decidable.

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