Generalized Cops and Robbers: A Multi-player Pursuit Game on Graphs

Abstract

We introduce and study the Generalized Cops and Robbers game (GCR), an N -player pursuit game in graphs. The two-player version is essentially equivalent to the classic Cops and Robbers (CR) game. The three-player version can be understood as two CR games played simultaneously on the same graph; a player can be at the same time both pursuer and evader. The same is true for four or more players. We formulate GCR as a discounted stochastic game of perfect information and prove that, for three or more players, it has at least two Nash Equilibria: one in positional deterministic strategies and another in non-positional ones. We also study the capturing properties of GCR Nash Equilibria in connection to the cop-number of a graph. Finally, we briefly discuss GCR as a member of a wider family of multi-player graph pursuit games with rather interesting properties. * The author thanks Steve Alpern and Pascal Schweitzer for several useful and inspiring discussions. pursuit in graphs is our own [12]. It is also remarkable that, while classic CR and many of its variants admit a natural game theoretic formulation and study, this has not been exploited in the CR literature.Regarding pursuit in Euclidean spaces, the predominant approach is in terms of differential games as introduced in the seminal book [10]. There is a flourishing literature on the subject, which contains many works involving multiple pursuers, but they are generally assumed to be collaborating [5,11,15,18,19]. The case of antagonistic pursuers has been studied in some papers [8,9] but the methods used in these works do not appear to be easily applicable to the study of pursuit / evasion on graphs.This paper is organized as follows. Section 2 is preliminary: we introduce notation, define states, histories and strategies and give a general form of the payoff function. In Section 3 we prove that, for any graph and any number of players, GCR has a NE in deterministic positional strategies; this result is applicable not only to GCR but to a wider family of pursuit games, as will be discussed later. In Section 4 we show that in the two-player GCR game: (i) the value of the game exists (essentially it is the logarithm of the optimal capture time) and (ii) both players have optimal deterministic positional strategies. Because of the close connection of GCR to the classical CR game, these results also hold for CR; while they have been previously established by graph theoretic methods, we believe our proof is the first game-theoretic one. In Section 5 we study the three-player GCR game and prove: (i) the existence of a NE in deterministic positional strategies; (ii) the existence of an additional NE in deterministic but non-positional strategies; (iii) various results connecting the classic cop number of a graph to capturability. In Section 6 we briefly discuss N -players GCR when N ≥ 4. In Section 7 we show that the ideas behind GCR can be generalized to obtain a large family of multi-player pursuit games on graphs. Finally, in Section 8 we summarize, present our conc...

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