The binary pathological liar game, as described by Ellis and Yan in [Ellis and Yan, 2004], is a variation of the original liar game, as described by Berlekamp, R enyi, and Ulam in [Berlekamp, 1964]... Show moreThe binary pathological liar game, as described by Ellis and Yan in [Ellis and Yan, 2004], is a variation of the original liar game, as described by Berlekamp, R enyi, and Ulam in [Berlekamp, 1964], [R enyi, 1961], and [Ulam, 1976]. This two person, questioner/ responder, game is played for n rounds for a set of M messages. The game begins by the responder selecting a message from the set M. Each round the questioner partitions the messages into two distinct subsets. The responder selects one subset, and elements not in the selected subset each accumulate a lie. Elements accumulating more than e lies are eliminated. The questioner wins the original game provided after the completion of n rounds there is at most one surviving message. The questioner wins the pathological game provided there is at least one surviving message. The focus here will be to generalize the pathological game from two subsets to q subsets with a focus on providing a winning condition for the questioner. The q-ary variant of the pathological liar game has been studied, with rst results in [Ellis and Nyman, 2009]. We let the number of rounds the game is played go to in nity, with e a linear fraction of n, and present an upper bound on the number of messages required by the questioner to win the q-ary Pathological Liar Game. The liar machine and linear machine as discussed by Cooper and Ellis in [Cooper and Ellis, 2010] have been adapted to t this generalization and are used to track the approximate progression of the game. We provide an upper bound on the initial number of chips by bounding the discrepancy between the actual progression of the game and the approximate progression of the game as described by the linear and liar machines respectively. A similar upper bound can be found in [Tietzer, 2011], with di erent elements in the argument. Using methods similar to those found in [Cooper and Ellis, 2010], we provide a partial order argument to show that the winning condition bound for one response strategy by the questioner transfers to all possible response strategies. Show less
