Strategies in a mis\`ere two-player tree searching game

In this paper, we analyse a misere tree searching game, where players take turns to guess vertices in a tree with a secret `poisoned' vertex. After each turn, the guessed vertex is removed from the tree and the game continues on the component containing the poisoned vertex, and as soon as a player guesses the poisoned vertex, they lose. We describe and prove the solution when the game is played on a path graph, both between two optimal players and between a player who makes their decisions uniformly at random and an opponent who plays to exploit this. We show that, with two perfect players, the solution involves different guessing strategies depending on the value of n modulo 4. We then show that, with a random and an exploitative player, the probability that the exploitative player wins approaches a constant (approximately 0.599) as n increases, and that the vertices one away from the leaves of the path are always optimal guesses for them. We also solve the game played on a star graph, and briefly discuss the possibility for extending the analysis to more general trees.