Question/Answer Games on Towers and Pyramids

Abstract

Question/Answer games [3] (Q/A games) are a generalization of the game introduced in [1,2]. They are motivated by the classical game of twenty questions and are a generalization of Rényi-Ulam Game. A k-round Q/A game, G = (D,(q ₁, ..., q _k )), is played on a rooted directed acyclic graph, D = (V,E). In the i-th round, Paul selects a set, Q _i  ⊆ V, of at most q _i non-terminal vertices. Carole responds by choosing an outgoing edge from each vertex in Q _i . At the end of k rounds, Paul wins if Carole’s answers define a unique path from the root to one of the terminal vertices in D. Arbitrary Q/A games are known to be PSPACE-complete [3], and k-round games are known to be Σ _2k − 2-complete [4]. In this paper we study Q/A games on two classes of graphs, towers and pyramids, respectively. We completely solve the problem of determining the winner for Q/A games on towers. We also solve an open problem on Q/A games on pyramids from [1,2]. Furthermore, we give some non-trivial lower and upper bounds for the rest of the cases for Q/A games on pyramids.