Minimizing the Cost of Guessing Games

Abstract

A two-player “guessing game” is a game in which the first participant, the “Responder,” picks a number from a certain range. Then, the second participant, the “Questioner,” asks only yes-or-no questions in order to guess the number. In this paper, we study guessing games with lies and costs. In particular, the Responder is allowed to lie in one answer, and the Questioner is charged a cost based on the content of each question. Guessing games with lies are closely linked to error correcting codes, which are mathematical objects that allow us to detect an error in received information and correct these errors. We will give basic definitions in coding theory and show how error correcting codes allow us to still guess the correct number even if one lie is involved. We will additionally seek to minimize the total cost of our games. We will provide explicit constructions, for any cost function, for games with the minimum possible cost and an unlimited number of questions. We also find minimum cost games for games with a restricted number of questions and a constant cost function. KEYWORDS: Ulam’s Game; Guessing Games With Lies; Error Correcting Codes; Pairwise Balanced Designs; Steiner Triple Systems