Quiz Show Games: Searching with Bimodal Hiding

2026-07-14

AI Summary

This study addresses the optimal allocation of rewards by a quiz show host under a fixed budget to strategically influence contestants’ answering order—a problem that models real-world search scenarios such as national security reconnaissance and nuclear facility inspections. The work extends the classical question-answering sequencing problem into a novel two-mode hidden-information game-theoretic framework and analyzes equilibrium behavior under three distinct reward mechanisms. For the first mechanism, a complete closed-form equilibrium solution is derived; the second reduces to a previously studied solvable game; and the third yields partial analytical progress through geometrically structured modeling. By integrating game theory, exponential strategy analysis, and geometric methods, this research achieves substantive theoretical and modeling advances.

Abstract

We consider a quiz show game in which a contestant is presented with a sequence of questions. Each time the contestant answers a question correctly, she receives a prize and proceeds to the next question; the probability of answering each question correctly is given. If the contestant answers a question incorrectly, she receives a consolation prize and the game ends. The contestant's problem of determining the optimal order in which to answer questions, for known fixed parameters, is a classic one studied in Kadane (1969), and admits a simple index-based solution. We consider game-theoretic versions of this problem in which a game show host can choose how to allocate a fixed prize budget. Our models are motivated by operational search problems in national security involving reconnaissance missions and inspecting for evidence of nuclear enrichment, as well as certain scheduling problems. We study three variants of the game, corresponding to different ways in which the host can distribute the prize money. For the first variant, we provide complete closed-form solutions, including equilibrium strategies and the value of the game. We reduce the second variant to a game solved in the literature. For the third variant, we obtain partial results by analyzing a more general game with a geometric structure.

Problem

• quiz show games
• bimodal hiding
• game theory
• optimal search
• prize allocation

Innovation

• quiz show games
• bimodal hiding
• game-theoretic optimization
• prize allocation
• equilibrium strategies