Quiz Show Games: Searching with Bimodal Hiding

Melike Baykal-G\"ursoy; Mingshi Yao; Thomas Lidbetter

Quiz-show search games with a host allocating a fixed prize budget admit closed-form equilibria in the first prize-budget variant, reduce to a known game in the second, and yield partial results via a geometric game in the third.

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2026-07-15 02:44 UTC pith:LAUXGYIE

load-bearing objection Abstract-only: clean extension of Kadane’s quiz-show index into three host prize-budget games, with claimed closed forms, a reduction, and partial geometric results. the 2 major comments →

arxiv 2607.12867 v1 pith:LAUXGYIE submitted 2026-07-14 cs.GT math.OC

Quiz Show Games: Searching with Bimodal Hiding

Mingshi Yao , Thomas Lidbetter , Melike Baykal-G\"ursoy This is my paper

classification cs.GT math.OC

keywords quiz show gamessearch gamesprize allocationequilibrium strategiesgeometric gamesinspection gamesscheduling

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies game-theoretic versions of the classic quiz-show problem in which a contestant chooses the order of questions with known success probabilities, while a host distributes a fixed prize budget. Three allocation rules are considered, motivated by operational search and inspection tasks such as reconnaissance and nuclear-enrichment checks. For the first rule the authors derive complete closed-form equilibrium strategies and the value of the game. The second rule reduces directly to a game already solved in the literature. The third rule is only partially solved by embedding it in a broader geometric game. A sympathetic reader cares because the models convert an index-based single-agent ordering problem into adversarial search settings whose solutions give explicit mixed strategies and payoffs usable for scheduling and inspection planning.

Core claim

When a host allocates a fixed prize budget under three distinct rules, the resulting zero-sum games against a contestant with known success probabilities possess, respectively, fully closed-form equilibria, a reduction to a previously solved game, and partial characterizations obtained from a more general geometric game.

What carries the argument

Bimodal prize allocation by the host (the three budget-distribution rules) together with the classic index ordering of questions; the first rule yields explicit equilibrium mixed strategies and game value, the second maps onto a known game, and the third is analyzed via a geometric game that generalizes the prize structure.

Load-bearing premise

That the host's three prize-budget rules and the contestant's fixed known success probabilities faithfully represent the operational search and inspection settings that motivate the models.

What would settle it

Compute or simulate the equilibrium value and strategies for the first prize-budget variant under a concrete small instance with known probabilities and budget; any systematic deviation from the claimed closed-form expressions falsifies the complete-solution claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Abstract-only: clean extension of Kadane’s quiz-show index into three host prize-budget games, with claimed closed forms, a reduction, and partial geometric results.

read the letter

The one thing worth knowing is that this takes Kadane’s classic single-player ordering problem for quiz questions and turns it into three zero-sum games in which the host allocates a fixed prize budget under different rules. They claim complete closed-form equilibria and the value for the first variant, a reduction of the second to a game already solved in the literature, and partial results via a more general geometric game for the third. The national-security and scheduling motivations are stated but not load-bearing for the math.

What is actually new is the host-side budget control and the three allocation variants. The contestant’s index is classical; the contribution is the game-theoretic layer. If the closed forms check out, that is useful applied work. The reduction is honest, and analyzing a geometric generalization for the partial case is a standard, legitimate move. They cite the prior literature cleanly and do not oversell the applications.

Soft spots are mostly availability: we have only the abstract, so the derivations, explicit strategies, and value formulas cannot be inspected. That keeps confidence low, but it is not evidence of a flaw. The modeling bridge to reconnaissance or nuclear inspection is purely motivational and does not affect the equilibrium claims. The third variant is only partial, which they state openly. Nothing in the abstract suggests circularity or forced results.

This is for search theorists and applied game theorists who care about inspection, scheduling, or budgeted allocation games. A reader who likes closed-form zero-sum solutions will get something concrete if the proofs hold. It is serious enough to deserve a real referee rather than a desk reject.

Send it to peer review. The claims are sharp enough that a specialist can verify them quickly; if they stand, it is a clean incremental contribution.

Referee Report

2 major / 0 minor

Summary. The manuscript studies zero-sum game-theoretic extensions of Kadane’s classic quiz-show ordering problem. A contestant faces a sequence of questions with known fixed success probabilities and chooses an order; a host allocates a fixed prize budget under three distinct distribution rules. The abstract asserts complete closed-form equilibrium strategies and game values for the first variant, a reduction of the second variant to a game already solved in the literature, and partial results for the third variant obtained by analyzing a more general geometric game. Motivating applications include reconnaissance and nuclear-enrichment inspection, as well as certain scheduling problems.

Significance. If the claimed closed-form equilibria, reduction, and geometric analysis are correct, the paper would supply exact solutions for a natural family of adversarial search/scheduling games that extend Kadane’s index result to an optimizing host. Parameter-free or low-parameter closed forms and an explicit reduction would be valuable contributions in combinatorial game theory and operational search. The geometric-game intermediate result may also be of independent interest. Significance cannot be fully assessed without the derivations and verification that the abstract promises but does not contain.

major comments (2)

Only the abstract is available for review. The central claims—complete closed-form equilibria and value for variant 1, a reduction of variant 2 to a known game, and partial geometric results for variant 3—are existence and derivation claims whose correctness cannot be checked without the body of the paper (proofs, equilibrium formulas, and any intermediate lemmas). No load-bearing gap can be confirmed or refuted from the abstract alone; the manuscript is therefore not yet refereable on soundness.
The abstract asserts that the host’s three prize-budget allocation rules and the contestant’s known fixed success probabilities capture operational settings (reconnaissance, nuclear inspection). That modeling bridge is motivational rather than load-bearing for the pure game-theoretic claims, but if the paper intends the applications as part of its contribution, the full text must make the correspondence precise (strategy spaces, information structure, and how prize allocation maps to search effort). Without the body this cannot be verified.

Circularity Check

0 steps flagged

No significant circularity; abstract-only review of pure game-theoretic closed forms and a literature reduction

full rationale

Only the abstract is available. It presents three well-posed zero-sum game variants of the classic Kadane quiz-show ordering problem, claims complete closed-form equilibria and value for the first prize-budget rule, a reduction of the second variant to a game already solved in the literature, and partial results for the third via a more general geometric game. No parameters are fitted to data, no prediction is renamed from a fit, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in. The sole external reference is the classical Kadane (1969) index, which is independent and not load-bearing for circularity. The modeling motivation (reconnaissance, nuclear inspection) is not used in any derivation. With no full text, no equation-level reduction can be exhibited; under the hard rules this yields score 0 and an empty steps list. The reader's suggested score of 2 is not supported by any quotable self-citation that is load-bearing.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

Abstract-only review. Free parameters, axioms, and invented entities cannot be exhaustively extracted. The ledger records only what the abstract itself makes load-bearing: known success probabilities, a fixed prize budget, three allocation rules, and the modeling link to search/inspection.

free parameters (2)

per-question success probabilities
Treated as known fixed parameters supplied to both players; their numerical values are inputs, not derived.
fixed prize budget
Host allocates a fixed total prize mass; the budget size is an exogenous parameter of the game.

axioms (3)

domain assumption Contestant success probabilities are known, fixed, and independent across questions.
Stated as given in the abstract; underpins both the classic index rule and the game extensions.
domain assumption Host and contestant play a zero-sum game over prize allocation and question order.
Implicit modeling choice that turns the classic problem into the three variants studied.
standard math Kadane (1969) index solution for the single-player ordering problem is correct.
Used as the baseline that the contestant side inherits.

pith-pipeline@v1.1.0-grok45 · 6115 in / 2141 out tokens · 15022 ms · 2026-07-15T02:44:38.912760+00:00 · methodology

0 comments

read the original 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.

Quiz Show Games: Searching with Bimodal Hiding

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)