Cut-and-choose games in topological spaces

2025/10/07 by Lucas Chiozini, Chiozini, Lucas, Tamás Csernák +3
Decision Sciences · Economics, Econometrics and Finance · #54A25 #54D10 #91A44 #FOS: Mathematics #Game Theory and Applications #Game Theory and Voting Systems #General Topology (math.GN)

paper · pdf · doi:10.48550/arxiv.2510.05754

openalex publication_date 2025/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study transfinite cut-and-choose games on T₀ spaces, introducing the \em point-separating number ps(X) and the \em set membership number sm(X) as the ordinal-valued invariants measuring the minimal length of a game in which a Seeker can determine a hidden point or subset. A central motivating question is which countable ordinals can occur as the value of ps(X), in particular whether any countable ordinal can arise. These invariants generalize Scott's T₀-pseudoweight ψw₀. We establish fundamental inequalities relating ps(X), sm(X), ψw₀(X), and |X|, including the sharp bounds |X|≤ 2^ps(X) and ψw₀(X)≤ 2^

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