2024/08/26 by Tonatiuh Matos-Wiederhold, Matos-Wiederhold, Tonatiuh, Luciano Salvetti +1
Decision Sciences · Economics, Econometrics and Finance · #Economic theories and models #FOS: Mathematics #Game Theory and Applications #Game Theory and Voting Systems #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2408.14624
openalex publication_date 2024/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An infinite game on the set of real numbers appeared in Matthew Baker's work [Math. Mag. 80 (2007), no. 5, pp. 377--380] in which he asks whether it can help characterize countable subsets of the reals. This question is in a similar spirit to how the Banach-Mazur Game characterizes meager sets in an arbitrary topological space. In a recent paper, Will Brian and Steven Clontz prove that in Baker's game, Player II has a winning strategy if and only if the payoff set is countable. They also asked if it is possible, in general linear orders, for Player II to have a winning strategy on some uncountable set. To this we give a positive answer and moreover construct, for every infinite cardinal κ, a dense linear order of size κ on which Player II has a winning strategy on all payoff sets. We finish with some future research questions, further underlining the difficulty in generalizing the characterization of Brian and Clontz to linear orders.