Constructing large peak-pit Condorcet domains

Abstract

AbstractWe present a new method of constructing Condorcet domains from pairs of Condorcet domains of smaller sizes (concatenation + shuffle scheme). The concatenation + shuffle scheme provides maximal, connected, copious, peak-pit domains whenever the original domains have these properties. It allows to construct maximal peak-pit Condorcet domains that are larger than those obtained by the Fishburn’s alternating scheme for all $$n\ge 13$$ n ≥ 13 alternatives. For a large number n of alternatives, we get a lower bound $$2.1045^{n}$$ 2 . 1045 n for the cardinality of the largest peak-pit Condorcet domain and a lower bound $$2.1890^{n}$$ 2 . 1890 n for the cardinality of the largest Condorcet domain, improving Fishburn’s result. We also show that all Arrow’s single-peaked domains can be constructed by concatenation + shuffle scheme starting from the trivial domain.