In an ordinary Condorcet cycle one can identify, for each candidate, a second candidate preferred, by a majority of voters, to the first.  In a Condorcet cycle of order 2 one can identify, for each pair of candidates, a third candidate preferred, by a majority of voters, to both.  We construct two Condorcet cycles of order 2.  The first, with 11 alternatives and 11 voters, improves the example of 15 alternatives and 15 voters given in [1].  The second, with 7 alternatives and 21 voters, shows that the lower bound on alternatives established in [4] and [3] (and independently in [1]) is sharp. Both our constructions use the method of horizontal rotation, introduced here, which generalizes the more typical form of rotation used to construct standard Condorcet cycles. The second example also makes use of a beautifully symmetric tournament constructed in [3].

William S. Zwicker^a (joint work with Davide P. Cervone^b)

keywords: Condorcet cycle of order 2, Condorcet winning set, tournament

[1] Elkind, E., Lang, J., and Saffidine, A., Condorcet Winning Sets, Soc Choice Welf 44, 493-517 (2015)

[2] Erdös, P., On a problem of graph theory, Math Gaz 47, 220-223 (1963)

[3] Graham, R.L. and Spencer, J.H., A constructive solution to a tournament problem, Can Math Bul 14, 45-48 (1971)

[4] Szekeres, E. and Szekeres,G., On a problem of Schütte and Erdös, Math Gaz 49, 290-293 (1965)

^aWilliam D Williams Professor of Mathematics Emeritus, Union College, New York; and Murat Sertel Center for Advanced Economic Studies, Istanbul Bilgi University

^bMathematics Department, Union College, New York