A Condorcet winner is an alternative that defeats every other alternative in a pairwise majority comparison. Such a winner need not exist, as cyclic preferences can produce voting paradoxes.
Let 
be the probability of a Condorcet winner when 
voters independently choose uniformly
random strict rankings of 
alternatives. Kim and Shin (2026) proved Kelly's two monotonicity
conjectures for odd electorates: 
for 
and odd 
, and 
for 
and odd 
. The restrictions exclude 
for odd 
and 
. The results do not settle the corresponding questions
for even electorates.
Kim and Shin (2026) report LLM assistance with proof exploration and computational checks.
