On the existence of an equilibrium in the split-the-difference mechanism over an uncountable set with a singular part

Broman (89) analyzes the mixed Nash equilibria of the split-the-difference mechanism over countable sets. She leaves as an open question whether there may be mixed equilibria over uncountable sets with singular parts. In this note, I propose such an equilibrium; the support of the strategies is the union of a countable set and an interval.