Before revision:  In other words, at least half of the voters "prefer" p over q, when an extra factor of \beta is taken in favor of p. Define \beta^*_{(X,d) = \sup\{\beta\mid \mbox{every finite multiset V in X admits a \beta-plurality point\}.
After revision:  In other words, at least half of the voters "prefer" p over q, when an extra factor of \beta is taken in favor of p. For \beta = \sup\{\beta\mid \mbox{every finite multiset V in X admits a \beta-plurality point\}.