Extending the V-sequence reduction to arbitrary independent distributions

Let $X_1,\ldots,X_n$ be independent random variables with arbitrary known distributions, where $n\ge2$. The optimal probability $V_n(X_1,\ldots,X_n)$ of choosing an overall maximum satisfies $V_n(X_1,\ldots,X_n)\ge(1-1/n)^{n-1}$, and this bound is attained. We obtain this known sharp bound by extending the reduction lemma of Allaart and Islas (2016) from continuous to arbitrary distributions. We construct a $V$-sequence whose optimal win probability does not exceed that of the original sequence, and then apply the bound for $V$-sequences. The construction first replaces atoms by uniform observations on strictly ordered intervals and applies the original continuous reduction lemma. The reduced distributions are determined by a common critical level and a common probability for splitting a possible atom at that level.