This is because both definitions are  equivalent. Obviously, \(p\)-strict dominance implies strict  dominance. To see the converse, suppose that \(m_i'\) is dominated by  \(m_i\) with respect to \(X\subseteq S_{-i}\). We show that for all \(q\in  \Delta(X)\), \(U_i(m_i,q) &gt; U_i(m_i',q)\) (and so \(m_i'\) is  \(p\)-strictly dominated by \(m_i\) with respect to \(X\)). Suppose that  \(q\in \Delta(X)\). Then,