From the authors' inroduction: ``It is well known that if two meromorphic functions \(w(z)\) and \(\hat w(z)\) take five values at the same points, then \(w(z)\equiv \hat w(z)\), and if \(w(z)\) and \(\hat w(z)\) share only four values, usually \(w(z)\not\equiv \hat w(z)\), but there exist some relations between \(w(z)\) and \(\hat w(z)\). Recently \textit{G. Gundersen} [J. Lond. Math. Soc., II. Ser. 20, 456-466 (1979; Zbl 0413.30025)] proved that if two meromorphic functions share three values, then the proportion of their characteristic functions is finite. On the \(\nu\)-valued algebroid functions \textit{G. Valiron} pointed out that if two \(\nu\)-valued algebroid functions \(w(z)\) and \(\hat w(z)\) take \(4\nu+1\) values at the same points with same multiple order, then \(w(z)\equiv \hat w(z)\). In [the first author, Sci. Sin. 14, 174-180 (1965; Zbl 0161.267)], we prove a uniqueness theorem which refined the result of Valiron. In the present paper we first prove that if two \(\nu\)-valued algebroid functions \(w(z)\) and \(\hat w(z)\) share \(4\nu\) values, then there exists some relations between \(w(z)\) and \(\hat w(z),\) and we construct two different \(\nu\)- valued algebroid functions sharing \(4\nu\) values. Secondly we prove that if two \(\nu\)-valued algebroid functions share \(2\nu +\lambda\) values with \(1\leq \lambda \leq 2\nu -1\), then the ratio of their characteristic functions is finite, and we give two \(\nu\)-valued algebroid functions sharing \(2\nu\) values, but the ratio of their characteristic functions is infinite. We also obtain some results concerning the multiplicity.''