Given a locally compact group \(G\), let \(L^\infty_0(G)\) be the Banach algebra of equivalence classes of essentially bounded functions vanishing at infinity, and \(A:=L^\infty_0(G)^*\) be the dual Banach algebra, equipped with the Arens product [see A. T.-M. Lau and J. Pym, J. Lond. Math. Soc., II. Ser. 41, No. 3, 445–460 (1990; Zbl 0667.43004)]. A continuous linear map \(T: A\to A\) is called a left multiplier if \(T(ab)=T(a)b\) for all \(a,b\in A\). A left multiplier is called compact if it is a compact operator. As the main result, the authors show that \(A\) admits a compact left multiplier if and only if \(G\) is compact. This complements studies of multipliers on the second dual algebras \(L^1(G)^{**}\) and \(M(G)^{**}\) by F. Ghahramani and A. T.-M. Lau (see [Math. Proc. Camb. Philos. Soc. 111, No. 1, 161–168 (1992; Zbl 0818.46050)] and subsequent work).