Summary: Given a search space \(S=\{1,2,\ldots ,n\}\), an unknown element \(x^{\ast }\in S\) and fixed integers \(\ell \geq 1\) and \(q\geq 1\), a \(q+1\)-ary \(\ell \)-restricted query is of the following form: which one of the set \(\{A_{0},A_{1},\ldots ,A_{q}\}\) is the \(x^{\ast }\) in?, where \((A_{0},A_{1},\ldots ,A_{q})\) is a partition of \(S\) and \(|A_{i}|\leq \ell \) for \(i=1,2,\ldots ,q\). The problem of finding \(x^{\ast }\) from \(S\) with \(q+1\)-ary size-restricted queries is called as a \(q+1\)-ary search game with small sets. In this paper, we consider sequential algorithms for the above problem, and establish the minimum number of average-case sequential queries when \(x^{\ast }\) satisfies the uniform distribution on \(S\).