Summary: We consider two-person zero-sum games with lack of information on one side given by \(m\) matrices of dimension \(m\times m\). We suppose the matrices to have the following ``symmetric'' structure: \(a^s_{ij}= a_{ij}+ c\delta^s_i\), \(c> 0\), where \(\delta^s_i= 1\) if \(i= s\) and \(\delta^s_i= 0\) otherwise. Under certain additional assumptions we give the explicit solution for finite repetitions of these games. These solutions are expressed in terms of multinomial distributions. We give the probabilistic arguments which explains the obtained form of solutions. Applying the central limit theorem we get the description of limiting behavior of value closely connected with recent results of \textit{B. De Meyer} [`Repeated games and multidimensional normal distribution'' and `Repeated games and the central limit theorem'', CORE Discussion Papers 8932 (1989) and 9303 (1993)].