The compound decision problem, called by Robbins [7], consists in the simultaneous consideration of n decision problems which have an identical formal structure. All the papers [1]--[11] except [8] treated the continuous compound decision problem. Robbins [8] considered the decision problem between the binomial distributions and the distribution which is given by a unit step function. The problem of convergence rate was treated by Hannah-Van Ryzin [4], Van Ryzin [13] and Samuel [10] in the continuous case, by Johns [6] in the two-action case and by Robbins [8] in the binomial case. In this paper, we shall t reat the compound decision problem between k + l (at most) ~z-variate multinomial distributions and present the construction of the "bes t" simple symmetric solution and the exact convergence rate of the asymptotically "good" non-simple solution, using the different approach from that for the continuous case. This is an extension of [8] and a modification of [1], [2], [5], [11] and [13] which treated the continuous compound decision problem involving multiple component decisions. We do not consider the sequential case (cf. [4], [5], [9], [10] and [11]). But, in that case, a similar argument can be made by modifying only the part of the estimation of the true parameter vector.