¹ Since the number of representatives allocated to each party must be a nonnegative integer, the actual q _i's will be slightly different from those defined in (1.1) and (1.2). This feature will be disregarded in the present note, but it should be stressed that the criterion used in Section 2—the minimization of the information expectation defined in eq. (2.4)—can be used to handle this problem. The approach is then to minimize the information expectation subject to the constraint that each q _i,- be a nonnegative multiple of 1/N, where N is the total number of parliamentary seats.

² Plus the case α = ∞ (i.e., allocating all seats to the ruling party), a procedure used in some of the democratically less advanced countries.

³ For a brief introduction to the informational concepts used in this section, see e.g. Chapters 1 and 2 of the author's Economics and Information Theory (Chicago: Rand McNally and Company, and Amsterdam: North-Holland Publishing Company, 1967)Google Scholar. The idea of using the entropy for this particular purpose is due to Mr. B. M. S. van Praag of the Econometric Institute in Rotterdam.

⁴ Even for n = 2 the entropy does not determine the q's uniquely, because (θ, 1–θ) and (1–θ, θ) have the same entropy for any θ in the interval (0, 1). It is obvious, however, that the present discussion loses much of its interest when n = 2 instead of n > 2.

⁵ This becomes particularly clear when I(q:p) is expanded according to powers of (q _i–p _i)/p _i. If we use natural logarithms (which is done throughout this paper) and disregard third and higher powers, the result is

provided that the p's and q's are pairwise sufficiently close to each other (so that the expansions converge). The right-hand side of this approximation is proportional to a chi-square with the p's as theoretical probabilities and the q's as observed frequencies.

⁶ To make the discussion more concrete, we will talk about parties and parliamentary seats rather than about both these and the geographic representations such as those of IFORS and the U.S. Senate.

⁷ Write p ₁ for the largest p _i, then q ₁ converges to 1 as α → ∞, so that H _q = 0 in the limit. If p ₁ =p ₂ and if all other p _i's are smaller, then q ₁ = q ₂ = ½ in the limit and hence H _q>0 even when α = ∞. In the case of a k-fold multiple maximum, p ₁ = … = p _k>p _k+1 ≥ … ≥p _n, the interval for H _q mentioned in Theorem 1 should be confined to (log k, log n).

⁸ By substituting log q _i for α log p _i – log C in the second line of (2.12) we obtain a result that can be written as follows:

which means that a small relative increase in α reduces the entropy H _q by k times this increase, where k is the variance of the distribution whose values are –log q ₁, … , –log q _n and whose probabilities are q ₁, …, q _n. Obviously, the mean of this distribution is the entropy H _q.

⁹ For details, see e.g. Economics and Information Theory (quoted in footnote 3), p. 7, or Koopman, B. O. and Kimball, G. E., “Information Theory,” Chapter 9 of Notes on Operations Research 1959, assembled by the Operations Research Center, Massachusetts Institute of Technology (Cambridge, Mass.: The Technology Press, M.I.T., 1959)Google Scholar.

¹⁰ This will of course require the teaching of logarithms at an early stage. Mathematics teachers should regard it as a challenge to teach a subject with immediate political science applications.