  Proportional representation (PR) has long been believed the ideal system for
the equality of individuals in apportioning the seats of a legislature body to
subgroups. We observe that PR implicitly assumes the (standard) number of
representatives is proportional to the population, a situation no longer
observed since 1820s. To address this issue, we suggest to formulate the
apportionment problem in a broader context by explicitly specifying a standard
function $f$ such that $f(p)$ is the standard, possibly fractional number of
representatives for population $p$, where PR assumes $f(p)\propto p$. For this
generalized apportionment problem, we give a population seat index (PSI)
$\frac{f^{-1}(s)}{p}$ for quantifying the contribution of an individual in
assigning $s$ seats to a population $p$, where $f^{-1}$ is the inverse of $f$.
With the PSI, we derive apportioning schemes with absolute and relative
individual equality. Particularly, for $s$ seats, populations $p_1, \ldots,
p_k$, and a standard function $f(p) = a + b p^\gamma$ with constants $a, b,
\gamma \ge 0$, the ideal, possibly fractional number of seats for subgroup $i$
is $a + \frac{(S-ka)p_i^{\gamma}}{\sum p_j^{\gamma}}$, not $\frac{Sp_i}{\sum
p_j}$ calculated by PR which works only for $a=0$, $\gamma=1$. Finally, since
real-world observations indicate a standard function $f \propto p^\gamma$ with
$\gamma < 1$, we conclude that PR represents individuals in less populous
subgroups less than individuals in more populous subgroups.
