  We study the electric dipole polarizability $\alpha_D$ in ${}^{208}$Pb based
on the predictions of a large and representative set of relativistic and
non-relativistic nuclear mean field models. We adopt the droplet model as a
guide to better understand the correlations between $\alpha_D$ and other
isovector observables. Insights from the droplet model suggest that the product
of $\alpha_D$ and the nuclear symmetry energy at saturation density $J$ is much
better correlated with the neutron skin thickness $\Delta r_{np}$ of
${}^{208}$Pb than the polarizability alone. Correlations of $\alpha_D J$ with
$\Delta r_{np}$ and with the symmetry energy slope parameter $L$ suggest that
$\alpha_D J$ is a strong isovector indicator. Hence, we explore the possibility
of constraining the isovector sector of thenuclear energy density functional by
comparing our theoretical predictions against measurements of both $\alpha_D$
and the parity-violating asymmetry in ${}^{208}$Pb. We find that the recent
experimental determination of $\alpha_D$ in ${}^{208}$Pb in combination with
the range for the symmetry energy at saturation density $J=[31\pm (2)_{\rm
est.}]$\,MeV suggests $\Delta r_{np}({}^{208}{\rm Pb}) = 0.165 \pm (0.009)_{\rm
exp.} \pm (0.013)_{\rm theo.} \pm (0.021)_{\rm est.} {\rm fm}$ and $L= 43
\pm(6)_{\rm exp.} \pm (8)_{\rm theo.}\pm(12)_{\rm est.}$ MeV.
