We use polarized inelastic neutron scattering to study low-energy spin excitations and their spatial anisotropy in electron-overdoped superconducting BaFe$_{1.85}$Ni$_{0.15}$As$_{2}$ ($T_c=14$ K). In the normal state, the imaginary part of the dynamic susceptibility, $chi^{primeprime}(Q,omega)$, at the antiferromagnetic (AF) wave vector $Q=(0.5,0.5,1)$ increases linearly with energy for $Ele 13$ meV. Upon entering the superconducting state, a spin gap opens below $Eapprox 3$ meV and a broad neutron spin resonance appears at $Eapprox 7$ meV. Our careful neutron polarization analysis reveals that $chi^{primeprime}(Q,omega)$ is isotropic for the in-plane and out-of-plane components in both the normal and superconducting states. A comparison of these results with those of undoped BaFe$_2$As$_2$ and optimally electron-doped BaFe$_{1.9}$Ni$_{0.1}$As$_{2}$ ($T_c=20$ K) suggests that the spin anisotropy observed in BaFe$_{1.9}$Ni$_{0.1}$As$_{2}$ is likely due to its proximity to the undoped BaFe$_2$As$_2$. Therefore, the neutron spin resonance is isotropic in the overdoped regime, consistent with a singlet to triplet excitation.
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