  In this work, we present new data on the $^{182,183,184}$W($\gamma,n$) cross
sections, utilizing a quasi-monochromatic photon beam produced at the NewSUBARU
synchrotron radiation facility. Further, we have extracted the nuclear level
density and $\gamma$-ray strength function of $^{186}$W from data on the
$^{186}$W($\alpha,\alpha^\prime\gamma$)$^{186}$W reaction measured at the Oslo
Cyclotron Laboratory. Combining previous measurements on the
$^{186}$W($\gamma,n$) cross section with our new $^{182,183,184}$W($\gamma,n$)
and ($\alpha,\alpha^\prime\gamma$)$^{186}$W data sets, we have deduced the
$^{186}$W $\gamma$-ray strength function in the range of $1 < E_\gamma < 6$ MeV
and $7 < E_\gamma < 14$ MeV.
  Our data are used to extract the level density and $\gamma$-ray strength
functions needed as input to the nuclear-reaction code \textsf{TALYS},
providing an indirect, experimental constraint for the
$^{185}$W($n,\gamma$)$^{186}$W cross section and reaction rate. Compared to the
recommended Maxwellian-averaged cross section (MACS) in the KADoNiS-1.0 data
base, our results are on average lower for the relevant energy range $k_B T \in
[5,100]$ keV, and we provide a smaller uncertainty for the MACS. The
theoretical values of Bao \textit{et al.} and the cross section experimentally
constrained on photoneutron data of Sonnabend \textit{et al.} are significantly
higher than our result. The lower value by Mohr \textit{et al.} is in very good
agreement with our deduced MACS. Our new results could have implications for
the $s$-process and in particular the predicted $s$-process production of
$^{186,187}$Os nuclei.
