  Ag$_5$Pb$_2$O$_6$ has attracted considerable attention due to its novel
nearly-free-electron superconductivity, but its electronic structure and
orbital character of the Cooper-pair electrons remain controversial. Here, we
present a method utilizing core-level photoemission to show that Pb 6$s$
electrons dominate near the Fermi level. We observe a strongly asymmetric Pb
4$f_{7/2}$ core-level spectrum, while a Ag 3$d_{5/2}$ spectrum is well
explained by two symmetric peaks. The asymmetry in the Pb 4$f_{7/2}$ spectrum
originates from the local attractive interaction between conducting Pb 6$s$
electrons and a Pb 4$f_{7/2}$ core hole, which implies a dominant Pb 6$s$
contribution to the metallic conduction. In addition, the observed Pb
4$f_{7/2}$ spectrum is not explained by the well-known Doniach-\v{S}unji\'{c}
lineshape for a simple metal. The spectrum is successfully generated by
employing a Pb 6$s$ partial density of states from local density approximation
calculations, thus confirming the Pb 6$s$ dominant character and
free-electron-like density of states of Ag$_5$Pb$_2$O$_6$.
