  A statistical analysis of stacked Compton$-y$ maps of quasar hosts with a
median redshift of $1.5$ using Millennium Simulation is performed to address
two issues, one on the feedback energy from quasars and the other on testing
dark matter halo models for quasar hosts. On the first, we find that, at the
resolution of FWHM=$10$ arcmin obtained by Planck data, the observed thermal
Sunyaev-Zeldovich (tSZ) effect can be entirely accounted for and explained by
the thermal energy of halos sourced by gravitational collapse of halos, without
a need to invoke additional, large energy sources, such as quasar or stellar
feedback. Allowing for uncertainties of dust temperature in the calibration of
observed Comton$-y$ maps, the maximum additional feedback energy is $\sim 25\%$
of that previously suggested. Second, we show that, with FWHM=$1$ arcmin beam,
tSZ measurements will provide a potentially powerful test of quasar-hosting
dark matter halo models, limited only by possible observational systematic
uncertainties, not by statistical ones, even in the presence of possible quasar
feedback.
