  The goal of this paper is to develop a generic framework for converting
modern optimization algorithms into mechanisms where inputs come from
self-interested agents. We focus on aggregating preferences from $n$ players in
a context without money. Special cases of this setting include voting,
allocation of items by lottery, and matching. Our key technical contribution is
a new meta-algorithm we call \apex (Adaptive Pricing Equalizing Externalities).
The framework is sufficiently general to be combined with any optimization
algorithm that is based on local search. We outline an agenda for studying the
algorithm's properties and its applications. As a special case of applying the
framework to the problem of one-sided assignment with lotteries, we obtain a
strengthening of the 1979 result by Hylland and Zeckhauser on allocation via a
competitive equilibrium from equal incomes (CEEI). The [HZ79] result posits
that there is a (fractional) allocation and a set of item prices such that the
allocation is a competitive equilibrium given prices. We further show that
there is always a reweighing of the players' utility values such that running
unit-demand VCG with reweighed utilities leads to a HZ-equilibrium prices.
Interestingly, not all HZ competitive equilibria come from VCG prices. As part
of our proof, we re-prove the [HZ79] result using only Brouwer's fixed point
theorem (and not the more general Kakutani's theorem). This may be of
independent interest.
