  A public decision-making problem consists of a set of issues, each with
multiple possible alternatives, and a set of competing agents, each with a
preferred alternative for each issue. We study adaptations of market economies
to this setting, focusing on binary issues. Issues have prices, and each agent
is endowed with artificial currency that she can use to purchase probability
for her preferred alternatives (we allow randomized outcomes). We first show
that when each issue has a single price that is common to all agents, market
equilibria can be arbitrarily bad. This negative result motivates a different
approach. We present a novel technique called "pairwise issue expansion", which
transforms any public decision-making instance into an equivalent Fisher
market, the simplest type of private goods market. This is done by expanding
each issue into many goods: one for each pair of agents who disagree on that
issue. We show that the equilibrium prices in the constructed Fisher market
yield a "pairwise pricing equilibrium" in the original public decision-making
problem which maximizes Nash welfare. More broadly, pairwise issue expansion
uncovers a powerful connection between the public decision-making and private
goods settings; this immediately yields several interesting results about
public decisions markets, and furthers the hope that we will be able to find a
simple iterative voting protocol that leads to near-optimum decisions.
