  Approximate Competitive Equilibrium from Equal Incomes (A-CEEI) is an
equilibrium-based solution concept for fair division of discrete items to
agents with combinatorial demands. In theory, it is known that in
asymptotically large markets:
  1. For incentives, the A-CEEI mechanism is Envy-Free-but-for-Tie-Breaking
(EF-TB), which implies that it is Strategyproof-in-the-Large (SP-L).
  2. From a computational perspective, computing the equilibrium solution is
unfortunately a computationally intractable problem (in the worst-case,
assuming $\textsf{PPAD}\ne \textsf{FP}$).
  We develop a new heuristic algorithm that outperforms the previous
state-of-the-art by multiple orders of magnitude. This new, faster algorithm
lets us perform experiments on real-world inputs for the first time. We
discover that with real-world preferences, even in a realistic implementation
that satisfies the EF-TB and SP-L properties, agents may have surprisingly
simple and plausible deviations from truthful reporting of preferences. To this
end, we propose a novel strengthening of EF-TB, which dramatically reduces the
potential for strategic deviations from truthful reporting in our experiments.
  A (variant of) our algorithm is now in production: on real course allocation
problems it is much faster, has zero clearing error, and has stronger incentive
properties than the prior state-of-the-art implementation.
