  The availability of vast amounts of data is changing how we can make medical
discoveries, predict global market trends, save energy, and develop educational
strategies. In some settings such as Genome Wide Association Studies or deep
learning, sheer size of data seems critical. When data is held distributedly by
many parties, they must share it to reap its full benefits.
  One obstacle to this revolution is the lack of willingness of different
parties to share data, due to reasons such as loss of privacy or competitive
edge. Cryptographic works address privacy aspects, but shed no light on
individual parties' losses/gains when access to data carries tangible rewards.
Even if it is clear that better overall conclusions can be drawn from
collaboration, are individual collaborators better off by collaborating?
Addressing this question is the topic of this paper.
  * We formalize a model of n-party collaboration for computing functions over
private inputs in which participants receive their outputs in sequence, and the
order depends on their private inputs. Each output "improves" on preceding
outputs according to a score function.
  * We say a mechanism for collaboration achieves collaborative equilibrium if
it ensures higher reward for all participants when collaborating (rather than
working alone). We show that in general, computing a collaborative equilibrium
is NP-complete, yet we design efficient algorithms to compute it in a range of
natural model settings.
  Our collaboration mechanisms are in the standard model, and thus require a
central trusted party; however, we show this assumption is unnecessary under
standard cryptographic assumptions. We show how to implement the mechanisms in
a decentralized way with new extensions of secure multiparty computation that
impose order/timing constraints on output delivery to different players, as
well as privacy and correctness.
