  We focus on a permutation betting market under parimutuel call auction model
where traders bet on the final ranking of n candidates. We present a
Proportional Betting mechanism for this market. Our mechanism allows the
traders to bet on any subset of the n x n 'candidate-rank' pairs, and rewards
them proportionally to the number of pairs that appear in the final outcome. We
show that market organizer's decision problem for this mechanism can be
formulated as a convex program of polynomial size. More importantly, the
formulation yields a set of n x n unique marginal prices that are sufficient to
price the bets in this mechanism, and are computable in polynomial-time. The
marginal prices reflect the traders' beliefs about the marginal distributions
over outcomes. We also propose techniques to compute the joint distribution
over n! permutations from these marginal distributions. We show that using a
maximum entropy criterion, we can obtain a concise parametric form (with only n
x n parameters) for the joint distribution which is defined over an
exponentially large state space. We then present an approximation algorithm for
computing the parameters of this distribution. In fact, the algorithm addresses
the generic problem of finding the maximum entropy distribution over
permutations that has a given mean, and may be of independent interest.
