Parimutuel Betting on Permutations

TL;DR

This paper introduces a polynomial-time computable market mechanism for permutation betting, providing a way to determine prices and approximate distributions over permutations efficiently.

Contribution

It proposes a novel proportional betting mechanism with polynomial-sized convex programming formulation and maximum entropy-based distribution approximation.

Findings

Convex program yields unique marginal prices for bets.

Marginal prices can be computed in polynomial time.

Maximum entropy approach provides a concise parametric distribution.

Abstract

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.