Log-time Prediction Markets for Interval Securities

TL;DR

This paper introduces two computationally efficient prediction market designs for continuous variables, enabling rapid, precise probability distribution recovery through interval securities and modular market structures.

Contribution

It presents the first logarithmic-time prediction market for continuous variables, using novel modular LMSR-based designs for efficient, flexible outcome trading.

Findings

Market operations run in logarithmic time relative to interval count

The first design is an exponentially faster LMSR variant using a binary tree structure

The second design employs parallel LMSRs for multi-resolution outcome expression

Abstract

We design a prediction market to recover a complete and fully general probability distribution over a random variable. Traders buy and sell interval securities that pay \$1 if the outcome falls into an interval and \$0 otherwise. Our market takes the form of a central automated market maker and allows traders to express interval endpoints of arbitrary precision. We present two designs in both of which market operations take time logarithmic in the number of intervals (that traders distinguish), providing the first computationally efficient market for a continuous variable. Our first design replicates the popular logarithmic market scoring rule (LMSR), but operates exponentially faster than a standard LMSR by exploiting its modularity properties to construct a balanced binary tree and decompose computations along the tree nodes. The second design consists of two or more parallel LMSR market makers that mediate submarkets of increasingly fine-grained outcome partitions. This design remains computationally efficient for all operations, including arbitrage removal across submarkets. It adds two additional benefits for the market designer: (1) the ability to express utility for information at various resolutions by assigning different liquidity values, and (2) the ability to guarantee a true constant bounded loss by appropriately decreasing the liquidity in each submarket.