APMM: Automated Parlay Market Maker

2026-07-13

AI Summary

Traditional prediction markets struggle to efficiently support parlay betting due to the sparsity of multi-outcome combinations, which leads to redundant information acquisition and high subsidy costs. This work proposes an Automated Parlay Market Maker (APMM) that employs a hierarchical parameterization scheme to enable state sharing and coherent pricing across parlays, thereby avoiding the exponential modeling redundancy inherent in treating each combination independently. Theoretical analysis shows that the market maker’s worst-case loss is O(M²) when trades concentrate on low-leg parlays, but improves to O(M) when trading activity is distributed across distinct clusters. Simulations based on the Logarithmic Market Scoring Rule (LMSR) and empirical validation using historical order flow from Kalshi demonstrate that the proposed mechanism significantly outperforms independent LMSR-based markets in realistic settings.

Abstract

Parlays - joint contracts on the simultaneous resolution of several events - are among the most heavily traded products in betting markets, but prediction markets have struggled to offer them natively. In this paper, we offer the full combinatorial family of parlays on top of $M$ binary events, as liquid markets, bounding the market maker loss for subsidizing the markets to $O(M^2)$. Any single parlay attracts few traders, so each is an inherently thin market, and the logarithmic market scoring rule (LMSR) is the natural mechanism for thin markets. But running a separate LMSR for each of the exponentially many parlays forces the market maker to pay for the same information many times over. We show that a market maker which automatically propagates information across related parlays avoids this redundancy. We make three contributions. First, we introduce the automated parlay market maker (APMM), which uses a \emph{hierarchical parameterization}: the state of each low-leg parlay is shared into every higher-leg parlay that contains it, so pricing one updates all the others coherently. Second, we show that when informed trading is concentrated in parlays with few legs, the market maker's worst-case loss is $O(M^2)$, and it falls to $O(M)$ when the traded parlays are also spread across different clusters rather than concentrated on a few. Third, we validate these bounds in simulation and on historical Kalshi order flow, confirming that real belief updates are dominated by low-leg changes and that APMM's advantage persists under real trading patterns.

Problem

• parlay
• prediction markets
• market maker
• information redundancy
• combinatorial contracts

Innovation

• Automated Parlay Market Maker
• Hierarchical Parameterization
• Information Propagation
• Combinatorial Prediction Markets
• Market Maker Loss Bound