ParlayMarket: Automated Market Making for Parlay-style Joint Contracts

Abstract

Prediction markets are powerful mechanisms for information aggregation, but existing designs are optimized for single-event contracts. In practice, traders frequently express beliefs about joint outcomes - through parlays in sports, conditional forecasts across related events, or scenario bets in financial markets. Current platforms either prohibit such trades or rely on ad hoc mechanisms that ignore correlation structure, resulting in inefficient prices and fragmented liquidity. We introduce ParlayMarket, the first automated market-making design that supports parlay-style joint contracts within a unified liquidity pool while maintaining coherent pricing across base markets and their combinations. Our main result is a convergence characterization of the resulting system. Under repeated trading, the AMM dynamics converge to a unique fixed point corresponding to the best approximation to the true joint distribution within the model class. We show that (i) parameter error remains bounded at stationarity due to a balance between signal and noise in trade-induced updates, and (ii) pricing error and monetary loss scale with this parameter error, implying that aggregate market-maker loss remains controlled and grows at most quadratically in the number of base markets. These results establish explicit limits on the information-retrieval error achievable through the trading interface. Importantly, parlay trades play a structural role in this convergence: by providing direct constraints on joint outcomes, they improve identifiability of dependence structure and reduce steady-state error relative to markets that rely only on marginal trades. Empirically, we show both in controlled simulations and in replay on historical Kalshi parlay data that this design achieves the intended scaling while remaining effective in realistic market settings.

Figures (7)

• Figure 1: ParlayMarket enables $2^M$ markets with $O(M^2)$ capital. Theoretical curves show the asymptotic loss scaling implied by the analysis in Section \ref{['sec:correlation-mm']}. Empirical curves are obtained from controlled simulation of correlated binary markets under the Gaussian score model described later in Sections \ref{['sec:objectives']} and \ref{['sec:evaluation']}, and confirm the same polynomial-vs.-exponential separation in practice.
• Figure 2: In request-for-quote mechanism, market makers short on the requested parlay. They have to constantly manage liquidity and risk.
• Figure 3: The RFQ trilemma.
• Figure 4: The ParlayMarket is the only non-oracle model whose loss decays exponentially with the number of base markets $M$, in the presence of parlays; all other non-oracle models plateau or grow.
• Figure 5: Price convergence over time (mean $\pm 1\sigma$ across 1000 simulations). LMSR market-price MAE vs. true probabilities.

Theorems & Definitions (13)

• Theorem 1: Geometric convergence
• Proposition 1: Strong convexity
• Proposition 2: Complete-parlay loss scaling
• Lemma 1: Quadratic sensitivity
• Proposition 3: Parlay-accelerated convergence
• proof : Proof of the liquidity-consistency bound in the Property 1 paragraph of Subsection \ref{['subsec:amm-architecture']}
• proof : Proof of the moment-matched target claim in Subsection \ref{['subsec:convergence']}
• proof : Proof of Proposition \ref{['prop:strong-convex']} (local strong-convexity form used in the convergence argument)
• proof : Proof of Theorem \ref{['thm:linear-conv']}
• proof : Proof of the $O(\log T)$ cumulative-regret statement following \ref{['eq:mu-eff']}