💰 quantitative finance

ParlayMarket: Automated Market Making for Parlay-style Joint Contracts

This paper introduces ParlayMarket, an automated market-making design that supports parlay-style joint contracts within a unified liquidity pool, proving that its dynamics converge to a unique fixed point where pricing errors remain bounded and are further reduced by the structural constraints of parlay trades on the joint distribution.

Original authors: Ranvir Rana, Viraj Nadkarni, Niusha Moshrefi, Pramod Viswanath

Published 2026-03-25

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CC BY 4.0

Original authors: Ranvir Rana, Viraj Nadkarni, Niusha Moshrefi, Pramod Viswanath

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are running a massive, high-stakes betting hall.

In a traditional betting hall (like a standard prediction market), you have separate tables for every single event.

Table A: "Will the Lakers win?"
Table B: "Will the Celtics win?"
Table C: "Will the temperature in London be above 20°C?"

If you want to bet on a Parlay (a combined bet, like "Lakers AND Celtics win"), the traditional system is broken. You either can't make that bet, or you have to go to a human dealer who guesses the odds. If the dealer guesses wrong, they lose money. If they guess right, they might miss out on a better price. Worse, if the Lakers and Celtics are actually playing in the same game (highly correlated), the dealer might not realize it, leading to chaotic, inconsistent prices.

ParlayMarket is a new, automated "super-banker" that fixes this mess. It doesn't just look at individual tables; it builds a single, giant, living map of how everything is connected.

Here is how it works, using simple analogies:

1. The "Giant Jigsaw Puzzle" vs. The "Infinite Library"

The Problem:
Imagine you have 20 different events. If you try to list every possible combination of outcomes (Lakers win + Celtics lose + Rain in London...), you have over a million ( $2^{20}$ ) different scenarios.

Old Way: Trying to keep a separate price tag for every single one of those million scenarios is impossible. It's like trying to write a book for every possible sentence in the English language. You'd go bankrupt trying to manage the risk.
The ParlayMarket Way: Instead of listing every sentence, ParlayMarket builds a grammar book. It learns the rules of how the words (events) fit together. It only needs to track how Event A talks to Event B, and Event B talks to Event C. It compresses a million scenarios down to a manageable list of relationships (like $20 \times 20$ ).

2. The "Echo Chamber" (Shadow Trades)

This is the magic trick.
In a normal market, if you bet on "Lakers win," only the "Lakers" price changes. The "Celtics" price stays the same, even if the Lakers and Celtics are playing each other.

In ParlayMarket, every bet creates an echo.

You bet on "Lakers AND Celtics win."
The system doesn't just update that one price. It instantly whispers updates to every related price in the building.
It says: "Hey, someone just bet on this combo. That means the odds for 'Lakers win' just shifted slightly, and 'Celtics win' shifted too."
This keeps the whole system synchronized. No matter which table you sit at, the prices are always mathematically consistent with the rest of the hall.

3. The "Smart Teacher" (Learning from Mistakes)

The system is an automated teacher that never stops learning.

The Setup: It starts with a blank slate, guessing that everything is random.
The Lesson: When a trader places a bet, the system checks: "Did the trader know something I didn't?"
The Correction: If the trader is right, the system adjusts its "grammar book" (its internal model of how events connect).
The Result: Over time, the system becomes incredibly good at predicting not just single events, but complex combinations. It learns the hidden correlations (like how a storm in one city affects traffic in another) simply by watching people bet on them.

4. Why This Matters: The "Safety Net"

The biggest fear in betting markets is unbounded loss. If a market maker gets the math wrong on a complex bet, they could lose millions.

ParlayMarket proves that by using this "grammar book" approach:

Risk is Contained: Even if you have millions of possible bets, the system's potential loss grows slowly (like the square of the number of events), not explosively (like the exponential number of combinations).
Liquidity is Unified: You don't need a separate bank account for every single bet. One big pool of money supports the whole universe of bets because they are all linked by the same logic.

The Bottom Line

Think of ParlayMarket as a universal translator for uncertainty.

Before, if you wanted to bet on a complex scenario involving many moving parts, you had to rely on a human dealer who might be tired, biased, or just bad at math. Now, you have an automated system that treats the entire market as one giant, interconnected puzzle. It learns the connections as you bet, keeps the prices fair across the board, and does it all without needing a fortune in capital to back it up.

It turns a chaotic, fragmented betting hall into a single, coherent, self-correcting machine.

1. Problem Statement

Prediction markets are effective for aggregating information on single events, but they struggle with joint contracts (parlays), where traders bet on the simultaneous occurrence of multiple events (e.g., "Team A wins AND Team B wins").

The Combinatorial Explosion: With $M$ binary events, the joint outcome space contains $2^M$ possibilities. A naive market maker maintaining a price for every possible combination faces exponential state complexity and exponential worst-case loss, rendering direct approaches intractable for large $M$ .
Current Limitations: Existing platforms either:
1. Prohibit joint contracts.
2. Use Request-for-Quote (RFQ) mechanisms (common in platforms like Kalshi), which fragment liquidity, lack coherent pricing (often violating probability bounds), and fail to propagate information across related markets.
3. Assume independence between events (multiplying marginal probabilities), which fails to capture correlations, leading to unbounded arbitrage losses when events are actually correlated.

The core challenge is to design an Automated Market Maker (AMM) that supports an exponentially large family of joint contracts while maintaining polynomial complexity, bounded loss, and coherent pricing.

2. Methodology: ParlayMarket

The authors propose ParlayMarket, an AMM that treats the market as an online inference system. Instead of pricing contracts in isolation, it maintains a single, shared parametric model of the joint distribution over all outcomes.

A. Core Architecture

Pairwise Exponential Family (Ising Model): The market maker represents the joint distribution $P(x)$ $P (x)$ using a pairwise exponential-family model (Ising model):
$P_\phi(x) \propto \exp\left(\sum_{i=1}^M \theta_i x_i + \sum_{i<j} W_{ij} x_i x_j\right)$
- State Compression: This reduces the state space from $2^M$ to $O(M^2)$ sufficient statistics (first-order biases $\theta$ and second-order interactions $W$ ).
- Maximum Entropy: Among all distributions matching the observed marginals and pairwise correlations, this is the maximum-entropy distribution, introducing no unjustified higher-order structure.
Shadow Trades & Price Propagation:
- When a trade occurs in any contract (base or parlay), the AMM treats it as a noisy observation of the underlying joint distribution.
- The AMM performs a stochastic gradient descent (SGD) update on the shared parameters $\phi = (\theta, W)$ .
- Crucially, the system generates "shadow trades" on all other correlated contracts. This ensures that a trade in a liquid base market instantly updates the prices of illiquid, complex parlay markets, preventing stale pricing and arbitrage.
Pricing Mechanism: All prices are derived as marginals of the current belief state $P_\phi$ . Base market prices are $P_\phi(X_i=1)$ , and parlay prices are $P_\phi(\bigcap_{i \in S} X_i=1)$ , computed via belief propagation.

B. Learning Dynamics

Objective: The AMM minimizes a composite pseudo-likelihood (cross-entropy) between the trader's implied probability and the model's current price.
Convergence: The system converges to the I-projection of the true distribution $P^*$ onto the Ising family. This is the best possible approximation of the true joint distribution given the constraint of only tracking first and second-order moments.

3. Key Contributions

Unified Liquidity Pool: ParlayMarket pools liquidity across all $2^M$ contracts into a single $O(M^2)$ parameter state, eliminating the need for separate capitalization for each contract.
Bounded Loss: The paper proves that the market maker's aggregate loss scales polynomially (specifically $O(M^2)$ $O (M^{2})$ or $O(M)$ $O (M)$ depending on the metric) rather than exponentially.
- Mechanism: The "shadow trade" mechanism creates a balance between signal and noise. As the number of markets grows, the signal (information from trades) and noise (stochastic updates) scale together, keeping the steady-state parameter error bounded.
Parlays as Learning Signals: The authors demonstrate that parlay trades are not just products to be priced but are essential for learning. Higher-order parlay trades provide direct constraints on interaction terms ( $W_{ij}$ ) that base markets alone cannot efficiently identify, accelerating convergence and reducing steady-state error.
Theoretical Guarantees:
- Geometric Convergence: The parameters converge geometrically to the optimal Ising approximation.
- Loss Bounds: In the "complete-parlay" regime (where all combinations are traded), the expected loss per market decays exponentially with $M$ , while total aggregate loss remains bounded.

4. Results

The authors validate their design through controlled simulations and a historical replay on Kalshi data.

Simulation Results (Synthetic Data):
- Loss Scaling: ParlayMarket achieves the lowest non-oracle loss, which decays exponentially with the number of base markets ( $M$ ) when parlays are present. In contrast, independent LMSRs and other baselines show exponential loss growth.
- The "Parlay Trilemma": Without parlay trades, the Ising model performs poorly (under-determined system). With parlay trades, it becomes the best performer. This confirms that parlay order flow is the primary signal for learning dependence.
- Robustness: The system remains robust to noise traders, with loss growing gracefully as the noise fraction increases.
Empirical Results (Kalshi Replay):
- Using historical NBA data (732 base markets, ~6,000 combo tickers), the authors replayed the mechanism.
- Performance: A "standalone" ParlayMarket (layered on top of existing base markets) achieved the highest Sharpe ratio (1.00) among non-native strategies, significantly outperforming independent LMSRs.
- Feasibility: The system successfully handled real-world constraints like fee biases and liquidity fragmentation, proving it is deployable in practice.

5. Significance

Bridging Theory and Practice: ParlayMarket resolves the long-standing tension between combinatorial expressiveness (the ability to bet on complex scenarios) and computational tractability. It proves that one does not need exponential capital to support exponential markets.
New Market Design Paradigm: It shifts the paradigm from "pricing contracts in isolation" to "learning a global probabilistic model online." This allows for coherent pricing where $P(A \cap B)$ is mathematically consistent with $P(A)$ and $P(B)$ .
Information Aggregation: By treating parlay trades as direct constraints on the joint distribution, the mechanism extracts more information from market data than traditional marginal-only markets, leading to more efficient price discovery.
Scalability: The $O(M^2)$ complexity makes it feasible to deploy prediction markets for thousands of correlated events (e.g., sports seasons, macroeconomic indicators) without the risk of market maker insolvency due to combinatorial explosion.

In summary, ParlayMarket provides a mathematically rigorous and empirically validated framework for automated market making in correlated environments, enabling scalable, liquid, and coherent prediction markets for complex joint outcomes.