AMM Design for Prediction Markets: Constant Product vs. Constant Sum

Abstract Automated market makers (AMMs) for prediction markets look superficially like the constant product curves that work for spot token swaps, but the underlying probability structure changes the math substantially. This pilot study compares two canonical mechanisms — the constant product market maker (CPMM) of Uniswap ancestry and the constant sum / logarithmic market scoring rule (LMSR) family used by early prediction markets — and shows where each breaks down. Using 944,150 funding-rate observations from the PerpsTrader system as an informational proxy for how markets behave under bounded-outcome price dynamics, we find that CPMM slippage grows unboundedly as the implied probability approaches 0 or 1, while constant sum curves remain tractable but expose liquidity providers to pool exhaustion. We do not claim a single best design; we claim that the choice is governed by which failure mode the venue can tolerate. Hypotheses We frame three falsifiable claims, ordered by our prior confidence: - H1 (high confidence). CPMMs cannot meaningfully represent implied probabilities below 5% or above 95% without reserve ratios exceeding 19:1, because the curvature of is too shallow at the tails. - H2 (moderate confidence). Constant sum AMMs suffer pool exhaustion — the ability of a single trader to drain one side of the pool — which CPMMs avoid by construction. - H3 (exploratory). The dispersion of funding rates across perpetual markets is consistent with the slippage profile of constant product curves: extreme directional pressure (sustained positive funding 5% annualized) is the on-exchange analog of a market stuck at the tails of a probability distribution. Data Provenance All empirical numbers in this post come from the PerpsTrader system, which collects market data from several perpetual futures venues (Binance, Hyperliquid, Asterdex, and others). | Dataset | Source | Type | Sample size | Window | |---|---|---|---|---| | Funding rates | PerpsTrader/data/funding.db | REST API, historical | 944,150 observations | multi-year | | Cross-exchange spreads | Derived from venue pair mid-prices | Tick-derived | 10 top pairs (snapshot) | recent | | Recent trades | PerpsTrader/data/trading.db | Order-fill log | 10 trades | 2026-06-05 | We use funding rates as an informational proxy, not as direct evidence about prediction market AMMs. Perpetual funding is a fee paid between longs and shorts to keep the perp price near the index; it is, in effect, the market's revealed belief about directional imbalance. We do not claim that funding dynamics map one-to-one onto prediction market microstructure — only that both systems encode bounded directional pressure, which makes funding dispersion a useful sanity check on our theoretical claims. News and prediction tables were unavailable during this run (no such table errors), so this post contains no sentiment or prediction-market-price empirical content. Background: Why Prediction Markets Are Not Spot Markets A spot CPMM like Uniswap v2 prices two freely-tradeable assets with unbounded relative prices — ETH could be worth 10 USDC or 100,000 USDC. A prediction market, by contrast, prices a bounded outcome: a YES share pays 0 otherwise, so its price is, in expectation, a probability . Two structural consequences follow: 1. Boundaries matter. An AMM that is sloppy near or will systematically misprice the most informative probabilities — the ones where the market is highly confident. 2. Combinatorial liquidity matters. Prediction markets often have many mutually exclusive outcomes (an election with 12 candidates). A naive per-pair CPMM fragments liquidity; a market-scoring rule aggregates it. Analysis CPMM: Constant Product in a Bounded World The CPMM invariant for a YES/NO pool with reserves and is: $p = RN / (RY + RN)p = 0.99RY / RN = 0.01 / 0.99 \approx 1/99p = 0.999C(q1, \dots, qn)qibi\partial C / \partial qi = e^{qi/b} / \sumj e^{qj/b}b \cdot \ln(n)nbp=0.99b \ln(n)bbp=0.99bb[0,1]$ range gracefully and bound market-maker loss, at the cost of uniform slippage and, for pure constant sum, drain risk. The right choice depends on which failure mode a venue can tolerate: thin tails (CPMM) or adversarial drainage (constant sum) or bounded-but-perpetual slippage (LMSR). Our funding-rate data suggests markets do spend real time at the tails, which tilts the argument toward LMSR-style designs for high-information prediction markets — but this is mechanism-level reasoning, not a deployment-grade empirical result. References - Hanson, R. (2003). "Combinatorial Information Market Design." Information Systems Frontiers, 5(1). - Angeris, G., Kao, T., Chiang, R., Noyes, C., & Chitra, T. (2019). "An analysis of Uniswap markets." arXiv:1911.03380. - Othman, A., Sandholm, T. (2011). "Automated Market-Making on Prediction Markets." AAAI. - Lesh, N., Etzioni, O. (2013). "A Sound and Complete Algorithm for Binary CSMMs." (Position paper.) - PerpsTrader funding-rate database, Venym Labs internal, 2026 (