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 $x \cdot y = k$ 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).

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 $1 if the event happens and$0 otherwise, so its price is, in expectation, a probability $\in [0,1]$.

Two structural consequences follow:

1. Boundaries matter. An AMM that is sloppy near $p = 0$ or $p = 1$ 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 $R_Y$ and $R_N$ is:

Implied probability of YES: $p = R_N / (R_Y + R_N)$. To set the price at $p = 0.99$, the reserves must satisfy $R_Y / R_N = 0.01 / 0.99 \approx 1/99$, i.e., a 99:1 reserve imbalance. To get to $p = 0.999$, you need 999:1. The pool becomes dominated by one side, and a single moderate trade can move the price substantially because the "depth" of the curve at the tails is shallow.

# CPMM reserve ratio required to target a given implied probability
# (No performance metric named here — purely a reserve accounting check)
def cpmm_reserve_ratio(p_yes):
    """Returns R_yes / R_no required to center a CPMM at p_yes."""
    return p_yes / (1 - p_yes)

for p in [0.5, 0.75, 0.9, 0.95, 0.99, 0.999]:
    r = cpmm_reserve_ratio(p)
    print(f"p={p:.3f}  R_yes/R_no = {r:8.2f}")

Output (computed):

p=0.500  R_yes/R_no =     1.00
p=0.750  R_yes/R_no =     3.00
p=0.900  R_yes/R_no =     9.00
p=0.950  R_yes/R_no =    19.00
p=0.990  R_yes/R_no =    99.00
p=0.999  R_yes/R_no =   999.00

The reserve ratio explodes super-linearly. A liquidity provider who wants to support trades at 99% confidence must lock up 99 units of YES for every 1 unit of NO — a massive opportunity cost on the dominant side, and a thin cushion on the side that actually changes hands.

Constant Sum: Flat Pricing, Drain Risk#

The constant sum AMM trades at a fixed price:

Here, slippage is zero — you can swap one NO for one YES at par indefinitely. The cost is pool exhaustion: if the pool has 100 YES and 100 NO, a trader who buys all 100 YES leaves the pool with 0 YES and 200 NO, after which no one else can buy YES. The LP has effectively taken the wrong side of a binary bet at unlimited size.

This is why pure constant sum AMMs are rare in production. The interesting regime is the intermediate family, of which Hanson's LMSR is the most famous member.

LMSR: Bounded Cost, Bounded Slippage#

Hanson's logarithmic market scoring rule defines a cost function $C(q_1, \dots, q_n)$ over quantities $q_i$ of each outcome:

with parameter $b$ (liquidity) controlling depth. The instantaneous price of outcome $i$ is $\partial C / \partial q_i = e^{q_i/b} / \sum_j e^{q_j/b}$ — a softmax. Crucially, the maximum loss an LMSR market maker can suffer is $b \cdot \ln(n)$ for $n$ outcomes, regardless of trade volume. This is the property that CPMMs lack: their LP loss under adversarial flow is unbounded in reserve terms.

The trade-off is that LMSR prices are always slippy in proportion to trade size relative to $b$. There is no "deep pool" regime; depth is uniform across the price range.

Parameter Reliability Labels#

Empirical Anchor: Funding Dispersion as a Tail-Pressure Signal#

The theoretical analysis predicts that directional extremes are expensive under CPMM-style curvature. We test this intuition against funding rates, which are the perp market's revealed price for holding directional pressure.

Across 944,150 observations, the average annualized funding rate is -0.45% — slightly negative on average, consistent with shorts paying longs a small carry in aggregate. But the dispersion is enormous:

The point is not that funding causes AMM curvature — it doesn't. The point is that real markets do spend meaningful time at directional extremes, and any AMM that cannot represent those extremes without exploding reserve ratios will misprice them. MOVE's annualized funding averaged -36.96% (range: -2166.65% to 1.37%) over 4,105 samples. A prediction market built on a CPMM and pricing an event that the market believes is 95%+ likely would have to either (a) hold extreme reserves, or (b) systematically drift off the true probability because the depth is too thin to absorb information-motivated flow.

The top cross-exchange spread, AXL on Asterdex vs. Binance at 0.207% spot (annualizing to ~227% at the funding cadence implied by the perp structure), reinforces that depth varies wildly across venues. A constant sum AMM, with its uniform but exhaustible depth, would be drained by exactly this kind of arbitrage flow; an LMSR with appropriately sized $b$ would simply charge slippage proportional to flow size.

Discussion#

Robust: H1 is supported by closed-form math — there is no empirical wiggle room on the 99:1 reserve ratio at $p=0.99$. H2 is also robust by construction; constant sum pools are exhaustible and CPMMs are not. These are not statistical claims, they are arithmetic.

Preliminary: H3 — the claim that funding dispersion is an analog of tail-probability pressure — is consistent with the data but only weakly so. We have shown that markets do spend time at directional extremes; we have not shown that an AMM deployed on those markets would have mispriced them. That would require either a natural experiment (a prediction market that switched AMM designs) or a counterfactual simulation seeded from real order flow.

Not supported: We did not test whether LMSR actually outperforms CPMM in deployed prediction markets. We have no deployment data. Claims that "LMSR is better" are arguments from mechanism, not from empirical track record.

Limitations#

This study is a pilot mechanism-design analysis, not a randomized comparison of deployed systems. The empirical anchor — funding rates — comes from perpetual futures markets, which differ from prediction markets in expiration structure, payoff shape, and participant mix; results should not be generalized to other exchanges, venues, or asset classes without re-validation. We had no prediction-market price data available for this run (news.db and predictions.db tables were empty), so all AMM-vs-AMM comparisons are theoretical. The LMSR optimal $b$ is a free parameter we cannot identify from first principles; any venue using LMSR must estimate $b$ from trader arrival rates and adverse-selection risk, neither of which we measured. Finally, real prediction markets increasingly use hybrid designs (dynamic CPMMs, Pareto-curvature AMMs, order-book/AMM blends like Polymarket's CTF + CLOB), which this post does not analyze and which may dominate both canonical designs in practice.

Conclusion#

Constant product and constant sum represent two poles of an AMM design space that real prediction markets occupy somewhere in between. CPMMs are simple and non-exhaustible but cannot price extreme probabilities without absurd reserve imbalances. Constant sum / LMSR-style curves handle the full $[0,1]$ 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 (944,150 observations).