Prediction Market AMM: How Bonding Curves Power Crypto Markets

A concrete scenario: a binary market quoting 62% with b = 2

Imagine a binary prediction market that pays $1 for a Yes outcome and $0 for No. The market currently quotes Yes at $0.62. The platform uses an LMSR-style bonding curve with liquidity parameter b = 2. For clarity:

Item	Value
Quoted implied probability (Yes)	62% ($0.62)
Liquidity parameter b	2
Payout if Yes	$1 per contract
Payout if No	$0 per contract

That quoted $0.62 is the marginal price — the AMM’s answer to the question “what would I pay to buy one more infinitesimal share right now?” An AMM doesn’t keep a stack of limit orders; it exposes a continuous price function (the bonding curve) and updates that marginal price as traders move the market.

For LMSR, starting with no outstanding shares the marginal price begins at 50%. Previous buys have shifted the outstanding Yes position so the current marginal price sits at 62%; in LMSR terms that corresponds to roughly 0.98 outstanding Yes units (with No at 0), as shown in the worked numbers below.

Running the numbers: marginal price, outstanding position, and the cost of buys

Two different prices matter here: the marginal price p (what the next tiny share costs) and the average price you pay for a finite-sized purchase (the integral of the curve over your size). Using the LMSR relationship, a marginal price of 0.62 corresponds to an outstanding Yes quantity q_yes ≈ 0.979 (with q_no = 0).

If you buy a truly infinitesimal share, you pay the marginal price $0.62 per unit. But discrete purchases move q_yes and therefore the marginal price during your trade, so a 1-unit purchase will cost more than $0.62 on average.

Concrete numbers from the current state:

• Current cost function value C(q) at q_yes ≈ 0.979 (q_no = 0) is about $1.935.
• After buying 1 full Yes contract (q_yes ≈ 1.979), the cost function value becomes about $2.610.
• Incremental cost to buy that 1 unit = 2.610 − 1.935 ≈ $0.675.

So:

• Marginal quoted price = $0.62 (price of the next infinitesimal share).
• Cost to buy a full 1-unit contract right now ≈ $0.675 (average price paid ≈ $0.675), because your trade moves the curve upward.

Smaller buys show smaller divergence:

• Buying 0.1 units from the same start raises the average paid to ≈ $0.63 per unit (total ≈ $0.063), close to the quoted marginal price.

These concrete deltas illustrate the practical lesson: the marginal price you see tells you what the next tiny share would cost, but larger purchases pay the integral of the curve and therefore often have a higher average per-share price.

What happens at each outcome for someone who buys now

If you buy at the marginal quote of $0.62 (i.e., you place a trade priced exactly at the marginal price for an infinitesimal unit), then:

• If the event resolves Yes, each contract you hold pays $1, so profit per marginal unit = $1 − $0.62 = $0.38 (before fees).
• If the event resolves No, that marginal unit is worth $0, so loss = $0.62.

If instead you buy a full 1-unit contract from the current curve (the discrete-cost case above):

• You pay ≈ $0.675.
• If the event resolves Yes your profit is $1 − $0.675 ≈ $0.325 (before fees).
• If No, you lose your $0.675.

The difference between $0.38 and $0.325 underscores why the distinction between marginal price and average trade price matters. Large buys drive the marginal price up while you are executing, raising your average cost.

How the maths (and market behaviour) shifts when the price or b changes

Liquidity parameter b and the shape of the bonding curve are the knobs that control sensitivity and capital needs.

• Curve steepness / liquidity parameter (b)

  • Small b (steeper curve): small trades move the marginal price a lot. Markets become more responsive to single trades but are easier to move and less capital-efficient for liquidity providers.
  • Large b (shallower curve): trades move the marginal price less. The market needs more capital to shift, lowering manipulation risk but slowing price response to new information.

• Fees and fee structure

  • Per-trade fees cushion liquidity providers and make tiny repeated trades less attractive.
  • Fees raise the breakeven cost for manipulation — if fees exceed expected profit from shifting the market, manipulation is less profitable.

• How liquidity is added or removed

  • On some platforms users can deposit capital to back the curve and earn fees; others use protocol collateral. Who supplies liquidity determines capital efficiency, fee distribution, and risk exposure.

• Price granularity and rounding rules

  • Real implementations use discrete share units, tick sizes, transaction batching and gas limits. These engineering choices mean the live marginal price will be a discretized approximation of the continuous math.

Practical reminder: changing b changes both how much capital is required to create a given price and how much a fixed trade moves that price. In our worked numbers above, the same trade sized identically would move the market more if b were smaller and less if b were larger.

Choosing the right market architecture: LMSR AMM, constant-product adaptations, or orderbooks

Different mechanisms fit different needs; the same AMM idea can feel quite different when implemented differently.

Feature	AMM (LMSR-style bonding curve)	AMM (constant-product adaptation)	Orderbook (limit orders)
Price source	Cost function derivative => continuous implied probability	Reserve ratio => continuous price from token pool	Best bid/ask from standing orders
Typical opening price	50% when balanced	Depends on initial pool composition	Set by the first orders
How price changes	Smooth, controlled by b	Depends on reserve ratio and pool depth	Stepwise as orders fill
Capital efficiency	Needs capital proportional to b; fees often earned	Efficient for token liquidity but fragile at extremes	Capital sits in orders until used
Price sensitivity to trade	Tunable by b	Tunable by pool size	Determined by order depth
Manipulation risk	Lower with large b; still present	Lower with large reserve; flash trades can exploit	Low with deep books; can be spoofed if thin
Best for	Continuous, predictable probability markets	Integrations with DeFi pools	Professional trading and fine-grained control

Use LMSR-style AMMs when you want a continuously available market with a single tunable sensitivity parameter (b). Use constant-product adaptations when you need to integrate prediction liquidity into broader token pools. Use an orderbook when you expect professional market makers to supply discrete liquidity and you need granular order control.

Common mistake to avoid: assuming the quoted marginal price equals the average price you will pay for a multi-unit buy. Large trades pay the integral of the curve, not just the current marginal.

Keep reading

• What Are Prediction Markets?
• How Do Prediction Markets Work?
• What Are Yes/No Contracts?
• How to Read Prediction Market Prices