Prediction Market Arbitrage Bounds: Dutch Book, Ladders & Fee-Adjusted Limits

Q: What is a Dutch book in prediction markets?

A set of trades that guarantees a profit whatever happens, possible when quoted prices are not a coherent probability distribution. For example, if the Yes and No asks sum to less than $1, buying both locks in the difference before fees.

Q: Should all outcome prices in a prediction market add up to 100%?

Before costs, yes. After taker fees, a complete set of n equally priced outcomes can sum to about 100 − 7(1 − 1/n) cents without being an exploitable arbitrage, at a 0.07 fee coefficient.

Q: What are Fréchet bounds?

The range of joint probabilities consistent with given marginals: P(A and B) must lie between max(0, a + b − 1) and min(a, b). Inside that band a joint contract's price reflects correlation, not arbitrage.

AbstractEvent-contract prices must form a coherent probability distribution or someone can construct a Dutch book against them. We list the four constraints that matter in practice (Yes + No = 1, complete sets sum to 1, ladders are monotone, joint outcomes obey Fréchet bounds) and show how taker fees proportional to P(1−P) widen each into a band. One result is worth memorising: buying a complete set of n equally priced outcomes as a taker costs about

7 (1 - 1/ n)

¢ in fees per set at Kalshi's published coefficient, so a 10-outcome market can show prices summing to 94¢ with no exploitable arbitrage.

Coherence: the Dutch book argument#

De Finetti's (1937) observation is that if someone's betting prices on a set of events violate the axioms of probability, a bookmaker can choose stakes that guarantee them a loss whatever happens: a Dutch book. For traders the statement runs the other way. If a set of quoted prices violates the axioms, there is a riskless trade, before costs. Four constraints cover almost every case seen in event markets.

Table 1. Coherence conditions for event-contract prices. a and b are the marginal probabilities of A and B.
Constraint	Statement	Typical violation	Trade
Complement	$P (Yes) + P (No) = 1$	Yes ask + No ask < 100¢	Buy both
Complete set	$\sum_{i = 1}^{n} P (A_{i}) = 1$	Outcome asks sum < 100¢, or bids sum > 100¢	Buy all / sell all
Ladder monotonicity	$K_{1} < K_{2} \Rightarrow P (X \geq K_{1}) \geq P (X \geq K_{2})$	Higher strike priced above lower	Buy lower, sell higher
Fréchet bounds	$max (0, a + b - 1) \leq P (A \land B) \leq min (a, b)$	Joint contract outside the band	Joint vs. marginals

Fees turn equalities into bands#

With a taker fee $ϕ (P) = θ P (1 - P)$ per contract ( $θ = 0.07$ in Kalshi's published schedule), buying a Yes/No pair at asks $a$ and $b$ is profitable only if

1 - a - b > θ [a (1 - a) + b (1 - b)]

(1)

At $a \approx b \approx \frac{1}{2}$ the right-hand side is $θ /2$ = 3.5¢; per-order rounding pushes it to 4¢ for single contracts. Our fee note tabulates the full band by price and size. The general lesson is that the no-arbitrage band is widest at 50¢ and shrinks toward the tails, the same shape as the fee.

Complete sets: the band grows with n#

For a market with $n$ mutually exclusive, exhaustive outcomes (who wins a primary, which bucket a data release lands in), buying one share of every outcome guarantees $1. If each outcome trades near $1/ n$ , the total taker fee for the set is

i = 1 \sum n θ \frac{1}{n} (1 - \frac{1}{n}) = θ (1 - \frac{1}{n})

(2)

Table 2. Taker fees for buying one complete set when every outcome trades at 1/n (large orders, θ = 0.07, rounding ignored).
Outcomes n	2	3	5	10	20	50
Fee per complete set	3.50¢	4.67¢	5.60¢	6.30¢	6.65¢	6.86¢
Asks can sum to as little as	96.5¢	95.3¢	94.4¢	93.7¢	93.3¢	93.1¢

Equation (2) approaches 7¢ as $n$ grows: the fee on a complete set never exceeds the fee coefficient. Prices summing to 95¢ in a 20-outcome market are therefore not an arbitrage for a taker. With uneven prices the sum is smaller (equation 2 is the maximum for a given $n$ ), and maker fees, typically a quarter of taker fees, shrink it further. A market maker resting bids on every outcome faces a very different band from a taker sweeping asks.

Ladders: monotonicity and implied distributions#

A ladder of contracts " $X \geq K_{j}$ " on one quantity (inflation, a temperature, a vote share) is a discretised survival function. Prices must be non-increasing in the strike, and adjacent differences are the market's probabilities for each bucket:

Pr (K_{j} \leq X < K_{j + 1}) = P (X \geq K_{j}) - P (X \geq K_{j + 1}) \geq 0

(3)

A monotonicity violation (the 3.1% strike priced above the 3.0% strike) is the cleanest arbitrage in event markets. Buy the lower strike, sell the higher; the payoff is never negative and is positive if $X$ lands between them. Net of two taker fees it needs roughly the band in equation (1), and the rules must be checked: if strikes resolve on different releases or rounding conventions, the "violation" may be a basis.

Ladders also give something more useful than arbitrage: the implied distribution. Its mean, variance and skew can be compared directly with a nowcast, and the bucket probabilities can be checked for coherence against related ladders (headline vs. core inflation, for example).

Joint outcomes and the Fréchet band#

When a venue lists a contract on "A and B" alongside contracts on A and B separately (a parlay, a combination of results, or "rate cut and inflation above X"), the joint price is constrained only loosely by the marginals:

max (0, a + b - 1) \leq P (A \land B) \leq min (a, b)

(4)

With $a = 0.6$ and $b = 0.5$ the joint can be anywhere from 10¢ to 50¢ without arbitrage. The independence price, 30¢, sits in the middle. Inside the band, the joint contract is a bet on the dependence between A and B, not an arbitrage, and pricing it requires a model of correlation. Outside the band the trade is mechanical: if the joint is quoted above $min (a, b)$ , sell the joint and buy the cheaper marginal.

Multi-leg contractsVenues increasingly list combinations of many legs. For

k

legs the lower Fréchet bound

max (0, \sum a_{i} - (k - 1))

is almost always zero, so arbitrage bounds say little. The price is a correlation model plus a margin, and market makers quoting them need a joint model of the legs.

Worked example: auditing an inflation ladder#

A ladder of "CPI year-on-year at or above K" contracts shows the quotes below. Bucket probabilities are adjacent differences of the mids (equation 3).

Table 3. A hypothetical ladder with one monotonicity violation (3.1% priced above 3.0%).
Strike K	Bid	Ask	Mid	Implied Pr(bucket starting at K)
2.7%	93¢	95¢	94¢	13%
2.8%	80¢	82¢	81¢	21%
2.9%	59¢	61¢	60¢	18%
3.0%	41¢	43¢	42¢	-1%
3.1%	42¢	44¢	43¢	34%
3.2%	8¢	10¢	9¢	9% (≥ 3.2%)

The 3.1% strike is priced above the 3.0% strike, implying a negative probability (-1%) for the 3.0–3.1% bucket. The trade is to buy 3.0% at its 43¢ ask and sell 3.1% at its 42¢ bid. That costs 1¢ net up front, plus taker fees of 3.42¢ per pair at the 0.07 coefficient. The payoff is +$1 if the print is exactly 3.0% and zero in every other state, since both legs pay or neither does. So the position is a 4.42¢ option on the 3.0% bucket. The neighbouring buckets are priced at 18% and 34%, and a smooth distribution through the ladder puts the 3.0% bucket at roughly 15–20%, so the option is worth far more than it costs. This is the typical shape of a ladder violation after fees: not a locked-in profit but a very cheap option, and only after checking that both strikes resolve on the same release with the same rounding.

A monitoring checklist#

For every binary pair, compute Yes ask + No ask and compare with the fee band at that price.
For every complete set, sum asks and bids and compare with equation (2) for its size.
For every ladder, check monotonicity of bids against asks across adjacent strikes.
For every listed joint contract, check the Fréchet band against the marginals' bids and asks.
Before acting on any violation, compare the resolution rules of every leg. Most persistent violations are differences in rules, not prices (see cross-venue arbitrage).

Questions#

What is a Dutch book in prediction markets?

A set of trades that guarantees a profit whatever happens, possible when quoted prices are not a coherent probability distribution. For example, if the Yes and No asks sum to less than $1, buying both locks in the difference before fees.

Should all outcome prices in a prediction market add up to 100%?

Before costs, yes. After taker fees, a complete set of n equally priced outcomes can sum to about 100 − 7(1 − 1/n) cents without being an exploitable arbitrage, at a 0.07 fee coefficient.

What are Fréchet bounds?

The range of joint probabilities consistent with given marginals: P(A and B) must lie between max(0, a + b − 1) and min(a, b). Inside that band a joint contract's price reflects correlation, not arbitrage.

References#

de Finetti, B. (1937). La prévision : ses lois logiques, ses sources subjectives. Annales de l'Institut Henri Poincaré, 7(1), 1–68.
Kalshi. Fee schedule (published PDF), accessed October 2026.

DisclosureThis note is research and education. It is not investment advice or an offer to trade, and nothing in it describes the firm's own results. Code and outputs: research/fees.py in our research repository, reproducible with a fixed seed.