Walk-the-Book Bet Sizing 2026 — Order Book Aware Kelly

Walk-the-book bet sizing — order book ladder with stacked liquidity at progressively worse prices — Photo via Picsum

Open any Polymarket market and you'll see a single ask price: maybe 23 cents for YES. That's the headline number. It is also a lie about your real entry cost if you're staking more than the trivial top-of-book size. The deeper your stake, the more you walk into the order book and pay worse prices for each additional unit. Walk-the-book bet sizing is the math that makes Kelly honest under this constraint.

Why the displayed ask isn't your real entry cost

An order book is a stacked queue of sell orders at progressively worse (higher) prices. The first level — the "top of book" — has whatever liquidity sits there. Maybe $500. Behind that there's another $2,000 a cent higher, then $5,000 two cents higher, and so on.

If you market-buy with a $300 stake, you fill entirely at the top — your effective price equals the screen ask. Easy. But if you market-buy with $5,000, you eat the top $500, then $2,000 of the next level, then $2,500 of the level after. Your average fill price is a volume-weighted blend of all three.

That blended fill is the number Kelly should be sizing off, not the displayed top-of-book ask.

Why this matters specifically for Polymarket

Polymarket's binary outcomes mean prices live between 0 and 1. A penny on a 20-cent market is a 5% relative difference in price. Walk-the-book penalties of 1-2 cents are not rounding errors — they can wipe out a 4-cent edge entirely.

Compare to equities, where a $100 stock with a 1¢ slippage is a 0.01% effect. Prediction markets are 10-100× more sensitive to walk-the-book costs because the absolute prices are small.

The fixed-point problem

Here's where the math gets interesting. Kelly says "bet f times your bankroll where f depends on your edge over the entry price." But the entry price depends on how much you bet, because of walk-the-book. So Kelly's recommendation is a function of itself.

Mathematically: f = (p − m_eff(f)) / (1 − m_eff(f)), where m_eff is a function of the stake S = bankroll × f. This is a fixed-point equation — you have to solve for f and m_eff together.

The clean way to do this is iterative. Start with a guess (e.g., the f you'd compute if walk-the-book didn't exist), compute the resulting blended price, recompute f at that price, iterate until the recommendation stabilizes. With proper damping it converges in 5-10 iterations.

A worked example

Bankroll: $100,000. Belief: 70% YES. Displayed ask: 50¢. Top-of-book size: $1,000. Order-book depth: $2k at +0.5¢, $5k at +1.0¢, $15k at +2.0¢.

Naive Kelly (ignoring depth): edge = 0.20, f = 0.40, recommended bet = $40,000.

Walk-the-book Kelly: at $40k stake you fill $1k at 50¢, $2k at 50.5¢, $5k at 51¢, then $15k at 52¢, then $17k at the worst-level price. Blended effective price ≈ 51.6¢. Recompute Kelly at that price: edge drops to 0.184, f drops to 0.38. Bet recommendation falls to ~$38k.

The recommendation iterates downward until it converges. The final recommendation might be $30k-$32k once the solver fully accounts for the fact that bigger bets keep raising the effective price.

That $8-10k difference is real money. Sizing off the screen ask would have left you overbet relative to your true edge.

What a walk-the-book solver actually does

Inside our Kelly Criterion calculator, the solver is a damped fixed-point iteration:

Initialize stake S₀ from naive Kelly at the screen ask
Compute m_eff(S₀) by walking the book until S₀ is filled
Compute new f and S₁ = bankroll × f at m_eff
Damp: S = 0.5 × S₀ + 0.5 × S₁ (prevents oscillation)
Repeat until |S_new − S| < $0.50
Final m_eff and f are the walk-the-book honest answers

Without the damping step the solver can oscillate between an over-large and under-large stake — a textbook trap when the function is steep. With damping, you get smooth convergence in under 10 iterations.

When walk-the-book matters most

Big bankrolls relative to market liquidity. If your stake is bigger than the visible top of book, walk-the-book is in play.
Tight edges. The smaller your edge, the more 1-2 cents of slippage hurts as a fraction of it.
Niche markets. Order books on small Polymarket markets can be thin even at the headline level.

If you're betting $50 on a heavily-traded election market, walk-the-book is irrelevant. If you're betting $5,000 on a niche geopolitical market, it's the difference between a profit and a loss.

Pros and cons of walk-the-book sizing

Pros

Honest accounting for actual fill price
Prevents the classic "I sized off the screen and got worse fills than expected" surprise
Built-in protection against over-betting in thin liquidity

Cons

Requires you to know order book depth before sizing
Slightly more compute than naive Kelly (negligible in practice)
Doesn't account for adversarial order placement during your fill

Frequently asked questions

How do I get order book depth for a Polymarket market?

Polymarket displays bid/ask spread and top-of-book size by default. For deeper levels, click into the market and the order book panel shows several levels. Note the offsets and sizes — these are your inputs to a walk-the-book calculator.

Does walk-the-book matter for small bets?

If your stake fits inside the top-of-book size, walk-the-book is a no-op — your effective price equals the screen ask. It only matters when you're sizing past the top level.

What's the difference between walk-the-book Kelly and naive Kelly?

Naive Kelly assumes infinite liquidity at the displayed price, which means it consistently over-recommends in real markets. Walk-the-book Kelly accounts for the fact that bigger bets raise your effective entry price, and arrives at a smaller, honest recommendation.

Bottom line

Walk-the-book bet sizing is what separates a toy spreadsheet from a calculator built for real Polymarket bankrolls. The math is a fixed-point iteration that converges in seconds. The result is a recommended bet size that respects the order book reality — typically 10-30% smaller than naive Kelly, but actually achievable at the price the math thinks.

Size Your Polymarket Bets Like a Pro

Try our free Kelly Criterion calculator — Bayesian shrinkage, walk-the-book sizing, 10k-trial Monte Carlo. Built for serious Polymarket traders. Auto-fills from live ProbBrain signals.

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