# The math

> The math behind being the house on Polymarket parlays, explained through worked examples: what a parlay is worth, why small per-leg edges compound into large total margins, expected value per ticket, variance and why any night can lose, bankroll sizing, the correlation trap, longshot pricing error, and pricing against competing makers. Nothing here needs more than multiplication.

Canonical URL: https://pre.game/the-math · Site index for agents: https://pre.game/llms.txt

## The only equation that matters

**profit ≈ margin × volume ± luck**

Margin is how far above fair value you price. Volume is how many tickets you fill. Luck is real but averages toward zero as volume grows — margin does not.

**Example:** average margin 5%, average at-risk $50/ticket, 60 tickets/week → expected weekly profit ≈ 5% × $50 × 60 = **+$150**. Any given week can miss; the average is the business.

## What a parlay is actually worth

A parlay pays only if every leg hits. If the legs are independent, its true probability is the legs **multiplied together**:

| Legs (each 60¢ / 60%) | Fair value | Hits about |
|---|---|---|
| 1 | 60¢ | 60% |
| 2 | 36¢ | 1 in 3 |
| 3 | 21.6¢ | 1 in 5 |
| 4 | 13¢ | 1 in 8 |
| 5 | 7.8¢ | 1 in 13 |

Five coin-leaning favorites in a row is already a longshot. Bettors feel 60%; math says 8%. That gap is the raw material of the parlay business.

## Why small edges get big

Margin multiplies exactly the way probability does:

**total margin = (1 + per-leg margin)^legs − 1**

| Legs at 2% per-leg margin | Total margin |
|---|---|
| 2 | 4.0% |
| 3 | 6.1% |
| 4 | 8.2% |
| 5 | 10.4% |
| 6 | 12.6% |

Nobody notices 2% on a leg. Everybody's ledger notices 12% on a 6-leg. This is why parlays are the most profitable product in sports betting.

## Expected value, per ticket

**EV per ticket = (your price − fair value) × payout size**

**Example (a real ticket):** fair 13.83¢, quoted 14.59¢ → edge 0.76¢ per $1 of payout (5.5%). Payout size $68.54 → EV ≈ **+$0.52** per fill.

Key habit: think in EV, not outcomes. A ticket that loses can still have been a great quote; a winner can have been a terrible one. You are paid for the price, not the result.

## Any night can lose

EV is the average; a night is one draw. The house wins small amounts often and pays out rarely-but-big — which means streaks in both directions.

**Example — 40 identical tickets ($10 stake, $68.54 payout, 13.8% true hit rate):**

- Collected: 40 × $10 = +$400
- Expected winners: ~5.5 → expected night ≈ **+$23**
- Unlucky night (3 extra winners): ≈ **−$183**
- Lucky night (3 fewer): ≈ **+$228**

Same book, same prices — a ±$200 swing is normal. Volume averages it; caps bound the bad tail.

## Sizing: how big should a ticket be?

The house survives by making sure no single outcome matters:

- **Per ticket: ~5% of bankroll.** One worst-case ticket dents you 5%; the edge rebuilds faster than that stings.
- **Open book: ~30% of bankroll.** If every open ticket paid at once, you lose a third, not the roll.
- **Per team: ~12% of bankroll.** Tickets sharing a team win together; cluster risk gets its own cap.

**Example — $500 bankroll:** $25/ticket, $150 open, $60/team. Catastrophic night = −$150 (30%), survivable by construction. These are recommendations — every number is user-set.

## Correlation: the trap that eats naive books

Multiplying legs assumes independence. Same-game legs move together.

**Example:** Yankees ML at 55% × Yankees over 4.5 team runs at 50% → naive price 27.5¢. But those outcomes are strongly linked; the true joint probability is ~35%. You just sold a 35¢ ticket for 27.5¢, and sharp bettors hunt exactly this. Pregame's engine refuses same-game and shared-team combos outright: when independence is in doubt, declining is free and mispricing is not.

## Longshots: where small errors are huge

Pricing error is relative to the price. The same 1¢ error is 1.7% on a 60¢ leg (noise) and **50% on a 2¢ parlay** (ruinous). A $10 stake at 2¢ is also $500 of payout. The desk's fair-price band exists for this: below your floor, it does not quote. Longshot appetite is a setting, not a default.

## Pricing against other makers

Other makers answer the same requests; the bettor takes the best price. Expected profit per request = margin × probability your quote wins the auction:

| Margin (illustrative) | Auction win rate | EV index |
|---|---|---|
| 2% | 60% | 1.2 |
| 5% | 35% | **1.75 ← sweet spot** |
| 9% | 10% | 0.9 |

Pregame's follow-the-market mode automates this: it tracks what winning quotes clear at per leg count and prices at your chosen percentage of that — undercut to fill more, price over to earn more per fill.

## A real ticket, worked end to end

The desk's first live fill, every number real:

| | |
|---|---|
| Legs at quote time | 62.5 × 58.5 × 60.5 × 62.5¢ |
| Fair value | 13.83¢ |
| Quoted (+5.5% margin) | 14.59¢, answered in 155ms |
| Stake collected | +$10.00 |
| Collateral locked | $58.54 |
| EV at fill | +$0.52 |
| Result | one leg went over → ticket died, stake kept |
| Net after settlement & fees | **+$9.59** |

The +$9.59 was luck. The +$0.52 was the business. Judge every ticket by the second number.

## Rules of thumb

- **Multiply to price:** independent legs multiply; so does margin. Long tickets carry big total edge.
- **Think EV, not outcomes:** you are paid for the price you set, not for how one night lands.
- **Volume is the variance cure:** more small tickets beats few big ones at the same total risk.
- **Cap by cluster:** ticket ~5%, open book ~30%, any one team ~12% of bankroll.
- **Never price correlation as independence:** decline instead.
- **Fear cheap tickets:** relative error explodes below a few cents. Use the price floor.
- **Margin is a market:** track what fills, undercut deliberately, and let the fill rate tell you when you're too greedy.