> For the complete documentation index, see [llms.txt](https://praxis-4.gitbook.io/praxis/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://praxis-4.gitbook.io/praxis/documentation/strategies-and-guides/mathematical-foundations.md).

# Mathematical Foundations

The mathematical models behind prediction pricing and expected value.

At first glance, prediction markets can look similar to gambling. You choose an outcome, place capital behind it, and with some probability receive a payout. But prediction markets have one significant element that gambling lacks — an **informational edge**.

Informational edge might sound like a negative thing. Insider trading always triggers controversy. But in reality, informational edge is the core engine of every prediction market.

People participate because they believe they understand something better than the market itself:

* Deeper research
* Faster information
* Superior analytics
* Domain expertise
* Stronger models
* Or simply conviction

Without this feeling of edge, prediction markets truly become gambling. And mathematics proves it.

***

## Without Edge, You're Gambling

If you repeatedly participate in a perfectly efficient 50/50 market with no edge, your expected return trends toward zero over time (before fees). Some predictions win, some lose, but statistically your results average out.

{% hint style="success" %}
If your understanding of probability is identical to the market's understanding of probability, you gain no long-term advantage.
{% endhint %}

This is where prediction markets become interesting. Unlike casinos, markets constantly expose probability pricing. Participants can identify situations where the market is mispricing reality and act on it. This transforms prediction markets from a gambling venue into a true financial instrument.

***

## "I Know This Team Will Win"

The simplest strategy is straightforward: choose the side you believe will happen. No deep market analysis. No pricing investigation. Just conviction.

In the short term, this can absolutely work. But over long periods, this approach becomes problematic.

Being correct about *direction* is not enough. Consider a coin flip market. If the market correctly prices both sides at 50/50, repeatedly choosing one side gives no statistical advantage. Short-term, you may double your position. Long-term, your expected return trends toward neutral — and fees eventually push it negative.

This becomes even worse when markets are imbalanced. Suppose you want to pick Heads, but that side costs $0.60 with Tails at $0.40. Buying the overpriced side is worse than not participating at all.

{% hint style="success" %}
Simply knowing which side is more likely to happen does not automatically create an edge. The key is whether the market probability differs from your estimated probability.
{% endhint %}

***

## Predicting Better Than the Market

{% hint style="success" %}
Long-term, value flows toward participants with better information.
{% endhint %}

Markets don't operate on "winning" or "losing" sides, they operate on probabilities. Your goal as a participant is not merely to predict outcomes. It's to identify where market pricing differs from reality.

### Example: Mispriced Probability

Imagine a simple coin that is 50/50 likely to land Heads or Tails but the market incorrectly prices them at $0.6 and $0.4, respectively.

There is an opportunity simply by buying Tails tokens. Even though you still lose 50% of the time, repeated participation becomes statistically profitable across many predictions.

This is an important concept: even with a positive expected return, some individual positions will lose because randomness remains. But over a sufficiently large number of predictions, statistical edge dominates.

### "Going with the Losing Side" Can Still Be Profitable

One of the most unintuitive aspects of prediction markets:

{% hint style="success" %}
Sometimes the profitable trade is going with the outcome most likely to lose.
{% endhint %}

Imagine a market with a real probability of 20/80 — but a market evaluation of 10/90. It becomes statistically profitable to purchase the undervalued 10% side.

This outcome still loses most of the time. But if payouts are large enough relative to actual probability, the position becomes mathematically valuable.

This is called **value trading**: in prediction markets, probability matters but pricing matters more.

***

## Position Sizing — How Much Should You Allocate?

Even with positive expected value, risk management matters. Committing your entire bankroll to every prediction means a single incorrect outcome can wipe you out completely.

Mathematics offers a well-known solution: the [**Kelly Criterion**](https://en.wikipedia.org/wiki/Kelly_criterion). It calculates the optimal fraction of capital to allocate based on your estimated probability and the current market pricing.

### Standard Form

$$
f^\* = \frac{bp - q}{b}
$$

| Symbol        | Meaning                                                                                   |
| ------------- | ----------------------------------------------------------------------------------------- |
| $$p$$         | Your estimated probability of a win                                                       |
| $$q = 1 - p$$ | Your estimated probability of a loss                                                      |
| $$b$$         | Net odds received per unit risked (e.g., if a $1.00 stake wins $1.00 net, then $$b = 1$$) |
| $$f^\*$$      | Optimal fraction of your bankroll to allocate                                             |

### Prediction Market Form

In prediction market terms, where prices directly encode probabilities, the formula simplifies to:

$$
F = \frac{p - c}{1 - c}
$$

| Symbol | Meaning                                                            |
| ------ | ------------------------------------------------------------------ |
| $$p$$  | Your estimated probability of the outcome (e.g., $$0.20$$ for 20%) |
| $$c$$  | Current market probability (token price)                           |

**Example:** Using the 10/90 market with a real probability of 20/80:

$$
F = \frac{0.2 - 0.1}{1 - 0.1} \approx 0.11
$$

The mathematically optimal allocation is roughly **11%** of your prediction bankroll for this position.

{% hint style="info" %}
If your estimated fair probability equals the market price, the formula outputs zero and confirming there is no edge and no reason to participate.
{% endhint %}

<details>

<summary>Why not bet everything on high-confidence positions?</summary>

The Kelly Criterion accounts for the risk of ruin. Even a very high-probability position can lose due to randomness. Betting a larger fraction than Kelly suggests increases variance without a corresponding improvement in long-run returns. A few large losses can eliminate the compounding effect that makes positive-EV strategies work over time.

Many participants use **fractional Kelly** (e.g., half-Kelly) to reduce variance further at a modest cost to expected growth rate.

</details>

<details>

<summary>What if the formula outputs a negative number?</summary>

A negative Kelly output means the market is pricing the outcome *above* your estimated fair probability (you believe the outcome is less likely than the market does). There is no edge on that side. The formula is telling you to stay out, or to consider whether the *opposite* side has value instead.

</details>

***

For a practical walkthrough of how these principles translate into actionable strategies, see [Base Strategies](/praxis/documentation/strategies-and-guides/base-strategies.md) and [PT/YT Trading Strategies](/praxis/documentation/strategies-and-guides/pt-yt-trading-strategies.md).

For an explanation of how Praxis prediction markets work mechanically (including AMM pricing and price impact), see [Prediction Markets](/praxis/documentation/how-it-works/prediction-markets.md).