A brief technical overview of Manifold Markets’ Dynamic Parimutuel (DPM) betting system

By Stephen Grugett

*Note: Manifold is no longer using DPM for binary markets but is using the generalized, multi-outcome version of DPM for free-response markets.*

Basic facts:

- Markets are structured around a question with a binary outcome.
- Traders can place a bet on either YES or NO. The trader receives some shares of the betting pool. The less likely the outcome, the more shares the trader receives per dollar.
- When the market is resolved, the traders who bet on the correct outcome are paid out of the final pool in proportion to the number of shares they own.

## Probability

If *y* is the total number of outstanding shares of YES, and *n* is the total number of outstanding shares of NO, the instantaneous implied probability of the market is given as

$$
P(y, n) = \frac{y^2} {y^2 + n^2}
$$

## Betting

We introduce a cost function *C* which captures the total amount wagered by all traders given the current shares of YES and NO:

$$
C(y,n) = \sqrt{y^2 + n^2}
$$

If a trader places a bet of $*b* on YES, they add $*b* into the YES pool and receives *s* shares of the final pool if YES is the outcome.

$$
pool_{Y_{new}} = pool_{Y_{current}} + b
$$

$$
b = C(y+s, n) - C(y, n)
$$

## Antes

The market creator chooses an initial probability *p* and an ante amount to initialize the betting pool.

$$
p = \frac{y_{start}^2}{y_{start}^2+n_{start}^2} \; \newline s.t. \; \; \sqrt{y_{start}^2+n_{start}^2}= ante \newline y_{start},n_{start} \gt 0
$$