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

By Stephen Grugett

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 number of shares depends on the current probability.
• 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. We also allocate an additional $100 of fake money based on their choice.

$$ p = \frac{y_{start}^2}{y_{start}^2+n_{start}^2} \; \newline s.t. \; \; \sqrt{y_{start}^2+n_{start}^2}= 100 + ante \newline y_{start},n_{start} \gt 0 $$

This fictitious starting capital is included in share purchase calculations but not payouts.