Modified Exchange Rate

Mathematics Behind Providence.

First define σ(t) to be a piecewise sigmoid function of time (t) (detailed in Sigma section):

• Let Nj denote the number of outcome j tokens minted to the swapper.

• Let Ni denote the number of outcome i tokens burned by the swapper.

• Let Eij denote the exchange rate between i and j outcome tokens.

• Let Si denote the total supply of outcome i tokens.

• Let Sj denote the total supply of outcome i tokens.

A term P is given by the reciprocal of the current unmodified exchange rate (Si / Sj).

And define a term a:

A function f(t) can be derived:

Combining this, the exchange rate between outcome i and j (Eij) is given:

Note that the reciprocal relation is given by Eji = σ(t) · p/f(t).

Figure 2: Visualization of the exchange rates Eij(t) and Eji(t) over time, based on σ(t) and f(t), assuming constant odds p and Te (expiry) of 10.

A small swap fee as a percentage of the outcome tokens burned will be collected and automatically liquidated into USDC post-settlement.