Position Tokens¶
Preliminaries¶
This document builds on Representation Of State.
PredictionSwap records economic state directly as an exposure vector.
Position tokens are views of that exposure.
This document describes how exposure maps to token balances.
Uniform and Residual Exposure¶
Outcome Space¶
Consider a mutually exclusive outcome space:
Exactly one of these outcomes will occur.
Exposure Decomposition¶
Given an exposure vector
it can be decomposed uniquely into:
Where¶
- \( c = \min(e_R, e_G, e_B) \)
- \( r = e - c(1,1,1) \)
By construction:
Interpretation¶
- The constant component \( c \) represents uniform exposure across all outcomes — this is free cash.
- The residual vector \( r \) represents outcome-specific exposure — this is shares.
This decomposition is unique.
Example¶
Starting Exposure¶
Alice's holdings are uniquely characterised by the exposure vector:
Alice's Free Cash¶
Alice's Shares¶
YES Tokens¶
Definition¶
YES token balances correspond directly to the residual exposure, or shares Alice has in each outcome.
From:
Token Balances¶
- YES–R = 8
- YES–G = 0
- YES–B = 5
YES tokens are mutually exclusive.
Each corresponds to exposure in exactly one outcome.
NO Tokens¶
Definition¶
A NO token corresponds to exposure in all outcomes except one.
For example:
- NO–G pays $1 if either \( R \) or \( B \) occurs.
Economically, one NO–G is equivalent to:
- 1 unit of YES–R
- 1 unit of YES–B
Applying to the Example¶
Alice has:
- YES–R = 8
- YES–B = 5
Since each NO–G requires one YES–R and one YES–B,
she has:
- NO–G = 5
NO tokens are overlapping views of exposure.
They are combinations of YES tokens.
Alice’s Token Balance¶
- YES–R = 8
- YES–G = 0
-
YES–B = 5
-
(NO–G = 5)
Interpretation¶
An exposure vector has a single canonical form.
YES tokens provide a unique decomposition of outcome-specific exposure.
NO tokens are derived overlapping views.