Position Tokens¶
Preliminaries¶
This document builds on Representation Of State.
PredictionSwap records economic state directly as a Conditional Account Value Vector.
Token balances are derived views of account value.
This document describes how token balances are derived.
Freecash and Position Value¶
Outcome Space¶
Consider a mutually exclusive outcome space:
Exactly one of these outcomes will occur.
Account Value Decomposition¶
Given a Conditional Account Value Vector
it can be decomposed uniquely into:
Where¶
- \( c = \min(e_R, e_G, e_B) \)
- \( r = e - c(1,1,1) \)
By construction:
This decomposition separates account value into two components:¶
- Freecash \( c \), which contributes equally across all outcomes
- Conditional Position Value \( r \), which represents outcome-dependent value generated by positions
This decomposition is unique.
Interpretation¶
The constant component \( c \) represents uniform value across all outcomes — this is freecash.
The residual vector \( r \) represents outcome-specific value contributed by positions — this is the Conditional Position Value Vector.
This decomposition is unique.
Example¶
Starting Account Value¶
Alice's holdings are uniquely characterised by the Conditional Account Value Vector:
Alice's Freecash¶
Alice's Shares¶
YES Tokens¶
Definition¶
YES token balances correspond directly to the Conditional Position Value, or the 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 position value in exactly one outcome.
NO Tokens¶
Definition¶
A NO token corresponds to position value 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 position value.
Alice’s Token Balance¶
- YES–R = 8
- YES–G = 0
-
YES–B = 5
-
(NO–G = 5)
Interpretation¶
A Conditional Account Value Vector has a single canonical form.
YES tokens provide a unique decomposition of the Conditional Position Value Vector.
NO tokens are overlapping views of YES tokens.