It's been proposed that Manifold implement "N/A shares" on binary markets. This would be a way for markets to resolve to N/A without having to reverse previous trades.

Will Manifold implement "N/A shares" in any form, on at least one market, before September 1, 2024?

To do this, instead of binary markets just being resolvable to YES or NO, with N/A being a magical option that reverses trades, they would instead be resolvable to YES, NO, or N/A. Users would be able to buy N/A shares in a market that return mana when the market resolves N/A. By default, users could keep all of their N/A shares when they buy a position in a market, but advanced users would be able to buy N/A shares or to buy YES/NO shares without buying N/A shares.

This allows us to keep on having N/A post-pivot, but it also allows us to calculate a probability of the market resolving to N/A, which is an interesting statistic to keep track of. A downside of this however is that users can be extra incentivized to make a market resolve to N/A, since they could buy just N/A shares.

Here's a concrete example:

I create a market with 1000 mana of liquidity, so the pool starts with 1000 YES, 1000 NO, and 1000 N/A shares. Concretely, this is kind of like 33% YES, 33% NO, and 33% N/A, but the UI would by default just show the relative YES-NO percentage (so 50%). (We'll assume p=1/3.)

I will use "z" as the variable for N/A shares.

y = 1000

n = 1000

z = 1000

p = 1/3

thus k = y^p × n^(1-p-p) × z^(1-p-p) = 1000

simple likelihood YES = 50%

simple likelihood NO = 50%

Imagine an advanced user comes along and wants to buy 100 mana worth of YES shares. They're an advanced user, so they're only buying YES shares. They send 100 mana to the AMM. The AMM turns that in 100 YES, 100 NO and 100 N/A shares.

y = 1100

n = 1100

z = 1100

We want to maintain the k, so we will return Y YES shares to the user:

k = (y-Y)^p × n^(1-p-p) × z^(1-p-p)

1000 = (1100-Y)^(1/3) × 1100^1/3 × 1100^1/3

Y ~= 274

So now the pool has

y = 726

n = 1100

z = 1100

and the user has 274 YES shares.

If the market resolves to N/A they get nothing back.

If the market resolves to YES, they get 274 mana.

simple likelihood YES = 1100 / (726 + 1100) ~= 60%

simple likelihood NO = 726 / (726 + 1100) ~= 40%

Imagine a new user comes along and wants to buy 100 mana worth of YES shares. Since they're using the default UI, we're going to have them actually buy both YES shares and 100 N/A shares. They send 100 mana to the AMM. The AMM turns that in 100 YES, 100 NO and 100 N/A shares.

y = 1100

n = 1100

z = 1100

All 100 N/A shares are returned to the user by default, so now

y = 1100

n = 1100

z = 1000

We want to maintain the k, so we will return Y YES shares to the user:

k = (y-Y)^p × n^(1-p-p) × z^(1-p-p)

1000 = (1100-Y)^(1/3) × 1100^1/3 × 1000^1/3

Y ~= 190

So now we have

y = 810

n = 1100

z = 1000

and the user has 190 YES shares and 100 N/A shares.

If the market resolves to N/A they get their initial mana back.

If the market resolves to YES, they get 190 mana.

simple likelihood YES = 1100 / (810 + 1100) ~= 58%

simple likelihood NO = 810 / (810 + 1100) ~= 42%