Hello market math nerds. I have a problem I can't figure out, and I need your help.

I had this idea about simplifying linked multiple choice. Instead of answers that have yes and no shares each, we could specify just yes shares for each answer.

I'm trying to prove for any set yes-only pools, there is one unique equivalent version with yes/no pools.

For example, for 3 answers with equal Yes pools:

8

8

8

This is equivalent to (Yes, No) pools of

4, 2

4, 2

4, 2

Where each answer has No / (Yes + No) probability = 2 / (4 + 2) = 1 / 3.

In addition to the probability sum constraint (all answers probability sums to 1), there's also the constant liquidity constraint (each answer has the same liquidity, which mean yes shares * no shares is constant across answers).

Here is my notion doc with scratch work: https://manifoldmarkets.notion.site/Multiple-choice-with-only-yes-shares-a849532cc4bc4d7192f1a4652c56c7b1?pvs=4

Can we solve the general case? If I have yes-only pools of n answers, can we compute the equivalent yes/no pools? And is the solution unique?

I will award a bounty of at least 10k mana (perhaps more) for a solution or proof that there is no solution.

Here are the equations in the 3-answer case:
A = Ay + Bn + Cn

B = By + An + Cn

C = Cy + An + Bn

Ap = An / (Ay + An)

Bp = Bn / (By + Bn)

Cp = Cn / (Cy + Cn)

1 = Ap + Bp + Cp

AyAn = ByBn = CyCn

If you want to learn more about our multiple choice mechanism, check out my Substack article: https://news.manifold.markets/p/multiple-choice-markets