# Binary Markets and N/A

Created by
J. F. Jurchen

Today on Manifold, most "binary" markets actually have three possible outcomes - YES, NO, and N/A. When a market resolves N/A, all trades are rolled back - traders get back as much as they put in, and everybody's profit is M\$0. This affects even traders with no position in the market! If I successfully arbitrage a market and make M\$50 of profits, and then a year later the market resolves N/A, the M\$50 is clawed back, and my balance could even go negative. This is confusing for traders ("why did my balance just go down by M\$50 when I didn't have an active position in any markets?") and can result in negative balances.

In this post, I'll be outlining an alternative approach that avoids clawbacks and seems mathematically more elegant, without changing user-facing behavior on the core Manifold site. The basic idea is to promote N/A to a first-class outcome under the covers, rather than a special case - all markets would be explicitly trinary.

## Primitives

The primitive operations on a binary market on Manifold are:

1. Mana -> share conversion. You pay Manifold M\$10 and get back 10 shares of YES and 10 shares of NO.

2. Share -> mana conversion. You give Manifold 10 shares of YES and 10 shares of NO, and get back M\$10.

3. Resolution. When you resolve YES, every YES share turns into M\$1 and every NO share turns into M\$0. The same is true for NO. When you resolve NA, all trades are rolled back.

4. Trading. Users can trade shares among themselves, for other shares or for mana. Note that buying YES shares directly should be mathematically equivalent to converting your mana into YES + NO shares, and then trading the NO shares for other traders' YES shares (including Manifold's AMM bot).

Under my proposed alternatives, the primitives change as follows:

1. When converting mana to shares, you pay M\$10 for 10 shares each of YES, NO, and NA.

2. When converting shares to mana, you must also give Manifold 10 shares of NA.

3. When resolving N/A, each NA share turns into M\$1 and all other shares turn into M\$0.

4. You can trade NA shares as well as YES/NO shares.

So every market is just a normal trinary market, and there's no special treatment for NA.

## UX

Now let's look at the things Manifold lets you do through the UI under the current system.

1. Buying shares. Let's say you buy 100 shares of YES for M\$10. You're down M\$10 cash, and you'll get M\$100 if the market resolves YES and M\$10 if it resolves N/A. You've effectively converted M\$10 into 10 shares of every outcome, then traded 10 shares of NO for 90 shares of YES.

2. Redeeming shares. You end up with 100 shares of YES and 100 shares of NO. Manifold gives you M\$100 cash, and will claw back M\$100 if the market resolves N/A.

3. Selling shares. Selling NO is equivalent to buying YES and redeeming, except you don't have to do as much math.

How could you reproduce these outcomes under my proposed alternative?

1. Buying shares. You'd spend M\$10 to buy 100 shares of YES and 10 shares of NA. You've effectively converted M\$10 into 10 shares of every outcome, then traded 10 shares of NO for 90 shares of YES. Making NA a first-class outcome shouldn't change the odds ratio between NO and YES, so there shouldn't be any change in the price the user gets for their shares here.

2. Redeeming shares. To replicate the current behavior, Manifold just needs to allow naked shorting of NA shares at redemption time. I have 100 YES shares and 100 NO shares, I borrow 100 NA shares from Manifold, I exchange all of those things for M\$100 cash, and I have to repay M\$100 if the market resolves N/A and M\$0 if it resolves YES or NO. (Note that in practice I probably have some NA shares already from buying those YES and NO shares - my net short/long NA position will be exactly equal to my profits/losses on the market, modulo fees.)

3. Selling shares. I'm pretty sure the "selling = buying + redeeming" equivalence is true under the new system (so selling 100 shares of YES for M\$60 is equivalent to "buy 100 shares of NO for M\$40, naked-short 100 shares of NA, and redeem for M\$100") and the numbers all work out the same, but I'm too lazy to prove it right now.

## Benefits

That's how to replicate the existing Manifold experience under a "first-class NA" model. Beyond being mathematically cleaner, it also means that power users could explicitly trade NA shares, perhaps in a separate "Manifold Pro" interface or only via the API.

• Advanced users don't need to worry about their profits getting clawed back - they can fully close out their position in a market by buying/selling NA shares as necessary

• Every market automatically gets a probability estimate for N/A resolution. For markets like "Conditional on B, will A happen?", you've effectively got odds for P(~B) as well as P(A|B) and P(~A|B). That's neat! Note that even if there are no advanced traders and this probability estimate for NA is way off, the currently-supported behaviors still work just fine - none of them care about the market price of NA, only about the ratio of YES to NO.

• If there's a robust market for NA shares, you could consider disabling the naked-short behavior described above. This would likely confuse non-power users ("why is my payout lower than I expected?"), but it prevents balances going negative, which is good for the Manifold economy.

## Conclusion

That's all I got

This is the approach that augur took!

Conditional markets would be P(A and B), P(~A and B), and P(B) ... from which you can derive P(A|B) by dividing P(A and B) / P(B)