This market asks about this section of this blog post on Futarchy.

The post describes a model under which Futarchy could be applied to aggregate private information. Here is my attempt at fleshing out this model into an explicit game:

1. There are 20 players

2. Each player receives a random bit privately (50%/50% chance of 0/1)

3. Each player starts with $10000

4. There are two futarchy markets, one each for Coin A and Coin B, each equipped with a CPMM starting with $100/$100 in each of the YES and NO pools.

5. There is a single "batch auction" phase in which traders get to trade in the markets.

6. Going into the batch auction, each player gets to place up to a total of $100 in limit orders on the four share types (YES and NO in the two markets). The orders must spend multiples of $1, and can only be placed at price levels which are integer percentages (a la Manifold).

7. The Batch auction then computes the market clearing price, accounting for the CPMM, share creation, and all the limit orders.

8. All trader (BUY orders below/SELL orders above) the market clearing price are filled at that price. Trades at MCP are also maximally partially filled (if there is a larger volume of orders at the market clearing price in one direction, then each trader's order of that type is filled in proportion to the volume of the order at that price).

9. The markets are then resolved according to futarchy (i.e. the higher priced market is resolved N/A, a market is chosen at random if both prices are equal)

10. If A is resolved, it resolves with a 60% chance of YES

11. If B is resolved it resolves with a (49+x)% chance of YES, where x is the number of 1 bits sampled.

How does this question resolve?

This question is a purely mathematical question asking if there is a Nash Equilibrium to the above game where the futarchy always makes the "correct" decision. i.e. it always chooses coin B when the coin B probability of heads is higher, and always chooses coin A when the probability of coin B heads is higher (for this, it doesn't matter what happens when the probabilities are equal).

Market to be extended until someone posts a proof in the comments one way or the other.