I read G.K. Chesterton’s essay “A Defence of Rash Vows” today, and it got me thinking what it would look like for a group of people to collectively make vows (which you could also call commitments, or challenges) that were challenging, but not too challenging. I came up with the following format:

The competition has a base entry fee of $X.

Contestants each set themselves n challenges (for integer n>0 of their choice), and contribute $X * n^2 to the prize pool.

A prediction market is made for each challenge, resolving YES if it’s completed before the competition is over and NO if not.

At some randomly chosen point early in the competition, a snapshot is taken of each market’s probability (after they’ve had time to settle, before contestants have had time to make much progress).

When the competition ends, the prize pool is paid out to everyone who completed a challenge, in proportion to the log snapshot probability of all challenges completed - in other words, in proportion to how surprising their achievements were.

For example, let’s say that Alice, Bob, Charlotte, Dennis, and Emily all complete one challenge. Their respective achievement snapshot odds were 1%, 2%, 10%, 50%, and 80%. The total prize pool is $100000.

In base 2 log odds (i.e. converted to info-theoretical bits of surprise), these are:

Alice: -6.64385619

Bob: -5.64385619

Charlotte: -3.321928095

Dennis: -1

Emily: -0.321928095

These are their relative shares of the prize pool, so they get paid as follows:

Alice: $39239

Bob: $33333

Charlotte: $19619

Dennis: $5906

Emily: $1901

These are rounded down, with $2 left over - the organiser might keep this as a fee.

Each contestant only completed one challenge here, but this generalises to arbitrary amounts of challenges per contestant - just sum the total bits of surprise (or, equivalently, the total payouts).

Thus, you’re incentivised to commit to completing challenges that are difficult (so you get a larger share) but not too difficult (so you actually stand a chance). The quadratic entry fee structure means you need to choose your challenge(s) carefully.

I’m thinking of running this on Manifold with a base entry fee of M$1000 ≈ $1USD. I’d love to hear any feedback/ideas you wonderful people have!