In the game Zendo, a master makes up an arbitrary, secret rule, and students compete with each other to be the first to guess it. The rule classifies configurations of pyramids into two types. We call a configuration of pyramids a 'koan'. An example rule: a koan 'has Buddha-nature' if it has a red pyramid pointing at a green one.
Play starts with the master setting out two example koans, one of each type, labeled accordingly; then the master picks a student to move first.
The student builds a new koan. Then, if that student wishes, all students separately and simultaneously guess the koan's type, and the master awards a green stone to each student who was correct. Whether there was a guessing round or not, the master labels the koan with its type.
Also, if the student whose move it is has a green stone, they may give it back to the master and guess the rule. A correct guess wins the game. For a wrong guess, the master builds and labels a koan that the guess classifies incorrectly. The student may keep choosing to guess for as long as they have stones. Play passes to the next student on the left.
So that's the game; there are a few more details of play and terminology in the official rules, e.g. the type labels are white and black stones, they use Zen terms instead of my boring English, etc. I've never played it -- I was just reading its very interesting design history last night, via Chris Okasaki.
This morning I thought of a variant: instead of the master picking an initial student, and instead of play passing to the next student on the left, there's an auction on each move for the right to move. (With real money or play money, whichever. A second-price auction with sealed bids seems the right thing, although you'd want to avoid the auction machinery overwhelming the actual play.) The winner of the auction pays the auction-assigned value to the previous mover, then moves. At the end, the student who guessed the rule gets awarded some money; this money was collected from all the players at the beginning of the round when the master made up the rule. I guess for the very first auction, before there's any mover to pay the auction-value to, the payment should go into the prize pot. (Maybe in subsequent rounds the first payment goes to the winner of the previous round.)
The rationale is to reward players for inventing insightful koans whose answer is likely to bring the solution closer. If you think of a 'good experiment', the value of the next move will be higher than you had to pay, and you'll make a profit. Also -- while it's not apparent from my description which deemphasized the "Mondo" and the "Buddha nature" and such -- I think it's hilarious to mix capitalism and Zen.
(This was suggested by Eric Baum's reinforcement learning auction ideas. I suppose you could make a game this way out of any competitive machine-learning algorithm.)
ETA: This has the obvious defect that the first mover can hold onto ownership for the rest of the game. There needs to be some cost to stop that, something like the ante in poker, I guess. Oops. If per-auction antes went to the master, that'd encourage the master to make overcomplicated rules and counterexamples, so that's not the answer. Hmm. Maybe just bite the bullet and say the house takes a cut? Vegas Zen Agora. But then it needs gambling! Argh! Zendo really does sound elegant, doesn't it?
So, another attempt:
We play with funny money managed by a neutral banker. At the start of a session each player gets a stash. Each move, the banker collects 'rent' from each player for the privilege of staying on the trading floor. There's an auction for the move; the price is paid to the previous mover. The new mover builds a koan, which the master labels. The mover then may guess the rule, once. A correct guess wins the set prize money, and ends the round. A wrong guess produces a counterexample from the master, and ends the move.
This variant eliminates the green guessing stones and the all-students-guess option, since they seem sort of redundant with the auction. We could potentially take out the student-built koans, too, and have them just guess the rule every move, but that may be going too far.