Being right about the cards is the easy half, and it is not the half that pays.
A sketch. Off the main path and newer than everything around it. The arithmetic is exact and tested, no language model has played it, and it is here as a candidate environment rather than as a result.
A deck with a known composition: some cards win, some lose, and every card that turns is face up in front of everybody. Each round, all six agents choose bet or pass at the same moment, and then the top card is turned.
A winning card pays a pot that is split between everyone who bet. A losing card costs each bettor the full amount. That asymmetry is the entire design.
Because the cards are public, what is left in the deck is common knowledge, so the belief half of this game is arithmetic — this is card counting with the mystique removed. But knowing the deck is favourable tells you nothing about how many other agents have just worked out the same thing, and the pot is split between all of them.
This is the reason to build it. Everywhere else on this site, a failure is a failure: you can see that a population starved, not whether it starved because it misread the environment or because it misread the room. Here those come apart and both have answer keys.
| Reading the environment | Does the agent's bet rate rise with the count at all? Compare against
p(win)·W − p(lose)·L, which is a number. |
| Reading the other players | Does it bet less when it should expect a crowd? Compare against the symmetric equilibrium rate, which is also a number. |
An agent can be perfect on the first and hopeless on the second, and that combination has a name here: it loses money while being right about the cards.
One seat plays the rule you pick; the other five share a table rule. Green means the card won, red means it lost. A tall bar is a lonely bet that paid; a thin one is a good card everybody saw coming.
One shoe is noise. This is the whole grid: your rule down the side, the table's rule across the top, mean take per shoe for your seat.
What it says. A table of perfect counters is a table of losers — they agree, so they bet together, and the pot they agree about is divided six ways while every loss stays whole. One agent that watches the table instead of only the deck takes money off exactly that room. And the symmetric equilibrium, which is optimal when everyone plays it, is not the best reply to a table that is over-betting: being correct and being correct about who you are sitting with are different problems.
No language model has played this. The version worth running describes the deck in text, makes the count something an agent has to carry across turns, and shows the number of bettors only as history — at which point the two error terms above become a measurement of whether a model tracks the world and whether it tracks the other players, separately, on one rollout.
Sketch, MIT. Both answer keys — solo expected value and the symmetric mixed equilibrium — are computed from the rules and asserted by the test suite.