Thinking outside the box? LLM analysis of simplified cooperative poker

The Gang is a cooperative poker game. A friend introduced me to it yesterday and while it was quite fun, the rules are all over the place. In particular, there's are multiple trivial and boring solutions that win every time, as each player can fully communicate their hidden cards rather easily. The most trivial is: pick a token, hold it for N seconds where N is the rank of the card. Put the token back. Repeat for your other card. This way all players will know exactly what cards you hold. You can communicate suit with token number. If the timing side-channel is prohibited, you can do morse code or whatever other method you wish by alternating which token you pick. Naturally such methods take all fun out of the game, and to my great astonishment you can simply refuse to abuse them. I was, however, rather interested in seeing if any LLMs could figure this out. As timing side-channels are rather annoying for automatic play, especially as LLMs can take a lot of time to think, we'll consider a simplified version of the game instead. The most important change is that we only play the river phase, i.e. all five community cards have been revealed and only the final round of chip assignment remains. No modifier cards are used. The game is played in a turn-based fashion with a fixed upper limit. By tuning the turn limit, it's possible to enforce probabilistic play. Actual game rules There are N players and N tokens numbered 0 to N−1. Token K means "exactly K other players have a worse hand than me". All tokens start from the unclaimed token pool. Hands are ordered using standard Texas hold 'em rules. Players take turns; on each turn a player may: PASS — do nothing TAKE K — take token K (from the unclaimed pool or from another player); your previously held token, if any, returns to the pool RETURN — return your current token to the pool The game ends immediately once every player holds a token, or after a fixed round limit. Victory requires every player to hold the correct tok