Hat guessing with the option to pass
Asked at Jane Street, Citadel
Three players are each fitted with a hat, colored red or blue by an independent fair coin flip. Each sees the other two hats but not their own. Simultaneously, each player either guesses their own color or passes. The team wins if at least one player guesses and every guess is correct (if everyone passes, or any guess is wrong, they lose).
No communication once hats are on; they may strategize beforehand.
What is the best achievable win probability? (Random individual guessing gives only .)
Show a hint
It's fine if the players are often wrong, as long as their wrong guesses "pile up" on the same few configurations while their correct guesses spread out. Focus on the two monochromatic cases.