[I created this as a sub-puzzle to "Strategy to beat the Casino reversed", that has been sitting unsolved for months. You may wish to read that puzzle first, but this is self-contained. My answer to that is rather long, and it occurs to me to break it up, into this partial answer, this puzzle here, and later my answer to this puzzle. EDITED: I've divided it into three puzzles, this one, and two followups: easy version and hard version.]
Below is a game that is played by a Master and Slave team versus Bob the Violent but Honest Psychopath. The Master and Slave team starts with a certain number of Mulligans. This number is decided by Bob, with certain restrictions, as we shall see. The main part of the game consists of four rounds in which all three players simultaneously play Red or Black on each round. Unless all three players play the same colour, the Master and Slave team loses one Mulligan... or else, if they have no Mulligans left, Bob will KIIIIIIILL THEEEEEEM BAHAHAAAHAH.
In addition to the four Red/Black rounds, the game has initial and final steps involving a certain Deck of cards, to be explained later. The point of these stages is that they allow us to chain strategies together to give a partial answer to the original puzzle (and also, a complete answer to "Strategy to beat the casino over the long haul").
The Master and Slave can agree upon strategies beforehand, but there is no communication allowed once the game starts, except as provided by the rules. It is for you to come up with strategies for them which guarantee their survival against Bob, who is bloodthirsty and very clever, but who will strictly follow the rules.
The game is this:
- Bob selects a card from the Deck, which he reveals to the Master.
- Bob reveals to the Master the Red/Black plays he will make in the four rounds. (And yes, Bob is Honest.)
- The Master then selects a card from the Deck (possibly the same card), which he reveals to Bob.
- Bob then decides to do one of two things:
- He grants them one Mulligan, and tells the Slave a Large amount of information about the Master's card; or
- He grants them two Mulligans, and tells the Slave a Small amount of information about the Master's card.
- Then they play four Red/Black rounds.
- Then the Slave decides to do one of two things:
- He guesses a Large amount of information about Bob's card (and the guess must be correct, on pain of getting KIIIIIIILLED); or
- He forfeits one of the team's Mulligans, then (correctly) guesses a Small amount of information about Bob's card.
To be quite clear: step (5) consists of four rounds, in each of the which, the Master, Slave and Bob simulatneously play either Red or Black. These are their two options in each round (and is not related to the Deck in steps (1),(3),(4) and (6)). Unless all three players play the same colour, the Master and Slave team loses one Mulligan. As seen above, the Master knows in advance what Bob will play, but cannot pass this information to the Slave directly. In the first round, the Slave only knows what Bob tells him in step (4). In subsequent rounds, the Slave also knows what was played in previous rounds.
In step (4), the Master also knows what Bob told the Slave.
And of course, in step (6), the Slave is only allowed to choose the latter option if they have a Mulligan left.
I have not yet said what the Deck is. And I have not yet said what Large and Small amounts of information mean. That is because it is up to you, dear puzzle solver! You can choose whatever Deck works for your answer. Of course, it must have the same meaning in all the steps. As I say, that is really the whole point of this, to "chain" answers to solve the original puzzle!
Example: in your answer, you might specify that the Deck is the standard deck of 52 cards; "Large amount" means you know exactly which card it is; and "Small amount" means you only know which suit the card is. This is just an example, if you use it, I'm pretty sure it will get them KIIIIIIILLED.
So, your task is: to come up with a (finite) Deck, and a definition for Large and Small amounts of information, and GUARANTEED (not probabilistic) survival strategies for this game.
You may wish to start by dividing the possibilities for Bob's four Red/Black rounds into odd and even parity.
Medium hint: the Deck I plan to use is
the top four ranks from a standard deck (16 cards). These correspond to the odd parity cases (3-1 split). The colour of the card is the majority colour, and the rank of the card is the position of the minority colour. The difference between clubs and spades, and between hearts and diamonds, is irrelevant (but becomes relevant in the followup puzzle).
And a larger hint
can be found by looking at the edit history of my linked answer