Master and Slave versus Bob: easy version

Question

Continuation of: "Sub-puzzle: Master and Slave versus Bob the Violent but Honest Psychopath"
Continued in: "Master and Slave versus Bob: hard version"

Previously I described a game that is played by a Master and Slave team versus Bob the Violent but Honest Psychopath. Here I describe two additional games with very similar structure to the previous game.

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The Main game is the one described previously, that is:

1. Bob selects a card from the Deck, which he reveals to the Master.
2. Bob (who is honest) reveals to the Master the Red/Black plays he will make in the four rounds of step (5).
3. The Master then selects a card from the Deck, which he reveals to Bob.
4. 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.
5. Then they play four Red/Black rounds. [To recap: this consists of four rounds, in each of the which, the Master, Slave and Bob simultaneously play either Red or Black. Unless all three players play the same colour, the Master and Slave team loses one Mulligan, or else if they have none left, Bob KIIIIIIILS THEEEEM.]
6. Then the Slave decides to do one of two things:
   • He guesses a Large amount of information about Bob's card [which must be correct on pain of getting KIIIIIIILLED]; or
   • He forfeits one of the team's Mulligans, then guesses a Small amount of information about Bob's card.

The Deck in steps (1),(3),(4),(6) is explained below.

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One new game is the Prologue and goes like this:

1. Bob selects a card from the Deck, which he reveals to the Master.
2. Bob reveals to the Master his three plays.
3. Nothing happens in step 3. (The Master does not select a card.)
4. Bob grants them three Mulligans. (Bob does not make a decision, the Slave gets no information about the Master's card, which does not exist.)
5. Then they play three Red/Black rounds.
6. Then the Slave decides to do one of two things:
   • He guesses a Large amount of information about Bob's card; or
   • He forfeits one of the team's Mulligans, then guesses a Small amount of information about Bob's card.

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The other new game is the Epilogue and goes like this:

1. Nothing happens in step 1. (Bob does not select a card.)
2. Bob reveals his three plays up front to the Master.
3. The Master then selects a card from the Deck, which he reveals to Bob.
4. Bob then decides to do one of two things:
   • He grants them zero Mulligans, and tells the Slave a Large amount of information about the Master's card; or
   • He grants them one Mulligan, and tells the Slave a Small amount.
5. Then they play three Red/Black rounds.
6. Nothing happens in step 6. (The Slave does not make a guess about Bob's card, which does not exist.)

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[Just as a reminder of the point of all this: these games can be "chained" together in a certain way, resulting in this game:

Step 0. Bob selects a positive integer N.

1. Nothing happens in step 1.
2. Bob reveals his 4N+2 plays up front to the Master.
3. Nothing happens in step 3.
4. Bob grants them N+2 Mulligans.
5. Then they play 4N+2 Red/Black rounds.
6. Nothing happens in step 6.

And there is no Deck in this game. This is effectively the game described here, with p=3/4. ]

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As before, I have not said what the Deck is, or what Large and Small amounts of information mean. Because that is up to you. Of course, it must have the same meaning in all three games (or they won't chain together).

So, your task is: to come up with a Deck, and a definition for Large and Small amounts of information, and GUARANTEED (not probabilistic) survival strategies for the Main game, Prologue, and Epilogue.

Of course you might be able to simply reuse one of the answers to the previous question (at the time of writing, only one correct answer is posted), in which case you just have to solve the Prologue and Epilogue for the answer you chose.

No further hints. I mean it when I say 'easy version'. You can do this :-)

Answer 1

... Anyone? ... Anyone? ...

I am reusing the Deck from my own answer to the previous question,

A 16-card deck, top four ranks of the standard deck. Only rank and colour matters here, there is no difference between clubs and spades, or between hearts and diamonds. So this is effectively an 8-card deck. A Large amount of information means knowing the rank and colour of the card, and a Small amount means knowing the colour of the card.

So the Main game is already solved in the previous answer. For the Epilogue, the Master's strategy in step (3) goes like this:

(Actually this is part of my answer to the hard version, where the difference between clubs and spades, and between hearts and diamonds, will matter. Here, as I say, only rank and colour matters.) It is a similar idea to the previous answer, here the colour of the card indicates the majority colour of Bob's plays, and the rank of the card indicates the position of the minority colour, K-Q-J for rounds 1-2-3, and A for nowhere (3-0 colour split).

And then in step (5), if the Slave has a Large amount of information about the Master's card,

he knows exactly what Bob will play, so they will make no mistakes.

And if the Slave has a Small amount,

they have one Mulligan, and he knows the majority colour that Bob will play. He can play that colour three times and they will make at most one mistake.

And there is nothing to do in step (6).

For the Prologue, there is nothing to do in step (3). For step (5),

They have three rounds and three Mulligans. The Master and Slave can simply ignore Bob's plays. They will make at most three mistakes.

For step (6),

The Master's plays tell the Slave the rank and colour of Bob's card (they can basically use the Epilogue table in reverse), so he has a Large amount of information.

So they are guaranteed to survive.

(This strategy for the Prologue does not work for the hard version.)

As I say, chaining these together with N copies of the Main game solves the original problem for 3N+3 rounds of out 4N+6 (3 rounds of out 6, 6 out of 10, 9 out of 14, and so on, converging on p=75%).

- Play the each game up to step (3). Start with the Epilogue, then the Nth copy of the Main game, ..., then the 1st copy of the Main game, then the Prologue. The Master's card in step (3) becomes Bob's card in step (1) of the next game.
- Play each game starting from step (4). Start with the Prologue, then the 1st copy of the Main game, ..., then the Nth copy of the Main game, then the Epilogue. The Slave's decision in step (6) becomes Bob's decision in step (4) of the next game.