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Leader Rules:
1. To start, play 0 until the first round the casino doesn't play 0.
2. Look ahead the next 3 rounds. Play the value that the casino plays in the majority of the 3 rounds. This tells the follower what to play in his step 2.
3a. If you will only win 2 of the 3 rounds, look ahead to the 3 rounds after and play the majority value in the round you will lose. Play the winning value in the other 2 rounds. Go to step 3 (a or b).
3b. If you will win all 3 rounds, play the winning value in all 3 rounds and keep playing it until you won't win. Go to step 2.

Follower Rules:
1. To start, play 0 until you lose a round.
2. Play the value played by the leader in the last round that was lost. Do >! this for 3 rounds.
3a. If you only won 2 of the 3 rounds, do step 2 again.
3b. If you won all 3 rounds, keep playing the value until you lose a round. >! Do step 2 again.

Leader Rules:
1. To start, play 0 until the first round the casino doesn't play 0.
2. Look ahead the next 3 rounds. Play the value that the casino plays in the majority of the 3 rounds. This tells the follower what to play in his step 2.
3a. If you will only win 2 of the 3 rounds, look ahead to the 3 rounds after and play the majority value in the round you will lose. Play the winning value in the other 2 rounds. Go to step 3 (a or b).
3b. If you will win all 3 rounds, play the winning value in all 3 rounds and keep playing it until you won't win. Go to step 2.

Follower Rules:
1. To start, play 0 until you lose a round.
2. Play the value played by the leader in the last round that was lost. Do this for 3 rounds.
3a. If you only won 2 of the 3 rounds, do step 2 again.
3b. If you won all 3 rounds, keep playing the value until you lose a round. Do step 2 again.

By using Joel Rondeau's strategy as a base, plus some kludgy patchwork, we can get a strategy that always gets at least 6 wins.

By using Joel Rondeau's strategy as a base, plus some kludgy patchwork, we can get a strategy that always gets at least 6 wins.

By using Joel Rondeau's strategy as a base, plus some kludgy patchwork, we can get a strategy that always gets at least 6 wins.

The only cases we need to patch are the ones in which groups 2 and 3 contain a 2/1 split, and group 4 contains a 1/1 split. Notice that in the original strategy Leader always takes a point if possible, so if he doesn't the second player can automatically assume the casino sequence to be of this form.

Case 1: Group 3 is of the form ABA
In this case, we will intentionally give the minority bit in round 1 (the second player will be able to tell you gave him the wrong bit after round 4). We will lose two of the three rounds in group 2, so we have two free bits of information: with the first of them, identify the first bit of group 3, and with the other one identify the first bit in group 4. This uniquely determines groups 3 and 4.

Case 2: Group 3 is AAB or BAA and Group 4 is 01
We will now roughly follow the initial strategy, with one exception: if group 3 is AAB, in the first round of group 2 that the majority bit is played by the casino deliberately play the other bit instead. If group 3 is BAA do the same, but in the second round. This, coupled with the value of A you will give as normal (in the round the minority bit is played), uniquely determines groups 3 and 4.

Case 3: Group 3 is AAB or BAA and Group 4 is 10
3a) Group 3 is AAB
Play rounds 1 to 5 as normal, but play B in round 6 instead. You only miss round 6 in this case, so the second player is able to deduce the correct play for the last 3 rounds.

3b) Group 3 is BAA
Play rounds 1 to 4 as normal, except that you should flip the answer to the losing round of R2-R4. Then play A in round 5. This is the only case in which you miss round 5, so the second player can determine that he should switch the bit for the next two rounds.

By using Joel Rondeau's strategy as a base, plus some kludgy patchwork, we can get a strategy that always gets at least 6 wins.

For reference, this is his strategy:

Leader Rules:
1. To start, play 0 until the first round the casino doesn't play 0.
2. Look ahead the next 3 rounds. Play the value that the casino plays in the majority of the 3 rounds. This tells the follower what to play in his step 2.
3a. If you will only win 2 of the 3 rounds, look ahead to the 3 rounds after and play the majority value in the round you will lose. Play the winning value in the other 2 rounds. Go to step 3 (a or b).
3b. If you will win all 3 rounds, play the winning value in all 3 rounds and keep playing it until you won't win. Go to step 2.

Follower Rules:
1. To start, play 0 until you lose a round.
2. Play the value played by the leader in the last round that was lost. Do >! this for 3 rounds.
3a. If you only won 2 of the 3 rounds, do step 2 again.
3b. If you won all 3 rounds, keep playing the value until you lose a round. >! Do step 2 again.

The only cases we need to patch are the ones in which groups 2 and 3 contain a 2/1 split, and group 4 contains a 1/1 split. If that doesn't happen, then Leader follows the original set of rules. In the remaining cases, we will need to do something that deviates from the original strategy as soon as possible, so that follower will have time to react to it. We will do that by sacrificing some points:

Case 1: Group 3 is of the form ABA
In this case, we will intentionally give the minority bit in round 1 (the second player will be able to tell you gave him the wrong bit after round 4). We will lose two of the three rounds in group 2, so we have two free bits of information: with the first of them, identify the first bit of group 3, and with the other one identify the first bit in group 4. This uniquely determines groups 3 and 4.

Case 2: Group 3 is AAB or BAA and Group 4 is 01
We will now roughly follow the initial strategy, with one exception: if group 3 is AAB, in the first round of group 2 that the majority bit is played by the casino deliberately play the other bit instead. If group 3 is BAA do the same, but in the second round. This, coupled with the value of A you will give as normal (in the round the minority bit is played), uniquely determines groups 3 and 4.

Case 3: Group 3 is AAB or BAA and Group 4 is 10
3a) Group 3 is AAB
Play rounds 1 to 5 as normal, but play B in round 6 instead. You only miss round 6 in this case, so the second player is able to deduce the correct play for the last 3 rounds.

3b) Group 3 is BAA
Play rounds 1 to 4 as normal, except that you should flip the answer to the losing round of R2-R4. Then play A in round 5. This is the only case in which you miss round 5, so the second player can determine that he should switch the bit for the next two rounds.

According to the new leader strategy, we can rewrite the follower rules. As some notation, ~A is the value that is not A (so ~1 is 0 and ~0 is 1). Rules added in bold:

1. To start, play 0 until you lose a round.
2. Play the value played by the leader in the last round that was lost. Do this for 3 rounds, unless 3d or 3e happens.
3a. If you only won 2 of the 3 rounds, do step 2 again.
3b. If you won all 3 rounds, keep playing the value until you lose a round. Do step 2 again.
3c. If you won only 1 round, go to Exception 1.
3d. If you played the same value as the Casino, but Leader didn't, and you're at or after Round 5, go to Exception 2.
3e. If you played the same value as the Casino, but Leader didn't, and you're before Round 5, go to Exception 3.

Exception 1. Look at the bits played in the last 2 rounds you lost. Let those be A and B. Play the next 5 rounds as: A ~A A B ~B.
Exception 2. Let the bit you played in the last round you lost be A. Play ~A until round 7, then play 1 0 in rounds 8 and 9.
Exception 3. Play the same bit you played in the last round until round 4. Let A be the bit Leader played in the last round that the Casino played a different bit than you did. If you had already won a round when you came to this exception, play ~A A A 0 1 for the last 5 rounds. Otherwise play A A ~A 0 1.