Coin Triplets

[BMAD]

Two players play the following game with a fair coin. Player 1 chooses (and announces) a triplet (HHH, HHT, HTH, HTT, THH, THT, TTH, or TTT) that might result from three successive tosses of the coin. Player 2 then chooses a different triplet. The players toss the coin until one of the two named triplets appears. The triplets may appear in any three consecutive tosses: (1st, 2nd, 3rd), (2nd, 3rd, 4th), and so on. The winner is the player whose triplet appears first.

What is the optimal strategy for each player? With best play, who is most likely to win?

Suppose the triplets were chosen in secret? What then would be the optimal strategy?

What would be the optimal strategy against a randomly selected triplet?

[bonanova]

Player 1 picks a triplet T₁.

Player 2 picks a triplet T₂.

Coin is tossed until T₁ or T₂ appears and wins.

Every triplet T₁ has an evil twin T₂ that will beat it on average.

Suppose T₁ = TTT.

If TTT is not the result of the first three flips, it will be preceded by H and lose to HTT.

TTT is the result of the first three flips 1/8 of the time.

Thus TTT loses to HTT with 7:1 odds.

Other pairwise triplet probabilities can be deduced similarly.
The table gives P₂'s winning probability for every triplet combination T₁, T₂.
The bottom two rows give the best and average winning probability for each T₁

What is the optimal strategy for each player?

P₁ should pick T₁= HTH, HTT, THH or THT.
The best that P2 can achieve then is a winning probability of .667,

That is the smallest best winning probability among the possible T₁ choices.

P₂ should look in the column headed by T₁ and choose the T₂ that has the highest winning probability.

With best play, who is most likely to win?

P₂ can always gain favorable odds, ranging from 2:1 to 7:1 depending on T₁

Suppose the triplets were chosen in secret?

I interpret this to mean P2 chooses T₂ without knowing T₁.

Nothing different for P₁. He should still choose HTH, HTT, THH or THT.

P₂ does not know T₁, so he should seek the T₂ with best average winning probability.
These numbers are given in the last column of the table.

The choices are either HTT or THH with average winning probability of .507.

HTT loses only to HHT; draws against itself, HTH, THH and THT; wins against HHH, TTH and TTT.
THH loses only to TTH; draws against itself, THT, HTT and HTH; wins against TTT, HHT and HHH.

What would be the optimal strategy against a randomly selected triplet?

I take this to ask P₁'s optimal strategy in the case that T2 will be chosen at random.

P₁ should look at the bottom row of the table to see P₂'s winning probability averaged over all the T2s.

P₁ should thus choose HTT or THH.

This gives T₂ the lowest average winning probability: .368.


P2\P1| HHH HHT HTH HTT THH THT TTH TTT | MAX | AVG |
---\-+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
HHH | --- .500 .400 .400 .125 .417 .300 .500 |.500 |.330 |
HHT |.500 --- .667 .667 .250 .625 .500 .700 |.700 |.489 |
HTH |.600 .333 --- .500 .500 .500 .375 .583 |.600 |.424 |
HTT |.600 .333 .500 --- .500 .500 .750 .875 |.875 |.507 |
THH |.875 .750 .500 .500 --- .500 .333 .600 |.875 |.507 |
THT |.583 .375 .500 .500 .500 --- .333 .600 |.600 |.424 |
TTH |.700 .500 .625 .250 .667 .667 --- .500 |.700 |.489 |
TTT |.500 .300 .417 .125 .400 .400 .500 --- |.500 |.330 |
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
MAX |.875 |.750 |.667 |.667 |.667 |.667 |.750 |.875 |
-----+-----+-----+-----+-----+-----+-----+-----+-----+
AVG |.545 |.386 |.451 |.368 |.368 |.451 |.386 |.545 |
-----+-----+-----+-----+-----+-----+-----+-----+-----+