Best Answer bonanova, 22 May 2013 - 12:00 PM

Player 1 picks a triplet T_{1}.

Player 2 picks a triplet T_{2}.

Coin is tossed until T_{1} or T_{2} appears and wins.

Every triplet T_{1} has an evil twin T_{2} that will beat it on average.

Suppose T_{1} = 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 P_{2}'s winning probability for every triplet combination T_{1}, T_{2}.

The bottom two rows give the best and average winning probability for each T_{1}**What is the optimal strategy for each player?****P _{1}** should pick

**T**=

_{1}**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 T_{1} choices.**P _{2}** should look in the column headed by T

_{1}and choose the

**T**that has the

_{2}__highest winning probability__.

**With best play, who is most likely to win?**

**P**can always gain favorable odds, ranging from

_{2}**2:1**to

**7:1**depending on T

_{1}

**Suppose the triplets were chosen in secret?**

I interpret this to mean P2 chooses T_{2} without knowing T_{1}.

Nothing different for **P _{1}.** He

**should still choose**

**HTH**,

**HTT**,

**THH**or

**THT**.

**P _{2}** does not know T

_{1}, so he should seek the T

_{2}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 P_{1}'s optimal strategy in the case that T2 will be chosen at random.

P_{1} should look at the bottom row of the table to see P_{2}'s winning probability __averaged__ over all the T2s.

P_{1} should thus choose **HTT** or **THH**.

This gives T_{2} 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 |

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