Let’s play a game. I’ve got a coin here, and I’m going to toss it repeatedly and record the sequence of outcomes: heads (H) and tails (T).
Here we go…
H T H H H T T H T H …..
That was fun.
Next, I’ll do that again, but before doing so I’ll ask you to make a prediction for a sequence of 3 tosses. Then I’ll do the same, making a different choice. So you might choose THT and then I might choose TTT. I’ll then start tossing the coin. The winner will be the person whose chosen triplet shows up first in the sequence of tosses.
So, if the coin showed up as in the sequence above, you’d have won because there’s a sequence of THT starting from the 7th toss in the sequence. If the triplet TTT had shown up before that – which it didn’t -then I’d have won.
Now, assuming we both play optimally, there are 3 possibilities for who this game might favour (in the sense of having a higher probability of winning):
- It favours no one. We both get to choose our preferred sequence and so, by symmetry, our chances of winning are equal.
- It favours you. You get to choose first and so you can make the optimal choice before I get a chance.
- It favours me. I get to see your choice before making mine and can make an optimal choice accordingly.
Which of these do you think is correct? Have a think about it. You might even have to decide what it means to play ‘optimally’.
If you’d like to mail me with your answers I’d be happy to hear from you. In a subsequent post I’ll discuss the solution with reasons why this game is important.