Lesson 14

The game from the previous step is fair: it is easy to see that the expected winnings of each player are equal to zero.

Now suppose there are not one but two strangers, and the game is arranged as follows. Each of the three players initially places two euros on the table, then writes “heads” or “tails” on a piece of paper, trying to guess the coin flip. Then the coin is flipped, and the six euros from the center of the table are evenly divided among those who guessed correctly. If no one guessed correctly, each player gets their two euros back. The diagram below shows how much you will receive in each of the eight possible cases. The first level shows whether the first player guessed correctly, the second and third levels show whether the second and third players guessed correctly, respectively.

It is easy to see that the expected winnings of the first player are equal to zero. And this aligns with intuition: could any of the players have a positive average gain? After all, they all know equally little about how the coin will land!

Problem. Will you agree to play this game with two strangers?