In a two people's game, you start with one dollar and you are betting 1 dollar coin at the very start of the game, then if you win you have double amount of money and you can choose of choose not to continue to bet all your money, if you lose, you lose all the money you own. This game is a fair game with the follwing rule, when the typical coin is tossed head you win and if you tossed tail you lose, the question is what is the expected return when played on the best strategy of this game?

But when i calculate the answer as follow when played this game infinite numbers of times, something is weird:

expected return = 1*(1/2)+2*(1/4)+4*(1/8)+...= infinity

is this a famous paradox?

  • 1
    $\begingroup$ >what is the best return when played on the best strategy of this game? Do you mean what is the expected return or what is the best return? You ask two different questions in your title and in your question. $\endgroup$ Feb 21, 2014 at 0:31
  • $\begingroup$ Note that it's only possible to play the game infinitely when the coin lands heads every time. In which case the return would indeed diverge to infinity! $\endgroup$
    – Tetrinity
    Feb 21, 2014 at 0:37

1 Answer 1


You're right, it is infinite, but finite-turn simulations show that the amount of money you earn in practice is drastically lower! This is known as the St. Petersburg Paradox.


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