# Repeated betting game with positive expected value

Consider the following basic repeated betting game:

• A player can enter the game with an amount of money x.
• The game consists of multiple rounds.
• In each round a fair coin is flipped. If it shows heads, the player's current balance triples to 3*x (i.e., they get twice the amount of money on top of their current balance). If it is tails, they lose everything.
• The player can decide before each round, if they want to stop and keep the winnings, or continue.
• The game ends if either the players stops, or the balance drops to zero.
• The player only gets a single, once in a lifetime chance to play the game.

This game confuses me. Basing a decision purely on maximizing the expectation value, it seems like in each round the best decision is to continue playing, because tripling the money with a probability of 0.5 simply has a positive expectation value. On the other hand, this strategy will force the player to continue playing indefinitely, and by definition the game will end with the player losing everything, i.e. losing their initial bet x.

My questions are:

• Is it somehow possible to incorporate the stopping criteria (game ends if you lose) into the mathematical model? Somehow it feels like math should give a better strategy then losing everything eventually, i.e., is there something more meaningful than the naive "expectation value maximization"?
• If math doesn't have a better answer, this may be a fundamental game theory dilemma. Does the game have a name in game theory, or are there any related dilemma I could read up on?
• Have you heard of the St Petersburg paradox? en.m.wikipedia.org/wiki/St._Petersburg_paradox – Bayesian May 29 at 10:20
• @Bayesian Thanks, no I didn't, very interesting. Not quite the same game, but definitely highly related. Still have to wrap my head around that. – bluenote10 May 29 at 10:34
• This is not a game theory problem; there is only one player here, no strategic interaction. – Giskard May 29 at 10:50
• This decision problem or "game" is not well-defined; what is the objective of the player? – Giskard May 29 at 10:54
• And what do you mean by "Is it somehow possible to incorporate the stopping criteria (game ends if you lose) into the mathematical model?" Yes? You can make this a rule as you outlined if you wish? – Giskard May 29 at 10:55