<p>Consider the following basic repeated betting game:</p>
<ul>
<li>A player can enter the game with an amount of money <code>x</code>.</li>
<li>The game consists of multiple rounds.</li>
<li>In each round a fair coin is flipped. If it shows heads, the player's current balance triples to <code>3*x</code> (i.e., they get twice the amount of money on top of their current balance). If it is tails, they lose everything.</li>
<li>The player can decide before each round, if they want to stop and keep the winnings, or continue.</li>
<li>The game ends if either the players stops, or the balance drops to zero.</li>
<li>The player only gets a single, once in a lifetime chance to play the game.</li>
</ul>
<p>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 <code>x</code>.</p>
<p>My questions are:</p>
<ul>
<li>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 &quot;expectation value maximization&quot;?</li>
<li>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?</li>
</ul>

CC BY-SA 4.0
2021-05-29T10:20:35.517Z
Have you heard of the St Petersburg paradox? https://en.m.wikipedia.org/wiki/St._Petersburg_paradox
Bayesian
CC BY-SA 4.0
2021-05-29T10:34:05.370Z
@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
CC BY-SA 4.0
2021-05-29T10:50:07.327Z
This is not a game theory problem; there is only one player here, no strategic interaction.
Giskard
CC BY-SA 4.0
2021-05-29T10:54:24.170Z
This decision problem or "game" is not well-defined; what is the objective of the player?
Giskard
CC BY-SA 4.0
2021-05-29T10:55:04.187Z
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
CC BY-SA 4.0
2021-05-29T10:56:09.857Z
Finally the [Kelly formula](https://en.wikipedia.org/wiki/Kelly_criterion) also seems relevant here.
Giskard
CC BY-SA 4.0
2021-05-29T11:14:19.093Z
@Giskard The objective is to maximize the amount won. What I meant by that is not changing the rules, but changing the mathematical model to incorporate the termination rule as well. For instance, can math give a clear answer like "the optimal number of rounds to play is 3" (which has a positive expectation value) instead of "the optimal number of rounds to play is ∞" (which definitely loses all money).
bluenote10
CC BY-SA 4.0
2021-05-29T11:18:40.077Z
"*The objective is to maximize the amount won.*" This is not a valid goal function in any game of chance; you have to take risk into account somehow. Please tell us a bit about your background so we can know what sources to recommend. Have you studied optimization theory, microeconomics or game theory?
Giskard
CC BY-SA 4.0
2021-05-29T11:39:12.850Z
@Giskard In fact I have, but ~15 years ago, and suffer from memory loss. How is that not a valid goal function? Because I have colloquially omitted the word "expected"? Another try: The objective function is the same as in the St Petersburg paradox.
bluenote10
CC BY-SA 4.0
2021-05-29T11:53:44.370Z
@Giskard To clarify, the risk is to lose the initial bet `x`.
bluenote10
CC BY-SA 4.0
2021-05-29T12:15:09.680Z
But how do you maximize expected value of the amount won in this problem when several strategies lead to undefined/infinite expected value? How do these compare to strategies with finite expected values?
Giskard
<expected-utility>
Repeated betting game with positive expected value
bluenote10
# Repeated betting game with positive expected value

(Asked by: username_0 on 2021-05-29)

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?

**Tags:** expected-utility

> **Comments:**
> **username_1**: Have you heard of the St Petersburg paradox? https://en.m.wikipedia.org/wiki/St._Petersburg_paradox  
> **username_0**: @username_1 Thanks, no I didn't, very interesting. Not quite the same game, but definitely highly related. Still have to wrap my head around that.  
> **username_2**: This is not a game theory problem; there is only one player here, no strategic interaction.  
> **username_2**: This decision problem or "game" is not well-defined; what is the objective of the player?  
> **username_2**: 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?  
> **username_2**: Finally the [Kelly formula](https://en.wikipedia.org/wiki/Kelly_criterion) also seems relevant here.  
> **username_0**: @username_2 The objective is to maximize the amount won. What I meant by that is not changing the rules, but changing the mathematical model to incorporate the termination rule as well. For instance, can math give a clear answer like "the optimal number of rounds to play is 3" (which has a positive expectation value) instead of "the optimal number of rounds to play is ∞" (which definitely loses all money).  
> **username_2**: "*The objective is to maximize the amount won.*" This is not a valid goal function in any game of chance; you have to take risk into account somehow. Please tell us a bit about your background so we can know what sources to recommend. Have you studied optimization theory, microeconomics or game theory?  
> **username_0**: @username_2 In fact I have, but ~15 years ago, and suffer from memory loss. How is that not a valid goal function? Because I have colloquially omitted the word "expected"? Another try: The objective function is the same as in the St Petersburg paradox.  
> **username_0**: @username_2 To clarify, the risk is to lose the initial bet `x`.  
> **username_2**: But how do you maximize expected value of the amount won in this problem when several strategies lead to undefined/infinite expected value? How do these compare to strategies with finite expected values?  

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## 0 answers

