♛ | I used to think it was irrational to buy a $1 lottery ticket, or play roulette -- a tax on people who are bad at math -- but now I don't. |

♔ | Um.. because it is a fun experience? |

♛ | No. I used to think the rational thing to do was maximize expected money. |

♔ | Why is that irrational? |

♛ | Well, there are times when maximizing expected money will probably leave me broke. Let's say someone offers to triple my bet if I win a fair coin toss. I expect to make $1.50 for every dollar I bet, so the "rational" thing to do is bet all my money. If I win, and they offer another go at it, I should bet everything again. In fact, if they offer 100 flips, I should bet everything on every flip, which will very likely leave me broke. |

♔ | True, though you could maximize your expected log-money. The log of zero is negative infinity, so you'd never risk all your money. Also, the log would value each new dollar less and less, which sortof makes sense, since one extra dollar when you have a billion dollars isn't worth much. |

♛ | Sure, that turns out to be the kelly criterion for this game. But what is special about the log? I mean, someone could offer so much money for winning 100 tosses in a row that I would still bet everything on that remote possibility -- everything except a penny, that is, to avoid having nothing. |

♔ | Hm.. What if you imagined that you were competing against someone else with the same opportunity, and you wanted to maximize your chance of ending up with more money than them? This would stop you from caring too much about remote possibilities, since you would want to do whatever maximized your money most of the time. This also sortof makes sense, since people like to have more money than other people. |

♛ | Yeah, that also turns out to be the kelly criterion for this game. But what is special about ending up with more money than someone else playing the same game? I mean, consider a game with a single coin that is biased slightly in my favor, and if I win, I get back slightly more than I bet. To have the best chance of having more money than my friend, who has the same opportunity, I should bet everything, but this means I'm expected to lose about half my money. |

♔ | Hm.. Well, you could maximize your expected utility. Von Neumann and Morgenstern suggest that if you're rational, your preferences for decisions like this will be consistent, and that that implies there will exist some function mapping dollars to utiles such that you are really maximizing your expected utility. |

♛ | That's an interesting reframing of the problem, but saying it is rational to have consistent preferences does not say what those preferences should be, i.e., they don't say what the mapping should be from dollars to utiles. I claim that this mapping is arbitrary. I can't think of any objective reason to prefer one mapping over another. |

As an example, it seems fine to value $36 a hundred times more than $1. After all, you can buy something with $36. You probably can't buy anything with just $1, not even 1/36th of something. Hence, it may make sense for you to put your single dollar down on red 27 in roulette for a small chance of winning $36, since you have a 1/38 chance of getting a hundred times as many utiles as you had with your $1, which is an expected gain in utility. |

## 5/24/12

### rational gambling

from an old blog post on glittle.org/blog.. decided to move it here instead of realgl.blogger.com because it seems a bit half-baked:

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