The rational choice depends on your utility function. Your utility function is unlikely to be linear with money. For example, if your utility function is log (X), then you will accept the first bet, be indifferent to the second bet, and reject the third bet. Any risk-averse utility function (i.e. any monotonically increasing function with negative second derivative) reaches a point where the agent stops playing the game.
A VNM-rational agent with a linear utility function over money will indeed always take this bet. From this, we can infer that linear utility functions do not represent the utility of humans.
(EDIT: The comments by Satt and AlexMennen are both correct, and I thank them for the corrections. I note that they do not affect the main point, which is that rational agents with standard utility functions over money will eventually stop playing this game)
Any risk-averse utility function (i.e. any monotonically increasing function with negative second derivative) reaches a point where the agent stops playing the game.
Not true. It is true, however, that any agent with a bounded utility function eventually stops playing the game.
In the St Petersburg Paradox the casino is offering a fair bet, the kind that casinos offer. It is generally an error for humans to take these.
In this scenario, the casino is magically tilting the bet in your favor. Yes, you should accept that bet and keep playing until the amount is an appreciable fraction of your net worth. But given that we are assuming the strange behavior of the casino, we could let the casino tilt the bet even farther each time, so that the bet has positive expected utility. Then the problem really is infinity, not utility. (Even agents with unbounded utility functions are unlikely to have them be unbounded as a function of money, but we could imagine a magical wish-granting genie.)
He’s not fighting the hypothetical; he merely responded to the hypothetical with a weaker claim than he should have. That is, he correctly claimed that realistic agents have utility functions that grow too slowly with respect to money to keep betting indefinitely, but this is merely a special case of the fact that realistic agents have bounded utility, and thus will eventually stop betting no matter how great the payoff of winning the next bet is.
The rational choice depends on your utility function. Your utility function is unlikely to be linear with money. For example, if your utility function is log (X), then you will accept the first bet, be indifferent to the second bet, and reject the third bet. Any risk-averse utility function (i.e. any monotonically increasing function with negative second derivative) reaches a point where the agent stops playing the game.
A VNM-rational agent with a linear utility function over money will indeed always take this bet. From this, we can infer that linear utility functions do not represent the utility of humans.
(EDIT: The comments by Satt and AlexMennen are both correct, and I thank them for the corrections. I note that they do not affect the main point, which is that rational agents with standard utility functions over money will eventually stop playing this game)
Not even that. You start with $1 (utility = 0) and can choose between
walking away with $1 (utility = 0), and
accepting a lottery with a 50% chance of leaving you with $0 (utility = −∞) and a 50% chance of having $3 (utility = log(3)).
The first bet’s expected utility is then −∞, and you walk away with the $1.
Not true. It is true, however, that any agent with a bounded utility function eventually stops playing the game.
Thanks for catching that, I stand corrected.
You are fighting the hypothetical.
In the St Petersburg Paradox the casino is offering a fair bet, the kind that casinos offer. It is generally an error for humans to take these.
In this scenario, the casino is magically tilting the bet in your favor. Yes, you should accept that bet and keep playing until the amount is an appreciable fraction of your net worth. But given that we are assuming the strange behavior of the casino, we could let the casino tilt the bet even farther each time, so that the bet has positive expected utility. Then the problem really is infinity, not utility. (Even agents with unbounded utility functions are unlikely to have them be unbounded as a function of money, but we could imagine a magical wish-granting genie.)
He’s not fighting the hypothetical; he merely responded to the hypothetical with a weaker claim than he should have. That is, he correctly claimed that realistic agents have utility functions that grow too slowly with respect to money to keep betting indefinitely, but this is merely a special case of the fact that realistic agents have bounded utility, and thus will eventually stop betting no matter how great the payoff of winning the next bet is.
This is a stupid comment. I would downvote it and move on, but I can’t, so I’m making this comment.
I agree with this, if “this” refers to your own comment and not the one it replies to.
Assuming that the total time it takes to make all your bets is not infinite, this results in
http://www.braingle.com/news/hallfame.php?path=logic/supertasks.p&sol=1