This… means you’re vulnerable to the Dutch Book described in the post. Why do you think otherwise?
I’m really not. You mean, “This means that according to my theory you’re vulnerable to the Dutch Book described in the post” Like I said though, I’m not accepting trades with negative utility, and being money pumped and Dutch Booked both have negative utility.
As for the “money pump” described in the post, I gain $23,999.98 if it happens as described. Also, there would have been no need to pay the first penny as the state of the switch was not relevant at that time. Also the game was switched from “34% for 24,000 and 33% for 27,000″ to “34% chance to play game 1, at which time you may choose”
Basically, this. The point of utility is that it’s linear in probability, which disallows a premium for certainty. If I know your utility for $27,000, and your utility for $24,000, and $0, then I can calculate your preferences over any gamble containing those three outcomes. If your decision procedure is not equivalent to a utility function, then there are cases where you can be made worse off even though it looks to you like you’re being made better off.
I agree that if you take the probability out of my utility function, then I am directly altering my preference in the exact same situation. Even so, there is in reality at least one difference: if someone is cheating or made a miscalculation, option 1A is cheat-proof and error-proof but none of the other options are. And I’ve definitely attached utility to that. This aspect would disappear if probabilities were removed from my utility function.
Like I said though, I’m not accepting trades with negative utility, and being money pumped and Dutch Booked both have negative utility.
You’ve expressed that 1A>1B, and 2B>2A. The first deal is “Instead of 2A, I’ll give you 2B for a penny.” By your stated preference, you agree. The second deal is “Instead of 1B, I’ll give you 1A.” By your stated preference, you agree. You are now two pennies poorer. So either you do not actually hold those stated preferences, or you are vulnerable to Dutch booking. (What does it mean to actually prefer one gamble to another? That you’re willing to pay to trade gambles. Suppose you hate selling things; then your preferences depend on the order you received things, which makes you vulnerable to the order in which other people present you options!)
Also the game was switched from “34% for 24,000 and 33% for 27,000” to “34% chance to play game 1, at which time you may choose”
What is the difference between those two games? The outcome probabilities are the same (multiply them out and check!). Or are you willing to pay hundreds of dollars (in expectation) to have him roll two dice instead of one?
Even so, there is in reality at least one difference: if someone is cheating or made a miscalculation, option 1A is cheat-proof and error-proof but none of the other options are.
But, don’t you have some numerical preference for this? If it were a certain 24,000 against a 33/34ths chance of 27 million, I hope you’d pick the latter, even if there’s some chance of the die being loaded in the second option. What this suggests, then, is that you need to adjust your probabilities- but if the probabilities are presented to you as your estimate after cheating is taken into account, then it doesn’t make sense to double-count the risk of cheating!
(One useful heuristic that people often have when evaluating gambles is imagining the person on the other side of the gamble. If something looks really good on your end and really bad on their end, then this is suspicious- why would they offer you something so bad for them? Keep in mind, though, that gambles are done both against other people and against the environment. If there’s gold sitting in the ground underneath you, and you have a 97% chance of successfully extracting it and becoming a millionaire, you shouldn’t say “hmm, what’s in it for the ground? Why would it offer me this deal?”)
You’ve expressed that 1A>1B, and 2B>2A. The first deal is “Instead of 2A, I’ll give you 2B for a penny.” By your stated preference, you agree. The second deal is “Instead of 1B, I’ll give you 1A.” By your stated preference, you agree.
Note that it becomes a different problem this way than my stated preferences (and note again that my stated choices (not preferences) were context-dependent) -- there is the additional information that the dealmaker had a good chance to cheat and didn’t take it. This information will reduce my disutility calculation for the uncertainty in the offer, as it increases my odds of winning 1B from [33/34 - good chance of cheating] to [33/34 - small chance of cheating]
You are now two pennies poorer.
Or 23,999.98 dollars richer.
So either you do not actually hold those stated preferences, or you are vulnerable to Dutch booking
If I did hold those preferences, I would not be vulnerable to Dutch booking, nor money pumping. Money pumping is infinite, whereas by giving me two pairs of different choices you can make me choose twice (and it’s not a preference reversal, though it would be exactly a preference reversal if you multiply the first choice’s odds by 0.34 and pretend that changes nothing).
For me to be vulnerable to Dutch booking, you’d have to somehow get money out of me as well. But how? I can’t buy game 1 for less than 24,000 minus the cost of various witnesses if I intend to choose 1A, and you can’t sell game 1 for less than 26,200. You’d have an even worse time convincing me to buy game 2. You can’t convince me to bid against either of the theoretically superior choices 1B and 2B. If you change my situation I might change my choice, as I already stated several conditions that would cause me to abandon 1A.
What is the difference between those two games?
Option 1A has a 0% chance of undetected cheating. Options 1B, 2A, and 2B all have a 100% chance of undetected cheating. In Game 3, you can pay to change your default choice twice, and the dealmaker shows a willingness to eliminate his ability to cheat before your second choice.
But, don’t you have some numerical preference for this?
Not currently. There would be a lot of factors determining how likely I think a miscalculation or cheating might be, and there is no way to determine this in the abstract.
I’m really not. You mean, “This means that according to my theory you’re vulnerable to the Dutch Book described in the post” Like I said though, I’m not accepting trades with negative utility, and being money pumped and Dutch Booked both have negative utility.
As for the “money pump” described in the post, I gain $23,999.98 if it happens as described. Also, there would have been no need to pay the first penny as the state of the switch was not relevant at that time. Also the game was switched from “34% for 24,000 and 33% for 27,000″ to “34% chance to play game 1, at which time you may choose”
I agree that if you take the probability out of my utility function, then I am directly altering my preference in the exact same situation. Even so, there is in reality at least one difference: if someone is cheating or made a miscalculation, option 1A is cheat-proof and error-proof but none of the other options are. And I’ve definitely attached utility to that. This aspect would disappear if probabilities were removed from my utility function.
You’ve expressed that 1A>1B, and 2B>2A. The first deal is “Instead of 2A, I’ll give you 2B for a penny.” By your stated preference, you agree. The second deal is “Instead of 1B, I’ll give you 1A.” By your stated preference, you agree. You are now two pennies poorer. So either you do not actually hold those stated preferences, or you are vulnerable to Dutch booking. (What does it mean to actually prefer one gamble to another? That you’re willing to pay to trade gambles. Suppose you hate selling things; then your preferences depend on the order you received things, which makes you vulnerable to the order in which other people present you options!)
What is the difference between those two games? The outcome probabilities are the same (multiply them out and check!). Or are you willing to pay hundreds of dollars (in expectation) to have him roll two dice instead of one?
But, don’t you have some numerical preference for this? If it were a certain 24,000 against a 33/34ths chance of 27 million, I hope you’d pick the latter, even if there’s some chance of the die being loaded in the second option. What this suggests, then, is that you need to adjust your probabilities- but if the probabilities are presented to you as your estimate after cheating is taken into account, then it doesn’t make sense to double-count the risk of cheating!
(One useful heuristic that people often have when evaluating gambles is imagining the person on the other side of the gamble. If something looks really good on your end and really bad on their end, then this is suspicious- why would they offer you something so bad for them? Keep in mind, though, that gambles are done both against other people and against the environment. If there’s gold sitting in the ground underneath you, and you have a 97% chance of successfully extracting it and becoming a millionaire, you shouldn’t say “hmm, what’s in it for the ground? Why would it offer me this deal?”)
Note that it becomes a different problem this way than my stated preferences (and note again that my stated choices (not preferences) were context-dependent) -- there is the additional information that the dealmaker had a good chance to cheat and didn’t take it. This information will reduce my disutility calculation for the uncertainty in the offer, as it increases my odds of winning 1B from [33/34 - good chance of cheating] to [33/34 - small chance of cheating]
Or 23,999.98 dollars richer.
If I did hold those preferences, I would not be vulnerable to Dutch booking, nor money pumping. Money pumping is infinite, whereas by giving me two pairs of different choices you can make me choose twice (and it’s not a preference reversal, though it would be exactly a preference reversal if you multiply the first choice’s odds by 0.34 and pretend that changes nothing).
For me to be vulnerable to Dutch booking, you’d have to somehow get money out of me as well. But how? I can’t buy game 1 for less than 24,000 minus the cost of various witnesses if I intend to choose 1A, and you can’t sell game 1 for less than 26,200. You’d have an even worse time convincing me to buy game 2. You can’t convince me to bid against either of the theoretically superior choices 1B and 2B. If you change my situation I might change my choice, as I already stated several conditions that would cause me to abandon 1A.
Option 1A has a 0% chance of undetected cheating. Options 1B, 2A, and 2B all have a 100% chance of undetected cheating. In Game 3, you can pay to change your default choice twice, and the dealmaker shows a willingness to eliminate his ability to cheat before your second choice.
Not currently. There would be a lot of factors determining how likely I think a miscalculation or cheating might be, and there is no way to determine this in the abstract.