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.
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.