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.
Choose between the following two options:
Which seems more intuitively appealing? And which one would you choose in real life?
Now which of these two options would you intuitively prefer, and which would you choose in real life?
The Allais Paradox - as Allais called it, though it's not really a paradox - was one of the first conflicts between decision theory and human reasoning to be experimentally exposed, in 1953. I've modified it slightly for ease of math, but the essential problem is the same: Most people prefer 1A > 1B, and most people prefer 2B > 2A. Indeed, in within-subject comparisons, a majority of subjects express both preferences simultaneously.
This is a problem because the 2s are equal to a one-third chance of playing the 1s. That is, 2A is equivalent to playing gamble 1A with 34% probability, and 2B is equivalent to playing 1B with 34% probability.
Among the axioms used to prove that "consistent" decisionmakers can be viewed as maximizing expected utility, is the Axiom of Independence: If X is strictly preferred to Y, then a probability P of X and (1 - P) of Z should be strictly preferred to P chance of Y and (1 - P) chance of Z.
All the axioms are consequences, as well as antecedents, of a consistent utility function. So it must be possible to prove that the experimental subjects above can't have a consistent utility function over outcomes. And indeed, you can't simultaneously have:
These two equations are algebraically inconsistent, regardless of U, so the Allais Paradox has nothing to do with the diminishing marginal utility of money.
Maurice Allais initially defended the revealed preferences of the experimental subjects - he saw the experiment as exposing a flaw in the conventional ideas of utility, rather than exposing a flaw in human psychology. This was 1953, after all, and the heuristics-and-biases movement wouldn't really get started for another two decades. Allais thought his experiment just showed that the Axiom of Independence clearly wasn't a good idea in real life.
(How naive, how foolish, how simplistic is Bayesian decision theory...)
Surely, the certainty of having $24,000 should count for something. You can feel the difference, right? The solid reassurance?
(I'm starting to think of this as "naive philosophical realism" - supposing that our intuitions directly expose truths about which strategies are wiser, as though it was a directly perceived fact that "1A is superior to 1B". Intuitions directly expose truths about human cognitive functions, and only indirectly expose (after we reflect on the cognitive functions themselves) truths about rationality.)
"But come now," you say, "is it really such a terrible thing, to depart from Bayesian beauty?" Okay, so the subjects didn't follow the neat little "independence axiom" espoused by the likes of von Neumann and Morgenstern. Yet who says that things must be neat and tidy?
Why fret about elegance, if it makes us take risks we don't want? Expected utility tells us that we ought to assign some kind of number to an outcome, and then multiply that value by the outcome's probability, add them up, etc. Okay, but why do we have to do that? Why not make up more palatable rules instead?
There is always a price for leaving the Bayesian Way. That's what coherence and uniqueness theorems are all about.
In this case, if an agent prefers 1A > 1B, and 2B > 2A, it introduces a form of preference reversal - a dynamic inconsistency in the agent's planning. You become a money pump.
Suppose that at 12:00PM I roll a hundred-sided die. If the die shows a number greater than 34, the game terminates. Otherwise, at 12:05PM I consult a switch with two settings, A and B. If the setting is A, I pay you $24,000. If the setting is B, I roll a 34-sided die and pay you $27,000 unless the die shows "34", in which case I pay you nothing.
Let's say you prefer 1A over 1B, and 2B over 2A, and you would pay a single penny to indulge each preference. The switch starts in state A. Before 12:00PM, you pay me a penny to throw the switch to B. The die comes up 12. After 12:00PM and before 12:05PM, you pay me a penny to throw the switch to A.
I have taken your two cents on the subject.
If you indulge your intuitions, and dismiss mere elegance as a pointless obsession with neatness, then don't be surprised when your pennies get taken from you...
(I think the same failure to proportionally devalue the emotional impact of small probabilities is responsible for the lottery.)
Allais, M. (1953). Le comportement de l'homme rationnel devant le risque: Critique des postulats et axiomes de l'école américaine. Econometrica, 21, 503-46.
Kahneman, D. and Tversky, A. (1979.) Prospect Theory: An Analysis of Decision Under Risk. Econometrica, 47, 263-92.