it is assumed that the utility scales with the monetary reward.
Not necessarily. It is assumed that receiving $24000 is equally good in either situation. Your utility function can ignore money entirely (in which case 1A2A is irrational because you should be indifferent in both cases). You can use the utility function which prefers not to receive monetary rewards divisible by 9: in this case, 1A>1B and 2A>2B is your best bet, giving you 100% and 34% chances to avoid 9s, rather than 0% chances. In general, your utility function can have arbitrary preferences on A and B separately; but no matter what, it will prefer 1A to 1B if and only if it prefers 2A to 2B.
As for the rest of your reply -- yes, it is true that real people use strategies ("heuristic" is the word used in the original post) that lead them to choose 1A and 2B. That's sort of why it's a paradox, after all. However, these strategies, which work well in most cases, aren't necessarily the best in all cases. The math shows that. What the math doesn't tell us is which case is wrong.
My own judgment, for this particular sum of money (which is high relative to my current income), is that choice 1A is correctly better than choice 2A, in order to avoid risk. However, choice 1B is also better than choice 2B, upon reflection, even though my intuitions tell me to go with 2B. This is because my intuitions aren't distinguishing 33% and 34% correctly.
In reality, faced with the opportunity to earn amounts on the order of $20K, I should maximize my chances to walk away with something. In the first case, I can maximize them fully, to 100%, which triggers my "success!" instinct or whatever: I know I've done everything I can because I'm certain to get lots of money. In the second case, I don't get any satisfaction from the correct decision, because all I've done is improve my chances by 1%.
In general, the heuristic that 1% chances are nearly worthless is correct, no matter what's at stake: I can usually do better by working on something that will give me a 10% or 25% chance. In this case, this heuristic should be ignored, because there is no effort spent making the improvement, and furthermore, there isn't really anything else I can do.
On the other hand, suppose that the amount of money at stake is $2.40 or $2.70. Suddenly, our risk-aversion heuristic is no longer being triggered at all (unless you're really strapped for cash), and we have no problem doing the utility calculation. Here, 1A<1B and 2A<2B is the correct choice.
The utility function has as its input only the monetary reward in this particular instance. Your idea that risk-avoidance can have utility (or that 1% chances are useless) cannot be modelled with the set of equations given to analyse the situation (the percentage is no input to the U() function) - the model falls short because the utility attaches only to the money and nothing else. (Another example of a group of individuals for whom the risk might out-utilize the reward are gambling addicts.) Security is, all other things being equal, preferred over insec...
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.