The rationality quiz that I posted on Tuesday has drawn a lot of comments from folks who think they can reconcile inconsistent answers by appealing to risk aversion. That’s surely incorrect. To see why, let’s start with another quiz.
Question 0: Which do you like better, dogs or cats?
Economists would not presume to declare either choice an irrational one. There’s no accounting for tastes.
Now I have two more questions for you:
Question 1. Which would you prefer:
- A dog
- A pet that has an 89% chance of being a dog and an 11% chance of being a cat
Question 2. Which would you prefer:
- An 11% chance of getting a dog
- An 11% chance of getting a cat
When John von Neumann and Oskar Morgenstern set out to axiomatize the notion of rationality, one of their axioms was (in essence) this: If you prefer dogs to cats, you’ll answer A to both questions. If you prefer cats to dogs, you’ll answer B to both questions.
It seems pretty hard to argue with this axiom. In Question 1, why would a dog lover ever take an unnecessary chance of winning a cat? Why would a cat lover settle for a dog? In Question 2, it’s almost impossible to imagine a dog lover choosing B or a cat lover choosing A.
Notice that none of this has anything to do with attitudes toward risk. It’s strictly about attitudes toward dogs and cats — and more specifically, about the consistency of those attitudes.
Now replace the dog with a million dollars, and the cat with a lottery ticket that gives you a 10/11 chance at five million dollars. Which do you like better, the “dog” or the “cat”? Again, either answer is fine. There’ s no accounting for tastes.
Now let’s revisit Questions 1 and 2:
Question 1: Which would you prefer?
- A “dog” (i.e. a million dollars)
- A prize that has an 89% chance of a being a “dog” (i.e. a million dollars) and an 11% chance of being a “cat” (i.e. a 10/11 chance of five million dollars).
[Notice that an 11% chance of being a 10/11 chance of five million dollars is the same thing as a 10% chance of being five million dollars.]
Question 2: Which would you prefer?
- An 11% chance of getting a “dog” (i.e. a million dollars)
- An 11% chance of getting a “cat” (i.e. a 10/11 chance of getting five million dollars).
If you’ve bought into the von Neumann-Morgenstern axiom, then rationality requires you to answer either A to both questions (if you’re a “dog” lover) or B to both questions (if you’re a “cat” lover).
This is a simple consistency requirement, which, again, has nothing to do with how you feel about risk. If you’re averse to risk, you should be a consistent “dog” lover. If you love risk, you should be a consistent “cat” lover. But no single attitude toward risk can justify acting like a dog lover half the time and a cat lover the other half.
Questions 1 and 2 here are, of course, the same questions I posed on Tuesday. If your answers (like many people’s) were A and B, then you’ve certainly violated the von Neumann-Morgenstern dog/cat axiom.
Several commenters have made the mistake of arguing that Question 1 involves a “sure thing” while Question 2 does not. Here’s exactly why that argument is wrong:
In either scenario, there’s an 89% chance your decision won’t matter. 89% of the time, you’re sure to win a million in Question 1 or sure to win zero in Question 2 — and there’s nothing you can do about that. Only the remaining 11% of the time does your decision have any effect on the outcome — so you might as well make your decision on the assumption that this is one of those 11 out of 100 times.
In that case, what you’re choosing between is a sure million versus a 10/11 chance of five million. That’s your choice in Question 1 and that’s your choice in Question 2. If you like “dogs” (i.e. sure things), then you should take the sure thing both times. If you prefer to gamble, you should take the gamble both times. In the only cases where your decision matters, Questions 1 and 2 both present you with exactly the same choice between a gamble and a sure thing. In each case, you should pick the one you prefer.
So if you gave inconsistent answers, you can’t justify them by risk aversion. Is there some other way to justify them? Maybe. I’ll post (at least) once more on this topic in the next several days.