One of the assumptions of von-Neumann and Morgenstern’s utility theory is continuity: if the decision maker prefers outcome A to outcome B to outcome C, then there is a number p in the unit interval such that the decision maker is indifferent between obtaining B for sure and a lottery that yields A with probability p and C with probability 1-p.
When I teach von-Neumann and Morgenstern’s utility theory I always provide criticism to their axioms. The criticism to the continuity axiom that I use is when the utility of C is minus infinity: C is death. In that case, one cannot find any p that would make the decision maker indifferent between the above two lotteries.
This morning during breakfast my younger son provided a different type of counter example to the continuity axiom. Every morning my son has a yogurt; a specific type of yogurt, that comes with a side bin full of chocolate balls, that you should pour into the yogurt before eating it. The balls are made of white chocolate, dark chocolate and milk chocolate.
My son prefers the white chocolate, and so he counts the number of white chocolate balls before he eats the yogurt; the higher the number, the happier he is. Today the older son had a bright idea: why don’t we call the producer and ask it to produce yogurts with only white chocolate balls! Brilliant. The younger kid opposed this idea: he prefer the current lottery, where he does not know the outcome, to obtaining the best outcome with probability 1.
So my son does not satisfy von-Neumann’s and Morgenstern’s axioms, and so all the deep theory that I use in raising the kids is useless.
The Leisure of the Theory Class 
7 comments
August 31, 2010 at 7:29 am
afinetheorem
Surely death does not give everyone negative infinity utility….I think most people would reveal their preference for their own death over that of their kids, for instance, and everyone would reveal preference for a quick death over a death by torture. To refute continuity, you would need to define “ultimate bliss” or its converse, and it is tough to imagine such things exist.
I’m a bit confused about how the son’s preferences violate continuity, as well. A preference for ambiguity can be modeled by retaining continuity and modifying some independence axiom. I suppose this is part of the larger philosophical point that axioms are in general not refutable, but rather only axiomatic systems are. That is, the Duhem-Quine thesis applies.
August 31, 2010 at 7:54 am
justacomment
The set of choices over which your son expresses his preference is the set of distributions of the distribution of chocolate balls that comes with this yogurt into three types of chocolate. His claim is that he prefers the current distribution (that governs the current random choice) to a situation where deterministically the outcome is all white chocolate with probability one.
Your observation that the higher the number of current balls, happier he is, seems to be a further preference relation on the outcomes of the random experiment. Under the current random setup, if he found a package with all whites, perhaps he would have been most thrilled. But you having this conversation this morning may have changed his point of view.
August 31, 2010 at 8:56 am
Greg Finley
I agree that there is no such thing as negative utility. We’re all going to die eventually, so actions that increase our immediate chance of death, while very bad, are not infinitely bad.
I’ve always liked Steven Landsburg’s headache example, which is described at the end of this article.
August 31, 2010 at 8:57 am
Greg Finley
Make that “I agree that there’s no such thing as infinitely negative utility.”
August 31, 2010 at 1:40 pm
Jonathan Weinstein
Agree with comments — negative infinite utility for death implies being too chicken to cross the road, let alone get in a car or plane.
September 3, 2010 at 2:48 pm
michael webster
Here is a more plausible example of the failure of continuity.
a= $2, b=$1, but c=being executed.
a>b>c, but there is no gamble between a and c which is indifferent to b.
But I am not aware of any extension to vm utility axioms which allows continuity to be violated.
It is appears to be a mere technical requirement for the mapping to be from the objects of choice to the real numbers.
If the mapping was to discrete intervals, for whatever reason. then continuity would not be required.
September 3, 2010 at 2:56 pm
Greg Finley
As with Landsburg’s example in my earlier comment, people might be willing to pay $1 to avoid a very very small chance of death (like 1 in a million).
So if you refuse to pay this $1, that’s like taking the gamble between a and c, is it not?