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.