It’s astonishing how expectation could change one’s view of an outcome of some events. The higher we go, the higher we fall. Does that mean then that it is better to have lower expectation? Let us consider an event with two possible outcomes 0 and 1. Suppose that a person is monitoring this event. He/she will be happy if the outcome is 1, and sad otherwise.

Of course, the lower the expectation he/she has that the outcome will be 1, the happier he/she will be when 1 occurs. Let x be one’s expectation and let f(x) be the satisfaction function that is decreasing on x. If the outcome is 1, then the person’s payoff is f(x), otherwise it is 0. The expected payoff for this person is then pf(x), where p is the probability that 1 will occur. Given a value of p, we want to maximize f(x) to maximize the payoff, that is by making x smaller. However, we don’t want x to be too small (we know that if we are hopeless and have extremely low expectation, we would also be miserable). Let us then define g(x), a misery function that is also decreasing on x. We can then optimize the person’s payoff by finding the intersection of f(x) and g(x). Let’s suppose that f(a) = g(a). Then for any x<a, if f(x) > g(x), then we choose to stay on f(x) for x<a and g(x) for x>a. If for x<a, g(x) > f(x), then we would choose to stay on f(x) throughout. We have to balance between the misery before knowing the outcome and the satisfaction after knowing the outcome. Of course, if the probability of 1 occurring is very low, then he/she would be better off not expecting anything at all (if you are going to be miserable, then it’s better to just get it over with).

I have no clue how I just came up with that… it doesn’t even make any sense.. lol