# Basic concept of utility

I'm new to the concept of utility and I'm struggling to understand an important idea. Say we have some bet. I don't understand how the utility of the expected value of the bet differs from the expected value of the utility of the bet. In my mind, both correspond to what I would expect to get out of the bet on average, scaled to take in account my preferences for different monetary values.

Perhaps a simple example showing how exactly the two ideas are different would help. Thanks!

• Your example is not completely clear, but typically the context would be something like this. – GeoMatt22 Apr 28 '17 at 0:42

Let's pick a simple utility function and a simple gamble. Our utility function will be $$U(X)=\sqrt{X}$$ Our lottery will pay 1 unit with 50% probability and 4 units with 50% probability. The expected return is: $$E(X)=.5*\times1+.5\times4=2.5.$$ The utility of the expected value is: $$U(E(X))=\sqrt{2.5}\approx{1.581139}.$$

The utility of each payoff is: $$U(1)=1\wedge{U}(4)=2.$$ The expected utility of the lottery is: $$E(U(X))=.5\times{U}(1)+.5\times{U}(4)=.5\times{1}+.5\times{2}=1.5.$$

The expected utility is the expectation of the satisfaction that would be received over the sample space of possible outcomes. The utility of the expected value may not happen unless one of the prizes happens to match it. The utility of the expected value is the satisfaction that would be received if the expected utility were actually received. It won't be received.

Wins are experienced through the emotion they generate, losses too. The expected utility is your expected satisfaction.

• re: your final sentence, which I like. For a while, lotteries in Australia were marketed as a 'ticket to dream', tacitly acknowledging the chance of winning being negligible. Hence, the utility of a loss was the emotion you experienced between paying for the ticket and the dream being over when you discovered you didn't win, and utility of a win was irrelevant in calculating the utility of the lottery ticket. – Robert de Graaf Apr 28 '17 at 4:26

The idea of utility is very simple. It converts the numerical outcome into the utility to you. Say, you get 10 apples or 5 oranges, or some combination of two. How would you compare apples to oranges? Easy, I offer you to take either of two choices, and you pick one. So, theory goes that there's some kind of a function $U(x_1,x_2,\dots,x_n)$ such that when you feed it a basket of goods $1,2,\dots,n$ with quantities $x_i$ of each good, this function returns a scalar value, which reflects the utility of this basket to you. You pick the one with a greater utility.

So, if I feed it $U(10,0)$ or $U(0,5)$, where the good 1 is apples, and the good 2 is oranges, it'll return two different numbers, the greater one will indicate which basket has more utility to you. YOu can combine goods in the same basket $U(1,1)$ - one apple and one orange.

So, in a nutshell, the utility function is a concept that allows us to compare apples to oranges. The theory states that you pcik the baskets based on their utility, which is calculated by plugging the basket descriptions into this utility function.