the intuitive difference between expected utility and utility of expected profit in a gambling game What is the intuitive difference between expected utility and utility of expected profit in a gambling game ? Which one is the "usefulness of the game" to a player ? 
 A: Example: 
There is a game with two possible results $P_1=100, P_2=0$. The probability for each result is $p=1-p=0.5$ The Player have the utility function $U(P_i)=\sqrt{P_i}$
$\text{Expected Utility}= E(U(P)) =0.5 \cdot \sqrt{ 100} +0.5\cdot \sqrt 0=5$
$\text{Utility of the Expected Value}=U(E(P))=\sqrt{0.5 \cdot 100+0.5 \cdot 0}=\sqrt{50}\approx7.071$
A: If the utility function has a uniform slope, that is, $U(x) = ax + b,$
then of course the expected utility is the same as the utility of the expected value of the game, because in general
$E(aX + b) = aE(X) + b$ for any real numbers $a$ and $b.$
This is called linearity of expectation, and it guarantees that
$$
E(U(X)) = E(aX + b) = aE(X) + b = U(E(X)).
$$
But if the utility function does not have this kind of linear nature,
then linearity of expectation does not apply, and we should be prepared for the possibility that $E(U(X)) \neq U(E(X)).$
In general, in mathematics, if we cannot prove whether two things are equal, then we can still consider it possible that they are equal, but we should also consider it possible that they are different.

Another way to get an intuitive idea about the difference between expected utility and utility of expected value is to first get some intuition about why "utility" matters in the first place.
The idea of utility is often introduced in an attempt to explain why someone might rationally decide not to play a game that has a positive expected return,
when the expected return is based on an almost certain chance of losing a large amount of money and a tiny chance of gaining an astronomically large amount of money.
One argument is that if you have a very great amount of money,
having ten times as much money will not make you ten times as happy.
But whatever the reason for distinguishing utility from profit is,
the non-linearity of the utility function means that the amount of utility you gain when you gain a dollar depends on how many dollars you had before the gain. Hence your $n$th dollar may be more (or less) valuable to you than your $n+1$st dollar when we measure utility.
This also implies that starting with a particular number of dollars,
losing a dollar may cost you more (or less) utility than the utility you would gain from gaining a dollar.
A tricky thing about utility is that if utility matters in reality, then in reality the gain or loss of utility from the game depends not just on how much you win or lose but also on how wealthy you were before the game started. In theoretical exercises, however, I think it is typical to
"recalibrate the utility scale" so that the utility of your wealth at the start of the game is zero.
Otherwise it does not really make sense to talk about the "utility" of the expected value of the game.
Now let's "re-center" the outcomes of a game so that the possible gains or losses are all relative to the expected value.
That is, if the outcome of the original game is a random value $X,$
we treat the game as the sum of two transactions, one in which you are certain to receive a payment equal to $E(X),$
and one in which you get a random payment of $X - E(X).$
A payment equal to a negative number means you are the one paying money to someone else.
The profits and losses of the second part of the modified game are perfectly balanced: the expected value of $X - E(X)$ is zero.
But now let's look at the gain or loss of utility.
If the first part of the game were the end of the story
(that is, if there were no random component), the utility of the game would just be the utility of the expected value, $U(E(X)).$
But when we get to the second part of the modified game, we have to account for the idea that a dollar lost may be worth more (or less) in terms of utility than a dollar gained.
The effect of the non-linearity of utility is that whichever direction has more utility per dollar--losing or gaining--the outcomes in that direction during the second part of the modified game will have their weight magnified relative to outcomes in the other direction.
This can unbalance outcomes that were perfectly balanced when we were just looking at the gain or loss of money.
The expected utility, $E(U(X)),$ should be the perfect balance point of all the possible utility outcomes. If the utility function has non-linearity that unbalances the utilities of the outcomes around $U(E(X)),$ however, then $U(E(X))$ is not the perfect balance point of the utility outcomes, and therefore $E(U(X))$ must be some other utility value.

Regarding the "usefulness of the game to the player,"
in a typical gambling game you do not automatically receive the expected value of the game and then the game is over.
Instead, you have multiple possible outcomes, each with its own utility.
Given the way that a utility function can "unbalance" the outcomes around
$U(E(X)),$ does it make sense to say that the utility of the game is the same as the utility of receiving a guaranteed payment of exactly $E(X)$?
Or does it make more sense to combine the utilities of the outcomes,
weighted by their probabilities, that is, take $E(U(X))$?
