Gotta' hate qualitative questions amirite?
Apologies for my terribly written and possibly (probably?) incorrect explanations in advanced.
I'd say, first note that both the prices would be the same when the utility function is linear (this can easily be seen, just make the utility function u=c or even u=a+bc where a and b are whatever), so it is something about the concave, decreasing slope, of the utility function that is causing this difference (in economic terms, the diminishing expected marginal utility), I guess a more mathematical way to see this is
this(part 1) and
this(part2) and
this(part3) (part 3 is a little vague, it will become clearer after reading the endnote probably) (might want to zoom in).
More lucidly I guess, consider comparing these two terms:
U1 : (1/2)Root(Money in possession if winning lottery) + (1/2)Root(Money in possession if losing lottery)
U2 : Root(Money in possession when not participating in lottery)
(ofc U1 is so because the buyer is assumed to care about his expected utility, this is the significance about that line regarding Von-Neumann Morgenstern)
When considering prices, we're trying to determine a value of price (say p) so that both options participating in the lottery (utility of which is U1) and not participating in the lottery (utility of which is U2) are feasible.
For this to happen we have to equate U1 and U2.
In the following explanation
this graph can be referred to for more clarity.
In the first case (selling the ticket)
U1 = (1/2)Root(25)+(1/2)Root(9)
U2 = Root(9+p)
We're increasing U2 by increasing p (blue line), or, increasing the compensation received for selling the ticket, to get to U1 (red line), because it is concave, each incremental p (or compensation) will give less and less utility.
In the second case (buying a ticket when you don't have one)
U1 = (1/2)Root(25-p)+(1/2)Root(9-p)
U2 = Root(9)
We're decreasing U1 by increasing p (purple line), or, decreasing the expected utility of taking part in the lottery by increase the price required to take part in the lottery, to get to U2 (green line)
Each increase in the price, actually decreases the expected utility more more (think of the general shape of a function of the form y=root(x), think of going "down" this slope, which is what we are doing each time we increase p).
As a whole then, in the second case, a lesser price is needed to convince the buyer that he is as good buying the ticket as he is not. This is because the consecutive increases in price, when being DEDUCTED from the compensation he/she gets and DECREASING expected utility have larger and larger impact on expected utility than do consecutive increases in price when being added to the compensation received and increasing expected utility (which have smaller and smaller impacts).
(P.S. note that this is only a partial explanation, actually, although in the second case the deductions are increasing with each p (I speak of the rate of change of the rate of change), the FIRST deduction in case 2, is smaller in magnitude than the FIRST addition in case 1 (see part3's image), however, because of decreasing marginal utility when making additions to compensation and increasing loss of marginal utility when making deductions to compensation, it doesn't take long for the deductions to "overtake" the additions and reach the end point faster) (the point I'm trying to make will be more clear if the same question is asked again but this time, instead of taking the lottery winning prize as 16 dollars, take it as 1 dollar or better yet, 0.5 dollars, in that case the max/min selling price would actually be less than the buying price)
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