It appears that a somewhat common interview question for quantitative jobs involves asking the interviewee to submit a maximum price he would be willing to pay to play a game of chance. This question is posed as an auction (sometimes sealed-bid) against other unknown players such that you will minimise your expected payoff.
These questions usually describe a game where the expected value and standard deviation are easy to determine, rather the intent is for the recipient to determine a satisfactory expected payoff. For the purposes of this question, ignore the recipient's value of time.
Here is a simple example of such a question:
A fair coin is tossed four times. If the coin comes up heads all four times the player wins $16, otherwise the game is over. The game is played only once. What is the maximum price you would pay to play this game?
In this case, the expected value of the game seems reasonably clear:
$$E[X] = \frac{1}{2^4} * \$16 = \$1.$$
But the game is played only once, and to pay $1 would be to accept zero expected payoff. So how might one approach determining their expected payoff? As I understand it, this is a question of risk appetites and might not have one clear approach. By risk I refer to the standard deviation of returns.
Note: While I see there have been many posts about these sorts of questions, it seems they often focus on calculating expected value and variance, or the post inadequately parameterises the question. I am hoping there might be a nice game-theory approach to answering these questions, where the focus is not on expected value but risk of playing the game.