Expected value of game involving 100-sided die The following question is from a Jane Street interview. 
You are given a 100-sided die. After you roll once, you can choose to either get paid the dollar amount of that roll OR pay one dollar for one more roll. What is the expected value of the game? (There is no limit on the number of rolls.)
P.S. I think the question assumes that we are rational.

All answers are correct, but maybe I can phrase it in a less complex manner. Say you in advance determine a value V, where if you roll V or some amount greater than V, you will stop the game. What is your expectation here? Given that you have rolled V or higher, you are uniformly likely to have rolled any value between V and 100. The expectation of this roll is 0.5*(V+100).
But how much did you pay to get to this point? On average, when an event occurs with probability p, where p is bounded between 0 and 1, then in expectation it takes 1/p independent trials to realize this event. The probability of rolling V or higher is (101-V)/100, so on average it will take 100/(101-v) rolls to achieve this outcome. Your expectation is gain - loss, which equals 0.5*(V+100) - 100/(101-V).
It is simple to see that by taking the derivative with respect to V and setting equal to zero, we can get the V that maximizes expectation. Assuming your opponent is perfectly rational, this is the only bid they will accept to play the game. As it turns out, we can solve the equation to get V equals approximately 87.
