I get far too many emails from potential graduate students who want to apply to Berkeley. I do not have the time to respond to everyone individually and so will put the responses to the most common queries here:
Whenever Carl and Dan go out to eat each Saturday, they take strict turns paying the tab. If Carl payed last time, Dan will pay this time and vice versa. But they don't bother about consciously keeping track of money and will order a big dish randomly from the menu and share it equally. Big dish prices are uniformly distributed on [5,10]
Abbey and Beth take a different approach to fairness. They too go out to eat and take strict turns paying the bill. But they order small dishes separately. Whenever one of them calculates that she is currently more than $10 behind in her fair share of payments, she will order the salad for $1. Otherwise she randomly picks a dish from the small dish menu where the prices are uniformly distributed on [2.5,5].
Carl and Dan believe that Abbey and Beth are being overly cautious and advise them to trust in the power of averaging to make things work out in the end. Abbey and Beth are skeptical and do not want to risk their friendship.
Assume that if at any time the balance of payments becomes skewed by more than M times the maximum price of a dish in one direction, then unconscious feelings of being cheated will manifest and the friendship will grow distant.