Recent developments in prophet inequalities

The classic prophet inequality states that, when faced with a finite sequence of non-negative independent random variables, a gambler who knows their distribution and is allowed to stop the sequence at any time, can obtain, in expectation, at least half as much reward as a prophet who knows the values of each random variable and can choose the largest one. Following this classic theorem from the 70s, many results have been obtained for several related optimal stopping problems. Moreover, the recently uncovered connection between prophet inequalities and posted price mechanisms, has given the area a new surge. We survey some new developments and highlight some compelling open problems.