Best Strategy for a die game You are allowed to roll a die up to six times. Anytime you stop, you get the dollar amount of the face value of your last roll.
Question: What is the best strategy?
According to my calculation, for the strategy 6,5,5,4,4, the expected value is $142/27\approx 5.26$, which I consider quite high. So this might be the best strategy.
Here,  6,5,5,4,4 means in the first roll you stop only when you get a 6; if you did not get a 6 in the first roll, then in the second roll you stop only when you roll a number 5 or higher (i.e. 5 or 6), etc.

Just work backwards. At each stage, you accept a roll that is >= the expected gain from the later stages:
Expected gain from 6th roll: 7/2
Therefore strategy for 5th roll is: accept if >= 4
Expected gain from 5th roll: (6 + 5 + 4)/6 + (7/2)(3/6) = 17/4
Therefore strategy for 4th roll is: accept if >= 5
Expected gain from 4th roll: (6 + 5)/6 + (17/4)(4/6) = 14/3
Therefore strategy for 3rd roll is: accept if >= 5
Expected gain from 3rd roll: (6 + 5)/6 + (14/3)(4/6) = 89/18
Therefore strategy for 2nd roll is: accept if >= 5
Expected gain from 2nd roll: (6 + 5)/6 + (89/18)(4/6) = 277/54
Therefore strategy for 1st roll is: accept only if 6
Expected gain from 1st roll: 6/6 + (277/54)(5/6) = 1709/324  
So your strategy is 6,5,5,5,4 for an expectation of $5.27469...
