(8 marks) Dynamic Programming. Consider the following game. You are given a fair five-sided die, with faces {0,1,2,3,4}. You can roll the die, and obtain a payoff equal to the value shown on the die, or you can choose to roll again. Note that if you choose to roll again, you can only get the new payoff or roll again. You can roll the die up to three times. You must pay $1.4 for the first roll, $1.05 for a second roll, and $0.85. for a third roll. You pay only for the rolls you choose to take, and get no money back if you move onto the next roll. What is your expected gain, assuming you were to behave optimally at each step? (Hint: this is somewhat similar to the marriage problem we looked at in lecture. Use dynamic programming; i.e., work backwards from the last roll!)