Collecting Coupons
Jane Street
You’re playing a game with one die. Whatever number you roll, you earn that many dollars (e.g., rolling a 6 earns you $6). After your first roll, you have the option to cancel it and roll a second time, with the second result being final. What’s the optimal strategy—when should you keep the first roll, and when should you reroll?
Answer
When you are asked for your strategy in playing a game, start by evaluating the expected payoffs of different types of strategies. We start by rolling the first die - this step is non-optional, so we get a number 1-6. Now we need a rule that says: depending on what your first roll was, either roll again or don't. When faced with a decision of whether to re-roll, we must compare the total expected payoff in the case we re-roll vs the case we keep our initial roll. Second die is rolled indpendently, so expected payoff of rolling again is $\frac{1}{6}\cdot(1+2+3+4+5+6)=3.5$ as all outcomes 1-6 are equally likely.
Now we know that rolling again will give us a payoff of 3.5 on average, and not rolling again leaves us with the payoff we got on the first roll. We should take which ever is bigger. In turn, our strategy is: if the first roll is smaller than 3.5 (1, 2 or 3), we roll again - as re-rolling gives us a higher result on average, whereas if the first roll is bigger than 3.5 (4, 5 or 6), we keep our first roll - as re-rolling will likely result in a smaller payoff