# Improving Judgement pill problem [closed]

Lets say there are 10 different pills that improve or impair judgement each assigned a rating (1-10) of increasing or impairing your judgement (1 stupid, 5 normal, 10 super smart)

You currently know of one pill that results in 8/10 judgement

Now each day you are one of these pills you earn a certian amount of money:

1/10: $$6 2/10:$$9
3/10: $$15 4/10:$$26
5/10: $$45 6/10:$$78
7/10: $$136 8/10:$$416
9/10: $$728 10/10:$$1274


You can only try one pill a day

What is the most profitable way to go about earning the most money in one week, one month, and a year?

Would this scale if there was 1000 pills assigned a rating of (1 stupid - 1000 really smart) pills each increasing the amount of money to make on the pill by 1.1 times for each (starting at 6$for the dumbest pill) You go about finding the other pills by taking one a day and marking down how much it improves your judgement (assuming you know how to do this accurately) Do you experiment 2 days out of the week and then for the other 5 you take the 8/10 pill until you find the 10/10 pill and then keep taking the 10/10 pill for the rest of the month when you find it? or do you take little risk and take the 8/10 pill every day? To summarize a few points : 1. You gain money depending on the last pill you took 2. After taking a pill, you know it's rating 3. You can only take one pill per day (or none?) 4. You want to maximize you gain after one (week/month/year) 5. You only know the 8/10 pill. • @bobble in this scenario there is no real downside of the higher pills (they result in higher profits). You go about finding the other pills by taking one a day and marking down how much it improves your judgement (assuming you know how to do this accurately). And in this hypothetical scenario yes. Commented Nov 2, 2023 at 5:21 • @bobble you do not have an infinite supply of each. In this scenario you start of with the 10 pills of which you know ONE of the pills results in 8/10 judgement (which can earn you$416 a day) Commented Nov 2, 2023 at 5:24
• Let me summarize a few points : 1. You have only one of each pill 2. You gain money depending on the last pill you took 3. After taking a pill, you know it's rating 4. You can only take one pill per day (or none?) 4. You want to maximize you gain after one (week/month/year) 5. You only know the 8/10 pill. Did i miss anything ? Commented Nov 2, 2023 at 8:18
• For context, this general problem of trying to maximize rewards by selecting options with unknown payout is call a multi-armed bandit problem - so named for the task of picking which slot machine to play without knowing their payout rates beforehand. The optimal solution is usually a mix of exploration and playing the best option you've found so far, but it depends on the time horizon (exploration is worth more if you have many trials remaining). Commented Nov 2, 2023 at 14:59
• I believe that the idea of this puzzle is good, but it's just not clear what the parameters are. Also, as answers are being presented and comments are being asked, the puzzle is changing. That's a good indicator that the puzzle should stop and be rewritten clearly. Commented Nov 2, 2023 at 15:59

For a week:

4 Days the 9/10 pill or 3 days the 10/10 pill is better then 7 days with the 8/10 pill.
How about the risk to not get the 9/10 or 10/10 pill?
P(get 9 or 10th pill in 4 days) = 1 - P(Get only 7th or lower pills) = 1 - (7/9 * 6/8 * 5/7 * 4/6) > 0.72
You should try to get the better pill in a week. The first 4 days take a unknown pill. If this pill is better then the 8/10 pill, stay with this pill.
If you couldn't find a better pill, try for the 10/10 pill on day 5. If you are unlucky (28%) keep trying it, but you lost money.

For a longer period:

After 3 days with the 10/10 pill and 8 days with all the other not 8/10 pills you earned more money than with the 8/10 pill every day for 11 days. (4895 versus 4576)
So take a pill, if it is the 10/10 pill stick with it, if it is not, take a unknown pill. Keep going until you find the 10/10 pill.

What is the most profitable way to go about earning the most money in one week, one month, and a year?

Eat all the pills on day 1. You'll gain $$6+9+\cdots+1274=2733$$ dollars right away.

• nice but I updated the question.. so that you can only try one pill a day Commented Nov 2, 2023 at 13:12