Hey everyone, I’ve been solving a probability game I saw on Tiktok and managed to solve the base case, but generalizing it seems impossible
The Base Game Imagine a game with 4 numbers drawn from a continuous uniform distribution U(0,100).
- I get to lock in exactly one number of my choice.
- The other three are generated completely at random.
- Once all four numbers are on the table, they are sorted. The Highest and Lowest numbers pair up into a team. The two Middle numbers pair up into a team.
The goal is to pick a number that maximizes the expected difference between my team's sum and the opposing team's sum.
Solution:
Picking the extremes (0 or 100) is a trap that yields an expected advantage of 0 (picking 100 -> other numbers will be around 25/50/75 so expected sum is the same). You can write the expected advantage function as E(x)=−2x3+3x2−x (where x is a percentage from 0 to 1). -> peak at x=21+63≈0.7886. So, picking roughly 79 is the mathematically optimal play.
Now, here is where I need help, how do I scale this up? What if instead of 4 numbers, there are 2n numbers?
- I pick exactly 1 number.
- 2n−1 numbers are generated randomly.
- They are sorted. Rank 1 pairs with Rank 2n, Rank 2 pairs with Rank 2n−1, and so on. ("rainbow" pairing).
- The Question: I want to maximize my team's sum minus the average sum of all the other teams. As n→∞, what percentile x should I choose? Does the optimal choice approach exactly 1.0, stay at ≈0.7886, or converge to a different limit entirely? (I made formulas for n=5 and n=6 but it doesn't seem to converge anywhere)
