Drawing a random number with chance of redrawing a second time. Best strategy that will never lose long term?
Q: We're going to play a game. You draw a random number uniformly between 0 and 1. If you like it, you can keep it. If you don't, you can have a do-over and re-draw, but then you have to keep that final result.
I do the same. You do not know whether I've re-drawn and I do not know whether you've re-drawn. Decisions are made independently. We compare our numbers and the highest one wins $1.
What strategy do you use?
I know the answer is 0.618..., tested this multiple times (0.5 will give the highest average value, but it won't guarantee you don't lose)
How do I come to this answer?
I was thinking I call the chosen threshold for A : "p", threshold for B: "q".
There is a (1-p)*(1-q) that they both take their first # : chance A wins = p-q+0.5
There is a (1-p)*q chance A takes his first #, B redraws a random # : chance A wins = 0.5p + 0.5
There is a p*(1-q) chance B takes his first #, A redraws a random # : chance A wins = 0.5-0.5q
There is a p*q chance they both redraw a #: chance A wins: 0.5
Total chance A wins:
(1-p)(1-q)(p-q+0.5) + (1-p)*q*(0.5p+0.5) + p*(1-q)*(0.5-0.5q)+0.5pq
I rewrote this and maximized this and after maximizing this, set p=q (equilibrium) and come out a different result. I ran tested above formula with some python programming and the formula is wrong. What is wrong with my reasoning here and how would this be done correctly? I've been trying to figure this out for a day. Thank you!
*individual probabilities for different possibilities seem to be correct based on testing in Python with 10 million runs each
- 416 views
- How to calculate how much profit a loan of a certain risk level (based on dice rolls) is likely to make?
- How do we define this choice function using mathematical notation?
- Stochastic Analysis question
- Find a number for 𝛼 so f(x) is a valid probability density function
- Probability and Statistics problem
- foundations in probability
- Questions for Statistics Project
- Central Limit Theorem question