Outreach module (grades 9-12)
Part 7: Random strategy lab
Problem Setup
Two envelopes contain X dollars and 2X dollars, where X is a positive integer. The envelopes are shuffled, and one is handed to you at random. You open it and observe A dollars. Before payment, you may decide to keep your envelope or switch to the other one. You are paid whatever is in your final envelope. In this module, we call it a win if your final envelope has the larger amount. Should you switch if your goal is to end with the larger amount?
In this lab, test a strategy that adds randomness to the decision. Compare it with always switch and always stay, and use the data to form a hypothesis.
Idea to test
The rule in this lab is simple: draw a random threshold \(T\). After you observe \(A\), switch if \(A < T\), otherwise keep your envelope.
Start in a mystery setup and compare this rule with always switch and always stay over many rounds. Then reveal the setup, or choose one yourself, and compare again.
When you have a hypothesis, go on to Part 8 for a mathematical walkthrough of the strategy.
In this lab, \(X\) is always a power of \(2\). This avoids giving away extra information from the parity of the number you observe.
In Mystery setup, the exact distribution is hidden. In Custom setup, you choose the highest power and the tilt. If the tilt is \(1\), the distribution is uniform. Larger tilt makes larger values more likely, and smaller tilt makes them less likely.
\[ X\in\{2^0,2^1,\dots,2^K\}, \qquad \mathbb{P}(X=2^i)\propto r^i. \]
So the highest power controls which powers of \(2\) are possible, and the tilt controls whether larger values are more likely or less likely.
Start with a mystery setup and compare the three strategies over many rounds. Then use single-round mode to inspect individual outcomes.
After that, reveal the hidden setup or choose your own. The goal is to decide whether adding randomness changes what is possible.
Win rate means the fraction of rounds where your final envelope had the larger amount.
| Setup stage | Strategy | Rounds | Win rate |
|---|---|---|---|
| Mystery | Random threshold | 0 | -- |
| Mystery | Always switch | 0 | -- |
| Mystery | Always stay | 0 | -- |
| Revealed | Random threshold | 0 | -- |
| Revealed | Always switch | 0 | -- |
| Revealed | Always stay | 0 | -- |
Observed amount (\(A\)): \(A = \text{--}\)
Open an envelope to see this round's summary.
Setup
Current setup
Mystery setup is active. Compare strategies first, then reveal the distribution.
Change the hidden setup or reveal its highest power and tilt.
Highest power means \(X\) can be \(1, 2, 4, \dots, 2^{\text{highest power}}\). Tilt \(=1\) is uniform.
Play
Draw \(T\), then switch when \(A<T\) and stay when \(A\ge T\).
Uniform mode: \(T\sim\mathrm{Unif}[T_{\min},T_{\max}]\).