Applications/How it relates to EA
This article's main application would be to help you decide whether you should:
- Ask for help with a project you're working on.
- Ask someone else to do the project for you.
- Allocate tasks across a team of people, whether it be for a school project, a new startup, or anything else.
- Other things that I haven't thought of. (If you do think of anything, please mention it in the comments.)
This ties to EA since many people in EA collaborate with and distribute work across co-workers, and since a key part of EA is trying to maximize good done, and one of the ways to do this is to choose the best person to do a given task.
Disclaimer
Although I do have a lot of experience in math, this article has not yet been thoroughly peer-reviewed yet (if you think you should peer-review it, please do.), and is subject to change. Do not take the things I say in this article at face value unless the stakes aren't that high.
How Much More Valuable is Your Time, Compared to Others?
TL;DR
This article is about how to decide who does what. (e.g. Should you send an email to your co-worker, who works faster than you, but is much busier?)
To figure out who should do what, with the objective of increasing the amount of good done, we look at:
the expected amount of good done by person 1[1] if person 1 does the task (a),
the expected amount of good done by person 2 if person 1 does the task (b),
the expected amount of good done by person 1 if person 2 does the task (c), and
the expected amount of good done by person 2 if person 2 does the task (d). (we only look at the amount of time where person 1 and 2 are affected by who does said task.)
Since the amount of good done (in the allotted time slot) if person 1 does the task is equal to , and the amount of good done (in the allotted time slot) if person 2 does the task is ,
person 1 does the task if:
person 2 does the task if:
It doesn't matter who does the task if:
Example and Details (Feel Free to Skip This Section)
Prerequisites
This article assumes that you know that =[2]
If you don't know this, a simple proof is as follows:
for some random variable X, E(X) is what you'd expect X to be. [2]
Now, for some constant C, we just need to show that .
This can be shown by imagining that we randomly generate some value for X, which we'll call . Now, let's multiply by c. We'll call this new variable . Then, we'll generate another value for X, which we'll call . Now, let's multiply by c. We'll call this new variable . We repeat this process a bunch of times.
We can then take the arithmetic mean of all the 's, resulting in , as trends towards infinity. By definition, this is .
Since we can rewrite each as , we can rewrite as, and, since , and , .
and, since, if and , then ,
And, since, by definition, , and .
And, since, if and , then , then:
.
Example/premise
For the sake of this example, let's say you and your friend are running an ice cream stand. (Bare with me!)
Details:
- It takes you:
🍫1 seconds to make chocolate ice creamyou can't make chocolate ice cream. (Which I'll be referring to as 🍫 from now on) .- seconds to make vanilla ice cream (Which I'll be referring to as 🍦 from now on),
- and seconds to make strawberry ice cream (Which I'll be referring to as 🍓 from now on).
- It takes your friend:
- seconds to make 🍫,
🍦2 seconds to make 🍦your friend can't make 🍦.- and seconds to make 🍓.
- The price of each flavor X can be expressed as 💵(X). (e.g., the price of 🍫 is 💵(🍫).)
- When there's a choice between serving someone who wants 🍓 and serving someone who wants 🍫OR[3]🍦, for whatever reason[4], they must choose 🍓.
- All you and your friend care about is money.
Currently, you're handling a large line of people who want 🍦, and your friend is handling a large line of people who want 🍫.
A group of people come in, saying they want 🍓.
The question is: should you serve the 🍓-ordering customers, or should your friend?
(From now on, I will write "serving the 🍓to the people who ordered 🍓" as "🟥".
Your friend quickly puts together a table, showing the pros and cons of each option.
| You serve 🍓(🟥) | Your freind serves 🍓[5](🟥) |
|---|---|
| people get served 🍓in seconds, with a gain of [6][7]. | people get served 🍓in seconds, with a gain of . |
| [2][8] customers who ordered 🍦 don't get served[9], leading to a loss of . | customers who ordered 🍫 don't get served, leading to a loss of . |
Since you might make🍦more efficiently or less efficiently if you serve 🍓 or if your friend serves 🍓, I point out that you serve 🍦at an updated rate, . Therefore[9], you can serve[10][11], as supposed to , if your friend serves 🍓. We can now express these two values in terms of 💵 by the following formulas: , and . Then, since , we can simplify these to and . We can then write the difference as .
| Similarly, for 🍫, we get: |
| Total (Using BIDMAS[12]) | |
| . | |
Now, we'll simplify both of these formulas. Feel free to skip this part.
Simplifying
.
Final formula
You should 🟥 if
[13]>
Your friend should 🟥 if