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A team of people are organizing who should do what.
A team of people organizing who should do which tasks. Image generated by DALL-E, and edited by me.

Applications/How it relates to EA

This article's main application would be to help you decide whether you should:

  1. Ask for help with a project you're working on.
  2. Ask someone else to do the project for you.
  3. Allocate tasks across a team of people, whether it be for a school project, a new startup, or anything else.
  4. 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:

  1. It takes you:
    1. 🍫1 seconds to make chocolate ice cream you can't make chocolate ice cream. (Which I'll be referring to as 🍫 from now on) .
    2.  seconds to make vanilla ice cream (Which I'll be referring to as 🍦 from now on), 
    3. and  seconds to make strawberry ice cream (Which I'll be referring to as 🍓 from now on).
  2. It takes your friend:
    1.   seconds to make 🍫,
    2. 🍦2 seconds to make  🍦 your friend can't make 🍦.
    3. and  seconds to make 🍓.
  3. The price of each flavor X can be expressed as 💵(X). (e.g., the price of 🍫 is  💵(🍫).)
  4. When there's a choice between serving someone who wants 🍓 and serving someone who wants  🍫OR[3]🍦, for whatever reason[4], they must choose 🍓.
  5. 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