Inequity and Randomness
Let’s play a simple game - an experiment of the mind.
Here is the setup and the rules of the game - don’t worry, they’re not complicated at all:
- We’re going to start off the game by gathering 1000 people together.
- We’re going to give each person 100 small pebbles in a cool, stylish drawstring bag. Perhaps it will be monogrammed. Who knows…
- We’re going to set up a timer that will, annoyingly, beep once every second.
- The game will last for 8 hours.
- With every annoying beep of the timer, each person is to - completely randomly - choose another person, and give them a pebble.
That’s it. Nothing more.
We won’t worry about how we’re going to physically place all of these folks in a room so truly random choices can happen. We’re not going to worry about feeding them lunch or giving them breaks. They won’t get bored or tired. This is, after all, an experiment of the mind.
Here’s my question, and I want you to really think about it: at the end of eight hours, what will the distribution of pebbles look like?
Remember: Each person randomly chooses someone else (not themselves) to give a pebble. On every turn, every person in the game gives away one pebble, but - because the choices are truly random, every person has the exact same chance (as every other person) of being the recipient of one or more of the 999 other pebbles that are given away.
[Scroll down to see the results]
So, when I first saw this demonstrated, I was stunned. My gut instinct had been to say, “Well, if everyone is chosen randomly, then everyone should have the same chance of getting a pebble added to their bag. Pebbles should end up being fairly evenly distributed…” I’m pretty sure if you’ve never seen this demonstrated before, you likely thought the same thing.
I couldn’t believe the results (click on the picture to expand):
That’s not just an isolated run. Every time this game is played, that’s the resulting distribution of pebbles. Some people end up with a crazy number of pebbles. Some people end up with none. The overall number of pebble held by the top few people changes… sometimes the bottom end people have a few more or less pebbles, but that’s the distribution. Every. Damned. Time.
If you want to play around, here’s some Python code that I used to simulate this simple game:
#!/usr/bin/env python3 import matplotlib.pyplot as plt import random from datetime import datetime # choose a random seed random.seed(datetime.now()) people =  # we begin with 1000 people, each with 100 pebbles for i in range(1000): people.append(100) # every second, each individual picks a random person and gives them one pebble for i in range(3600 * 8): for p in range(len(people)): r = random.randint(0,len(people) - 1) # Thou shalt not choose thyself... while r == p: r = random.randint(0,len(people) - 1) # if they've got a pebble to give, give it if people[p] > 0: people[p] -= 1 people[r] += 1 # sort the results, lowest to highest people.sort() #plot the results fig, ax = plt.subplots() ax.plot(people) ax.set(xlabel='Individuals', ylabel='Pebbles') fig.tight_layout() plt.show()
[You’ll need matplotlib installed to make the pretty graph…]
Here’s the thing: this is about as simple a game as you can imagine. There’s really only one rule. Theoretically, it’s about as fair a game as can be made - all of the choices involved are truly fair and random.
It’s just that the end results are anything but fair…
I have a couple of take-aways from this game:
- We’re really, REALLY bad at understanding the outcome of even brain-dead simple rule systems.
- There’s something fundamental going on here. If this is the distribution every time, something interesting is driving it.
I couldn’t help but draw a parallel between the graph resulting from this simple game, and this graph (click on the picture to expand):
This graph shows U.S. Household Income by Percentile for 2020. The data comes from IPMS CPS and is 1999 data adjusted to 2020 dollar values using CPI data.
I’m not saying that there are any conclusions that can be drawn from the fact that those two graphs look remarkably similar - in fact, I’ll go so far as to say that such conclusions would likely be invalid in any number of ways. All of that aside, however, this does have me thinking about many, many things.
When a simple, stupid game involving the truly random movement of pebbles can mimic the broader inequities found in our society - you really need to pause and reflect on life.
What I will say is this. The spoken and unspoken rules that drive the data that makes up that second graph are far more complex, far more entangled, and far more difficult to understand than the single rule that drives the data resulting from our game - and most if not all of you were unable to predict the results of that single, simple rule.
Many people right now are so sure that they understand the rules that describe our social and economic world - they understand the implicit and explicit mechanisms that drive who fails and who succeeds, and yet I’m pretty positive they don’t - especially if they’re the ones with a bag full of pebbles.
I’m absolutely not claiming to understand anything much about the stated and unstated rules in this world. I’ll even categorically state that I don’t understand those rules. But I will say this: there’s a whole lot of randomness to who wins and who loses in life. Far more randomness than most people are comfortable accepting.
All of the inequity in the result of our game arises from a perfectly fair system; in the simulation, the people who win are randomly different every time. Imagine the results if you threw in even the smallest amount of systemic unfairness…
Try to remember that, and try to be kind.
Owner, Principal Consultant
Bad Wolf Security, LLC
April 19, 2021