Unfair Fairness

How equitable rules can lead to inequitable results

I’ve always been intrigued by the potential game mechanism that I’m about to describe, but I’ve never seen it in a game.

It’s a type of bag-building. Here’s how it works: There are three different color tokens – red, blue, and yellow. At the start of the game, you put two of each token into a bag – so six total, two of each color.

Image generated by Dall-E 2. I couldn’t get it to put in the right number and color of cubes, so use your imagination.

On each turn you randomly draw a token. You then get to do whatever the color lets you do, and then you return that token plus an additional token of the same color. So if on the first turn I draw a blue token I get to do a blue action, and then I put two blue tokens back in the bag. Now there are 7 tokens in the bag – 2 red, 2 green, and now 3 blue.

Rinse and repeat.

The really interesting thing about this mechanism, in my opinion, is that in the early rounds the composition of the bag, in terms of the percentages of each color of token, can vary greatly. However, once you get up to around 20-30 chips in the bag it’s very hard to change.

Let’s say you’ve got 30 chips in the bag, and 50% are blue, 30% are red and 20% are yellow. The chances that you’ll draw a chip of a particular color are exactly those percentages – 50% chance you draw a blue chip. And even if you draw one of the rare yellow chips, adding another yellow chip won’t appreciably change the percentages.

So the color mix starts out exactly equal. It can vary a lot early on, but later on settles down into a stable pattern.

The neat thing is that you can’t predict at the start which color will end up having the most chips. At the start there is an equal number of each in the bag – but it is an unstable equilibrium. Anything that adds more chips of one of the colors will make it more likely that chip is chosen in the future and create a self-perpetuating mix. One game you may get lots of blue chips, another red may be dominant, and then in a third the chip distribution may stay roughly equal.

This model is typically used in economics to explain the perpetuation of market share. If I have 60% of the market in chocolate chip cookies, that will tend to remain the same, all other things being equal. I will get more shelf space and more market presence, and that will lead to more people buying my cookies. Market share in all goods can shift, but for some it takes a lot more effort or a big shift in consumer sentiment.

This is an example of a system where the rule is fair and doesn’t favor any one color – pull a chip and place another one of the same color in the bag – however it ultimately leads to unbalanced results which can be very hard to change. If the blue chips have become dominant in the bag, simply applying the same rule, even though it’s color-agnostic, will not change the composition of the bag. A different rule is required if we want to get close to color balance in the bag.

In addition to appreciating the applications of this mechanism to potential game designs, I also see this as a metaphor for diversity in a variety of spheres of human activity, including the gaming world. I firmly believe that a more diverse gaming community will create better experiences for all of us, as we get to enjoy different perspectives and takes on the fundamental human activity of ‘play’.

If you’re choosing an artist for your project, or a panelist for a convention session, the rule you may choose to apply is to simply ask the most prominent people, or the people who have done work on existing games that you enjoy or are popular. And that is 100% a fair and equitable rule. You are not applying gender or racial bias when you apply that rule. However, like in our bag example, fair rules applied fairly can lead to unfair results. And that’s even if everyone is operating in good faith, which isn’t the case in the real world.

If we want to change the color mix in our bag we will need to take some steps that look unfair, but in the long run will bring things back into balance and create a more diverse community.

Of course, the situation we find ourselves in now in the United States was caused by unfair rules, not fair ones. From the original sin of African slaves to the treatment of Chinese immigrants in the 1800’s and the Chinese Exclusion Act, to Japanese Internment camps during WW2 and Jim Crow laws, to the fact that women in the US couldn’t have a credit card until the mid 70’s - it’s been anything but fair. I understand that my ‘chips in a bag’ analogy is simplistic and a poor reflection of reality.

However, I do think that what it shows is that even if we all magically become 100% unbiased in how we relate to people, that we can’t necessarily just treat everyone fairly and have inequity disappear. We need to take active steps to change the status quo.