Learning from Games

Last week’s article introduced how sets and subsets could fit into a decision. With that background, in this post I want to share what fluency looks like when working with sets and subsets. I use this fluency daily to be efficient in my decision-making, and I hope that sharing my thought patterns can help you do the same.

The Game “Set”

The card game Set is a useful framework for how we think about defining a set or subset. In the game, there are 4 characteristics of each card: the color, the shape, the pattern, and the number of objects on the card. To construct a set, you have to identify three cards where, for each of those characteristics, either all three are the same or all three are different. For example, you could have all green, all diamonds, all solid fill, and then a card each with 1, 2, and 3 diamonds on them.

In my experience playing the game, the best trick was to choose any two cards and identify what the third card would have to be to create the set. Any two cards would tell you which characteristics needed to be the same, and which needed to be different, so it was easy to identify if it was out of not. If that third card is available, you’ve found a set. If it is not on the table, you should pick a different pair of cards to make a set with. The other strategy I use successfully starts by fixing a single characteristic (e.g., green cards) and seeing if any groupings appear within that collection. These two strategies can be generalized to principles which apply to sets in the wild as well.

The first principle I summarize as “if you have two elements in a set, you can compare them to see what is similar and what is different.” The next natural step is that things that are different can be identified as subsets of the original set. So, if we have a card with a single green diamond and solid fill and a second card which is a single red diamond with solid fill, we can compare them and say “the color is different.” We can split our set of cards by color!” Just like we learn in grade school, comparing and contrasting is a great way to recognize patterns. If all you see is those two cards, you may have no idea that the diamond shape is also a distinguishing feature, but some other pair of cards would teach you that fact.

The second strategy is useful because looking at everything can be overwhelming. So, my second principle says “looking at similar elements of a set can help you identify patterns that are hard to see in the overall chaos.” Four characteristics is a lot to keep track of at once. If you want to make the game simpler, for example to let a 4 year old play it, taking out one of the criteria makes it a lot easier to navigate.

In the game, either strategy allows you to find all “sets” according to the rules of the game, as long as you give it enough attempts. On the other hand, if you only look at one color at a time every time, you miss sets that have all different colors. This is perhaps a third principle that different patterns emerge depending on how you choose to break your sets down into subsets. If you look only at the green cards on the table, the differences between cards looks larger. But you miss the full variety of options available to you.

The Key Sets of a Decision Problem

In a decision problem (colloquially defined, where you are making a choice), there are always 3 key sets to consider:

• The set of viable decisions.

• A set that is sufficiently larger than the set of viable decisions that you can use the techniques above to identify that viable subset.

• The set of decisions you choose to make.

There may be many subsets other than these three as you consider different views on the problem. But these are also always the three core steps. In optimization, we identify the decision variables which is equivalent to the middle step. Then we layer on the constraints to get the set of viable decisions. Lastly, we define an objective function (goal) and identify the best decision from the point of view of that objective, of viable options. Incidentally, the fact that operations research approaches decision problems this way is part of what gives me confidence that my framework is correct.

You might have noticed I put the three sets out of order. Last post I spent a lot of my discussion on the idea of “ruling out options, but also not ruling out too many options.” This is why I think the set view of decision problems is so vital. If you had an oracle that could tell you “this is the best decision for you” and you trusted said oracle, you actually wouldn’t need to think about the rest of the sets.

But how do we know if we trust the oracle? What makes their answer right? For their answer to be right, we would have to agree that it is viable, and then conclude no other viable options are better. But practically speaking, the biggest mistake people make is missing viable options. If you forget to consider a great option, you’ll never recognize its greatness. It is rare that you can look at a single option handed to you by an oracle and be sure it is the best. It is through comparison to other decisions that we see how much better it is than the alternatives.

Once again, more to come later. And if you, dear reader, have found anything recent particularly interesting and want to learn more, please reply to this email / post and let me know what it is!