# Mathematical Proof for Diversity

Last week, Lisa and I joined some friends for a lecture put on by the Santa Fe Institute. SFI is known for its quirkiness, and this was the first of their lectures I had attended, so I didn’t know quite what to expect. The speaker was Scott Page, who is at SFI but also at Michigan. He is a computational economist, using methods like agent-based modeling to study economics and how they impact societies.

Page spoke about diversity and how it plays a role in problem solving and prediction. However, he approached the topic from a mathematical perspective. He began with some anecdotes which “demonstrate” that diversity is good for problem solving, predicting, etc. He told of one story, in particular, from an old English fair in which people were trying to guess the weight of a steer. The average guess was withing 1 pound of the right answer! These kinds of observations are what has led to the formal study of diversity and the role it plays in groups.

The short answer is that diversity often helps. If all of the people in a group are “smart” relative to a problem (that is, they know something about the topic; they are not completely ignorant about it, like a non-mechanic trying to fix a car), then it is better to have a diverse group, including what Page called the “pinhead”, rather than a whole bunch of people who are all the very best, but are all similar. Invariably, in computer models, the diverse group always solves problems better than the “better” group. This is because they have more tools at their disposal, as the pinhead has some tools the genius does not have. By working together, they can solve a wider range of problems than if the group only had geniuses.

Related to this is the predictive quality, and there is actually a mathematical equation relating the error a crowd makes in predictions to the diversity of the crowd. If the diversity is greater, the average prediction of the crowd has a smaller error. This has been demonstrated by looking at expert predictions for, for example, sports (NFL, NBA) drafts, comparing each expert’s pick with the average guess. In almost all cases, each individual expert had more error in their predictions than did the average prediction.

I asked Page about cases where diversity hurts and he pointed out that irreversible processes, such as cooking, are cases were diversity hurts. If I throw chili peppers into the soup, it doesn’t matter what tools you have, you can’t undo what I did. If my peppers ruined the soup, it is ruined no matter how many people are helping. The military, he pointed out, is an interesting case: you want diversity in planning, to come up with the best plan, but you don’t in operations, as you want people to follow the plan already made up. They need to be more single-minded in operations.

Thus, the two adages: “Two heads are better than one” and “Too many cooks spoil the stew” are both right. It is only now, though, that math and science can begin to tell us under which conditions one or the other applies. This is fascinating stuff!

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