I've been reading The Paradox of Choice, a book about more choices not always being better. There's also a one hour talk by the author. It made me think about how we might model choices, either for understanding our own behavior or for writing simulation games. The author of the book argues that at an abstract level people understand the benefit of additional choices but ignore their costs, whereas in practice people are affected by those costs, albeit not always in a rational way.

To model the benefit of choice, I'm going to say that if there are N-1 choices, and you are presented with 1 additional choice, your benefit has increased. By how much? It's only a benefit if the new choice was better. Since there are now N choices, let's say the probability the new one is better is 1/N. If it is better, by how much is it better? I think you can build an expectation function based on the distribution (for example, a gaussian distribution), but I'm going to be simplistic and say the benefit is constant. The new item is always 1 unit of value better than the old one. In practice I think the benefit decreases as the number of choices goes up, so I'm being generous here. So the added benefit of the Nth choice is the probability it's better multiplied by the amount it's better: 1/N * 1. To determine the total benefit, we have to sum from choice 1 to choice N, and we end up with something approximately equal to the logarithm: ln(N).

To model the cost of choice, I'm going to say that you have to make the comparison between the new item and the old items, even if the new item isn't better. You might compare to each of the old items, giving a cost of N, or maybe you only compare to the best of the previous items, giving a cost of 1. I'm going to be generous here and say the added cost is just 1. The total cost then is 1 for each new item, or a total of N.

So now we have a model in which both the benefits and costs go up as the number of choices increases. Each additional choice brings smaller and smaller benefits but larger and larger costs. Here's a plot of what this might look like:

Initially having choices greatly adds to your well-being. However, the rising costs eventually overtake the diminishing benefits, and the total value of having choices goes down. This seems to be the main message of the book: that additional choices do not always make us better off.

When I defined the model, I decided to be generous. The incremental benefit is 1 in my model, but it's probably decreasing as the number of choices goes up. This means the total benefit is lower than in my model. The incremental cost is 1 in my model, but it's probably increasing as the number of choices goes up, because people at some level will compare to all the alternatives, not just one. This means the total cost is higher than in my model. So the graph above is optimistic; in reality it probably drops even faster.

Note that the graph has no scale. That's because I think the costs and benefits will depend a great deal on the situation. When buying toothpaste, the benefit of more choices is pretty limited. But when choosing a job or spouse, it makes a much larger impact on your life. The main point is that additional choices will eventually not be worth the cost of evaluating them, so at some point you should just make your decision and not worry about it anymore.

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