## RE: Sum-of-Squares Utilitarianism

Posted by Michael Dickens on April 28, 2010

This is a response to a comment on my post about Median Utilitarianism. I am posting this as a blog post rather than a comment because (a) it’s rather long, (b) it is interesting, and (c) I haven’t been posting enough lately.

The entire comment is reproduced below:

I’m working on a separate problem, but it ties so closely into this post of yours that it immediately reminded me of it, and I wanted to run the idea by you.

Okay, the basic idea is, intuitively (at least, to me), we want to maximize total utility in the system, but we also want maximize the balance of utility in the system. I think this may be what you were getting at with the “median utility” idea.

Well, I think there’s an easy mathematical way of looking at it. I may be off on this. The idea just popped into mind, so I may be overlooking a huge flaw. It may be totally stupid. But, here goes.

Say your system includes only two people, A and B. Higher utility ratings are better. Say that A has 100 utility and B has 0 utility. The sum utility there is 100. Same as if A and 0 utility and B has 100 utility. Well, what about when A has 50 utility and B has 50 utility? That’s still a sum utility of 100. In other words, total utility alone does not differentiate between one person getting completely shafted and both people having an equal share. My hunch is, our genetic wiring tells most of us that the situation in which A has 50 and B has 50 is preferable to the other two situations. (At least, that’s how I feel.) However, maximizing for sum of utility isn’t a total loss–it at least ensures that the system has the highest possible utility, even as it doesn’t ensure an even distribution of utility.

Say that we instead take the sum of squares of utilities. In this case, we actually want the lowest sum. So, if A has 100 and B has 0, that’s a sum-of-squares of 10,000. Likewise if A has 0 and B has 100. If A has 50 and B has 50, that’s a sum-of-squares of 5,000. We want the lowest sum of squares, so we prefer the perfect balance over one of the two people getting perfectly shafted. However, since we are aiming for the lowest sum, the obvious way of optimizing the system is to remove all utility. Clearly, we don’t want that.

How about combining the two? We seek to maximize the sum of utility (so we have the most utility in our system) while at the same time minimizing the sum-of-squares of utility (so we have the best balance of utility in our system).

(What’s the gain? Well, mean utility is just scaled sum of utility, and thus, has the same problems as sum of utility. Median utility is a partial solution, but still allows for the “it’s okay for the majority to totally shaft the minority” problem, since it’ll be a member of the majority that determines the median. Maybe there’s still a way of gaming the system I proposed, but I think it would be a lot harder.)

I’m not sure how to turn the system I proposed into something useful in day-to-day decision-making. It’s just something that flashed into mind while working on something else. However, I thought you might find it interesting.

(Geeky aside: Minimizing the sum-of-squares of utility could be thought of as “minimizing the ‘error’ in the distribution of utility.”)

I think that that’s a clever idea. My idea about Median Utilitarianism was more of a conceptual idea than a position that I actually support; I am a proponent of Sum Utilitarianism.

The traditional idea of 50 + 50 being better than 0 + 100 is incorrect, but is based on an interesting relationship. If there is some thing X which makes you happy, then having twice as much X does not make you twice as happy. So if we are talking about how much of X we should have, then utility is increased if we do 50 + 50 instead of 0 + 100. I think this is where the intuitive idea comes from. But if we are talking about actual utility, then 50 + 50 is exactly the same as 0 + 100.

If we use something like the Veil of Ignorance (which places a veil over people so that they know nothing about their own selves, allowing them to make objective decisions), then everyone should agree that 50 + 50 is equivalent to 0 + 100 because your expected return is the same.

Another thing to take into account is the possibility of propagating utility. If one person has very low utility and the other has very high utility, it is possible that the person with low utility will find it more difficult to increase his utility. At the same time, the person with high utility will find it difficult to increase her utility, because it is already relatively high. To put this to a real-life example: the extremely poor have extreme difficulty in lifting themselves out of poverty, because they can’t afford an education or something like that; also, the extremely rich (although wealth does not necessarily imply happiness, let’s admit it, we all want to be rich) cannot increase their utility much further — even if wealth directly led to happiness, would someone like Bill Gates be all that much happier than your average millionaire? Both of them can afford virtually anything they could want, so the difference is not all that significant.

But along the same vein, people with a moderate level of utility are the most able to increase their happiness. Economically, the middle class are the most able to increase their own utility.

This is not to imply that wealth equals happiness; rather, I am using it to illustrate a point. Actual happiness is not nearly as concrete as wealth.

Another interesting possibility (that I do not endorse) is the idea of Product Utilitarianism. Instead of taking the sum of everyone’s utility, you take the product. 50 * 50 is a lot more than 1 * 100 (and definitely more than 0 * 100), so this would encourage everyone having a high utility while at the same time keeping utility rather balanced. The first problem I thought of is that the addition of one person with even a relatively low utility will increase the overall utility substantially, which means that overpopulation will continue for far longer than it ought to.

The second problem is that the product, unlike the sum, is heavily dependent on what units you’re using. If “utility points” are always between 0 and 1, then Product Utilitarianism states that there should only be one person alive at any given time. If “utility points” are always negative, then to maximize utility there always needs to be an even number of people. What units you’re using makes a big difference.

There are a lot of different methods with which to measure utility. It’s a lot of fun to speculate about different methods; however, I don’t think that that’s a good way to actually find a good basis for utility calculations. Not to say that we shouldn’t do it, but I just don’t think it’s the best way to decide what to do — how do we know which type of calculation is the right one? The best way to do it, I think, would be to start with some moral foundation and try to work from there. Actually finding this foundation is a whole different story.

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