Utility Curves

This is the first post in the "Mathematics of Utilitarianism" series.

Note to the reader: Here begins a series of posts on the mathematics of utilitarianism that are/will be scattered throughout this site. The posts are written in the style of a textbook; perhaps it eventually will become one. I'm writing all this because I don't know of any existing work on the subject. If I find something, this effort might come to an abrupt halt. If you know of any existing work, please tell me about it.
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The idea of felicific calculus came first from Bentham. He based it on seven properties: intensity, duration, certainty, propinquity, fecundity, purity, and extent. I prefer to simplify this by only considering utility, and then treat it more like a real calculus. I represent utility by the character :). With that, we can, for example, show a "utility curve" of someone's life as a graph of utility ( :) ) vs time (t):

Figure 1 shows someone who's utility increases steadily after birth, levels off during midlife, then steadily decreases until death. Utility is assumed to be equal to zero before birth and after death, which is a reasonable approximation for our purposes here but could be explored in greater detail. Indeed, exactly when human life begins remains somewhat of an open question, with answers usually being somewhere between conception and birth. When utility becomes nonzero is a subtle physiological question I am not at present qualified to answer. I am also assuming that there exists no relevant pre-life or after-life. If such phenomena turn out to exist, they could be easily built into the model.

An interesting question is the existence of negative utility. This would be a cognitive state so unpleasant, the individual would prefer being dead. Crude intuition suggests that such states do indeed exist. A simple example is of people who want to commit suicide: they believe that them being dead is preferable to them being alive. Yet people often try to talk them out of committing suicide. Are they correct in doing so?

Utility curves and basic calculus provide a straightforward framework for answering this question. For simplicity's sake, let's assume that the only individual who's utility matters is the one considering suicide so that we only need to look at one person's utility curve. Factoring in other individuals' utility isn't difficult, but we'll save that for later.

Let's look at a case in which the person would be better off not committing suicide:

The person is born and then has a good life at first (a). Then, things get bad (b). Finally, things get good again, until the person's death (c). How do we know that suicide would be a bad choice? Because region (c) is bigger than region (b), meaning that even when things get bad, they would eventually get good enough to more than make up for it.

Here's another utility curve:

This one was designed to tell the utility story of someone who had an ordinary early life, then was abruptly captured and tortured for an extended period of time, until finally that person committed suicide. The inspiration for this story is prisoner suicide at Guantanamo Bay Naval Base and from innocent people mistakenly captured and sent to Guantanamo. Please note I have no idea if innocent prisoners are being tortured there. My intention here is merely to give an example of using utility curves to tell a story.

Next post in the "Mathematics of Utilitarianism" series: Utility Curves For Multiple Individuals