Total Lifetime Utility

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

The earlier post on Utility Curves introduced the idea of representing an individual's utility as a function of time. Here we take a closer look at the total amount of utility and individual experiences over its lifetime (or over some other period of time). Then, we combine it with our work on Utility Curves For Multiple Individuals to arrive at a good model for the total utility of the whole entire universe, or at least the universe as we know it.

Recall Figure 2 from the Utility Curves post. Let's repost it here, slightly modified:



In the original post, we noted briefly that "the region (c) is bigger than the region (b)" but didn't specify exactly what we meant. The regions here refer to the area between the utility curve and the dashed line at utility=0. Mathematically, this is an integral. Thus, (a), (b), and (c) are found by the following integrals:





Similarly, the individual's total lifetime utility can be found by integrating from birth (t1) to death (t4):

If we want to (and we will), we can express the individual's total lifetime utility by integrating from the beginning of time (t0) to the end of time (t_infinity):

This gives us the same value because we're assuming that utility equals zero both before birth and after death, as discussed after Figure 1 in the Utility Curves post.

Combining this work on total lifetime utility with our earlier work on Utility Curves For Multiple Individuals, we can now easily represent the total utility of the entire universe (as we know it) by adding up every individual's total lifetime utility. There are two equivalent ways of expressing this mathematically:



In Equation 4, we first calculate each individual's total lifetime utility (knowing that each invidual will be born after the beginning of time and will die before the end of time), and then add up each individual's total lifetime utility. In Equation 5, we add up each individual's "instantaneous" utility, and then integrate over all instants. Both approaches are equivalent, and both give the total utility of the universe as we know it.

Why only "as we know it"? Because this model assumes that time is continuous and individuals are discrete. This ignores any possible quanitization of time which if it exists is too small to matter here. This also ignores the possibility of utility existing beyond our Earthly notion of individuals. Perhaps utility could be considered within an individual (for example, at different points within a brain), in which case it might be a continuous function of space or of some other dimensions, and an appropriate integral would need to be added to the model. Or perhaps some other, more exotic form of utility exists on this planet or beyond which requires alternative mathematical treatment. That would be quite interesting, but for our purposes here, not relevant.

Next post in the "Mathematics of Utilitarianism" series: Modeling Choice With Utility Curves