Modeling Choice With Utility Curves

Previous post in the "Mathematics of Utilitarianism" series: Total Lifetime Utility

Our work on utility curves strives not only to characterize the world's (or universe's) utility but also to help us make decisions which affect it. Following the principles of utilitarianism, we strive to choose those actions which maximize utility across all individuals and over all time. We therefore assign a separate total lifetime utility curve to each choice and pick the one with the largest total. This is often easier said than done, but for now we'll just take a look at how utility curves can be used to handle these choices.

Let's start with a simple example, one that's probably familiar to all of us. There are two options: Option (a) offers higher utility at first but lower total utility; Option (b) offers lower utility at first but higher total utility. Examples include (a) eating ice cream vs. (b) eating vegetables; (a) saving money vs. (b) spending it; (a) consuming resources unsustainably vs. (b) consuming resources sustainably. Here's a sample utility curve graph showing this choice:


The solid curve shows the utility between t1 (birth) and t2 (decision time) under both (a) and (b). Because this utility is the same in both cases, it can be ignored in our analysis. The dashed curve shows the utility between t2 and t3 (death under choice (a)). The dotted curve shows the utility between t2 and t4 (death under choice (b)). We can tell from looking at the graph that the area under the dotted curve is larger than the area under the dashed curve, which means that totally lifetime utility is greater with (b) than with (a). More formally,

Thus, while the individual might be tempted to choose (a) because it will be better in the short-term, (b) is actually the better choice.

Another insightful example is what I call "Critics vs. Warren Buffett". We'll return to this example often, as it is both interesting to analyze and very important. Buffett, the legendary investor (and a personal favorite character), was often criticized for never giving any of his vast fortune to charity. (See for example this: "the mind reels at how much good he could have done had he given away just a little bit of his money in the years since he became the richest (and then second-richest) man in the world.") Buffett's argument was that, given his rare ability to increase his wealth at an unusually high rate, society was better off with him earning as much as he could until his death and then giving that much more money to charity. If he's right, then the corresponding utility curves might look like this:

Here, U0 represents the "background" societal utility, i.e. society's utility without Buffett's donations. From this graph (assuming it's accurate), we can see that Buffet's critics should be a little patient- or at least we could until he silenced them with his recent donation: As with the previous example, total utility (area under the curve) is initially higher under one choice (a) but eventually becomes higher under the other (b).

We're making a few big assumptions here: (1) The more money Buffett donates to charity, the more society's utility increases; (2) Society's background utility does not change with time; (3) After Buffett donates to charity, utility goes up for some time but eventually returns to the background level. My sense is that (1) is valid, but that (2) and (3) are not: (1) The world is unlikely to ever not be able to put Buffett's money to good use; (2) Society's background utility changes as world population changes (or, more accurately, world utility-adjusted-population), and (3) given Buffett's taste in charities (which I think is excellent), societal utility would likely end up at a higher level.

Another interesting and important example which we'll analyze repeatedly is climate change, or more specifically, the impact of climate change and our response to it on utility. For a discussion of this based on real world data, see the post Utility Cost Of Greenhouse Gas.

Here, we consider a simple model in which there are two options: (a) Don't respond to climate change and (b) Respond to climate change. In crude terms, (a), often called the "business as usual" (BAU) scenario, is what those who do not consider climate change to be a problem would advocate; (b) is what those who do consider climate change to be a problem would advocate.

For now, like with the Buffett example, we're going to use some big assumptions: (1) Background utility is constant except when affected by climate change or our response to it; (2) We only have two choices; (3) We know what will happen under both choices. None of these assumptions are valid: (1) because many things other than climate change affect utility; (2) There are many ways we can respond to climate change; (3) We are not certain what will happen under either circumstance. However, these assumptions are not entirely unreasonable. Using them, the choices look like:

In this representation, t1 is the decision time. In the model, if we don't respond to climate change (a), utility stays constant until some future time t2 (dashed curve), at which point climate change kicks in and utility falls to a very low level (dotted curve). If we do respond (b), utility will decrease somewhat at t1 but then level off at a higher level than under choice (a) (solid curve). Like in the previous examples, total utility (area under the curve) is initially higher under one choice (a) but eventually becomes higher under the other (b). So, under this model, we should respond to climate change, even if we don't like it at first. While this model is crude, it does reflect environmental economics's findings. (See the post Reviewing The Stern Review.)

For a final example, we'll look at a scenario often cited as one in which utilitarian decision making is used effectively: medical triage. Here we have two patients, (a) and (b). Both are injured at the same time (t1). At time t2, the available doctor is able to treat one of the patients. Then, at time t3, the doctor is able to treat the other patient. The doctor must decide which patient to treat first. We are assuming that both patients take the same amount of time to treat; expanding this model to the general case is straightforward. We are also assuming that both patients start out at the same utility level and, upon treatment, end up back at that original treatment level:

Here, patient (b) loses much more utility by time t1. However, (b)'s condition is stable, whereas (a) will die before time t2 if left untreated. Thus, the doctor should treat (a) at time t1 and (b) at time t2.

Next post in the "Mathematics of Utilitarianism" series: Handling Uncertainty