Handling Uncertainty

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

If we knew the shape of the utility curve corresponding to each available choice, our work would be much easier: We could simply calculate the total utility of each choice and pick the one with the biggest number. However, we often don't know choices' corresponding utility curves. Here we explore scenarios in which choices could lead to different outcomes, each having some chance of happening.

To guide us through, we'll use the "Responding to climate change" example introduced in the post Modeling Choice With Utility Curves. Here we have two choices: (a) Don't respond to climate change or (b) Respond to climate change. Let's say that if we respond to climate change, we know what's going to happen: Our utility will immediately diminish by a relatively small amount. However, let's also say that if we don't respond to climate change, we don't know what's going to happen. We'll consider two possible outcomes: (a) Utility will remain at a baseline level until some time in the future, at which point climate change strikes, causing a large drop in utility, and (c) Utility remains at a baseline level indefinitely, which corresponds to a scenario in which climate change either doesn't happen or won't affect utility if it does. For more on this matter, see Wikipedia: Global warming controversy. Here's a utility curve graph representing these three scenarios:

Here, the choice occurs at time t1. If we choose not to respond to climate change, then at time t2, climate change either lowers utility or it doesn't. We have no choice in whether or not climate change lowers utility, so the split between (a) and (c) is fundamentally different from the split between (a) and (b). Instead of choosing between the two curves, we assign a probability that each curve occurs. With that, one approach to deciding which choice to make at time t1 is to compare the utility of (b) to the expected value of the utility from (a) and (c). At any point in time, this expected utility is given by:

Here, Pa and Pc are the respective probabilities of (a) and (c) occuring and Ua and Uc are the respective utility levels of (a) and (c) if they do occur. Note that Pa + Pc = 1 = 100%, meaning that if we choose not to respond to climate change, there is a 100% chance that either utility drops to a low level (a) or remains at the baseline indefinetely (c).

We can rewrite this equation to show the expected utility as a function of time:

Now, recalling lifetime utility integrals from the post Total Lifetime Utility, we can calculate the total expected utility from (a) and (c):

Similarly, we can calculate the total utility from (b):

Thus, if Utot-ac is greater than Utot-b, then we should choose (a)/(c), meaning we should choose not to respond to climate change; if Utot-b is greater than Utot-ac, then we should choose (b), meaning we should choose to respond to climate change; if Utot-b is equal to Utot-ac, then it doesn't matter which we choose.

In reality, there is more than one possible scenario for utility if we choose to respond to climate change and more than two possible scenarios if we choose not to respond. Indeed, there is effectively an infinite number of possible scenarios in each case, corresponding to every possible way the universe could play itself out. Each of these scenarios has a certain probability of happening. How can we handle this?

First, we note that any scenarios which result in the same amount of utility can be lumped together. For example, if I'll be just as happy wearing a blue shirt tomorrow as I will wearing a red shirt, then I don't have to consider each case separately (assuming no one else's utility is affected by this either). In our climate change example here, since we are only comparing total lifetime utilities and not instantaneous utilities, any scenarios with the same total lifetime utility can be lumped together, regardless of the shape of the underlying utility curve (utility vs. time). Thus, we can plot probability density functions (PDFs) of the two choices:

Here, U0tot is the total utility that would come if we remained at our baseline utility level U0 (that is, if (a) occurs). Thus, this plot shows that if we choose to respond to climate change (b), then we'll probably have some total utility a little bit less than U0tot, but there's some chance of having either a utility much less than U0tot or much greater than U0tot. Also, if we choose not to respond to climate change, there's almost a 50% chance of total utility being around U0tot, almost a 50% chance of total utility being much less than U0tot, and a small chance of total utility taking some other value. We can recover our expected total utilities by integrating over total utility:

Here, Uex-tot-ac is the expected total utility under choice (a)/(c) (not responding to climate change) and Uex-tot-b is the expected total utility under choice (b) (responding to climate change).

While we can compare these expected total utilities and pick the choice with the higher value, we might not want to do so. The discussion of why this might be the case is very much the same discussion held by investors making financial decisions. In particular, we might be risk averse, meaning that we might be willing to make a choice with a lower expected total utility if means we're more likely to avoid an outcome with a very low total utility. In our climate example, this could mean choosing to respond to climate change even if not responding to climate change has a higher expected total utility, in order to avoid the scenario in which climate change significantly diminishes total utility (a).

In investing, making these decisions is a matter of style. In other words, there is no one right answer. Some investors are more willing to take risks: They accept the possibility of greater loss in return for a higher expected return. But what does utilitarianism instruct us to do?

Others may disagree, but I believe this is a point in which utilitarianism breaks down as an exact, rigorous science. I believe we are completely justified in making choices with lower expected total utility in order to avoid the risk of an outcome with very low total utility. But either way, I certainly believe we should take the science as far as we can, recognize the nature (i.e., the total utility probability distribution) of the available choices, and pick what we believe to be the best option, however risk averse we decide to be.

This is currently the last post in the "Mathematics of Utilitarianism" series.