Expected Value Politics

5 minute read


In this post, I want to argue something crazy: that the major policy positions of both parties are rational.

Decision Theory

I want to start with a quick diversion into decision theory, and then come back to politics. There are a number of different rational ways to approach decisions with uncertain payoffs, but the one that I want to focus on is based on the expected value of a decision. The basic idea is that there is a set of possible outcomes, and different decisions have different probabilities of each outcome. The expected value of any given decision is defined as \(\sum_{i=1}^n P_i*x_i\). In other words, the probability of each outcome is multiplied by its benefit (or cost).

Choosing a Movie

As a simple example, let’s say you are deciding between watching two different movies: one which you’ve already seen and liked, and one which you know very little about. The formula for expected value requires numbers, so we’ll get a little Utilitarian here, and give numbers to how much you value each movie. For the first movie, you know that it will give you two “benthams” of enjoyment. For the second movie, there’s a 20% chance that you’ll love it, and get 4 benthams of enjoyment, a 60% chance it will be OK, and give you 1 bentham, and a 20% chance that it will be terrible, and give you -1 bentham.

For the first movie, your expected value is \(1 * 2 = 2\) benthams. The second movie is more complex. It is \(.2 * 4 + .6 * 1 + .2 * -1 = 1.2\). According to the expected value decision criteria, you should watch the first movie, since you get a guaranteed 2 benthams, while you only “expect” to get 1.2 benthams from the second. While we may not literally calculate these probabilities, I think that many of us heuristically pursue a similar strategy toward decision-making.

An important point is that any many situations like our example, the probabilities and the benefits are not empirically determined, and are up for debate. Perhaps my wife will argue that the new movie has an actor that I really like, and that I should bump up the probability of loving it to 30% and move down the probability of hating it to 10%.

Gun Control Example

I believe that public policy decisions can be viewed through this lens. I am in favor of gun control, and have been thinking about why many people who I know and respect are opposed to gun control. The kneejerk view of those who oppose us is that either a) they are ignorant or b) they are irrational. In some cases, either or both of those assumptions may be true :), but in many cases, we may simply be approaching the expected value calculation with different assumptions.

In this case, let’s simplify things, and think of outcomes along three dimensions: crime rate, accidental shootings, and recreational enjoyment. Let’s calculate expected values for gun control along each dimension, for a gun-lover and a gun-hater.

Crime rates: The gun-lover is likely to believe that crime rates will increase when law-abiding citizens give up their guns. Perhaps she believes that there is an 80% chance that gun control will cost society 1000 benthams, and a 20% chance that it will cost 100 benthams. The gun-hater is more optimistic about the ability of the state to remove firearms, and thinks that there is an 80% chance of a societal gain of 500 benthams, and a 20% chance of a gain of 1000 benthams.

Accidental shootings: Both parties agree that accidental shootings would go down, and agree that there is about a 50% chance of a gain of 1000 benthams and a 50% chance of a gain of 500 benthams.

Recreational enjoyment: Our gun-hater thinks that guns are terrible, and that people should be doing productive things instead of destructive. He thinks that gun control has a 75% chance of increasing recreational enjoyment by 500 benthams, and a 25% chance of increasing it by 100 benthams. The gun-lover, well, loves guns. She thinks that gun control has a 75% chance of reducing overall enjoyment by 1000 benthams, and a 25% chance of reducing it by 500 benthams.

Equally Rational Opponents

So, what’s the final calculation?

For the gun-lover, the expected value of gun control is:

\[.8*-1000+.2*-100+.5*1000+.5*500+.75*-1000+.25*-500\\ = -945\]

For our gun-hater, the expected value is:

\[.8*500+.2*100+.5*1000+.5*500+.75*500+.25*100\\ = 1570\]

So, for our gun-lover, the obviously rational choice is to stop gun control, while for the gun-hater, just as obvious, and just as rational, is the need for gun control.

Note that while social scientists (like me) work to try to find empirical evidence of what these parameters should be, they will never be exact. The world is too complex to know with certainty whether gun control in a given country at a given time will increase or decrease crime, and by how much. Reasonable people may have widely varying estimates. This problem is even more pronounced for things like recreational enjoyment, where the value of shooting a gun is wildly different for different people. The uncertainy surrounding expected value calculations means that there are many situations in which two opposing decisions can both be rational.

Stay Humble

So, what does this mean? More than anything, I think it means that we should recognize that our own beliefs are based on uncertainties and assumptions. Looking at things from an expected value perspective can help you to both quantify the value of a decision, but also to recognize where you are choosing to plug in numbers that may not be empirically justified, and which others could reasonably disagree with.