Measuring Free Will: Ian Durham at the 6th FQXi Meeting

It feels a bit odd blogging about myself, but here goes...

For most of the history of modern science the debate over free will has been largely left to the realm of philosophy. Indeed, the debate is as old as philosophy itself. But, increasingly, free will has gained in importance in certain areas of science. For example in quantum mechanical tests of Bell's inequalities, it is assumed that the "choices" made regarding the settings on the measurement devices are "freely" made by whatever is operating those devices (FQXi member Nathan Argaman argues that this is merely freedom in the variables themselves, but this is a debatable point). This led mathematicians John Conway and Simon Kochen to develop a free will theorem that posits that if we have free will, then, subject to certain assumptions, so do elementary particles. Specifically they take free will in this context to be the fact that our choices do not depend on the past. Their theorem is often interpreted as showing that science is incompatible with determinism.

Nobel laureate physicist and FQXi member Gerard 't Hooft disagrees with this sentiment. In his view, the "choices" in a Bell test are ultimately determined from the entire history of the universe up until the point at which the choice is ultimately made. This is sometimes referred to as superdeterminism. FQXi member Anton Zeilinger strongly opposes this view as antithetical to the entire scientific enterprise as he sees it as undermining falsifiability.

What is notable about both these views is that neither is particularly favorable toward free will as a concept. While Conway and Kochen certainly argue against determinism, by concluding that elementary particles must have free will if we do, it seems as if they are, through the apparent absurdity of that conclusion, arguing against free will.

This debate is far from purely academic. The modern legal system in many countries assumes that human beings have some level of free will while modern science seems to be pushing back on this assumption. Understanding the actions of our fellow humans entails a better understanding of consciousness and, by extension, free will. The fact is, whether or not we actually have free will, we behave as if we do. But what are the characteristics of this behavior? What does it mean to have (or behave as if we have) free will and can we quantify it?

This is a question that I addressed at the recent FQXi conference in Tuscany.

I began with a simple example. Suppose that I open my refrigerator in order to have a snack and am presented with the option of having either carrots or peppers. What would it mean from a behavioral standpoint for either choice to be seen as being "free"? In other words, what do we really mean when we say a choice is "free"?

For one thing, we need to have the confidence that whatever choice we make, that choice needs to have a high probability of actually occurring. That is, suppose that I choose to have a carrot. For that choice to really be free I need to have the confidence that at some point between making the choice and actually eating the carrot, it doesn't spontaneously turn into a pepper or a potato or a chair.

If the latter happened with any regularity, we would simply throw up our hands and stop making choices altogether because we could never be certain of the outcomes. Choices could just as easily be made by throwing darts at a wall. Outcomes would be completely random regardless of what we thought we chose. This is not free will. As philosopher Robert Nozik pointed out, "An action's being non-determined is not sufficient for it to be free-it might just be a random act."

This also suggests that we weigh our options against one another. Again, if we didn't, we could accomplish the same thing by throwing darts at a wall. This, in turn, suggests that the options we didn't choose generally don't change once we've made our choice (and started to act on it) either. Again, if it routinely did, we would simply give up in frustration.

None of this necessarily means that our choices are about fitness. I may simply "be in the mood" for carrots rather than thinking that carrots are somehow a more optimal choice than peppers at the time. But the point is that I have the time and ability to think about them. That is, I have "read" them into my memory and then contemplated them. In order to do so, however, the number of all possible choices must be processed in a finite amount of time or we have to consciously eliminate some without really processing them.

In order to quantify all of this in some way, we nevertheless need to recognize the fact that, at the most fundamental level, the universe is random. Whether or not we actually have free will, it is clear that there is a deterministic element to our behavior concerning choices, namely that those choices have a high likelihood of occurrence once chosen. But that level of determination has to arise from something more fundamentally random. How is this possible? A simple example provides a hint.

One of the more popular games at casinos around the world--and one of the oldest games of chance known--is the game of craps. The game of craps is relatively simple. It simply involves rolling a pair of dice and betting on the outcome. But not every outcome of the roll of a pair of dice is equally probable. Anyone who has ever played a board game knows that a roll of 12 ("double boxcars") or 2 ("snake eyes") is much less common than other rolls. But why is that?

The roll of each individual die is assumed to be completely random. That is, each number from 1 to 6 is assumed to be equally likely to arise. In fact, to try to ensure this is the case in the long run, casinos paint the dots on their dice rather than use dice whose dots are little holes as is common in most dice. This is to ensure that the center of gravity is as close to the physical center of the die as possible which helps keep the die balanced.

So if each number on a single die is equally likely to occur over the long run, how is it that a roll of 12 or 2 is less common than, say, a roll of 7 (which happens to be the most common)? The answer is that there are more combinations that give a 7 than give a 2 or 12. In fact, there is only one way to roll a 2 or 12--both dice have to be 1 or both dice have to be 6 respectively. But there are six ways to get a 7. So the probability of getting a 7 is higher because there are more configurations that lead to a 7.

While a 7 is more likely than a 2 or a 12, it isn't that much more likely in the grand scheme of things. Most people have rolled a 2 or a 12 while playing a board game at some point in their lives. We call the number rolled on the pair of dice (between 2 and 12) the macrostate and we call the number rolled on each die individually the microstate. But there are physical systems with macrostates that are overwhelmingly more likely than nearly every other macrostate such that the system almost always ends up in this state. Examples include two-state paramagnets and interacting Einstein solids (I discussed some of these in a recent FQXi essay). In other words, it would be like having a pair of dice that, despite the individual dice ending up on a random number each roll, the pair would always end up on the same overall roll. This isn't a strange quantum behavior. It's simply how combinations work when the number of combinations is very, very large.

The point of all this is to show that it is possible to get a nearly deterministic macrostate from a large collection of entirely random microstates. In fact, in statistical mechanics, this is known as the thermodynamic limit. There is nothing mysterious about it.

How does this help with modeling free will? Let's go back to the refrigerator again. What am I actually choosing when I choose carrots over peppers? The fact is that the carrots and peppers are in different places in the refrigerator. As such, the act of getting a carrot out of the refrigerator is an entirely different process than the act of getting a pepper. So when we make a choice, we are choosing between two different processes. Each of these processes has a probability distribution such that one macrostate is overwhelmingly more likely than any other--the chance that the act of choosing a carrot leads to eating a pepper is vanishingly small. That is, the variances of the probability distribution for each process is very small. This seems in line with the what we might think of as a "free choice" since it ensures that our choice is very likely to occur.

If all of our choices have probability distributions, we can represent the ensemble of possible choices with what is called a mixed distribution. There are a number of interesting properties of such distributions including something known as the Mahalanobis distance which can be construed as measuring just how distinct a pair of choices is. The Mahalanobis distance between two different carrots is going to be smaller, for example, than the corresponding Mahalanobis distance between a carrot and a pepper since the carrots are more "alike". One assumes that the freer the choice, the larger the Mahalanobis distances to all the other possible choices.

So one possible measure of "free choice" would be some function (which I call the zeta-function) of the minimum Mahalanobis distance between that choice and any other choice in the ensemble. Likewise it would be proportional to a time function that ensures that the full ensemble of choices can be read into our memory and processed in a finite amount of time. The zeta-function should also be inversely proportional to the variance of the choice (the smaller the variance, the greater the freedom as I just described).

The zeta-function is a measure of the freedom of an individual choice. It is safe to say that sometimes our choices are not free. Many actions in our life are instinctual. But it's safe to say that if a certain number of our choices are free we might say we have free will. So a measure of free will itself, which I call the Z-function, would be some function of the collection of zeta-functions for all the choices we have.

There are a lot of details that can't be captured by a blog post or a simple twenty minute talk (a paper is forthcoming). Certainly there are valid criticisms of such a model. But my aim was really to stimulate discussion and hopefully research into formal models of free will as it is becoming increasingly important in science, particularly in areas like quantum foundations and consciousness studies. So one can view this work as me throwing down the gauntlet and challenging the scientific community to start thinking about this in some depth. Hopefully I will have at least succeeded in that.