w_daniel_hillis's picture
Physicist, Computer Scientist, Co-Founder, Applied Invention.; Author, The Pattern on the Stone
Possibility Spaces: Thinking Beyond Cause and Effect

One of the most widely-useful (but not widely-understood) scientific concepts is that of a possibility space. This is a way of thinking precisely about complex situations. Possibility spaces can be difficult to get your head around, but once you learn how to use them, they are a very powerful way to reason, because they allow you to sidestep thinking about causes and effects.

As an example of how a possibility space can help answer questions, I will use "the Monty Hall problem," which many people find confusing using our normal tools of thought. Here is the setup: A game-show host presents a guest with a choice of items hidden behind three curtains. Behind one is a valuable prize; behind the other two are disappointing duds. After the guest has made an initial choice, the host reveals what is behind one of the un-chosen curtains, showing that it would have been a dud. The guest is then offered the opportunity to change their mind. Should they change or stick with their original decision?

Plausible-sounding arguments can be made for different answers. For instance, one might argue that it does not matter whether the guest switches or not, since nothing has changed the probability that the original choice is correct. Such arguments can be very convincing, even when they are wrong. The possibility space approach, on the other hand, allows us skip reasoning about complex ideas like probabilities and what causes change. Instead, we use a kind of systematic bookkeeping that leads us directly to the answer. The trick is just to be careful to keep track of all of the possibilities. 

One of the best ways to generate all the possibilities is to find a set of independent pieces of information that tell you everything you could possibly need to know about what could happen. For example, in the case of the Monty Hall problem, it would be sufficient to know what choice the guests is going to make, whether the host will reveal the leftmost or rightmost dud, and where the prize is located. Knowing these three pieces of information would allow you to predict exactly what is going to happen. It is also important that these three pieces of information are completely independent, in the sense that knowing one of them tells you nothing about any of the others. The possibility space is constructed by creating every possible combination of these three unknowns.

In this case, the possibility space is three-dimensional, because there are three unknowns. Since there are three possible initial choices for the guest, two dud options for the host, and three possible locations for the prize, there are initially 3x2x3=18 possibilities in the space. (One might reasonably ask why we don't just call this a possibility table. In this simple case, we could. But, scientists generally work with possibility spaces that contain an infinity of possibilities in a multidimensional continuum, more like a kind of physical space space.) This particular possibility space starts out as three-dimensional, but once the guest makes their initial choice, twelve of the possibilities become impossible and it collapses to two dimensions.

Let's assume that the guest already knows what initial choice they are going to make. In that case they could model the situation as a two-dimensional possibility space, one representing the location of the prize, the other representing whether the host will reveal the rightmost or leftmost dud. In this case, the first dimension indicates which curtain hides the prize (1, 2 or 3), and the second represents the arbitrary choice of the host (left dud or right dud), so there are six points in the space, representing the six possibilities of reality. Another way to say this is that the guest can deduce that they may be living in one of six equally-possible worlds. By listing them all, they will see that in four of these six, it is to their advantage to switch from their initial choice. 

  Host reveals left dud Host reveals right dud
Prize is behind 1 2 revealed, better to stick 3 revealed, better to stick
Prize is behind 2 3 revealed, better to switch 3 revealed, better to switch
Prize is behind 3 2 revealed, better to switch 2 revealed, better to switch

Example of a two-dimensional possibility space, when guest's initial Choice is 1


After the host makes his revelation, half of these possibilities become impossible, and the space collapses to three possibilities. It will still be true that in two out of three of these possible worlds it is to the guest's advantage to switch. (In fact, this was even true of the original three-dimensional possibility space, before the guest made their initial choice.)

This is a particularly simple example of a possibility space where it is practical to list all the possibilities in a table, but the concept is far more general. In fact one way of looking at quantum mechanics is that reality actually consists of a possibility space, with Schrödinger's equation assigning a probability to each possibility. This allows quantum mechanics to explain phenomena that are impossible to account for in terms of causes and effects. Even in normal life, possibility spaces give us a reliable way the solve problems when our normal methods of reasoning seem to give contradictory or paradoxical answers. As Sherlock Holmes would say,  "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth."