We’ve seen before on this blog that we often take probability theory to be the core of rational belief. What we haven’t seen before is that our beliefs tend to violate probability theory not just in random ways, which is what you would expect if you thought we were making random mistakes, but systematically, in a particular way. We call the tendency to violate probability theory in these ways cognitive biases, since they seem to arise more out of the shape of our cognition than from a single mistake.
One such bias discovered by Tversky and Kahneman is the conjunction fallacy. In the abstract, this is the tendency for humans to say that the probability of a proposition A&B is greater than that of A. This is of course a violation of the probability axioms. The famous example they use to test this is that of Linda.
Subjects in an experiment are given a description of a fictional character, Linda. The description makes it very likely she is a feminist, and makes her less likely to be a bank teller. Subjects are then asked to make judgement about how likely different things are. In particular, they are asked which is more like, that she is a bank teller, or that she is a bank teller and a feminist.
As you probably suspect from the lead in, most people make the judgement that it is more likely that she is a bank teller and a feminist than that she is a bank teller. To get an intuition for why this cannot be correct, here is a helpful diagram:
We can see this in every case where the Linda is both a bank teller and a feminist that she is also a bank teller, but not the other way around. So the probability that she is somewhere in the blue has to be at least as high as that she is in the intersection of blue and green, since the former includes the latter as a sub-case.
In the paper we will look at today Pothos and his co-authors are interested in two main questions:
- What is a good descriptive account of why people make judgment?
- Is the judgment actually in fact rational, at least in some circumstances?
In order to investigate these questions they use a different type of probability called quantum probability theory.
***The original paper can be found here.***
In what follows I will provide a more schematic summary of their analysis, skipping many of the mathematical details — they are in the paper if you are curious.
The main case of possible irrationality that Pothos et al. is the conjunction fallacy we just discussed, though they hope that quantum probability theory can help to understand others as well.
They first go through a few possible explanations of the conjunction fallacy that people have proposed in the literature — maybe participants don’t understand the word “and” as a conjunction, maybe “bank teller” is understood as “bank teller and not a feminist” in the context of the other questions, etc. — but note that, on balance, even when researchers try to correct for such possibilities the effect still remains.
This leaves us with their two questions I mentioned above. This is where they introduce quantum probability. Even though it has the word “quantum” in it, they are not saying that the brain might run on quantum physics or anything like that. Rather, they are talking about a specific type of mathematical function — a quantum probability function — which was developed to model quantum mechanics. They wonder if it can also be used to model parts of human cognition.
One important feature of quantum probability theory is that it is inherently contextual. Contextuality here is a specific technical term; intuitively, it means that “the same question is isolation has to be treated as a different question compared to when processed in the context of other, incompatible ones” (p. 6).
This is important for at least two reasons. The first is that we can see how this might shed light on the Linda case. In the Linda case something is causing people’s judgments to differ from those given by classical probability theory — perhaps contextuality, in conjunction with their cognition being well-described by quantum probability theory, is what is driving this. Furthermore, quantum probability theory can actually allow for the kind of conjunction effect we observe, given contextuality:
In [quantum probability theory], Prob(Y&X) > Prob(Y) can be appropriate. So, we have a situation where, superficially, the same judgement is correct according to the one probabilistic framework ([quantum probability theory]), but incorrect according to another ([classical probability theory]).p. 4
This is an important point for the authors — we can predict this judgement using quantum probability theory. This seems like a point in favour of this account of our cognition.
I want to highlight, however, two points that the authors raise that for me made this less of a point in favour of quantum probability theory. The first is that
Importantly, the idea that contextuality can change the meaning of superficially identical questions can be expressed classically too, since we can keep track of different meanings, through conditionalization.p. 6
So quantum probability theory is not unique in being able to represent contextuality, and in fact classical probability theory does just fine. The second point is that
incompatibility in [quantum probability theory] requires specific assumptions regarding the relevant questions, which in turn guide predictions for the [conjunction effect].p. 5
So it isn’t the case that we just assume quantum probability theory, and then we get a prediction about a conjunction effect in the Linda case — we have to do empirical work to first figure out whether there is or is not a conjunction effect, and then we can use quantum probability theory to represent why this might occur. This is of course very reasonable — it would be deeply surprising if the mathematics gave us the structure of our cognition in such a strong way. However, I think that is also makes it less surprising that we can use quantum probability theory to predict this effect, given that this is exactly the kind of thing we are trying to model.
However, Pothos et al. do somewhat address these two concerns:
In [quantum probability theory], incompatibility goes hand in hand with specific principles for relating different probability spaces and informing the exact difference between [questions]…Thus, the [quantum probability theory conjunction effect] approach can be extended in a reasonably constrained way to cover related findings…Instead, in [classical probability theory], contextuality requires what is typically a post hoc conditionalization and so is less appealing theoretically.p. 7
This is nice. It means that quantum probability theory gives us conditional constraints on how other types of judgments would be made, that we can then go out and empirically test. This is in contrast to classically probability theory, that is flexible enough to describe conjunction effects, but doesn’t constrain what we expect to see otherwise.
Pothos et al. then summarize a few studies, and conclude that overall the evidence for a quantum probability theory of our judgment is well supported. I lack enough knowledge in this field to make a strong judgment here about how plausible I find their analysis. However, they then move on to their second question, which I find actually more interesting — can this be rational?
One standard of rationality that Pothos et al. bring up is the Dutch Book, which we have actually looked at before. A kind of minimal standard for rationality is that your beliefs shouldn’t lead you to accept bets which you will always lose, no matter how the world turns out. This kind of gamble is called a Dutch Book. If this were a case, it would reflect a kind of inconsistency in your belief. Thus we can think about the kind of belief systems that allow one to avoid a Dutch Book, and take these as the ones that mean this minimal standard.
A very famous result due to de Finetti is that people whose beliefs conform to the classical probability theory axioms are indeed invulnerable to a Dutch Book. This classical probability theory is rational according to this minimal criterion.
The authors further note, however, that
It is surprising that [quantum probability theory] probabilities satisfy all the above properties required of betting rations too, that is, [quantum probability theory] decisions are consistent with the [Dutch Book] criterion in exactly the same way as [classical probability theory] ones are, in principle at least…This last qualification is a key aspect of this paper, since [quantum probability theory] concerns questions that are incompatible. In the outside macroscopic worlds, incompatible questions are contextual ones. If there are no contextual questions in the outside world, then it becomes irrelevant to consider the implications from [quantum probability theory] for rationality.p. 9
Okay, so here we have it. Relative to their own representations, agents that use quantum probability theory to make judgments are invulnerable to a Dutch Book. However, as they note, this just pushes the question one step further — does the world actually have contextuality like this?
(At this point in the paper the authors also take a section to describe an online experiment they conducted to see whether people were in fact treating the Linda-style questions in a contextual way. Their experiment seemed to weakly support their hypothesis that people treat these questions contextually. You can read that section for yourself — I am still more curious about whether or not this can be rational, regardless of whether people do or do not do it.)
In section 5 of their paper Pothos et al. then go through a few examples that they take to illustrate how the world may be contextual in some ways. If so, this would allow quantum probability theory to satisfy our minimal test of rationality, at least in our world.
They start off by noting it can sometimes be easier computationally to work with quantum probability theory, using it as a shortcut. However, this is insufficient for their purposes:
The adoption of contextual representations for outside world questions by analogy at best provides a bounded rationality justification for using such representations, but cannot address the key questions regarding normative prescription and rationality: Do certain questions in the outside world warrant the contextuality required by the [Dutch Book] criterion to absolve the [conjunction effect] of fallacy?p. 14
They offer three cases that they think this is the case. Let us examine each in turn:
The first class concerns situations where a decision maker shares with a judge (who judges the outcome for e.g. a bet) an assumption that two questions are contextual. If both the judge and a decision maker share the same contextual representations the decision maker with e.g. avoid a [Dutch Book] from a [conjunction effect].p. 14
The authors do not spend much time on this class, and rightfully so. It isn’t very impressive — this is more of a shared mental representation effect, for example, “if we interact a lot with fellow humans whose representations are contextual, then quantum-like reasoning by analogy may be an appropriate strategy” (p. 14). Instead, let’s turn to the examples that they think show there are questions for which the outside world is contextual:
The second class of contextual questions concerns the situations where a previous question can disturb a system (and so affect subsequent questions); i.e., the disturbing influence of the previous question produces the context.p. 15
They give an example of this, that I want to quote in full:
More importantly, questions can have a disturbing influence on macroscopic systems, thus creating contextual pairs of questions, for which a [conjunction effect] would be rational. Consider a baby, Eve. Her parents bet about whether Eve will be sleeping on her front or back at two times, 10pm and 11pm; call these variables and . They determine how Eve is sleeping by checking on her, but this wakes her up and she moves around before going back to sleep. Now suppose Eve tends to sleep on her front with a probability of .90, stationary over the night. Then, and , when there is no checking. But , since the 11pm question in the conjunction assumes a prior check at 10pm and measuring at 10pm disturbs Eve and makes her likely to roll over.
An observer wishing to make a decision regarding baby Eve could construct a single probability space representation with (assuming the 11pm question is evaluated after a prior check, but not the 10pm one). But this will lead to incorrect judgments, since it misses important structure in the problem. The correct representation of the questions is contextual, one has to employ either [classical probability theory] and conditionalization or the more detailed framework for contextuality of [quantum probability theory], and a [conjunction effect] is rational.p. 15
The judgement about the probabilities make sense to me, but this being an example of a conjunction effect doesn’t. This is something that bothered me throughout the paper but is obvious here, and even more obvious in the next class of questions they consider. For something to be a conjunction effect, I want the probability of “A and B” to be greater than that of just “A”, where A and B are both propositions, and furthermore the same propositions. Otherwise it really feels like we are equivocating. It looks something like a conjunction effect, but really it isn’t — it is then kind of odd to say that a conjunction effect can be rational, when there isn’t really anything like that going on.
In the above example with Eve, we aren’t talking about the probability of her being in a certain state after we check her — we are just talking about the probability that she is in a certain state. Bringing in this checking dynamic into our fundamental epistemic judgments doesn’t seem at all like a fine enough description of things. At its core, the conjunction in a proposition doesn’t describe a temporally ordered procedure, it is simply a normal junction. Furthermore, if people interpret it like that, then this goes back to something I mentioned earlier in this summary — the possibility that people are misunderstanding “and”. The authors wanted to rule out that case, but here it looks like it is creeping back in.
Let’s look at the final class they give. These are questions that can have different meaning depending on the context. They give the following example:
Consider a biology student, Peter, who overhears his parents discussion. Dad asks mum, ‘How likely is it that Bubbles will suffer from a heart attack?’ (H). Peter’s default assumption is that Bubbles refers to his pet fish, as he is very fond of it and often talks about it. Since fish’s hearts lack coronary vessels, they are unlikely to develop coronary disease and suffer from heart attacks. Hence, Prob(H)~very small. Suppose instead dad asks mum “How likely is it that Bubbles will go to the supermarket and how likely is it that Bubbles will suffer from a heart attack?” (S&HS). The first question in this conjunction makes it clear that Bubbles refers to Peter’s granddad, whose nickname is Bubbles because he likes scuba diving. It is straightforward to fill in details so that a CE is plausible (e.g., Bubbles the grandad may have high cholesterol and find going to the supermarket frustrating). In this examplep. 16
Prob(H) < Prob(S&HS) is clearly sensible. The Heart question has two separate meanings, one for which heart attacks are extremely unlikely (heart of a fish) and another for which they are fairly likely (heart attacks are a common cause of premature death in humans, especially amongst high cholesterol individuals).
This doesn’t so much feel like contextuality to me, either at the level of the world or of propositions, but rather contextuality about how Peter interprets the sounds that his parents are making. But the sounds themselves aren’t the propositions — they are they expressions of a proposition. It would feel weird to me to encode these two different propositions, that might be conveyed by the same linguistic expression in different contexts, as one thing in the the basic structure of our probability space. This is in fact why . They not really the same question, but different ones. I could see pushing them together somehow and then using some math to look at the relationship in a way that is useful for modeling people in a descriptive way, but this doesn’t really seem like a good way to justify a conjunction effect rationally — they are just different propositions.
Thus, even though I find some of the empirical evidence they give persuasive that something like quantum probability theory could be useful for modelling our cognition, I have yet to be convinced that a conjunction effect is not a conjunction fallacy.