3.1. The “Linda” Issue
Bounded rationality is a wide and deeply studied field of research. Tversky and Kahneman worked on many of its aspects. In particular, they [
21,
22] defined and described the so-called “conjunction fallacy” through interesting experiments. The most studied is perhaps the “Linda issue.” In their work, they submitted to a group of subjects personality sketches of a woman called Linda followed by a set of possible occupations or avocations.
This was the description “L” of Linda: Linda is 31 years old, single, outspoken and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Three of the proposed occupations/avocations were:
Linda is active in the feminist movement (F),
Linda is a bank teller (T),
Linda is active in the feminist movement and a bank teller (F & BT).
As reported by Tversky and Kahneman, “the description of Linda was constructed to be representative of an active feminist (F) and unrepresentative of a bank teller (BT)” [
22]. In fact, the result reported that the most probable description order (85% of the population investigated) is: F > F & BT > BT.
Interestingly, beyond the confirmation of F > BT, it is impressive that the great majority of subjects also rank the conjunctions (F & BT) as more probable than their less representative constituent (BT). This is a non-rational behavior within the theory of decision making because it violates the classical probability (CP), according to which the probability of a conjunction (P [F & BT]) cannot exceed the probabilities of its constituents (P [BT]). Tversky and Kahneman named this phenomenon the “conjunction fallacy.”
A strong body of scientific literature made the proposal of using quantum probability (QP) to explain the results of this experiment [
1,
7,
23,
24], showing that QP can account for P [F] > P [F&BT] > P [BT] where CP cannot.
3.2. The DMM and the “Linda” Issue
Let us go back to the DMM and let us consider the “concept” of Linda as a feminist (F) as a sparse committed minority at “+1” and the “concept” of Linda as a bank teller (BT) as a compact committed minority at “−1.” Of course, the case of Linda as F & BT occurs when both committed minorities are active. In a sense, such a view is similar to the one proposed in the QP models where the “concepts” of F and BT were considered as vectors on multidimensional subspaces [
1]. Using the DMM perspective, we can use the same vector representation to express the intensity of the opinions of committed minorities (we take “sparse” and “compact” just as two paradigmatic examples). In the same way, note that the concept of “F & BT” is a superposition: a vector superposition in the QP perspective, and a DMM minority superposition within our criticality perspective.
The “lookout birds” image affords another attractive interpretation based on looking at the “concepts” of “F” and “BT” as indications of a source of food for the flock. Let us interpret the “F” concept as a large source of food to the right of the flock: a sparse minority sees it and says “turn right” (namely, set the state to +1), thereby making the flock turn accordingly. Let us interpret the “BT” concept as a small source of food to the left, and a compact minority sees it and says “turn left” (namely, set the state to +1). Let us consider the case where both sources of food are present, with the sparse minority perceiving the large one and the compact minority perceiving the small one. According to the results of the numerical simulation, the flock will turn in a direction between “left” and “right.” Think of the direction of the flock as
, i.e., the global DMM status (see
Appendix A). In the case of a sparse committed minority only, it is a value next to 1, in the case of a compact minority, it is a value next to −1 and when both the committed minorities are present, it is a value between the two extremes.
Let us go back to the DMM and consider it in terms of probability. The DMM at criticality, with the “pure” concept of “F” (with a sparse committed minority at +1), will dynamically activate all of its lattice nodes (at +1). Most of the time (i.e., 99.7% of the time—see the previous section and
Appendix A), the majority of the nodes will be “+1” so that once the system moves to a supercritical stage, its final global state variable will be fixed at +1 with probability
p = 0.997. On the other hand, facing the “pure” concept of BT (with a compact minority at −1), the majority of DMM nodes will be in the state “−1” 95.60% of the time, so that in a supercritical state, the final global variable will be −1 with a probability of 95.60%. When both committed minorities are present (F & BT), the DMM will yield “+1” 95.71% of the time (going to the supercritical state, the final global variable will be +1 with a probability of 95.71%). The three cases are reported in
Figure 8 as point F, BT and F & BT and it should be noted that the F & BT concept has a probability between F and BT.
It should be noted that this superposition (F & BT) leads the DMM to a “dynamic state” which can occur at criticality only (it cannot be seen in a supercritical state). In fact, in the supercritical state, the DMM will always collapse to +1 or −1 because F & BT is not a “pure” dynamical state but a superposition of states. This is in line with quantum cognitive models (e.g., see [
1]). A possible interpretation is that in the brain, the concept of F and the concept of BT occur. Of course, one subject can think about both of them at the same time, but it is a superposition of known concepts and not a new one, so it seems hard to think of F & BT as a stable cognitive supercritical state. F & BT could be seen as a kind of “artifact” (maybe as a bi-stable figure: one can deal with it knowing that it is made up of two figures, but one can “see” just one of them at a time). Note that, besides disappearing at a supercritical stage, at criticality, it behaves as a dynamic state in the sense that it is related to its own particular DMM dynamics, just as F and BT are. It seems noteworthy that the superposed state F & BT “lives” at criticality only, just as in the QP model [
1] it lives as a coherent state.
Now consider a 4th dynamical state and let us call it Linda (L). Note that that L is a brand new dynamical state, and at criticality, it behaves just as F, BT and F & BT (L is related to its own particular DMM dynamics). Therefore, the DMM, coping with L, will activate a dynamical process of the same kind as that activated to deal with F, BT and F & BT. As a consequence, the system at criticality will spend a certain percentage of its time on the value +1 and −1, referring to L, and in the supercritical regime it will collapse with the corresponding probabilities, as we have discussed for the other states.
Tversky and Kahneman [
21,
22] described the new state (L) as different from F, BT or F & BT and, in their experiment, asked people to order the probabilities that L is like F, BT or F & BT. This is a fundamental point: they compared L versus F, BT and F & BT (they did not compare F, BT and F & BT!). The formal question is to compare the probability of similarity between L and F, versus L and BT and versus L and F & BT. Stating P(F) > P(F & BT) > P (BT) could be misleading because they did not find that. Instead, they found that, according to our formalism: |P(L) − P(F)| > |P (L) − P(F&BT)|> |P (L) − P(BT)|. In Kahneman’s words [
2]: “… the judgments that subjects made about the … Linda problem substituted the more accessible attribute of similarity (representativeness) for the required target attribute of probability.”.
Therefore, what we have to do is evaluate the distance between L and F, BT and F & BT. In particular, Kahneman and Tversky found that almost all the subjects said Linda is F; indeed, they built the experiment so that Linda the feminist is the description that best fits the description given. Therefore, it is very likely that the probability describing L is a point very close to the point F (see
Figure 8). Given that, it could be easily understood that the second closest point is F & BT, not BT, in line with the well-known “conjunction fallacy.” In fact, the distance |L − F| is very short, thus the second closer state must be F & BT and finally BT, in third place. Therefore, our model is clearly in line with the experimental result of the “conjunction fallacy.” Moreover, such an approach seems to evaluate the closeness between the state from the initial information and the state from the various questions, in a way analogous to the influential and well-known “representativeness heuristic,” suggested by Kahneman and Tversky [
25].
It should be noted that this comparison occurs at criticality because any decision (or comparison or evaluation) in DMM occurs at the criticality stage only, as shown in the previous section. We do not make a comparison between definite states, because they are properties of the supercritical condition. These properties are outcomes, and we aim at evaluating the intelligent “decisions” behind them.
We try to explain these concepts with an image: suppose you are sitting on dock of a bay, looking at the sky. You are just under the route of migrating birds so you can see flocks of birds coming from the south to north again and again (i.e., year after year). You know that in the area there is a farmer seeding his fields and the birds like it as food.
When the farmer seeds food 1 in the left field (field F), the flock turns left 99.7% of the time.
When the farmer seeds food 2 in the right field (field BT), the flock turns right 95.6% of the time.
When the farmer seeds both of the fields (fields F & BT), the flock turns left 95.71% of the time (because field 1 is larger or maybe they prefer food 1).
Now, in the new season, you can see the flock’s directions (e.g., turn left/right percentage) but you do not know what fields and foods have been chosen by the farmer. Suppose the flock turns left in new percentage, e.g., 98.0% (let us call this behavior “L”), what are the probabilities that the flock has seen field F, F & BT or BT? If you list them, of course you will find the same order of probabilities.
The Linda description (L state), being based on dynamics at criticality, is very close to the F state. As a consequence, the second closest state must be F & BT.
The described approach to bounded rationality, involving a DMM and having criticality as a core element, leads us to introduce the term criticality-induced bounded rationality (CIBR) to identify it.
3.3. Superposition in the “Linda” Issue
The “bounded rationality” outcome found by Tversky and Kahneman [
21,
22] can be described with QP or with CIBR and not with CP. The main issue is the experimental result that P [F & BT] is more probable than the single probability P [BT], in conflict with the theoretical laws of CP.
The reason why QP [
1] yields an interpretation of F & BT fitting the outcomes of psychological experiments is that QP allows us to interpret F & BT as a superposition of two different states rather than as a classical sum. CIBR yields the same benefit as QP of making F and BT live together in one dynamical superposition of states, bypassing the limits of CP. This is reminiscent of QP with a noteworthy difference.
In fact, the superposed state of F and BT can be clearly seen in the lattice of a DMM so that a clearly definite probability is observed and evaluated in space and time. Instead, in QP models [
1], BT and F are described only as separated projections: their superposed state cannot be observed (of course, because it is a quantum superposition, so to speak, a model of coherent state) and must be described through conjectures. Pothos and Busemeyer write: “As it is impossible to evaluate incompatible questions concurrently, quantum conjunction has to be defined in a sequential way, …that is, we have Prob(F ∧ then BT) = Prob(F)·Prob(BT|F)” [
1]. In other words, the QP model cannot directly compare the case F & BT (which is a superposed state) because it has to be considered as a sequence of “collapsed” states of F and BT. Moreover, in order to evaluate this in a sequential way, in QP, other assumptions must be made, and the same authors write “an additional assumption is made that in situations such as this, the more probable possible outcome is evaluated first.” On the contrary, the CIBR superposition of the states F and BT corresponds to a well-defined cluster of decisions made at criticality.
QP is based on the superposition of F and BT, a condition inaccessible to observation if the wave function collapse is not involved. We note that the wave function collapse is still an open problem for quantum mechanics. The adoption of CIBR allows us to use both quantum mechanical concept of state superposition, at criticality, with no limits on observation, and the counterpart of wave function collapse, in the supercritical condition.
In other words, in QP, one can deal with a collapsed state only (F and BT) and the superposed state is inaccessible, whereas with the CIBR approach, one can work with a collapsed state (e.g., in the supercritical stage) as well as with the superposed state (e.g., in the critical stage). If one considers a superposed state in QM, e.g., a qubit, it could have a value of
α and
β as a point in the Bloch sphere, but one will never know where it is in a definite moment, while on a DMM lattice, one can always know where the point is; only in CIBR can one see what happens in the decision process, in space and time. Note that CIBR does not necessarily imply a DMM, which is just an instance. Our approach can be defined as a non-linear stochastic theory, as are the approaches mentioned by Breuer and Petruccione [
26], to account for the wave function collapses in quantum mechanics.
QP and CIBR can both correctly describe the Linda issue because they seem to operate with superposition in a similar way, as opposed to the CP approach. In particular, CIBR seems to offer a better way because of its clearer approach, unlike the inaccessible quantum coherent state of QP.
3.4. Failures of Commutativity in Decision Making
Another interesting issue raised within the “bounded rationality” domain is the “failures of commutativity” in decision making, whereby asking the same two questions in different orders can lead to changes in responses. As an example, consider the questions “Is Clinton honest?” and “Is Gore honest?” and the same questions in a reverse order. When the first two questions were asked in a Gallup poll, the probabilities of answering “yes” for Clinton and Gore were 50% and 68%, respectively. The corresponding probabilities for asking the questions in the reverse order were 57% and 60% [
27]. Such order effects are puzzling according to CP theory because they seem to violate the commutativity laws. The QP approach, as is known, can describe the experimental result, suggesting that the quantum approach accounts for the fact that thinking about one “concept” (e.g., Clinton being honest) changes the basis when the second one (e.g., Gore being honest) is evaluated.
Let us consider this issue using the CIBR perspective. If we just ask: “Is Clinton honest?”, we must consider that the complex system has to make a “decision.” Thus, the system will move from its undercritical stage to criticality when the decision is made and then it will move to a supercritical stage when the decision is fixed (the system will collapse), just a phase change as described above. It is likely that it will start from a subcritical “random” layout of nodes with, probably, p [+1] = 0.5 and p [−1] = 0.5.
On the other hand, if we make the question “Is Clinton honest?” just after asking “Is Gore honest?” the starting layout will be the one left by the previous answer (or, best, decision) or at least be influenced by it. In other words, the starting layout of the second question for CIBR could not be a “neutral” one (p [+1] = 0.5 and p [−1] = 0.5) but of course related to the outcome layout of the previous answer. For instance, suppose the first question drove the system to a supercritical “+1” state. Now, for the second question, the system goes back to the critical phase (in order to make a second decision) from the +1 supercritical branch; the critical “decision” will not start from a random layout but from a layout with, somehow, a clear and definite majority of +1 elements so it is likely to be influenced by that.
Actually, it could be argued that in the “long term” (after one hundred thousand iterations), it is expected that the CIBR with a committed minority at criticality will reach a definite (a definite number of +1 and −1 elements) independently from the initial condition, being mainly influenced by the committed minority. Indeed, it is also expected that, at least in the early iterations, that the story of the system would be different, depending on the starting layout.
In order to verify this, we carried out two different simulations:
Scenario A: a committed minority at −1 in a sea of +1 elements (i.e., all the elements not in the committed minority are in the state +1);
Scenario B: a committed minority at −1 in a sea of −1 elements (i.e., all the elements not in the committed minority are in the state −1).
Simulation results strongly agree with our prediction (see
Appendix B): there is an initial interval that is different according to the initial starting layout so it can lead to different outcome probabilities.
Therefore, the failures of commutativity in decision making (whereby asking the same two questions in different orders can lead to changes in responses) could be due to the different CIBR dynamics. When we ask: “Is Clinton honest?”, the system at criticality leads to a layout (stage 1) which “collapses” to a definite supercritical state (stage 2). Just after that, the second question “Is Gore honest?” will move the system again to a critical state in order to make a second decision, but coming from a supercritical state where almost all elements have a definite value −1 (or +1), thus changing the outcome probability of the second question, having different outcomes at least in the first iterations.
It should be pointed out that this difference becomes evident only when there are very different initial layouts (e.g., when coming from different supercritical phases). Notably, this is consistent with the QP interpretation of different bases cited above.