Busemeyer, Jerome R.

Pothos, Emmanuel M.

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Emmanuel M. Pothos

Jerome R. Busemeyer

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Formalizing Heuristics in Decision-Making: A Quantum Probability Perspective

Emmanuel ePothos

Jerome R Busemeyer

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Formalizing heuristics in decision making: a quantum probability perspective

Formalizing heuristics in decision-making: a quantum 
probability perspective
Emmanuel M. Pothos1* and Jerome R. Busemeyer2
1 Psychology Human Sciences, Swansea University, Swansea, UK
2 Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN, USA
*Correspondence: e.m.pothos@swansea.ac.uk
One of the most influential research pro-
grams in psychology is that of Tversky and 
Kahneman’s (1973, 1983) on heuristics and 
biases in decision-making. Two character-
istics of this program are, first, compelling 
empirical demonstrations that in some 
decision-making situations naïve observers 
violate the rules of classic probability (CP) 
theory and, second, that corresponding 
behavior can be explained with simple heu-
ristics. Tversky and Kahneman’s work has 
led to a vast literature on what is the basis for 
psychological process in decision-making. 
Note that their work, however impactful, 
has not settled the debate of whether CP 
theory is suitable for modeling cognition 
or not. CP models have attracted enormous 
interest and they often do provide excel-
lent coverage of cognitive processes (e.g., 
Oaksford and Chater, 2007; Griffiths et al., 
2010; Tenenbaum et al., 2011).
The idea of heuristics is appealing. First, 
they are simple. The assumption that human 
cognition is based on heuristics partly avoids 
the computational intractability problems 
which plague some formal approaches (cf. 
Sanborn et al., 2010). Second, they often 
allow an understanding of one process in 
terms of theory developed for other cogni-
tive processes. Consider the representative-
ness and availability heuristics. According 
to the representativeness heuristic, judg-
ments of frequency are driven by similarity 
and according to the availability heuristic 
by the ease of identifying related instances 
in memory. Thus, with these heuristics, an 
explanation for decision-making becomes 
one of similarity or memory. Third, heu-
ristics often have strong empirical support. 
Tversky and Kahneman’s approach has been 
to motivate explanations based on heuristics 
by providing compelling demonstrations for 
violations of the standard approaches (in 
decision-making, CP theory). Other propo-
nents of heuristic approaches have argued 
that heuristic schemes lead to better results 
(e.g., Gigerenzer and Todd, 1999).
There is nothing wrong with heuristic 
approaches. But, there is a sense in which 
theoreticians have a bias for cognitive mod-
els based on formal frameworks, whether it 
is Bayesian probability, formal logic, or the 
quantum probability (QP) theory, which 
we discuss (cf. Elqayam and Evans, 2011). 
The properties of formal frameworks are 
interconnected. For example, all expres-
sions in classical probability theory are 
based on a handful of axioms. Thus, one 
cannot accept the psychological relevance 
of one expression, but reject another: they 
are all related to each other. By contrast, 
heuristics, however successful, are some-
what interchangeable. Postulating the 
relevance of the representativeness heu-
ristic does not necessitate the relevance 
of the availability heuristic (Pothos and 
Busemeyer, 2009a).
The QP research program in psychol-
ogy partly originated as an attempt to rec-
oncile people’s violations of CP theory in 
decision-making situations with formal 
theory and examine whether it is pos-
sible to express formally some of the key 
heuristics in decision-making. QP theory 
is a theory for assigning probabilities to 
observables (Isham, 1989). Physicists 
are happy to employ CP theory in most 
cases but they believe that, ultimately, QP 
theory is the more appropriate choice. CP 
theory works by defining a sample space 
and expressing probabilities in terms of 
subsets of this space. A key property of 
this approach is the commutative nature 
of events and subsequent order independ-
ence for probabilities assigned to the joint 
events. QP is a geometric approach to prob-
ability. Events correspond to different sub-
spaces and probabilities are computed by 
projections to these subspaces (note that 
projections have been discussed before in 
psychology; Sloman, 1993). Crucially, this 
makes probability assessment potentially 
order and context dependent and, e.g. (a 
suitable definition of), conjunction can fail 
commutativity. This and related interfer-
ence effects lead to interesting predictions 
from QP theory.
In the famous Linda experiment (Tversky 
and Kahneman, 1983), participants are told 
about Linda, who sounds like a feminist and 
are then asked to judge the probability of 
statements about her. The important com-
parison concerns the statements “Linda 
is a bank teller” and “Linda is a feminist 
and a bank teller.” The first statement is 
extremely unlikely. The second statement 
is a conjunction of the first statement and 
another one. Thus, according to CP theory, 
P(bank teller) ≥ (bank teller ∧ feminist). But, 
results violate CP theory, as most partici-
pants consider the statement “Linda is a 
bank teller and a feminist” as the more 
probable one (this is called the conjunction 
fallacy). Tversky and Kahneman’s explana-
tion was that cognitive process is not based 
on CP theory, rather, participants employ a 
representativeness heuristic. They consider 
Linda as a very typical feminist, so that the 
characterization “bank teller and feminist” 
is probable, regardless of the bank teller 
part. One could also invoke an availability 
heuristic (as Tversky and Koehler, 1994 later 
did), whereby the statement “bank teller and 
feminist” activates memory instances simi-
lar to Linda.
Figure 1 illustrates the QP theory expla-
nation of the conjunction fallacy. The state 
vector is labeled as Psi and corresponds to 
what participants learn about Linda from 
the story. One 1D subspace corresponds 
to Linda being a feminist and another to 
a bank teller. We compute the probability 
for each possibility by projecting the state 
vector onto the corresponding subspace 
and squaring the length of the projection. 
If participants are asked to evaluate the 
probability that Linda is a just bank teller or 
just a feminist this is very unlikely and likely 
respectively. In QP theory, conjunction has 
to be typically defined as a sequential opera-
tion, i.e., Prob(A ∧ B) ≡ Prob(A ∧ then B). 
www.frontiersin.org November 2011 | Volume 2 | Article 289 | 1
OpiniOn Article
published: 15 November 2011
doi: 10.3389/fpsyg.2011.00289
cooperating because they imagine the other 
person is willing to cooperate as well (Shafir 
and Tversky, 1992, called this idea wishful 
thinking). One could also apply Tversky 
and Shafir’s (1992) suggestion that viola-
tions of the sure-thing principle can arise 
from a failure of consequential reasoning 
(this idea was put forward for the two-stage 
gambling task). In the known-defect situ-
ation there is a good reason to defect and 
likewise for the known – cooperate situa-
tion. But, in the unknown conditions it is 
as if the (separate) good reasons for defect-
ing under each known condition cancel out 
(Busemeyer and Bruza, 2011, Chapter 9)!
Pothos and Busemeyer (2009a,b) cre-
ated a quantum and classical model for 
violations of the sure-thing principle. Both 
models assumed that the state vector in the 
unknown case is a convex combination of 
the states in the known-defect and known-
cooperate cases. Then, there is a process of 
evolving the state according to the relative 
payoff for different options and the cogni-
tive dissonance principle. In both the QP 
and the CP case, the probability of defect-
ing is determined by this evolved state. But, 
in the classic case, whatever the process of 
evolution, the evolved representation (vec-
tor) is still a convex combination of the 
known-defect, known-cooperate cases, 
which means that the CP model is always 
constrained by the law of total probabil-
ity. By contrast, in the QP case probabili-
ties are determined from the state vector 
by a squaring operation. For example, 
|a+b|2 = a2 + b2 + a*b + b*a The last two 
terms are interference terms and they can 
be negative, so that |a + b|2 < a2 + b2, violat-
ing the law of total probability. Thus, the 
QP model allows an expression of the idea 
that individually perfectly good reasons or 
causes (high a2, high b2) can partly cancel 
each other out. Note, further, that although 
the utility representation in the quantum 
model is simple (there is a utility parameter, 
analogous to that in more standard decision 
models, like Kahneman and Tversky’s, 1979, 
prospect theory), the possibility of interfer-
ence effects would allow, e.g., a consistent 
preference for a risky option, over the sure-
thing (i.e., a stable risk preference).
These are promising results for QP the-
ory. Its features which make us optimistic 
are that probability assessment is context- 
and order-dependent, so that earlier com-
ponents in a process can affect later ones. 
similarity between the initial representation 
(the initial information about Linda) and 
the representation for a bank teller. From a 
quantum theory perspective, representative-
ness, being a similarity process, is subject to 
chain and context effects, and this is exactly 
what happens in the Linda example. An 
alternative perspective is that seeing Linda 
as a feminist increases availability for other 
related information about Linda, such that 
Linda might be a bank teller. Briefly, this 
is the quantum theory explanation for the 
conjunction fallacy (Busemeyer et al., 2011).
Quantum probability theory has been 
applied in other decision-making situa-
tions (e.g., Trueblood and Busemeyer, 1992; 
Atmanspacher et al., 2004; Khrennikov, 
2004; Aerts, 2009). We next consider an 
application which illustrates a differ-
ent aspect of the theory. According to the 
sure-thing principle, if you intend to do 
A when B is true and you intend to do A 
when B is not true, then you should still 
intend to do A if you do not know if B is 
true or not. The sure-thing principle fol-
lows from the law of total probability in CP 
theory, P(A) = P(A ∧ B) + P(A ∧ not B). 
Surprisingly, Shafir and Tversky (1992) 
reported violations of the sure-thing prin-
ciple in a prisoner’s dilemma task. In their 
experiment, the matrix of payoffs was set 
up so that participants preferred to defect, 
knowing that the other person had already 
defected and knowing that the other person 
had cooperated. However, many partici-
pants reversed their judgment and decided 
to cooperate, when they did not know the 
other player’s action. Such a finding can be 
partly explained with cognitive dissonance 
theory (e.g., Festinger, 1957), accord-
ing to which people change their beliefs 
to be consistent with their actions. Thus, 
if participants have a cooperative bias, in 
the “unknown” condition, they might be 
Assume that in decision-making the more 
probable statement is evaluated first (this 
means that more probable statements are 
more likely to be included in the decision-
making process; cf. Gigerenzer and Todd, 
1999). Then, the probability computation 
involves projecting first to the feminist ray 
and then to the bank teller ray. The first pro-
jection is fairly large, we knew this already. 
The critical point is that from the feminist 
ray, there is now a sizeable projection onto 
the bank teller ray. Thus, whereas the direct 
projection to the bank teller one was small, 
the indirect projection (via the feminist ray) 
is much larger. Such a scheme can account 
for violations of the conjunction fallacy 
(and many other related empirical results; 
Busemeyer et al., 2011).
What is the implication about psycho-
logical process implied in the quantum 
theory model? In classical probability 
theory it has to be the case that Prob(bank 
teller ∧ feminist) = Prob(feminist ∧ bank 
teller) ≤ Prob(bank teller). But in QP the-
ory, when considering possibilities which are 
represented by subspaces at oblique angles as 
in Figure 1, the assessment of any possibility 
is dependent on the assessment of previous 
possibilities. In the case of the conjunctive 
statement in the Linda problem, assessing 
the possibility that Linda is a bank teller 
depends on the previous consideration that 
Linda is a feminist. Clearly, the Linda story 
makes it very unlikely that Linda is a bank 
teller. But, feminists can have all kinds of dif-
ferent professions and, even though being a 
bank teller is perhaps not the most likely one, 
it is still a plausible profession. Therefore, 
once a participant has accepted that Linda 
is a feminist, it becomes easier to think of 
various professions for Linda, including 
that of a bank teller. That is, according to 
the quantum model, accepting Linda as a 
feminist, allows the system to establish a 
Figure 1 | A simple explanation for how QP theory can account for the conjunction fallacy.
Pothos and Busemeyer Quantum probability in decision-making
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Citation: Pothos EM and Busemeyer JR (2011) Formalizing 
heuristics in decision-making: a quantum probabil-
ity perspective. Front. Psychology 2:289. doi: 10.3389/
fpsyg.2011.00289
This article was submitted to Frontiers in Cognition, a 
specialty of Frontiers in Psychology.
Copyright © 2011 Pothos and Busemeyer. This is an open-
access article subject to a non-exclusive license between the 
authors and Frontiers Media SA, which permits use, dis-
tribution and reproduction in other forums, provided the 
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