Decision theory with a state of mind represented by an element of a Hilbert space: The Ellsberg paradox

Abstract

In this paper, we propose an interpretation of the Hilbert space method used in quantum theory in the context of decision making under uncertainty. For a clear comparison we will stay as close as possible to the framework of SEU suggested by Savage (1954). We will use the Ellsberg (1961) paradox to illustrate the potential of our approach to deal with well-known paradoxes of decision theory.

Full text

Decision Theory with a Hilbert Space as
Possibility Space ∗
J¨urgen Eichberger
Alfred Weber Institut, Universit¨at Heidelberg
Hans J¨urgen Pirner
Institut f¨ur Theoretische Physik, Universit¨at Heidelberg
July 19, 2017
Abstract
In this paper, we propose an interpretation of the Hilbert space method
used in quantum theory in the context of decision making under uncer-
tainty. For a clear comparison we will stay as close as possible to the
framework of SEU suggested by Savage (1954). We will use the Ellsberg
(1961) paradox to illustrate the potential of our approach to deal with
well-known paradoxa of decision theory.
Keywords: Decision theory, uncertainty, Ellsberg paradox, quantum the-
ory, Hilbert space, possibility space
JEL Classification codes: C00, C44, D03, D81
1 Introduction
Subjective Expected Utility (SEU) theory (or Bayesian decision theory) in the
spirit of Savage (1954) has become the almost uncontested paradigm for decision
theory in Economics and Statistics.
In Economics a decision problem is described by a set of states of the world
s∈S, a set of outcomes (or consequences)ω∈Ω and a set of actions a∈A.
States in Sare assumed to be mutually exclusive and to be observed perfectly
after realizations, e.g., for a set of states S:= {s1, s2}state s1could mean ”it
rains”’ and state s2”‘it does not rain”. Outcomes are the ultimate objects
which a decision maker values, e.g., sums of money, consumption bundles, work
conditions, etc. In this sense, these sets contain semantic information.
In its most general form, actions a, or acts as Savage (1954) calls them, are
mappings from states to consequences, a:S→Ω.Hence, the set of potential
∗We would like to thank the Marsilius Kolleg at the Universit¨at Heidelberg. Without the
stimulating discussions with researchers from all disciplines of the humanities and sciencies at
this forum the research in this paper would not have been possible.
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actions is the set of all such mappings: A:= {a|a:S→Ω}.A decision maker
will choose an action a∈Abefore knowing which state s∈Swill be realized.
Once a state sis realized (revealed) the decision maker who chooses action a
obtains outcome a(s).
According to the behavioristic approach of economics, one assumes that
only choices of actions can be observed. Hence, Savage (1954) assumes that
every decision maker’s preferences over these actions can be characterized by
a complete preference order on A. A decision maker is assumed to choose a
rather than bif and only if a”is weakly preferred to” b, a b.
In economic applications, working with objective functions that contain
parameters which one can give semantic interpretations simplifies the analy-
sis. Therefore economists have searched for conditions on the preference order
,axioms as Savage (1954) calls them, characterizing an objective function
V:A→Rwhich represents preference in the sense that abif and only if
V(a)≥V(b).
In a seminal contribution, Savage (1954) provides eight axioms on prefer-
ences which guaranteed the existence of (i) a utility function u: Ω →Rand
(ii) a probability distribution pon Ssuch that the Subjective Expected Utility
(SEU ) functional V(a) := RSu(a(s)dp(s) represents this preference order .
Since the probability distribution pon Sis derived from the subjective prefer-
ences of the decision maker over action in Ait is called a subjective probability
distribution.
From a normative point of view, the SEU model has become widely ac-
cepted as the ”rational” way to deal with decision making under uncertainty.
As a descriptive model of behavior, however, the theory has been less successful.
Laboratory experiments have confronted decision theory with behavior contra-
dicting the assumption that subjects’ beliefs can be modeled by a subjective
probability distribution and that their valuations of random outcomes can be
represented by expected utility. Psychological factors have been put forward to
explain this phenomenon (e.g., Kahneman and Tversky 1979). Ellsberg (1961)
which shows that people systematically prefer choices for which information
about probabilities is available to uncertain risks. Such behavior violates the
basic rules of probability theory.
In response to the Ellsberg paradox and many other discrepancies between
behavior observed in laboratory experiments and behavior suggested by SEU
theory, Schmeidler (1989) and Gilboa and Schmeidler (1989) started off an ex-
tensive literature which tries to accommodate these paradoxes by relaxing the
axioms, mostly the independence axiom, proposed by Savage (1954). Machina
and Siniscalchi (2014) provide an excellent survey of this literature. All these
approaches remain, however, firmly rooted in the Savage framework of states,
acts, and consequences.
In this paper, we propose an interpretation of the Hilbert space method used
in quantum theory in the context of decision making under uncertainty. This
new approach departs from the Savage paradigm of states as ”a description of
the world, leaving no relevant aspect undescribed” ( Savage 1954, p. 9). In
quantum mechanics the state of a system cannot be verified directly in contrast
to measurements which can be observed. For the sake of a clear exposition, we
will stay in our exposition as close as possible to the framework of SEU suggested
by Savage (1954). We will use the two-color urn version of the Ellsberg (1961)
paradox to illustrate the potential of our approach to deal with well-known
2

paradoxes of decision theory.
This article illustrates how one can implement an abstract Hilbert space as
possibility space in decision theory. We propose mathematical methods common
in quantum physics. The difficulties applying quantum mechanics to decision
theory come from the distinct terminologies used in both disciplines. Hence,
we will pay special attention to concepts in both fields which share names but
not meanings. In order to avoid misunderstandings it is necessary to develop a
dictionary which relates concepts from decision theory and quantum mechanics.
Quantum mechanics uses a mathematical formalism that quantifies possibilities
in the form of wave functions. Wave functions are elements of a Hilbert space
and represent probability amplitudes. The Hilbert space does not provide prob-
ability distributions directly. We will argue in this paper that mapping concepts
from quantum mechanics to the theory of decision making under uncertainty
promises a framework in which one can deal with the interrelation between sub-
jective beliefs and objective information about events in a non-Bayesian way.
Applying quantum mechanic methods to describe decision making behavior
of humans does not mean that ”quantum physics determines human decisions”.
Hence, it would be misleading to speak of ”quantum decision theory”. The
fact that electromagnetic waves and water waves can be described by the same
mathematical methods does not mean that electromagnetic waves consist of
water.
Though we hope that our proposed interpretation opens new ways for the
analysis of decision making under uncertainty, our focus in this paper is on a
suggested interpretation of the quantum mechanics framework in the context of
decision theory. Hence, we will not exploit the full power of these methods. It
appears, however, straightforward to extend our approach to more general state
spaces. Yet, such generalizations will be left for further research.
1.1 Literature
There is a small literature which tries to apply methods of quantum mechanics
to decision making in Economics. All these papers use the same mathematical
framework of a Hilbert space of wave functions. The main differences lie in the
interpretation of the objects of the theory. In particular, one needs to give the
”wave functions” and their entanglement and superpositions an interpretation
in an decision context. Moreover, quantum mechanics has no notion of choice
over actions or acts. Hence, differences concern mainly these aspects of the
theory.
In a series of papers1, Diederik Aerts, Sandro Sozzo and Jocelyn Tapia pro-
pose a framework for modelling decision making under ambiguity by quantum
mechanic methods. Aerts, Sozzo, and Tapia (2012) applies a version of this
approach to the three-color urn version of the Ellsberg paradox. Similar to
the approach suggested in this paper, these authors interpret the elements of
a complex Hilbert space as determinants of the probabilities and actions as
projectors from the basic elements of the Hilbert space to outcomes or payoffs.
In contrast to this paper however, they obtain entanglement by projecting the
pre-probabilities of the extreme compositions of the urn.
1Aerts and Sozzo (2011), Aerts and Sozzo (2012b), Aerts and Sozzo (2012a), Aerts, Sozzo,
and Tapia (2012), Aerts, Sozzo, and Tapia (2013).
3

In a second group of papers2, Vyacheslav Yukalov and Didier Sornette model
decision making under uncertainty also by a Hilbert space of wave functions.
Yukalov and Sornette (2011) apply their model to several paradoxes of decision
theory, in particular to an analysis of the disjunction and the conjunction effect.
Their interpretation of wave functions as ”intentions” or ”intended actions” is
quite different from the interpretation suggested in this paper.
2 Individual decision making and the mathe-
matics of quantum mechanics: a suggested in-
terpretation
In this section, we will describe economic decision in terms of the mathematical
concepts provided by quantum mechanics. The most fundamental difference to
Savage‘s SEU approach sketched in the introduction is the fact that states of the
world are no longer observable after they are realized. In quantum mechanics,
”states of the world” in the Savage sense are orthonormal vectors of a general
Hilbert space. General elements of the Hilbert space, which in physics are
wave functions, represent states of the world, because they encode all kind
of information about the decision context, including probability information
about the orthonormal basis states. Most importantly, these states will never
be revealed or observed directly. What can be observed however, in a general
state are the probabilities of the basis states. These probability distributions
over basis states can be obtained by projections of the general state onto the
basis states. As we will show in this section, actions will also contain projectors
on the basis states.
For simplicity, we choose as starting point a finite dimensional Hilbert space
H:= C|Ω|spanned by a complete orthonormal set of basis states Ψω:= |ω >∈
H. In the following we will use interchangeably the symbols Ψ and the Dirac
notation |Ψ>for the elements of the Hilbert space. General states of the world
are linear combinations of these basis states |β >=PΩ
ω=1 βωΨω∈H. Each
state contains several complex amplitudes. Projectors of the composite state on
basis states Pω:= |ω >< ω|give the probability of finding the basis state ωin
the complex state |β >. Note, however, that the complex coefficients of a state
|β > contain more information than probabilities over outcomes.
An action αis a mapping Aα:= P
ω
αωPωfrom composite states to outcome-
specific utility payoffs, i.e., a weighted sum of projection operators which map
a composite state |β > into another state of the Hilbert space P
ω
αωPω|β >.
The average utility U(α, β) from an action operator Aαin a state |β > ∈H
is obtained by evaluating the expectation value of the action in a composite
state. In this sense, the state of the world |β > allows to identify a probability
distribution over outcomes in Ω. The expected utility of action αin the state
|β > is:
U(α, β) :=< β|Aα|β >=X
ω
u(α, ω)|βω|2.(1)
2Yukalov and Sornette (2010), Yukalov and Sornette (2011), Yukalov and Sornette (2012).
4

For a basis state |ω >, the action αreveals the utility obtained from the
action if the outcome ωwould occur with certainty:
u(α, ω) := U(α, ω) =< ω|Aα|ω >=αω.
In any state β, decision makers are assumed to choose among a set of actions
Aaccording to their expected utility arg max{U(α, β)|α∈ A}.
Remark 2.1 In the ”‘quantum”’ description one has to differentiate clearly
between wave functions or vectors and the operators or matrices, It is a lin-
ear theory, therefore all operations are simple and do not allow functions of
functions.
The following example illustrates this framework.
Example 2.1 Consider an urn containing 100 balls numbered from 1 through
to 100. Basis states of the world correspond to situations where a ball with a
particular number is drawn from this urn. Hence, the basis states Ψωcorrespond
to states where a ball with a particular number is drawn. General composite
states are complex-valued weighted sums of the basis states |β >=PωβωΨω
and contain information about the relative frequencies of the outcomes. In this
case, the Hilbert space H=C100.
Betting 100 Euros on a ball with an even number ωbeing drawn from the urn
is an action Aαe=P
ω
αe
ωPωwith
αe
ω=100 for ωeven
0for ωodd .
The action aeof betting on an even number induces a utility vector3over out-
comes yielding in state |β > the expected utility
U(αe, β) :=< β|Aαe|β >=X
ω∈Ω
ωeven
100|βω|2.(2)
Similarly, a bet on the odd numbers Pαo=P
ω
αo
ωPωwith αo
ω=100 for ωodd
0for ωeven
yields an expected utility of U(ao, β) =< β|Aαo|β >. We assume that, in a
state β, a decision maker chooses among actions α∈ A := {αe, αo}according
to whether U(αe, β)≶U(αo, β ).
The following table summarizes the basic framework of a decision problem:
3In this example, we consider the special case of a linear utility payoff function, i.e., of risk
neutrality. It would be straightforward to assume a non-linear utility of payoffs.
5

•Hilbert space H:= C|Ω|,
•Basis states Ψω:= |ω >∈H,
(complete set of normalized wave functions )
•Composite states of the world |β >:= Pωβω|ω >∈H,
(also normalized)
•Probability of an outcome ωin β:< β|Pω|β >=|βω|2,
•Actions Aα:= P
ω
αωPω=P
ω
αω|ω >< ω|,
•Payoffs of action α|α >=Aα|β >=P
ω
αωPω|β >∈H,
•Expected payoff of action αin state β:U(α, β ) :=< β|Aα|β > .
The expected payoff of an action αin a composite state β, U (α, β),can be
expanded as follows:
U(α, β):=< β|Aα|β >=< β|X
ω
αωPω|β >
=X
ω
αω< β|Pω|β >=X
ω
αω|βω|2.
In this framework, an action αassigns payoffs (in terms of utility) to the
outcomes of a composite state.
As in Anscombe and Aumann (1963) a state will be associated with a proba-
bility distribution over outcomes. In contrast to Anscombe and Aumann (1963),
however, we do not assume that states can be observed directly. Instead, the
possible states of the world determine the probabilities over outcomes by projec-
tion operators whose expectation values give the probabilities of the outcomes.
A decision maker’s information about possible probability distributions over
outcomes is encoded in the states of the world.
The following example illustrates these concepts in the context of a simple
portfolio choice problem.
Example 2.2 (Portfolio choice) Consider two assets, a stock aand a bond
b. The stock pays a state contingent payoff (ruq0, rdq0).The bond pays off the
same rin each state.
price quantity payoff
state ustate d
stock: q0a ruq0rdq0
bond: 1 b r r
Basis states are Ω = {u, d}reflecting the per unit return in each state, ru> rd.
The Hilbert space Hspanned by the two basis states |u > and |d > contains
6

composite states |β >=βu|u > +βd|d > which capture all information available
in state β. An investor chooses a portfolio α= (a, b)subject to a budget con-
straint q0a+b=W0,where W0denotes the initial wealth of the investor. For
an investment α= (a, b)and a state β, the investor expects an expected utility
of
U(α, β) = u(α, u)|βu|2+u(α, d)|βd|2
=u(rua+rb)|βu|2+u(rda+rb)|βd|2.
Given state β, a decision maker will choose an action α∗from a set Awhich
yields the highest expected utility U(α, β),i.e., U(α∗, β )≥U(α, β) for all α∈ A.
Notice, however, that, in contrast to classical economic decision theory, the
optimal choice α∗will depend on βthe state which determines the expected
utility.
Remark 2.2 Since states βin the Hilbert space Hcannot be observed directly,
observing the action chosen by a decision maker will in general not reveal the
state βcompletely.
2.1 Subjective states of mind
The main conceptual contribution of our paper is the assumption that a decision
maker’s subjective attitudes towards the available information can be modelled
by a subjective state of mind. States of mind are elements of the Hilbert space
Hwhich represent subjective information about the environment and the de-
cision makers attitude towards such information. In particular, the subjective
state of mind interacts with the actual and potential states of the world4. In
analogy to to a subjective prior in Bayesian decision making, the subjective
state of mind weighs the information contained in the states and assesses their
relevance for the choice of actions. Other than a subjective Bayesian prior dis-
tribution, however, the notion of a state of mind captures also psychological
features of information processing and distortion. A state of mind represents
the decision maker’s perception of the uncertainty. As we will explain in detail
below, the decision maker’s perception of uncertainty will change in the light of
new information. Such information is also encoded in elements of the Hilbert
space and will interact with the subjective state of mind. This interaction al-
lows us to use the power of quantum mechanics to model ”entanglements” and
”superpositions”.
Formally, a state of mind ΨMis a composite state |M > in H. It is associated
with the projector PM=|M >< M |which projects any state Ψβon to ΨM:
PM|Ψβ>=cβ|M > . (3)
Suppose a decision maker with a state of mind ΨMis confronted with in-
formation of another state Ψβ.Although Ψβrepresents a normalized state, the
moderated state of mind PM|Ψβ>is no longer normalized. In general the co-
efficient cβis complex, but the absolute magnitude |cβ|of the projection can be
4Yukalov and Sornette (2011, p. 290) introduce the concept of a ”space of mind” as
the direct product of ”mode spaces, which can be thought of as a possible mathematical
representation of the mind” in order to model mental aspects of a decision maker. This
construct appears to be quite different from our notion of a mind state.
7

used as a measure of distance in information of state |β > from the subjective
”mind state”. The more distant the state |β > is from the mind state |M > the
smaller the length of the projection (c.f. Fig.1).
Figure 1: Schematic picture showing the effect of the projection of a composite
state on the mind state. The length of the projection indicates its similarity
with the mind state (in general, the coefficients cwill be complex numbers)
To assess the information about the basis states |i > contained in an ar-
bitrary state of the world when the mind state is present, one evaluates the
expected value of the product of the projectors PMPiPM. For a composite state
|Ψβ>one gets:
< β|PMPiPM|β > =< β|M >< M|Pi|M >< M|β > (4)
=|cβ|2< M|Pi|M > (5)
where
cβ=< M|β > . (6)
As a result one obtains the probability for the basis state |i > in the mind
state ΨMmultiplied with the absolute square of the amplitude |cβ|. The weight
|cβ|2indicates the extent to which the state |β > agrees with the mind state
|M >. For |β >=|M > the decision maker’s state of mind will be consistent
with the actual state information, |cβ|2= 1.
This framework allows us to study how the subjective state of mind ΨM
interacts with actual information. The projections on to the basis states can
be interpreted as a core belief and the possible distances |cβ|as measures of
ambiguity regarding the information contained in other states.
In the following section we will illustrate how this general framework can
model behavior in the two-state Ellsberg paradox.
8

3 The Ellsberg experiment
In a seminal paper published in 1961, Ellsberg (1961) suggested the following
thought experiment. Consider two urns each containing 100 balls (see Fig.2).
The balls are either black or white. It is known that the first urn contains exactly
50 black and 50 white balls. For the second urn however, the composition of
the colors is unknown.
Figure 2: The two Ellsberg urns: Urn 1 on the left contains well defined pro-
portions of black and white balls, whereas for urn 2 on the right the proportions
are unknown.
Subjects hold bets on the color of the balls to be drawn from the urns. A bet
on black ”b” yields 100 Euro if the ball drawn from the urn is black. Otherwise
the payoff is zero. Similarly, betting on white ”w”, the subject earns 100 Euro
if a white ball is drawn and nothing otherwise. Table 1 summarizes this choice
problem.
Urn 1 Urn 2
50 50 100
Black White Black White
b(bet on black) 100 0 100 0
w(bet on white) 0 100 0 100
Holding a bet first on black (b) and then on white (w), subjects had to choose
the urn on which they wanted to bet.
A large number of subjects, about 60 percent, choose to bet on the draw
from Urn 1 for both bets5. These choices clearly contradict the assumption that
these subjects were maximizing SEU.
Denoting by pu(B) the probability of the event Bthat a black ball is drawn
from urn u,u= 1,2,the expected utilities of a bet on color Bfrom Urn
5The fact that subjects prefer to bet on the urn with the known proportion of colors could
be confirmed in many repetitions of the Ellsberg experiment. Oechssler and Roomets (2015)
reviewed 39 experimental studies of the Ellsberg experiments. They report a median number
of 59 percent of ambiguity averse subjects across all studies which they reviewed.
9

uis Uu(b) = pu(B)u(100) + [1 −pu(B)] u(0) and a bet on Wfrom Urn uis
Uu(w) = pu(B)u(0) + [1 −pu(B)] u(100).Obviously, choosing Urn 1 for both
bets leads to a contradiction since
•U1(b)> U2(b) implies p1(B)> p2(B),
•U1(w)> U2(w) implies p1(B)< p2(B).
Thus, the assumption of the decision maker maximizing SEU fails. No-
tice that for this conclusion it does not matter what utilities u(100) and u(0)
are attached to outcomes, i.e., independent of risk attitude. We only require
u(100) > u(0).
3.1 Modelling the Ellsberg experiment by a Hilbert space
The special feature of the Ellsberg experiment, where subjects face the same
bets on the same type of urn, lies in the fact that the information about the
composition of the urn is precise in case of Urn 1 and imprecise in case of Urn
2. We interpret this experimental arrangement in such a way that the precision
of information is reflected in two mind states of the decision maker, i.e., the
two urns exist separately in the imagination of the experimenter. We therefore
model them as two mind states in the same Hilbert space.
The basis states |ω > of the Hilbert space consist of the two exclusive cases
(outcomes of the experiment) ΨW=|W >, i.e. ”a white ball is drawn from the
urn” and ΨB=|B > i.e., ”a black ball is drawn from the urn” . These basis
states are exclusive i.e. orthogonal and normalized, i.e., < B|W >=< W |B >=
0 and < B|B >=< W |W >= 1. Composite states |β >=βB|B > +βW|W >
of the Hilbert space Hspanned by these basis states are characterized by the
complex weights β= (βB, βW).
The information of the decision maker regarding the composition of the urns
is encoded in the respective composite states (wave functions6) for the different
urns. Urn 1, the urn with the known proportion of 1
2black and white balls, is
described by the composite wave function Ψ1
Ψ1=1
√2(ΨB+ ΨW)
and Urn 2, for which the composition is unknown, is described by the wave
function Ψ2(x)
Ψ2(x) = xΨB+p1−x2ΨW
Without reducing the generality of our further arguments we can choose x
real, any choice of complex phases in Ψ2can be absorbed in the final phase of
the mind state.
The parameter xcontains the information about the composition of Urn 2.
Since it is known that there is a finite number of 100 balls in Urn 2, we assume
that the number of possible states Ψ2(x) of the Hilbert space His finite.
The probabilities of the outcomes, a white ball is drawn from the urn Wor
a black ball is drawn from the urn B, are obtained by the projection operators
Pω, the sum of which yields the identity operator:
6Griffiths (2002) (p. 203) calls these wave functions ”pre-probabilities”.
10

PB=|B >< B|, PW=|W >< W |,with PB+PW= 1.
The probabilities of the outcomes are the expectation values of these projection
operators evaluated with the respective composite state. The expectation val-
ues of the projection operators assign probabilities to the outcomes given the
information of a specific state of the world, Ψ1or Ψ2(x):
•Urn 1: p1(B) =<Ψ1|PB|Ψ1>and p1(W) =<Ψ1|PW|Ψ1>,
•Urn 2: p2(B) =<Ψ2(x)|PB|Ψ2(x)>and p2(W) =<Ψ2(x)|PW|Ψ2(x)> .
The two bets are modelled by the operators AB=u(100)PB+u(0)PW
and AW=u(0)PB+u(100)PW. Hence, given the information about Urn 1
contained in Ψ1,one obtains the following expected utilities from these actions:
U(b, 1) = <Ψ1|AB|Ψ1>=1
2u(100) + 1
2u(0),
U(w, 1) = <Ψ1|AW|Ψ1>=1
2u(0) + 1
2u(100).
Similarly, given the information regarding Urn 2, one has
U(b, x) = <Ψ2(x)|AB|Ψ2(x)>=x2u(100) + 1−x2u(0),
U(w, x) = <Ψ2(x)|AW|Ψ2(x)>=x2u(0) + 1−x2u(100).
Given this information about the urns and no subjective processing of informa-
tion, the Ellsberg puzzle would remain unresolved. A choice of betting on Urn
1 in both cases would yield for any x∈ R,
•U(b, 1) > U(b, x) =⇒1
2> x2,
•U(w, 1) > U (w, x) =⇒1
2< x2.
3.2 Solution of the paradox: a subjective mind state
Applying the notion of a mind state represents the decision makers assessment
of the situation given her information. In case of Urn 2, all the person knows
is the total number of balls in the urn. Hence, her mind state is a subjectively
determined general state from the Hilbert space H
ΨM2(y, d) := y|B > +eid p1−y2|W >
which is characterized by the parameters (y, d).In general, different decision
makers will have different initial states of mind ΨM2(y, d).
All decision makers are however confronted with the same information about
Urn 2
Ψ2(x) = xΨB+p1−x2ΨW
11

with the unknown parameter7x∈ R.
We assume that the mind state ΨM2(y, d) distorts the perception of the basis
states of the world represented by the projectors Pω:= |ω >< ω|,(ω=B, W ),
i.e., the operators to draw a black or a white ball,
PB→PM2PBPM2,
PW→PM2PWPM2,
where
PM2=|M2>< M2|
=y2e−idyp1−y2
eidyp1−y21−y2
is the projector of the mind state. For the mind state, the amplitudes reflect
the subjective distortions. Hence,
hΨ2(x)|PM2PBPM2|Ψ2(x)i=<Ψ2(x)|M2>< M2|PB|M2>< M 2|Ψ2(x)>
=y2|cx|2.
with
y2=< M2|PB|M2>
|cx|2=|< M2|Ψ2(x)>|2
=1−x2−y2+ 2x2y2+ 2xyp(1 −x2)(1 −y2) cos d
One sees how the mind state overwrites the state Ψ2(x) of Urn 2, and mod-
ifies the resulting probability with the overlap probability of the two states.
This overlap probability contains an interference term which is visible from the
trigonometric function. It encodes the reliability the person assigns to his/her
estimate of this probability. Similarly, one obtains the probability of Wfrom
the distorted state as
hΨ2(x)|PM2PWPM2|Ψ2(x)i=<Ψ2(x)|M2>< M2|PW|M2>< M 2|Ψ2(x)>
= (1 −y2)|cx|2.
with |cx|2as above and
1−y2=< M2|PW|M2>
In the case of Urn 1, the decision maker knows the composition of the urn.
Hence, her mind state should be one of subjective certainty
|ΨM1>=|Ψ1>=1
√2(ΨB+ ΨW)
7In case of the information, we will abstract from distortions of the amplitude.
12

where the state of mind |ΨM1>coincides with the actual information |Ψ1>.
Hence, we have
<Ψ1|PM1PωPM1|Ψ1>=|<ΨM1|Ψ1>|2<ΨM1|Pω|ΨM1>=1
2
for ω=Wand for ω=B.
Given the action projectors of the two bets AB=u(100)PB+u(0)PWand
AW=u(0)PB+u(100)PW,one obtains as the conditions for Ellsberg behavior:
•for the bet on Black b, Urn 1 is preferred if
U(b, 1, M 1) > U(b, x, M 2)
with
U(b, 1, M 1) = hΨ1|PM1ABPM1|Ψ1i
=1
2[u(100) + u(0)] ,
U(b, x, M 2) = |cx|2y2u(100) + 1−y2u(0),
•for a bet on White w, Urn 1 is preferred if
U(w, 1, M 1) > U(w, x, M 2)
with
U(w, 1, M 1) = hΨ1|PM1AWPM1|Ψ1i
=1
2[u(100) + u(0)] ,
U(w, x, M 2) = |cx|2y2u(0) + 1−y2u(100).
For u(0) = 0 and u(100) = 1,we obtain the following conditions:
1
2>|cx|2y2,
1
2>|cx|21−y2.
Assuming no effect of the interference term d=π
2,i.e., cos d= 0,Fig.3 shows
the regions for the Ellsberg choices.
On the horizontal axis we consider values of x∈[0,1] and on the vertical
axis of y∈[0,1].In this diagram we consider the case without interference due
to the complex conjugate since d=π
2implies cos d= 0.In this case, |cx|2sim-
plifies to 1−x2−y2+ 2x2y2.The grey ( in color blue) area contains (x, y)
combinations which correspond to Ellsberg behavior. Integrating, one obtains
an area of approximately 63 percent. In our interpretation, the parameter y
13

0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
y
Figure 3: The figure shows for fixed d=π
2the region 1
2>|cx|2y2and 1
2>
|cx|21−y2as grey ( in color blue) shaded area.
captures the decision maker’s initial disposition in regard to Urn 2. The pa-
rameter x, which is measured as the unknown actual probability of the draw of
a black ball, captures the objective circumstances of the situation. Projecting
x, √1−x2onto the mind state y, e π
2ip1−y2yields a mind state which is
affected by the actual situation.
In their review of 39 experimental studies of the Ellsberg paradox, Oechssler
and Roomets (2015) find a wide range of measured ambiguity across the various
studies. These different experimental contexts which include the different ways
of actually choosing the proportion on the balls in the unknown urn reflects the
kind of objective experimental environment which meets the individual states
of minds of the subjects. The resulting behavior is influenced by both factors,
the subjective mind state yand the objective situation x. Measurements such
as the observed behavior and the stated predictions regarding the proportions
of the colors should reflect these parameters. This will be discussed in the next
section.
4 Discussion and Conclusion
The Ellsberg paradox (Ellsberg (1961)) is usually interpreted as a manifestation
of the decision maker’s ambiguity about the unknown proportions in Urn 2.
Faced with a choice between bets on draws from an urn with a known probability
distribution and bets on an urn with an unknown probability distribution people
prefer to bet on draws from the urn with the known probability distribution.
In the two urn problem individual subjects select for all bets the urn where
14

white and a black balls are equally distributed. Such behavior is inconsistent
with maximization of expected utility for any subjective probability distribution
regarding the composition of Urn 2. In the literature on decision making under
uncertainty, most of the suggested solutions of the paradox assume ”ambiguity
aversion” of the decision maker, i.e., the decision maker evaluates bets on Urn
2 according to the worst case of all possible compositions.
The model which we put forward in this paper goes beyond this psychological
explanation by ”pessimism” and invokes the framework of a Hilbert space as
possibility space which captures various aspects of the choice situation including,
but not being confined to, information about the composition of the urns. The
Hilbert space contains the two possible draws from the urns, a black ball or
a white ball, as basis states. Based on the information about the urns, one
constructs wave functions for the respective urn states from these basis states.
These wave functions may contain complex amplitudes entangling the basis
states. Hence, there is more information in these wave functions than in real
probabilities.
In applications to a decision-theoretic context, a crucial role is played by the
subjective mind states which reflect the mental perception (consciousness) of
the subjects in the experiment. The precise information about the content of
Urn 1 we represent by a mind state equal to a preprobability without complex
amplitude. For the second urn, however, the mind state represents other possi-
bilities than the one given by the unknown objective composition of the second
urn. In this case the parameters describe the influence of beliefs, information,
and other context variables on the state of the second urn. The single complex
phase in the mind state summarizes all possible effects of complex amplitudes
for the second-urn wave functions.
For the three parameters (x, y, d), the condition for choices corresponding to
the typical Ellsberg behavior defines a subspace of the feasible space of the three
parameters. Fig. 4 illustrates the entanglement due to the complex amplitude
for the parameter space of our model [0,1] ×[0,1] ×[0, π].
In a first attempt to summarize the results about ambiguity-averse behav-
ior across the 39 Ellsberg experiments in their review, Oechssler and Roomets
(2015) obtain an average ratio remp of approximately 57 percent ambiguity
averse subjects8,
remp ∈[57.2%,57.9%].
Using the ratio of the volume of the subspace of parameters leading to
ambiguity-averse choices to the total volume as a naive measure for the per-
centage of ambiguity averse decisions in the Ellsberg two-urn problem yields
the ratio rquant,
rquant = 58.1% ±0.5%.
Though one obtains a surprisingly similar proportion, we do not claim that
aggregating over all parameter values is an appropriate measure for ambiguity
8The interval for the empirical result does not reflect a statistical error. It is derived from
the average over all existing experiments and over a restricted list omitting experiments with
the highest and lowest results.
15

Figure 4: The figure shows the grey (in color blue) region in the three parameter
space of x, y, d connected to the objective and subjective second urn state with
1
2>|cx|2y2and 1
2>|cx|21−y2
16

averse behavior in our model. A more careful analysis may restrict the range of
reasonable mind states and their complex amplitude (y, d).
The Hilbert space method which we suggest in this paper increases the num-
ber of parameters in general. On the one hand, this allows us to explain results
which are paradoxical in classical decision theory, on the other hand, this exposes
the model to the danger of arbitrariness. In especially designed experiments,
however, one could ask more questions in order to study the role of these ad-
ditional parameters separately. E.g., in the two urn problem one could ask the
subjects for their estimate of the number of black and white balls in the second
urn. This would allow one to fix one of the parameters of the wave function of
the mind state, thus providing a more stringent test of the model.
In Economics and most of the Social Sciences, classical decision theory un-
der uncertainty in the framework of Savage (1954) has been extremely fruitful,
allowing economists to develop new fields such as information economics, con-
tract theory, and financial markets analysis. Human choice behavior as observed
in experiments, however, has cast some doubts on the general accuracy of the
Savage approach as a description of actual behavior. Yet most existing gener-
alizations of the SEU model maintain the basic framework of actions mapping
states into consequences.
The main conceptual difference between the Savage framework and the
Hilbert space approach advanced in quantum mechanics is the notion of a
”state”. Savage (1954) defines ”a state (of the world)” as ”a description of
the world, leaving no relevant aspect undescribed” and ”the true state of the
world” as ”the state that does in fact obtain, i.e., the true description of the
world” (p. 9). In the Savage context, the true state is revealed unambiguously,
e.g., in the Ellsberg paradox the ” true state of the world” is represented by the
color of the ball drawn from an urn.
In quantum mechanics a ”state of the system” cannot be observed directly.
What is observed are measurements. In general, a state of a system contains
more information than can be observed in one measurement. Thus, the Hilbert
space approach takes into account that states of the world may be too complex
to be described or observed in its entirety. From this perspective, the state
of the world in the Ellsberg example comprises the complete environment of
two urns, the composition of these urns, the information about the urns, the
psychological interpretations of the subjects, etc. The observed ”color of the
ball drawn from the urn” is a measurement.
Even in the experimental environment of a laboratory, the (Savage) notion
of a state of the world has proved to be quite elusive. Framing and nudging are
well-known phenomena which contradict the assumption that there is a ”state of
the world” which can be treated as unrelated to the context of the experiment.
Hence, studying complex Hilbert spaces which generate entanglements between
subjective beliefs, information, and other aspect of the environment may be a
useful exercise. We view the possibility space as a first step towards such new
concepts.
Finally, classical decision theory is based on the maximization of utility
and probability theory which allows for information dynamics according to the
Bayesian updating rule. The Hilbert space approach allows also for a dynamic
evolvement. Quantum mechanics has been successful because Erwin Schr¨odinger
introduced a dynamics in the Hilbert space which determines the evolution of
states over time. In neurophysiology, we are far from understanding this dy-
17

namics. Exploring an Hilbert space of states may present a first step towards
understanding the evolvement of the human perception of a complex environ-
ment.
References
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Appendix
A: A short primer on notation and computations
The basis wave functions in the two dimensional Hilbert space are represented
by Dirac kets |>:
ΨB=|B > and ΨW=|W > .
They may be written as two dimensional column vectors:
|B >=1
0and |W >=0
1.
Two normalized wave functions characterize the two urns:
Ψ1= 1
√2
1
√2!,
Ψ2(x) = x
√1−x2.
The parameter xreflects the unknown number of black and white balls in Urn
2.
For the mind state, however, complex amplitudes matter. Hence, the ”mind
state” |ΨM>will be represented as a superimposition of the black and white
basis elements. Here we use a complex amplitude. The phases of the other
states can be absorbed in the phase dfor the final result:
ΨM=y
eidp1−y2.
Complex conjugation
The elements of the dual space are obtained by complex conjugation and
transposition. For the black and white states, complex conjugation does not
matter. Hence,
< B|= (1,0) and < W |= (0,1) .
For the mind state, however, one obtains
<ΨM|=y, e−id p1−y2.
19

Notice the complex conjugation in the latter case.
Projection operators
Projection operators have the property P2=P. They correspond to observ-
ables and will be realized in measurements, like drawing a black ball or drawing
a white ball. The projection operator for drawing a black ball PB=|B >< B|
is given by the outer product (tensor product) of the vector |B > of the original
Hilbert space with the vector < B|in the dual Hilbert space, i.e.,
PB=1 0
0 0 .
Similarly, the projection operator PWon white has the form:
PW=0 0
0 1 .
The probability to obtain a black ball in urn 1 can be obtained from the
expectation value of the projection operator on Black with the wave function of
Urn 1:
p1(B) = <Ψ1|PB|Ψ1>
=1
√2,1
√2 1 0
0 0  1
√2
1
√2!=1
2.
It involves the multiplication of three matrices: one row vector of length 2
for <Ψ1|, the 2x2 matrix for the projection operator, and the column vector
of length 2 for |Ψ1>. The probability of drawing a black ball from Urn 2 is
computed similarly as
p2(B) = <Ψ2(x)|PB|Ψ2(x)>
=x, p1−x21 0
0 0  x
√1−x2=x2.
Replacing the matrix PBby the matrix PW,one obtains the respective
expressions for the probabilities of drawing a white ball from the urns. In
general, the state of a system can only be described by such measurements.
The mind state projects the state of the urns onto the mind state by the
projection operator PM,
PM=|ΨM>< ΨM|
=y2e−idyp1−y2
eidyp1−y21−y2.
The subjective probability of drawing a white ball from Urn 2 given the mind
20

state ΨMis obtained as
hΨ2(x)|PMPWPM|Ψ2(x)i
=<Ψ2(x)|ΨM>< ΨM|PW|ΨM>< ΨM|Ψ2(x)>
=hx, p1−x2 y
eidp1−y2i
| {z }
<Ψ2(x)|ΨM>
hy, e−idp1−y2  0 0
0 1   y
eidp1−y2i
| {z }
<ΨM|PW|ΨM>
hy, e−idp1−y2  x
√1−x2i
| {z }
<ΨM|Ψ2(x)>
=1−x2−y2+ 2x2y2+ 2xyp(1 −x2)(1 −y2) cos d(1 −y2).
All expressions in the main paper are computable by these matrix operations.
21

B: Diagrams
The following diagrams show the parameter regions of Ellsberg behavior, i.e.
the overlap of the regions which represent
•the preference for betting on black (b) in Urn 1: 1
2>|cx|2y2,
and
•the preference for betting on white (w) in Urn 1: 1
2>|cx|21−y2,
for different values of the parameter d. Below the diagrams, we give the ratios
of the grey (in color blue) areas relative to the total areas in percentages.
d=0
4π d =1
4π
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
y
Grey (in color blue) area: 30% Grey (in color blue) area : 41%
d=2
4π
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
y
Grey (in color blue) area: 63%
d=3
4π d =4
4π
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
y
Grey (in color blue) area: 74% Grey (in color blue) area: 76%
22

Highlights:
•The Hilbert space method used in quantum theory is applied to decision
making under uncertainty.
•States of the world may be too complex to be described or observed in its
entirety.
•The potential of the approach to deal with well-known paradoxa of decision
theory is demonstrated in the context of the Ellsberg two-urn paradox.
23