Handelshochschule Leipzig (HHL)

Transcription

Decision Theory in the Presence
of Uncertainty and Risk
Pierfrancesco La Mura
HHL Working Paper No. 68
Comments are welcome and may be sent to
plamura@hhl.de
© Copyright: 2005
Jede Form der Weitergabe und Vervielfältigung
bedarf der Genehmigung des Herausgebers
Decision theory in the presence of risk and
uncertainty
Pierfrancesco La Mura
Department of Microeconomics and Information Systems
Leipzig Graduate School of Management (HHL)
April 2, 2005
1
Introduction
John von Neumann (1903-1957) is widely regarded as a founding father of many
fields, including game theory, decision theory, quantum mechanics and computer
science. Two of his contributions are of special importance in our context. In
1932, von Neumann proposed a mathematical foundation for quantum mechanics, based on a calculus of projections in Hilbert spaces. In 1944, together with
Oskar Morgenstern, he gave the first axiomatic foundation for the expected utility hypothesis (Bernoulli 1738). In both cases, the frameworks he contributed to
establish are still at the core of the respective fields. In particular, the expected
utility hypothesis is still the de facto foundation of fields such as finance and
game theory.
The von Neumann - Morgenstern axiomatization of expected utility was
immediately greeted as simple and intuitively compelling. Yet, in the course
of time, a number of empirical violations and paradoxes (Allais 1953, Ellsberg
1961, Rabin and Thaler 2001) came to cast doubt on the validity of the hypothesis as a foundation for the theory of rational decisions in conditions of risk and
uncertainty. In economics and in the social sciences, the empirical and prescriptive shortcomings of the expected utility hypothesis are generally well known,
but often tacitly accepted in view of the great tractability and usefulness of the
corresponding mathematical framework. In fact, the hypothesis postulates that
preferences can be represented by way of a utility functional which is linear in
probabilities, and linearity makes expected utility representations particularly
tractable in models and applications.
Interestingly, the empirical violations and paradoxes which motivated the
introduction of the quantum mechanical framework in physics bear a certain
resemblance with some of the violations and paradoxes which came to challenge
the status of expected utility in the social sciences. In physics, quantum mechanics was introduced as a tractable and empirically accurate mathematical
framework in the presence of such violations. In economics, the importance of
1
accounting for violations of the expected utility hypothesis has long been recognized, but so far none of its numerous alternatives has emerged as dominant, for
reasons which range from lack of mathematical tractability to the ad-hoc nature
of some axiomatic proposals. Motivated by these considerations, we would like
to introduce a decision-theoretic framework which accommodates the dominant
paradoxes while retaining significant simplicity and tractability. As we shall see,
this is obtained by weakening the expected utility hypothesis to its projective
counterpart, in analogy with the quantum-mechanical generalization of classical
probability theory.
The structure of the paper is as follows. The next section briefly reviews the
von Neumann-Morgenstern framework. Sections 3 and 4, respectively, present
Allais’ and Ellsberg’s paradoxes. Section 5 introduces a mathematical framework for projective expected utility, and our main result. Section 6 contains a
brief discussion. Sections 7 and 8 show how Allais’ and Ellsberg’s paradoxes, respectively, can be accommodated within the new framework. Section 9 discusses
applications and extensions, then concludes.
2
von Neumann - Morgenstern expected utility
Let S be a finite set of outcomes, and ∆ be the set of probability functions
defined on S, taken to represent risky prospects (lotteries). Next, let % be a
complete and transitive binary relation defined on ∆×∆, representing a decisionmaker’s preference ordering over lotteries. As usual, indifference of p, q ∈ ∆ is
defined as [p % q and q % p] and denoted as p ∼ q, while strict preference of p
over q is defined as [p % q and not q % p], and denoted by p Â q. The preference
ordering is assumed to satisfy the following two conditions.
Axiom 1 For all p, q, r ∈ ∆ with p Â q Â r, there exist α, β ∈ (0, 1) such that
αp + (1 − α)r Â q Â βp + (1 − β)r.
Axiom 2 For all p, q, r ∈ ∆, p % q if, and only if, αp + (1 − α)r % αq + (1 − α)r
for all α ∈ [0, 1].
A functional u : ∆ → R is said to represent % if, for all p, q ∈ ∆, p % q if
and only if u(p) ≥ u(q).
Theorem 3 Axioms 1 and 2 are jointly equivalent to the existence of a functional u : ∆ → R which represents % such that, for all p ∈ ∆,
X
u(p) =
u(s)p(s).
s∈S
2
3
The “double-slit paradox”of decision theory
The following paradox is due to Allais (1953). First, please choose between:
A: A chance of winning 4000 dollars with probability 0.2 (expected value
800 dollars)
B: A chance of winning 3000 dollars with probability 0.25 (expected value
750 dollars).
Next, please choose between:
C: A chance of winning 4000 dollars with probability 0.8 (expected value
3200 dollars)
B: A chance of winning 3000 dollars with certainty.
If you chose A over B and D over C then you are in the modal class of
respondents. The paradox lies in the observation that A and C are special cases
of a two-stage lottery E which in the first stage either returns zero dollars with
probably (1 − α) or, with probability α, it leads to a second stage where one
gets 4000 dollars with probability 0.8 and zero otherwise. In particular, if α is
set to 1 then E reduces to C, and if α is set to 0.25 it reduces to A. Similarly,
B and D are special cases of a two-stage lottery F which again with probability
(1 − α) returns zero, and with probability α continues to a second stage where
one wins 3000 dollars with probability 1. Again, if α = 1 then F reduces to
D, and if α = 0.25 it reduces to B. Then it is easy to see that the [A Â B,
D Â C] pattern violates the Independence axiom, as E can be regarded as a
lottery αp + (1 − α)r, and F as a lottery αq + (1 − α)r, where p and q represent
lottery C and D respectively and r represents the lottery in which one gets
zero dollars with certainty. Why should it matter what is the value of α? Yet,
experimentally one finds that it does.
Allais’ paradox bears a distinct resemblance with the double-slit paradox in
physics. Imagine cutting two parallel slits in a sheet of cardboard, shining light
through them, and observing the resulting pattern as particles going through
the two slits scatter on a wall behind the cardboard barrier. Experimentally,
one finds that when both slits are open, the overall scattering pattern is not
the sum of the two scattering patterns produced when one slit is open and the
other closed. The effect does not go away even if particles of light are shone
one by one, and this is paradoxical: why should it matter to an individual
particle which happens to go through the left slit, when determining where to
scatter later on, with what probability it could have gone through the right slit
instead? In a sense, each particle in the double-slit experiment behaves like a
decision-maker who violates the Independence axiom in Allais’ experiment.
3
4
Ellsberg’s paradox
Another disturbing violation of the expected utility hypothesis was pointed out
by Ellsberg (1961). Suppose that a box contains 300 balls of three possible
colors: red (R), green (G), and blue (B). You win if you guess which color will
be drawn. Do you prefer to bet on red or on green? Many respondents choose
red, on grounds that the probability of drawing a red ball is known (1/3), while
the only information on the probability of drawing a green ball is that it is
between 0 and 2/3. Now suppose that you win if you guess which color will
not be drawn. Do you prefer to bet that red will not be drawn (R) or that
green will not be drawn (G)? Again many respondents prefer to bet on R, as
the probability is known (2/3) while the probability of G is only known to be
between 1 and 1/3.
The pattern [R Â G, R Â G] is incompatible with von Neumann - Morgenstern expected utility, which only deals with known probabilities, and is
also incompatible with the Savage (1954) formulation of expected utility with
subjective probability as it violates its Sure Thing axiom. Hence, the paradox
suggests that subjective and objective uncertainty (risk) should be handled as
distinct notions.
5
Projective expected utility
Let S+ be the positive orthant of the unit sphere in Rn , where n is the cardinality of the set of relevant outcomes S := {s1 , s2 , ..., sn } . Then von Neumann
- Morgenstern lotteries, regarded as elements of the unit simplex, are in oneto-one correspondence with elements of S+ , which can therefore be interpreted
as prospects in which the probabilities are fully known. Observe that, while
the projections of elements of the unit simplex (and hence, L1 unit vectors) on
the basis vectors can be naturally associated with probabilities, if we choose to
model von Neumann - Morgenstern lotteries as elements of the unit sphere (and
hence, as unit vectors in L2 ) probabilities are naturally associated with squared
projections. The advantage of such move is that L2 is the only Lp space which is
also a Hilbert space, and the special properties of Hilbert space, and in particular
its associated calculus of projections, shall yield tractability to the representation. We call such prospects, whereas the only uncertainty is objective, pure
lotteries, and reserve the name mixed lotteries for convex combinations (mixtures) in Rn , which we interpret as situations of subjective uncertainty about
the actual pure lottery faced. Let M be the convex closure of S+ in Rn . Observe
that pure lotteries cannot be obtained as non-trivial mixtures, and that is why
we refer to them as pure.
Next, let hx|yi denote the usual inner product in Rn .
Axiom 4 (Born’s Rule) There exists an orthonormal basis (z1 , ..., zn ) such that,
for all x ∈ M and all si ∈ S,
2
psi (x) = hx|zi i .
4
Axiom 4 requires that there exist an interpretation of elements of M in terms
of uncertainty and risk as perceived by the decision-maker. In the von Neumann
- Morgenstern treatment, Axiom 4 is tacitly assumed to hold with respect to
the natural basis in Rn . By contrast, the preferred basis postulated in Axiom
4 is allowed to vary across different decision-makers, capturing the idea that
the relevant dimensions of risk and uncertainty may be perceived differently
by different subjects, whereas the natural basis represents those dimensions as
perceived by the modeler or an external observer.
In our context, orthogonality captures the idea that two events or outcomes
are mutually exclusive (for one event to have probability one, the other must
have probability zero). The preferred basis captures which, among all the possible ways to partition the relevant uncertainty into a set of mutually exclusive
events or outcomes, leads to a meaningful set of payoff-relevant outcomes to
which well-defined utility values can be assigned. Axiom 4 postulates that each
lottery is evaluated by the decision maker solely by the risk and uncertainty
that it induces on the payoff-relevant dimensions. Once an orthonormal basis is
given, each mixed lottery x P
uniquely corresponds to a function p(x) : S → [0, 1].
If x is a pure lottery, then s∈S ps (x) = 1 and p(x) can therefore be regarded
2
as a probability function. For a general mixed lottery x ∈ M , 1/ kxk can be
interpreted as a measure of uncertainty, and
qsi (x) :=
1
2
hx|zi i
||x||2
as subjective probability. We interpret the vector p(x) as the degree of belief
assigned to each outcome by lottery x. Let B be the set of all p(x), for x ∈ M,
and let % be a complete and transitive preference ordering defined on B ×B. We
postulate the following two axioms, which mirror the ones in the von Neumann
- Morgenstern treatment.
Axiom 5 (Continuity) For all x, y, z ∈ M with p(x) Â p(y) Â p(z), there exist
α, β ∈ (0, 1) such that αp(x) + (1 − α)p(z) Â p(y) Â βp(x) + (1 − β)p(z).
Axiom 6 (Independence) For all x, y, z ∈ M , p(x) % p(y) if, and only if,
αp(x) + (1 − α)p(z) % ap(y) + (1 − α)p(z) for all α ∈ [0, 1].
Observe that the two axioms impose conditions solely on beliefs, and not on
the underlying lotteries. In particular, note that a convex combination αp(x) +
(1 − α)p(y), where x and y are pure lotteries, also corresponds to a pure lottery,
since the convex combination of two probability functions remains a probability
function. Hence, the type of mixing that Axioms 5 and 6 are concerned with
is objective, while subjective mixing is captured by convex combinations of the
underlying lotteries, such as αx + (1 − α)y, which is generally not a pure lottery
even when both x and y are.
5
Theorem 7 Axioms 4-6 are jointly equivalent to the existence of a symmetric
matrix U such that u(x) := x0 U x for all x ∈ M represents %.
Proof. Assume that Axiom 4 holds with respect to a given orthonormal
basis (z1 , , zn ). By the von Neumann - Morgenstern result (which applies to any
convex mixture set, such as B) Axioms 5 and 6 are jointly equivalent to the
existence of a functional u which represents the ordering and is linear in p, i.e.
u(x) =
n
X
u(si )psi (x) =
i=1
n
X
2
u(si ) hx|zi i ,
i=1
where the second equality is by definition of p as squared inner product with
respect to the preferred basis. The above can be equivalently written, in matrix
form, as
u(x) = x0 P 0 DP x = x0 U x,
where D is the diagonal matrix with the payoffs on the main diagonal, P is
the projection matrix associated to (z1 , , zn ), and U := P 0 DP is symmetric.
Conversely, by the Spectral Decomposition theorem, for any symmetric matrix
U there exist a diagonal matrix D and a projection matrix P such that
x0 U x = x0 P 0 DP x
for all x ∈ M. But this is just expected utility with respect to the orthonormal
basis defined by P . Hence, the three axioms are jointly equivalent to the existence of a symmetric matrix U such that u(x) := x0 U x represents the preference
ordering.
6
Properties of the representation
The representation introduced in the previous section generalizes von NeumannMorgenstern expected utility in three directions. First, subject-specific uncertainty (from mixed states) and objective uncertainty (from pure states) are
treated as distinct notions. Second, within this class of preferences both Allais’
and Ellsberg’s paradoxes are accommodated. Finally, the construction easily
2
extends to the complex unit ball, provided that hx|yi is replaced by | hx|yi |2
in the definition of p, in which case Theorem [??] holds with respect to a Hermitian (rather than symmetric) payoff matrix U , and the result also provides
axiomatic foundations for decisions involving quantum uncertainty.
Let ei , ej be the degenerate lotteries assigning probability 1 to outcomes si
and sj , respectively, and let ei,j be the lottery assigning probability 1/2 to each
of the two states. Observe that, for any two distinct si and sj ,
1
1
Uij = u(eij ) − ( u(ei ) + u(ej )).
2
2
6
It follows that the off-diagonal entry Uij in the payoff matrix can be interpreted as the discount, or premium, attached to an equiprobable combination of
the two outcomes with respect to its expected utility base-line, and hence as a
measure of risk aversion along the specific dimension of risk involving outcomes
si and sj .
Observe that the functional in Theorem 7 is quadratic in x, but linear in p.
If U is diagonal, then its eigenvalues coincide with the diagonal elements. In
von Neumann - Morgenstern expected utility, those eigenvalues contain all the
relevant information about the decision-maker’s risk attitudes. In our framework, risk attitudes are jointly captured by both the diagonal and non-diagonal
elements of U . Furthermore, in our setting attitudes towards uncertainty are
captured by the concavity or convexity (in x) of the quadratic form x0 U x, and
therefore, ultimately, by the definiteness condition of U and the sign of its
eigenvalues. Convexity corresponds to the case of all positive eigenvalues, and
captures the idea that risk is ceteris paribus preferred to uncertainty, while
concavity captures the opposite idea.
7
Example: objective uncertainty
Figure [??] presents several examples of indifference maps on pure lotteries
which can be obtained within our class of preferences for different choices of U .
The first pattern (parallel straight lines) characterizes von Neumann - Morgenstern expected utility. Within our class of representations, it corresponds to
the special case of a diagonal payoff matrix U. All other patterns are impossible
within von Neumann - Morgenstern expected utility. The representation is sufficiently general to accommodate Allais’ paradox from section [??]. In the context
of the example of section [??], let {s1 , s2 , s3 } be the states in which 4000, 3000,
and 0 dollars are won, respectively. To accommodate Allais’ paradox, assume a
slight aversion to the risk of obtaining no gain (s3 ):
s
U= 1
s2
s3
s1
13
0
−1
s2 s3
0 −1
.
10 −1
−1 0
Let the four lotteries A, B, C, D be defined, respectively, as
unit vectors in S+ :
 √



 √


0.2
0.8
√0
a =  √0  ; b =  √0.25  ; c =  √0  ; d = 
0.8
0.75
0.2
the following

0
1 .
0
Then lottery A is preferred to B, while D is preferred to C, as
u(a) = a0 U a = 1.8,
7
8
u(b) = b0 U b = 1.634,
u(c) = c0 U c = 9.6,
u(d) = d0 U d = 10.
8
Example: subjective uncertainty
In the Ellsberg puzzle, suppose that either all the non-red balls are green (R =
100, G = 200, B = 0), or they are all blue (R = 100, G = 0, B = 200), with equal
subjective probability. Further, suppose that there are just two payoff-relevant
outcomes, Win and Lose. Then the following specification of the payoff matrix
accommodates the paradox:
U = W in
Lose
W in Lose
1
0 .
0
0
In fact,
let
µ p
¶
µ p
¶
1/3
2/3
p
p
r =
, r =
be the lotteries representing R and R,
2/3
µ ¶1/3 µ ¶
1
0
respectively, and let w =
,l=
be the lotteries corresponding to a
0
1
9
sure win and a sure loss, respectively. Further, let g := 21 r + 12 l and g := 12 w + 21 r
be the lotteries representing G and G, respectively.
Then
u(r) = r0 U r = 1/3,
u(r) = r0 U r = 2/3,
u(g) = g 0 U g = 1/6
u(g) = g 0 U g ≈ 0.62201,
and the paradox is accommodated.
Conversely, if the payoff matrix is given by
U = W in
Lose
W in Lose
0
0 ,
0
−1
the opposite pattern emerges, and subjective uncertainty is preferred to risk.
9
Multi-agent decisions and equilibrium
Within the class of preferences characterized by Theorem 7, is it still true that
every finite game has a Nash equilibrium? If the payoff matrix U is diagonal we
are in the classical case, so we know that any finite game has an equilibrium,
which moreover only involves objective risk (in our terms, this type of equilibrium should be referred to as “pure”, as it involves no subjective uncertainty).
For the general case, consider that u(p) is still continuous and linear in p ∈ B,
and therefore the main steps in Nash’ proof (proving that the best response correspondence is non-empty, convex-valued and upper-hemicontinuous in order to
apply Kakutani’s fixed-point theorem) still apply in our case. Then any finite
game has an equilibrium even within this larger class of preferences, even though
the equilibrium may not be pure (in our sense): in general, an equilibrium will
rest on a combination of objective randomization and subjective uncertainty
about other players’ decisions.
10
Applications and extensions
What epistemic conditions are implicit in our construction?
What are good measures of risk and uncertainty?
Can we increase the empirical accuracy of the CAPM?
11
References
Maurice Allais (1953), “Le Comportement de l’Homme Rationnel devant le
Risque: Critique des postulats et axiomes de l’École Americaine”. Econometrica
21, 503-546.
10
Daniel Bernoulli (1738), “Specimen theoriae novae de mensura sortis”. In
Commentarii Academiae Scientiarum Imperialis Petropolitanae 5, 175-192. Translated as “Exposition of a new theory of the measurement of risk”. Econometrica
22 (1954), 123-136.
Daniel Ellsberg (1961), “Risk, Ambiguity and the Savage Axioms”. Quarterly Journal of Economics 75, 643-669.
John von Neumann (1932), Mathematische Grundlagen der Quantenmechanic. Berlin, Springer. Translated as Mathematical foundations of quantum
mechanics, 1955, Princeton University Press, Princeton NJ.
John von Neumann and Oskar Morgenstern (1944), Theory of Games and
Economic Behavior. Princeton University Press, Princeton NJ.
Matthew Rabin and Richard H. Thaler (2001), “Anomalies: Risk Aversion”.
Journal of Economic Perspectives 15, 219-232.
Leonard J. Savage (1954), The Foundations of Statistics. Wiley, New York,
NY.
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Nr. 48
Prof. Dr. Manfred Kirchgeorg / Eva Grobe / Alexander Lorbeer (2003),
(noch nicht erschienen) 56 Seiten
Einstellung von Talenten gegenüber Arbeitgebern und regionalen
Standorten : eine Analyse auf der Grundlage einer Befragung von
Talenten aus der Region Mitteldeutschland
Nr. 49
Prof. Dr. Manfred Kirchgeorg / Alexander Lorbeer (2002),
60 Seiten
Anforderungen von High Potentials an Unternehmen – eine Analyse
auf der Grundlage einer bundesweiten Befragung von High Potentials und
Personalentscheidern
Nr. 50
Eva Grobe (2003) 78, XXVI Seiten
Corporate attractiveness : eine Analyse der Wahrnehmung von
Unternehmensmarken aus der Sicht von High Potentials
Nr. 51
Prof. Dr. Thomas M. Fischer / Dr. Petra Schmöller /
Dipl.-Kfm. Uwe Vielmeyer (2002) 35 Seiten
Customer Options – Möglichkeiten und Grenzen der Bewertung von
kundenbezogenen Erfolgspotenzialen mit Realoptionen
Nr. 52
Prof. Dr. Thomas M. Fischer / Dipl.-Kfm. Uwe Vielmeyer
(2002), 29 Seiten
Vom Shareholder Value zum Stakeholder Value? – Möglichkeiten und
Grenzen der Messung von stakeholderbezogenen Wertbeiträgen
Nr. 53
Carsten Reimund (2002), 34 Seiten
Internal Capital Markets, Bank Borrowing and Investment: Evidence from
German Corporate Groups
Nr. 54
Ansätze zur Erklärung des Prozesses der Formulierung von
Entscheidungsprozessen
Nr. 55
Prof. Dr. Wilhelm Althammer / Susanne Dröge (2002), 27 Seiten
International Trade and the Environment: The Real Conflicts. - PdfDokument
Nr. 56
Prof. Dr. Bernhard Schwetzler / Maik Piehler (2002), 35 Seiten
Unternehmensbewertung bei Wachstum, Risiko und Besteuerung –
Anmerkungen zum „Steuerparadoxon“
Nr. 57
Prof. Dr. Hagen Lindstädt (2002), 12, 5 Seiten
Das modifizierte Hurwicz-Kriterium für untere und obere
Wahrscheinlichkeiten - ein Spezialfall des Choquet-Erwartungsnutzens
Nr. 58
Karsten Winkler (2003), 25 Seiten
Getting Started with DIAsDEM Workbench 2.0: A Case-Based Tutorial
Nr. 59
Karsten Winkler (2003), 31 Seiten
Wettbewerbsinformationssysteme: Begriff, Anforderungen,
Herausforderungen
Nr. 60
Prof. Dr. Bernhard Schwetzler / Carsten Reimund (2003), 31 Seiten
Conglomerate Discount and Cash Distortion: New Evidence from
Germany
Nr. 61
Prof. Pierfrancesco La Mura (2003), [10] Seiten
Correlated Equilibria of Classical Strategic Games with Quantum Signals
Nr. 62
Prof. Manfred Kirchgeorg (2003), 26 Seiten
Markenpolitik für Natur- und Umweltschutzorganisationen
Nr. 63
Stefan Wriggers (2004), 32, XXV Seiten
Kritische Würdigung der Means-End-Theorie im Rahmen einer
Anwendung auf M-Commerce-Dienste
Nr. 64
Prof. Pierfrancesco La Mura / Matthias Herfert (2004), 18 Seiten
Estimation of Consumer Preferences via Ordinal Decision-Theoretic
Entropy
Nr. 65
Prof. Bernhard Schwetzler (2004), 31 Seiten
Mittelverwendungsannahme, Bewertungsmodell und Unternehmenswertung bei Rückstellungen
Nr. 66
Prof. Manfred Kirchgeorg / Lars Fiedler (2004), 47 Seiten
Clustermonitoring als Kontroll- und Steuerungsinstrument für
Clusterentwicklungsprozesse - empirische Analysen von Industrieclustern
in Ostdeutschland
Nr. 67
Prof. Dr. Manfred Kirchgeorg / Christiane Springer (2005),
39 , XX Seiten
UNIPLAN LiveTrends 2004/2005 : Effizienz und Effektivität in der Live
Communication ; Eine Analyse auf Grundlage einer branchenübergreifenden Befragung von Marketingentscheidern in Deutschland
Nr. 68
Prof. Pierfrancesco La Mura (2005),
Decision Theory in the Presence of Uncertainty and Risk
(Erscheint demnächst)

Handelshochschule Leipzig (HHL)

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