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arXiv:1209.2204v1 [q-fin.GN] 10 Sep 2012
HOW IS NON-KNOWLEDGE REPRESENTED IN ECONOMIC THEORY?
EKATERINA SVETLOVA∗and HENK VAN ELST†
1Fakult¨
at I: Betriebswirtschaft und Management, Karlshochschule International University
Karlstraße 36–38, 76133 Karlsruhe, Germany
September 10, 2012
Abstract
In this article, we address the question of how non-knowledge about future events that influence eco-
nomic agents’ decisions in choice settings has been formally represented in economic theory up to date.
To position our discussion within the ongoing debate on uncertainty, we provide a brief review of histor-
ical developments in economic theory and decision theory on the description of economic agents’ choice
behaviour under conditions of uncertainty, understood as either (i) ambiguity, or (ii) unawareness. Ac-
cordingly, we identify and discuss two approaches to the formalisation of non-knowledge: one based on
decision-making in the context of a state space representing the exogenous world, as in Savage’s axiomati-
sation and some successor concepts (ambiguity as situations with unknown probabilities), and one based on
decision-making over a set of menus of potential future opportunities, providing the possibility of deriva-
tion of agents’ subjective state spaces (unawareness as situation with imperfect subjective knowledge of all
future events possible). We also discuss impeding challenges of the formalisation of non-knowledge.
1 Introduction
The recent economic crisis once again drew attention to the insufficient ability of modern economic theory to
properly account for uncertainty and imperfect knowledge: neglect of these issues is argued to be one of the
reasons for the failure of the economic profession in the difficult times of 2007–2009; cf. The Economist (2007)
[20], Colander et al (2009) [10], Taleb (2010) [72], Akerlof and Shiller (2009) [2], and Svetlova and Fiedler
(2011) [70]. Next to the voices from the inside of the profession, there is the related criticism from neighbouring
disciplines such as, e.g., economic sociology; cf. Beckert (1996) [3], and Esposito (2007, 2010) [25,26]. The
impression arises that economists are utterly ignorant: they supposedly do not pay (enough) attention to the
issues which the rest of the world consider to be most crucial for economic life. We asked ourselves if this
ignorance is indeed a part of scientific practice in economics. Is it correct that nobody has properly tackled the
issue of true uncertainty and imperfect knowledge since Knight (1921) [45] and Keynes (1921) [42] during the
post-WW I twentieth century?
In this article, we aim to arrive at a more differentiated judgement. Based on a review of the literature, we
classify the developments in economics and decision theory that refer to uncertainty and imperfect knowledge.
We identify three major directions that deal with these issues in economics, specifically risk,uncertainty as
ambiguity, and uncertainty as unawareness. However, it should be stressed that our goal is not a detailed
classification of approaches per se, but answering the question of how non-knowledge has been represented
formally in economic theory to date. This task requires, however, some detailed detection work, because non-
knowledge has not been an explicit issue in economics yet.
Surely, there is knowledge economy, cf. Rooney et al (2008) [58], where knowledge is treated as a resource
or a desirable asset. Also, knowledge is an important topic in information economics, as pioneered by Stigler
∗E–mail: esvetlova@karlshochschule.de
†E–mail: hvanelst@karlshochschule.de
2 HISTORICAL DEVELOPMENTS
2
(1961) [67], Akerlof (1970) [1], Spence (1973) [66], and Stiglitz (1975, 2002) [68,69], where it is considered
to be one of the tools to maximise profit. Generally, in economics, knowledge is considered as a good that is
commonly available in principle (and should be used); the opposite — non-knowledge — is treated implicitly
as a lack of information. In philosophy and the social sciences, the situation is not very different, though there
are interesting recent attempts to overcome “theoretical preoccupations that underlie the study of knowledge
accumulation,” McGoey (2012) [49, p 1], and to develop an agenda for the social and cultural study of igno-
rance; cf. McGoey (2012) [49] and Proctor (2008) [56]. Ignorance should be treated “as more than ‘not yet
known’ or the steadily retreating frontier,” Proctor (2008) [56, p 3], and should be separately accounted for as
a strategic resource and the source of economic profit and progress; cf. Knight (1921) [45] and Esposito (2010)
[26]. In economic theory, there have been occasional voices pleading for more attention to “true uncertainty”,
understood as the principle impossibility offoreseeing all future events that may occur in theexogenous world,
cf. Davidson (1991) [14] and Dequech (2006) [17], and to “unknown unknowns”, cf. Taleb (2007) [71] and
Diebold et al (2010) [19]. However, non-knowledge has not become an independent issue of any significant
interest or importance for economists so far. Thus, to find out how ignorance is formalised in the approaches
considered here, we have to uncover first which aspects of decision-making are treated (often indirectly) as
unknown, and which mathematical instruments are used to represent them.
Our focus is on the principle non-knowledge of future events in the exogenous world, which is the primary
source of uncertainty. After providing, in Section 2, a brief historical overview to position the approaches
considered within the ongoing debate on uncertainty, we are concerned with the formal mathematical represen-
tation of ambiguity in Section 3, and of unawareness in Section 4. Accordingly, we identify and review two
approaches to the formalisation of non-knowledge in the literature: one based on economic agents’ decision-
making in the context of a state space representing the exogenous world, as in Savage’s (1954) [61] axioma-
tisation and some successor concepts (ambiguity as situations with unknown probabilities), and one based on
decision-making over a set of menus of potential future opportunities, providing the possibility of derivation
of agents’ subjective state spaces (unawareness as situation with imperfect subjective knowledge of all future
events). Due to the large number of papers written on this topic, we have to be selective and, hence, cannot
provide an exhaustive overview. We particularly draw attention to the last-mentioned line of research, namely
uncertainty as unawareness, as it represents an exciting attempt to formalise “unknown unknowns” by radically
departing from the mainstream paradigm of Savage’s axiomatisation. Finally, in Section 5, we discuss the
impending challenges and tasks of formalisation of non-knowledge in economics. We believe that without a
detailed understanding of how non-knowledge has been represented in economics so far, no serious research
agenda for studying ignorance as an independent part of economic theory can be developed. We hope that this
article provides one of the first useful steps towards such an agenda.
2 Historical developments
Though there has not been an explicit discussion on non-knowledge in economic theory, this issue permanently
turns up in relation to the topic of uncertainty. We identified three branches in the literature on decision-making
of economic agents under conditions of uncertainty — risk, ambiguity and unawareness — and, in what follows,
present those three directions and discuss the issue of knowledge versus ignorance in relation to each of them:
(i) risk: in formal representations, possible states and events regarding the exogenous world and their re-
spective probabilities are known to all economic agents; they agree on the probability measure to be
employed in calculations of individual utility,
(ii) uncertainty I – ambiguity: in formal representations, possible states and events are known but their
respective probabilities are not known to the agents; each of them employs their own subjective (prior)
probability measure in calculations of individual utility,
2 HISTORICAL DEVELOPMENTS
3
(iii) uncertainty II – unawareness: in formal representations, possible states and events are known only in-
completely to the agents; there is ignorance among them as regards relevant probability measures for
calculations of individual utility.
This classification goes back to the work on uncertainty by Knight (1921) [45], Keynes (1921, 1937) [42,43],
Shackle (1949, 1955) [62,63], and Hayek (1945) [37], who tightly connected the discussion of uncertainty
with two kinds of knowledge, or rather ignorance: specifically, with imperfect knowledge of future events
(uncertainty II), and with knowledge or non-knowledge of probability measures relating to future events (un-
certainty I). Though the detailed depiction of the historical development of those concepts would go far beyond
the scope of this paper, we consider it important to highlight the main ideas in this development in order to pro-
vide a topical frame for our discussion on the conceptualisation of non-knowledge in contemporary economic
theory.
Generally, the authors mentioned differentiate between epistemological and ontological uncertainty.Epis-
temological uncertainty is related to situations where economic agents lack the knowledge necessary to con-
struct adequate probability measures. According to Knight (1921) [45], e.g., theoretical, i.e., a priori probabili-
ties on the one hand, and statistical probabilities on the other, are based on a valid fundament of knowledge: the
law of large numbers, or statistical grouping. The a priori probability can be predicted using counting princi-
ples and a completely homogeneous classification of instances (e.g., by rolling dice), the statistical probability
describes the frequency of an outcome based on a classification of empirical events or instances, given repeated
trials. Knowledge is understood in both cases as (empirical) information that allows for the classification of
possible outcomes. These two kinds of probability (a priori and statistical) can be measured, and in this sense
are known and unanimously agreed upon by all agents involved in decision-making processes (the situation of
risk). Hence, such probability measures can be reasonably referred to as objective.
However, Knight suggests that these two categories do not exhaust all possibilities for defining a probability
measure; he adds “estimates”, or subjective probabilities. Quoting Knight (1921) [45, p 225]: “The distinction
here is that there is no valid basis of any kind for classifying instances. This form of probability is involved in
the greatest logical difficulties of all ....” Knight refers to this last situation as a situation of uncertainty (ibid
[45, p 233]); uncertainty can be defined as absence of probable knowledge. In the situation of risk, probabilities
represent the measurable degree of non-knowledge; in the uncertainty situation, this degree is immeasurable,
and in this sense probabilities are not known. Keynes (1921) [42] also suggested a concept of immeasurable
probabilities as logical relationships, and argued in his 1937 paper — in unison with Knight — that economic
agents lack a valid basis to devise probability measures. In his definition uncertainty exists, e.g., in the case
of predicting the price of copper or the interest rate 20 years hence (Keynes (1937) [43, p 113]): “About these
matters there is no scientific basis on which to form any calculable probability whatever. We simply do not
know.” Probabilities are used by economic agents as a convention that enables them to act (ibid [43, p 114]);
at the same time, though probabilities are widely applied, they represent the agents’ ignorance rather than their
(scientific) knowledge.
Interestingly, in the later literature this issue was taken up by Ellsberg (1961) [21], who, in his experiments,
distinguished between situations with known probability measures over some event space (when the color and
number of the balls in an urn are known to agents; thus, they can form probabilities), and situations with
unknown probability measures (agents know only the colors of balls but not the exact number of balls of each
color; thus, they deal with the ignorance of probability). Ellsberg demonstrated empirically that people tend
to prefer situations with known probability measures over situations with unknown probability measures; he
explicitly referred to situations with unknown probability measures as ambiguous and named the phenomenon
of avoiding such situations “ambiguity aversion” (corresponding to the term “uncertainty aversion” coined by
Knight (1921) [45]).
It must be noted that the discussion about measurability of probabilities in economic life, as well as about
their objective vs subjective character, was severely influenced and pulled in one particular, for a long time un-
contested, direction by the line of argumentation due to Ramsey (1931) [57], de Finetti (1937) [27], and Savage
(1954) [61]. Ramsey and de Finetti reacted to Knight’s and Keynes’ concepts of uncertainty as situations with
2 HISTORICAL DEVELOPMENTS
4
immeasurable probabilities with the axiomatisation of subjective probabilities: they demonstrated that subjec-
tive probabilities can always be derived from the observed betting behaviour of economic agents, rendering
the whole discussion about measurability and objectivity of probabilities seemingly obsolete. Adopting these
results, Savage generalised the theory of decision under risk, i.e., the expected utility theory as conceived of
originally by Bernoulli (1738) [4] and von Neumann and Morgenstern (1944) [53]. While the expected utility
concept as an element of risk theory was based on objective probability measures, Savage combined expected
utility theory and the subjective probability approach of Ramsey and de Finetti to deliver a new variant of an
axiomatisation of decision under conditions of uncertainty — subjective expected utility theory. This concept
was perfectly compatible with the Bayes–Laplace approach to probability theory and statistics where subjective
prior probabilities can always be assumed to exist and adjusted in the process of learning. The crucial feature of
Savage’s probabilistic sophistication is the principle neglect of the Knightian distinction between risk and un-
certainty, as Savage’s concept presupposes that even if an objective probability measure for future events is not
known, it can always be assumed that economic agents behave as if they apply an individual subjective (prior)
probability measure to estimating the likelihood of future events; and these probability measures can in princi-
ple be derived a posteriori from an axiomatic model on the basis of empirical data on agents’ choice behaviour.
By this theoretical move, the immeasurability (and thus the knowability) issue is eliminated. The question of
the validity of the subjective degrees of beliefs foundation, or of the origin of subjective probabilities, is beyond
Savage’s model, as these are built into the as-if-construction from the outset.
However, the Knightian distinction continued to bother economists and — especially after Ellsberg’s (1961)
[21] paper — a new branch of research appeared in the literature that endeavoured to re-introduce uncertainty,
understood as absence of perfect knowledge of relevant probability measures, into economic theory. The most
prominent attempt was delivered by Gilboa and Schmeidler (1989) [30]. In the next section, we will introduce
the basic elements of their axiomatisation of decision under uncertainty in terms of non-unique probability
measures, and contemplate how non-knowledge is represented in this concept. At the same time, the attentive
reading of Knight, Keynes and Shackle suggests that the issue of uncertainty is not restricted to the question
whether probabilities can be meaningfully defined or measured. There is a more fundamental issue of onto-
logical uncertainty which is concerned with the principle unknowability of what is going on in an economic
system; it goes beyond the scope of epistemic uncertainty.
Note that in the framework of epistemic uncertainty, knowledge that is relevant for the derivation of a mean-
ingful probability measure is generally treated as information; compare the respective definition by Epstein and
Wang (1994) [22, p 283], who define risk as a situation “where probabilities are available to guide choice, and
uncertainty, where information is too imprecise to be summarized adequately by probabilities.” It is interesting
that also beyond the borders of economic theory — in the IPCC (2007) [41] report — the Knightian distinction
between risk and uncertainty is understood as an epistemic one: “The fundamental distinction between ‘risk’
and ‘uncertainty’ is as introduced by economist Frank Knight (1921), that risk refers to cases for which the
probability of outcomes can be ascertained through well-established theories with reliable complete data, while
uncertainty refers to situations in which the appropriate data might be fragmentary or unavailable.” (... ) The
clear relation “information (empirical data) – probabilities” is presupposed. The lack of knowledge, in this
case, can be theoretically removed by becoming more skillful in calculating, or by collecting more information.
However, it should be stressed that Knight (as well as Keynes and Shackle) did not conceive of ignorance
as lack of information but rather as ontological indeterminacy, the “inherent unknowability in the factors”,
see Knight (1921) [45, p 219]. Shackle (1955) [63] relates the genuinely imperfect knowledge about future
events to the absence of an exhaustive list of possible consequences of choices. Traditional probability theory
assumes that the list of consequences over which probability is distributed is an exhaustive list of possible
outcomes, or, in Shackle’s terms, hypotheses. However, so Shackle, if there is a residual hypothesis, that is, the
list of possible consequences is incomplete, the probability model runs into trouble. By adding a hypothesis
to the list of possible hypotheses, each corresponding probability of the previously known hypotheses has
to be revised downwards; see Shackle (1955) [63, p 27]. If five possible hypotheses are considered and a
sixth hypothesis is added, and additivity of probabilities is assumed, the probability of each of the initial five
2 HISTORICAL DEVELOPMENTS
5
hypotheses is subsequently lower. This objection applies to both approaches, namely the frequentist approach
to probability theory on the one hand, and the Bayes–Laplace approach which deals with belief-type subjective
(prior) probability measures on the other, because neither can incorporate a residual hypothesis, or the principle
non-knowledge of future states. Thus, referring to the genuinely imperfect knowledge about future events,
Shackle (but also Knight and Keynes) expressed doubts whether probability theory in general is sufficient to
account for decision under uncertainty, and whether it should be the central issue after all.
By far more important than the issue of devising suitable probability measures seems to be the non-
knowledge of possible future states of the exogenous world and of related outcomes. Only if we manage
to account properly for this imperfect knowledge, can we conceptualise properly human decision-making, or,
in the words of Shackle (1959) [64, p 291], a non-empty decision. Crocco (2002) [11] explains: “An empty
decision is the mere account of a formal solution to a formal problem. It is that situation where a person has
a complete and certain knowledge about all possible choices and all possible outcomes of each choice. It is a
mechanical and inevitable action,” or, in the words of Heinz von F¨orster (1993) [28, p 153], every decidable
(or perfectly known) problem is already decided; true decisions always presuppose genuine undecidability. In
this sense, Savage’s concept is rather concerned with empty decisions, because it presupposes situations with
full knowledge of possible events, acts and outcomes, rendering agents’ choices just a mechanical application
of the personal utility-maximisation rule.
In economics, genuine undecidability should enter theory. Most economic decisions are truly undecidable
because they take place under conditions of imperfect knowledge of the situation to be faced, which is in the
sense of the American pragmatist philosopher John Dewey (1915) [18, p 506] a genuinely “incomplete situa-
tion”: “something is ‘there’, but what is there does not constitute the entire objective situation.” This “means
that the decision-maker does not have complete knowledge of the following: (a) the genesis of the present
situation, (b) the present situation itself, or (c) the future outcomes that remain contingent on the decisions that
are made in the present situation;” see Nash (2003) [51, p 259]. According to Dewey (1915) [18], the situation
is underdetermined, unfinished, or not wholly given.
This principle non-knowledge can be explained, so Shackle (1949, 1955) [62,63], by the character of eco-
nomic decisions, which he considers to be non-devisible, non-seriable, and crucial experiments. Non-devisible
experiments imply only a single trial; non-seriable experiments are not statistically important even in the aggre-
gate; an example of a seriable experiment is fire insurance: although no reasonable probability can be assigned
to an individual house to burn down, if there are sufficiently many events, a (statistical) probability will emerge.
Most importantly, economic decisions are crucial experiments: they inevitably alter the conditions under which
they were performed (this definition applies to all strategic situations, e.g., chess play, but also financial mar-
kets). Within the genuinely social context of economic life, economic events are rather endogenous to the
decision processes of agents and are dependent on the actions and thinking of other market participants. There
are path dependencies and reflexivity; cf. Soros (1998) [65]. In general, a meaningful approach to decision-
making should take into account that the future is principally unknowable, due to ontological features of the
exogenous world such as openness, organic unity, and underdeterminacy. These are features which are typi-
cally attributed to complex systems; cf. Keynes et al (1926) [44, p 150]: “We are faced at every turn with the
problems of Organic Unity, of Discreteness, of Discontinuity — the whole is not equal to the sum of the parts,
comparisons of quantity fail us, small changes produce large effects, the assumptions of a uniform and homo-
geneous continuum are not satisfied.” In such a system, not all constituent variables and structural relationships
connecting them are known or knowable. Thus, in an open and organic system, some information is not avail-
able at the time of decision-making, and cannot be searched, obtained or processed in principle. Surprises, or
unforeseen events, are normal, not exceptional. The list of possible events or states is not predetermined and
very little, or nothing at all, can be known about the adequate probability measure for this radically incomplete
set of future events.
These considerations require a more sophisticated distinction of decision-making configurations, namely a
distinction that goes beyond the usual risk vs uncertainty as ambiguity debate. As Dequech (2006) [17, p 112]
puts it: “Even though the decision-maker under ambiguity does not know with full reliability the probability
3 UNCERTAINTY AS AMBIGUITY: NON-KNOWLEDGE OF PROBABILITY MEASURES
6
that each event (or state of the world) will obtain, he/she usually knows all the possible event .. . . Fundamen-
tal uncertainty, in contrast, is characterized by the possibility of creativity and non-predetermined structural
change. The list of possible events is not predetermined or knowable ex ante, as the future is yet to be created.”
What Dequech calls “fundamental uncertainty” (or “true uncertainty” in terms of some post-Keynesians (e.g.,
Davidson (1991) [14]) enters the recent debate in the economic literature under the label of “unawareness”.
The unawareness concept, as introduced by Kreps 1979 [46], Dekel et al (1998, 2001) [15,16], and Ep-
stein et al (2007) [23], presupposes a coarse (imperfect) subjective knowledge of all possible future events.
This concept criticises Savage’s (1954) [61] axiomatisation and suggests a radical departure from it. Savage’s
axiomatisation is characterised by the in principle observability and knowability of all possible future events.
These events belong to the primitives of the model and are assumed to be exogenous and known to all eco-
nomic agents. In Savage’s model, the (compact) state space representing the exogenous world the agents are
continually interacting with is “a space of mutually exclusive and exhaustive states of nature, representing all
possible alternative unfoldings of the world”; see Machina (2003) [48, p 26]. The exhaustiveness criterion
is very restrictive and basically precludes non-knowledge of future states on the part of the agents. Machina
(2003) [48, p 31] continues: “When the decision maker has reason to ‘expect the unexpected’ [or the residual
hypothesis in terms of Shackle — the authors], the exhaustivity requirement cannot necessarily be achieved,
and the best one can do is specify a final, catch-all state, with a label like ‘none of the above’, and a very
ill-defined consequence.” Obviously, true uncertainty as imperfect knowledge of possible future states of the
exogenous world is not an element of Savage’s model. The pioneers of the unawareness concept depart from
Savage’s axiomatisation by replacing the state space in the list of primitives by a set of menus over actions
which are the objects of choice. This theoretical move allows for dealing with unforeseen contingencies, i.e.,
an inability of economic agents to list all possible future states of the exogenous world.
We now turn to give a more formal presentation of the two main concepts of uncertainty we discussed so
far: uncertainty as ambiguity and uncertainty as unawareness.
3 Uncertainty as ambiguity: non-knowledge of probability measures
All decision-theoretical approaches to modelling an economic agent’s state of knowledge regarding future de-
velopments of the exogenous world, the ensuing prospects for an individual’s opportunities, and the agent’s
consequential choice behaviour under conditions of uncertainty employ an axiomatic description of the char-
acteristic properties of observable choice behaviour and derive a quantitative representation of an agent’s pref-
erences in decision-making. Uncertainty in this context is generally interpreted as ambiguity perceived by an
agent with respect to unknown probabilities by which future states of the exogenous world will be realised.
In these approaches the standard assumption of neoclassical economics of an agent whose choices are fully
rational is being maintained. The main issue of modelling here is to put forward a set of primitives which can
be observed in principle in real-life settings, as well as a minimal set of axioms describing exhaustively the
interconnections between these primitives, to provide the conceptual basis for (in general highly technically
demanding) mathematical proofs of representation theorems. Most approaches in the literature propose an ex-
pected utility (EU) representation of an agent’s preferences in terms of a real-valued personal utility function
which is an unobservable theoretical construct, thus following the quantitative game-theoretical tradition of
von Neumann and Morgenstern (1944) [53]. A related issue is the question to what extent an agent’s choice
behaviour can be reasonably viewed as influenced by a set of personal subjective probabilities regarding the
(unknown) future states of the exogenous world. We begin by briefly reviewing the central aspects of the ax-
iomatic approach taken by Savage (1954) [61] to describe one-shot choice situations — the subjective expected
utility (SEU) framework, which attained the prominent status of a standard model in decision theory.
The primitives in Savage (1954) [61] are
(i) an exhaustive set of mutually exclusive future states ωof the exogenous world which an agent cannot
actively take an influence on; these constitute a state space Ωwhich is assumed to be continuous, com-
3 UNCERTAINTY AS AMBIGUITY: NON-KNOWLEDGE OF PROBABILITY MEASURES
7
pact, and can be partitioned into a finite number of pairwise disjoint events; possible events A, B, . . . are
considered subsets of Ω, with 2Ωthe set of all such subsets of Ω,
(ii) a finite or infinite set of outcomes xcontingent on future states ω, forming an outcome space X, and
(iii) a weak binary preference order (“prefers at least as muchas”) defined overthe agent’s objects of choice
— a set of potential individual acts fan agent may consciously take in reaction to realised future states ω
of the exogenous world, yielding predetermined outcomes x—, describing their personal ranking of
available options; these acts form a space F.
In more detail, an act is defined as a (not necessarily real-valued, continuous) mapping f:Ω→Xfrom the
set of future states Ωto the set of possible outcomes X, so the set of acts available to an agent at a given instant
in time, in view of known future states ωbut of unknown probabilities, is F=XΩ. There is no additional
structure needed in this model regarding measures or topology on either space Ωor X, except for continuity
and compactness of Ω. An observable weak binary preference order over the set of acts is given by ⊂ F×F,
intended to reflect an agent’s subjective beliefs regarding future states ω, and the usefulness of acts the agent
may take in response to ensuing states.
Savage introduces a minimal set of seven axioms (P1 to P7) to characterise the theoretical nature of this
preference order over acts (and, by implication, related outcomes), which are commonly referred to in the
literature as weak order resp. completeness, sure-thing principle, state-independence, comparative probability,
non-triviality, Archimedean, and finitely additive probability measures; cf. Nehring (1999) [52, p 105] and
Gilboa (2009) [29, p 97ff]. These axioms constitute the foundation of a representation theorem proved by
Savage which states that an agent’s (one-shot) choice behaviour under conditions of uncertainty may be viewed
as if it was guided by (i) a real-valued personal utility function U:X→Rthat assigns subjective value to
specific outcomes x∈X, and (ii) a single finitely additive subjective probability measure µ: 2Ω→[0,1] on
the space of all possible future events 2Ω. In particular, an agent’s choice behaviour may be modelled as if for
the acts favailable to them they strive to maximise a real-valued EU preference function V:F→R, defined
by
V(f) := ZΩ
U(f(ω)) µ(dω).(1)
Hence, in this setting an act f∈Fis weakly preferred by an agent to an act g∈F, iff V(f)≥V(g).
The elements of Savage’s SEU model may be schematically summarised in terms of a decision matrix of
the following structure (here for a partition of the continuous and compact Ωinto a finite number nof pairwise
disjoint events):
probability measure µP(ω1)P(ω2). . . P (ωn)
acts F\states Ωω1ω2. . . ωn
f1x11 x12 . . . x1n
f2x21 x22 . . . x2noutcomes X
.
.
..
.
..
.
.....
.
.
,(2)
where 0≤P(ωi)≤1and PiP(ωi) = 1 (and generally: µ≥0and RΩµ(dω) = 1). Note that, formally,
Savage’s framework reduces an agent’s situation of decision under uncertainty, in the Knightian sense of not
knowing the probability measure associated with (Ω,2Ω)a priori, to a manageable situation of decision under
risk by introducing a single subjective Bayesian prior probability measure as a substitute. This is to say, every
single economic agent possesses for themselves a unique probability measure which they employ in their indi-
vidual calculations of utility; a probability measure is thus known to every individual from the outset, but there
is no reason whatsoever that these measures should coincide between agents.
Savage’s main claim is that his framework can be used to explicitly derive for an arbitrary economic agent
who makesrational choices inparallel (i) aunique subjective probability measure µover (Ω,2Ω), and (ii) a per-
sonal utility function Uover F(unique up to positive linear transformations), from observation of their choice
3 UNCERTAINTY AS AMBIGUITY: NON-KNOWLEDGE OF PROBABILITY MEASURES
8
behaviour in practice. For the sequel it is worth mentioning that Savage’s numerical SEU representation (1) can
be interpreted to fall into either of the categories of ordinal or additive EU representations.
Various authors have criticised Savage’s SEU model for different reasons, where in particular the claim is
that one or more of his axioms are regularly being violated in real-life situations of (one-shot) choice. Bewley
(1986,2002) [5,6], for example, points the finger to the completeness axiom P1 in that he considers it unrealistic
to assume that all agents have a clear-cut ranking of all the acts available to them, when it need not necessarily
be clear from the outset which acts comprise the complete set F. In his work he therefore proposes an axiomatic
alternative to Savage’s SEU model which discards the completeness axiom in favour of an inertia assumption
regarding the status quo of an agent’s personal situation.
More prominent still is Ellsberg’s (1961)[21] empirical observation that in situations of choice under un-
certainty rational agents need not necessarily act as subjective expected utility maximisers: given the choice
between a game of chance withknown probabilities of the possible outcomes and the identical game of chance
where the probabilities are unknown, the majority of persons tested exhibited the phenomenon of uncertainty
aversion by opting for the former game. Ellsberg showed that this kind of behaviour correspond to a violation
of Savage’s sure-thing principle axiom P2.
A possible resolution of this conflict was suggested in the multiple priors maxmin expected utility (MMEU)
model due to Gilboa and Schmeidler (1989) [30], which takes uncertainty aversion explicitly into account by
stating that under conditions of uncertainty an agent need not have to have a unique subjective prior prob-
ability measure µ, but rather an entire set Πworth of such measures πfrom which they select in making
decisions according to the maxmin principle. In this sense, Gilboa and Schmeidler take an explicit attempt at
formalising Knightian uncertainty in problems of decision-making, interpreted as situations with in principle
unknowable probability measures over (Ω,2Ω). The degree of an agent’s ignorance is encoded in the generi-
cally unconstrained cardinality of the set of Bayesian priors Π: no criteria are formulated according to which
an agent assesses the relevance of any particular probability measure that is conceivable for a given situation
of decision-making. Non-knowledge regarding the likelihood of future events here is linked to the number of
elements included in the individual set Πthat is employed in an agent’s individual calculation of utility and so
is represented in a more comprehensible fashion than in Savage’s framework.
Nevertheless, the primitives of the MMEU model are unchanged with respect to Savage’s SEU model.
Based on a minimal set of six axioms (A1 to A6) referred to resp. as weak order, certainty-independence, conti-
nuity, monotonicity, uncertainty aversion and non-degeneracy, the representation theorem Gilboa and Schmei-
dler (1989) [30] prove employs a real-valued preference function V:F→Rdefined by the minimum expected
utility relation
V(f) := min
π∈ΠZΩ
(Ef(ω)U) dπ , (3)
with Π⊂∆(Ω)a non-empty, closed and convex set of finitely additive probability measures over (Ω,2Ω), and
U:X→Ra non-constant real-valued personal utility function. Again, an act f∈Fis then weakly preferred
by an agent to an act g∈F, iff V(f)≥V(g).
Since its inception, Gilboa and Schmeidler’s MMEU model has enjoyed a number of applications in the
econometrical literature; e.g. in Epstein and Wang (1994) [22] on intertemporal asset pricing; Hansen et al
(1999) [33] on savings behaviour; Hansen and Sargent (2001, 2003) [34,35] on macroeconomic situations;
Nishimura and Ozaki (2004) [54] on a job search model; and Epstein and Schneider (2010) [24] on implica-
tions for portfolio choice and asset pricing. Rigotti and Shannon (2005) [59], who propose an approach to
formalising uncertainty in financial markets on the basis of Bewley’s (1986,2002) [5,6] idea of discarding Sav-
age’s completeness axiom P1, contrast their findings on the impact of uncertainty on equilibrium configurations
in decision-making processes with corresponding consequences arising from an MMEU perspective.
The strongest criticism to date of Savage-type state space models of decision-making under conditions
of uncertainty was voiced at the end of the 1990ies by Dekel et al (1998) [15]. They showed that given one
considers it unrealistic for an economic agent to beaware of allpossible future states ωof the exogenous world,
4 UNCERTAINTY AS UNAWARENESS: NON-KNOWLEDGE OF COMPLETE STATE SPACES
9
a standard state space model is incapable of consistently incorporating the dimension of an agent’s unawareness
of future contingencies. The basis of the formal treatment of the issue at hand are information structures referred
to as possibility correspondences. A possibility correspondence amounts to a function P:Ω→2Ωthat maps
elements ωin some state space Ωto subsets thereof, so that P(ω)is interpreted as the set of states an agent
considers possible when the realised state is ω. In this picture, an agent “knows” an event E∈2Ωat a state ω
provided P(ω)⊆E. Hence, given a possibility correspondence P, a knowledge operator K: 2Ω→2Ωis
determined by
K(E) := {ω∈Ω|P(ω)⊆E}for all E∈2Ω;(4)
K(E)represents the set of states in Ωfor which an agent knows that event Emust have occurred. According
to Dekel et al, it is commonplace to assume that such a knowledge operator features the properties of (i) ne-
cessitation, meaning K(Ω) = Ω, and (ii) monotonicity, meaning E⊆F⇒K(E)⊆K(F). In addition, an
unawareness operator may be defined as a mapping U: 2Ω→2Ω, so that U(E)is to be regarded as the set of
states in Ωwhere an agent is unaware of the possibility that event Emay occur. With these structures in place,
a standard state space model is represented by a triplet (Ω, K, U ).
To obtain their central result, Dekel et al require a minimal set of only three axioms which characterise
the nature of the operators Kand U: these demand that for every event E∈2Ω, (i) U(E)⊆ ¬K(E)∩
¬K(¬K(E)), called plausibility,1(ii) K(U(E)) = ∅, called KU introspection, and (iii) U(E)⊆U(U(E)),
called AU introspection. Given a standard state space model (Ω, K, U )satisfies these three axioms, the theorem
proven by Dekel et al (1998) [15, p 166] states that in such a setting (a) “the agent is never unaware of anything,”
provided Ksatisfies the necessitation property, and (b) “if the agent is unaware of anything, he knows nothing,”
provided Ksatisfies the monotonicity property. This result renders standard state space models void as regards
the intention of formally capturing an agent’s unawareness of subjective contingencies in a non-trivial way.
The work by Dekel et al (1998) [15], in particular, triggered a series of papers written during the last
decade, which aspire to include an agent’s unawareness of future subjective contingencies in a coherent model
that continues to employ a kind of EU representation of an agent’s manifested preferences in situations of
choice under conditions of uncertainty. We turn to highlight the, in our view, most important papers of this
development next.
4 Uncertainty as unawareness: non-knowledge of complete state spaces
Since the status ofpossible future states ωof the exogenous world as a primitive in a decision-theoretical model
on an agent’s choice behaviour under conditions of uncertainty is questionable due to the lack of a convincing
operational instruction for observation of such states, a number ofauthors have dropped the state space Ωfrom
the set of primitives altogether and turned to focus instead on the description of an agent’s preferences when
they are unaware of some future subjective contingencies which take a direct influence on future outcomes
such as the pay-offs of certain actions. In the papers to be considered in the following, the conceptual line of
thought pursued in which originated in the work by Kreps (1979) [46], the primitives underlying this alternative
approach comprise in general
(i) a (typically finite) set Bof alternative opportunities, actions, or options; a generic element in this set will
be denoted by b,
(ii) a (typically finite) set Xof all conceivable non-trivial menus compiled from elements in B, with a
generic element denoted by x; note that X= 2B\{∅},
(iii) a weak binary preference order defined over the agent’s objects of choice, presently menus in X.
1The symbol ¬denotes complementation.
4 UNCERTAINTY AS UNAWARENESS: NON-KNOWLEDGE OF COMPLETE STATE SPACES
10
The setting conceived of in this approach considers a two-stage choice process in which an agent will initially
(“now”) choose a particular menu x, from which, contingent on subsequently ensuing states ωof the exogenous
world, they will choose a specific element bat an unmodelled later stage (“then”).2Hence, two kinds of
(weak) binary preference orders need to be introduced: an “ex ante preference” (preference “now”) over the
set X,⊂ X×X, and an “ex post preference” (preference “then”) over Bcontingent on a realised state ω,
∗
ω⊂B×B; cf. Dekel et al (2001) [16]. Generally, authors then proceed to formulate minimal sets of axioms
for the ex ante preference order , on the basis of which they prove representation theorems for modelling an
agent’s choice behaviour under conditions of uncertainty in the sense that the agent is unaware of some future
subjective contingencies. A particularly interesting feature of some of the works to be discussed in the sequel is
the possibility to derive in principle an agent’s subjective state space regarding future subjective contingencies
from observed choice behaviour, given some form of EU representation of the agent’s preference relation is
employed. This aspect is key to a meaningful representation of non-knowledge in economic theory. It is also
seen as an intermediate step towards derivation of an agent’s subjective probability measure regarding choice
behaviour under conditions of uncertainty on the basis of empirical data.
Kreps (1979) [46], in his pioneering paper, considers an agent with a “desire for flexibility” as regards
decision-making, the choice behaviour of which, however, may not satisfy “revealed preference”. He for-
malises these properties of an agent’s envisaged choice behaviour in terms of the following two axioms: for all
x, x′, x′′ ∈X,
x⊇x′⇒xx′,(5)
and
x∼x∪x′⇒x∪x′′ ∼x∪x′∪x′′ ,(6)
with ∼denoting the indifference relation on X. Note that in the literature the axiom (5) is often referred to as
the monotonicity axiom. Kreps, in his discussion, does not make explicit an agent’s uncertainty regarding un-
awareness of (some) future subjective contingencies. Rather, it is implied by the agent’s “desire for flexibility”.
He continues to prove that, given a “dominance relation” on Xdefined by
x≥x′if x∼x∪x′,(7)
and the axioms stated before, an agent’s preferences on Xcan be sensibly described as if they were “maximiz-
ing a ‘state dependent utility function of subsequent consumption’” in terms ofa formal real-valued preference
function V:X→R, defined by
V(x) := X
s∈S
max
b∈xU(b, s).(8)
Here Sdenotes the unobservable finite subjective state space ofan agent’s personal tastes, with generic element
s, and U:B×S→Ris the agent’s unobservable state-dependent real-valued utility function of alternative
opportunities available in the finite set B. Kreps points out that this representation is principally ordinal in
character. The bottom-line of Kreps’ approach is that the set of state-dependent ex post utilities {U(·, s)|s∈
S}, expressing the agent’s beliefs on potential future pay-offs, can be interpreted as an agent’s implicitly given
coherent subjective state space which describes their uncertainty regarding ex post choices over the set B, and
so can be legitimately used as a model of unforeseen contingencies (cf. Kreps (1992) [47]).
However, as Dekel et al (2001) [16, p 892, p 896f] emphasise, Kreps’ implied subjective state space
{U(·, s)|s∈S}of an agent is far from being determined uniquely, since the axioms he proposed prove not to
be sufficiently restrictive for this purpose. It is this feature in particular, which these authors set out to overcome
in their own work. To accomplish this goal, Dekel et al (2001) [16] extend Kreps’ analysis in two respects. On
the one-hand side, here the agent’s objects of choice are, in the spirit of von Neumann and Morgenstern (1944)
2As will be described in the following, in some of the works to be reviewed the elements of choice at stage “then” can be more
complex objects than simply elements b∈B.
4 UNCERTAINTY AS UNAWARENESS: NON-KNOWLEDGE OF COMPLETE STATE SPACES
11
[53], sets of lotteries ∆(B)defined over finite sets of future alternative opportunities B, on the other, the as-
sumption of an agent’s strict preference for flexibility is relaxed to also allow for a preference for commitment
in instances when this appears valuable. The latter feature introduces the possibility of an agent’s view “ex
ante” to differ from their view “ex post”. Tocontinue with the primitives: Dekel et al take the set ∆(B)to cor-
respond to a set of probability measures over B; a generic lottery in ∆(B)is denoted by β. Subsets of ∆(B)
are referred to as menus x, with Xdenoting the set of all non-empty subsets of ∆(B).Xis endowed with a
Hausdorff topology and constitutes the formal basis of an agent’s binary ex ante preference order, ⊂ X×X.
The two-stage choice process of Kreps (1979) [46] remains qualitatively unchanged: the agent chooses a menu
x∈X“now”, and a lottery β∈x“then”.
Dekel et al’s different kinds of representations of an agent’s ex ante preference order over menus x
of lotteries correspond to triplets (Ω, U, u), comprising the following three common elements: a non-empty
(exogenous) state space Ωserving merely as an index set to label ex post preferences over ∆(B), a state-
dependent real-valued personal utility function U: ∆(B)×Ω→R, and a real-valued personal aggregator
function u:RΩ→R. The aggregator function is a rather special feature in Dekel et al’s analysis. It is given
the role of translating an agent’s ex post utility levels of menus xinto corresponding ex ante values, making the
strong assumption that, in the model proposed, an agent has a coherent view of all future utility possibilities of
menus xavailable to them. The ex post preference order ∗
ωover ∆(B), given a state ω∈Ω, can be viewed
as being encoded in the utility function U(·, ω). In consequence, Dekel et al define an agent’s subjective state
space as the set P(Ω, U ) := {U(·, ω)|ω∈Ω}.
On the basis of an ex ante preference-characterising minimal set of seven axioms (A1 to A7), referred to
as weak order, continuity, non-triviality, indifference to randomisation (IR), independence, weak independence
and monotonicity, resp., Dekel et al (2001) [16] prove existence theorems for three kinds ofEU representations,
all of which can be cast in the form of a real-valued preference function V:X→Rdefined by
V(x) := u sup
β∈x
U(β, ω)!ω∈Ω!,(9)
with U(·, ω)an EU affine function in line with von Neumann and Morgenstern (1944) [53], i.e., for all β∈
∆(B)and ω∈Ω,
U(β, ω) := X
b∈B
β(b)U(b, ω).(10)
The main results following from the proofs of the EU representation theorems for the binary ex ante preference
order are: (i) uniqueness of an agent’s subjective state space P(Ω, U )related to their binary ex post pref-
erence order, as well as essential uniqueness of the associated aggregator function u, (ii) the size of an agent’s
subjective state space P(Ω, U )can be interpreted as a measure of their uncertainty about future subjective
contingencies, while the associated aggregator uindicates whether such contingencies trigger a preference for
commitment or rather for flexibility, (iii) ordinal EU representations offer the smallest subjective state space
P(Ω, U )possible for any ordinal representation, and (iv) existence of an additive EU representation when
in particular the standard independence axiom due to von Neumann and Morgenstern (1944) [53] holds; the
former is given by a real-valued preference function V:X→Rsuch that (up to monotone transformations)
V(x) = ZΩ
sup
β∈x
U(β, ω)µ(dω),(11)
with µa (non-unique) finitely additive probability measure on (Ω,2Ω). This last result, providing a representa-
tion in line with “standard” approaches, raises ideas on the possibility of identification of an agent’s probability
measure over their subjective state space P(Ω, U ), analogous to one of the central outcomes of Savage’s (1954)
[61] SEU model. However, it is the state-dependence of an agent’s ex post preference which renders this ob-
jective currently quite unrealistic.
4 UNCERTAINTY AS UNAWARENESS: NON-KNOWLEDGE OF COMPLETE STATE SPACES
12
Dekel et al’s (2001) [16, p 894] approach contains an inherent interpretational difficulty, which these au-
thors briefly address: the model represents an agent with an at least partially incomplete concept of future
subjective contingencies “now” with an agent with complete knowledge of all utility possibilities of menus
“then”; does the model, nevertheless, deal consistently with an agent’s non-knowledge of (some) future sub-
jective contingencies? Dekel et al do not see a need for full commitment to this issue, but leave this point
by resorting to the idea of an “as if” representation of their model. However, to put their model to the test,
they call for the identification of a concrete Ellsberg-type example of an agent’s choice behaviour which is in
contradiction with (some of) their axioms; cf. Dekel et al (2001) [16, p 920].
This challenge was met in the work by Epstein et al (2007) [23], in which they focus on criticising Dekel
et al’s additive EU representation in particular. The main argument Epstein et al give states that an economic
agent who is aware of their incomplete knowledge of future subjective contingencies, and in particular is averse
to this personal state of affairs, will feel a need to hedge against this uncertainty by randomisation over options
available to them, thus providing a case of violation of Dekel et al’s independence axiom. In addition, these
authors argue that the impossibility of fully describing all future contingencies relevant to an agent may lead
to the failure of quantifying an agent’s uncertainty about utilities “then” in terms of just a single probability
measure, as Dekel et al do in their additive EU representation. In Epstein et al’s (2007) [23, p 359] view,
Dekel et al’s model therefore precludes a consistent representation of incompletely known future subjective
contingencies and ambiguity about an agent’s preferences “then”.
To overcome the conceptual problems of Dekel et al’s model — in particular, to capture the ambiguity due
to an agent’s incomplete knowledge of future subjective contingencies, and their induced tendency for hedging
against it —, Epstein et al (2007) [23] propose two alternative axiomatic models of an agent’s ex ante choice
behaviour. These can be considered modifications of Dekel et al’s approach in the following sense. The first
model maintains the assumption of the IR axiom to hold, while the independence axiom is being relaxed; in
the second model both the IR and the independence axioms are dropped, and the primitives of the model are
extended to include random menus. The two models exhibit a qualitative difference as regards the status of ex
post ambiguity that an agent finds themself exposed to “then”. In the first model, an agent “now” expects to
gain complete knowledge “then” of a state realised in the meantime, i.e., before they choose a lottery βfrom
the ex-ante-preferred menu x; hence, ex post ambiguity is resolved. However, “now” the agent is uncertain
about their actual preferences “then”. In the second model, on the other hand, an agent “now” reckons that
even “then” their knowledge of all relevant contingencies will remain incomplete, leaving their preferences
“then” somewhat vague due to the lack of a complete view of all of the options available to them. In the present
work this circumstance is modelled in terms of a restricted set of utility functions (over lotteries) with unknown
likelihoods. Ex post ambiguity persists, making hedging against uncertainty “then” (and related potentially
unfavourable outcomes) a valuable tool.
In model I, Epstein et al (2007) [23] implement an agent’s need for hedging by following the ideas of Gilboa
and Schmeidler (1989) [30] on uncertainty aversion in that, via introducing a mixing operation defined over
menus, an axiomatisation of a multiple-priors utility representation of an agent’s ex ante preferences is pro-
posed. Starting from a minimal set of eight axioms requiring (weak) order, monotonicity, IR, non-degeneracy,3
preference convexity, worst, certainty-independence, and mild continuity, the corresponding representation the-
orem states that an agent’s ex ante choice behaviour amounts to maximising a real-valued preference functional
VMP :X→R, given by
VM P (x) := min
π∈ΠZN
max
β∈xU(β) dπ(U),(12)
with Πa (non-unique) convex and compact set of Borel probability measures on the space Nof specifically
normalised ex post utility functions; cf. Epstein et al (2007) [23, p 365].
For their model II, in order to provide a formal basis for dealing with persistent coarseness “then” of an
agent’s perception of future subjective contingencies, Epstein et al (2007) [23] enlarge the set of an agent’s
3In the literature the names non-degeneracy and non-triviality are used synonymously for one of the axioms.
5 DISCUSSION AND OUTLOOK
13
objects of choice to also include random menus of lotteries. This thus yields a set of Borel probability measures
∆(X)defined over menus in X. A generic element in ∆(X)is denoted by P. Proposing a minimal set
of six axioms comprising (weak) order, continuity, non-degeneracy, first-stage independence, dominance, and
certainty reversal of order to hold, a representation theorem is proved for an agent’s binary ex ante preference
order over random menus in ∆(X)to the extent that, in this model, an agent’s choice behaviour corresponds
to maximising a real-valued preference functional VP C : ∆(X)→R, given by
VP C (P) := ZX"ZKcc (N∗)
max
β∈xmin
U∈U U(β)µ(dU)#dP(x),(13)
with µ∈∆(Kcc(N∗)) a Borel probability measure over the set Kcc(N∗)of closed, convex and comprehensive
Hausdorff-topology subsets of the compact space of specifically normalised ex post utility functions N∗(cf.
Epstein et al (2007) [23, p 366]), which is unique up to linear transformations. U ⊂ N∗denotes the subset of
normalised ex post utility functions conceived of by an agent “now”, which, however, to them in that instance
have unknown likelihoods as regards realisation “then”. In this respect, N∗\U may be interpreted as relating
to an agent’s unawareness (or non-knowledge) “now” of possible subjective contingencies “then”.
5 Discussion and outlook
Now, having discussed the state of the art in the literature, we should question if the formal representation of
non-knowledge in economic theory has been satisfactory so far. Moreover, in what follows we sketch some
promising directions of research which we did not address in detail in this paper.
As highlighted in section 2, true uncertainty and, thus, genuine non-knowledge about the future, are features
of the situation in whichan agent is unaware of all future contingencies, not (just) due to their limited ability to
calculate, or to search for information, but due to the very nature of any economic system. The major insight
of Knight, Keynes, Shackle, and some Post-Keynesians, was that economic systems are open and organic
unities that are genuinely indeterminate; every decision situation is incomplete because it undergoes a constant
change while people decide and act and, by doing so, influence the set of relevant variables; hence, the major
characteristics of the decision situation — first of all, the future states that are possible and conceivable —
cannot be sufficiently determined; they are unknown.
We already mentioned the following concrete reasons for the indeterminacy of decision situations: (i) the
big world issue (i.e., the indefinite, non-exhaustive, number of possible future states), (ii) the endogeneity of
the decision situations, i.e., the dependence of future outcomes on decisions which are prepared and made in
the present, and (iii) the social contingency which is typical for economic systems, where the indeterminacy in-
creases due tothe dependence of an agent’s decisions on what other agents decide. Are those issues adequately
reflected in the ambiguity and unawareness approaches, which we discussed in this paper?
5.1 Big world issue
Savage’s (1954) [61] axiomatisation was often criticised for its restrictive assumption of the “small” world: the
list of possible events is presupposed to be exhaustive (though Savage [61, p 16] himself referred to such an
assumption as “preposterous”). Some of the follow-up concepts discussed in our paper differ in their treatment
of this issue.
The uncertainty as ambiguity approaches we mentioned continue to employ Savage-type state spaces as
primitives, which are continuous, compact, and can be partitioned into a finite number of mutually exclusive
events, while there is an uncountable number of different states. Although in principle no additional structure
is needed, some authors like Epstein and Wang (1994) [22, p 206] assume on the state space the existence of a
metric and a particular (“weak convergence”) topology, suggesting that one can construct an indefinite number
of different subsets of the state space; the boundaries of such subsets are not entirely clear. The question arises
5 DISCUSSION AND OUTLOOK
14
if, in the end, such mathematical structures make everything possible and thinkable, thus offering a loophole
for the assumption that the list of possible events is not exhaustive. It is worthwhile mentioning here that the
formal handling, but even more so providing compelling interpretations, of a potential infinitude of possibilities
or states regularly proves a delicate issue in most (if not all) areas of applied mathematics and statistics; see,
e.g., Hawking and Ellis (1973) [36].
In the uncertainty as unawareness models, in contrast, the big world issue, which relates to the state space
representing the exogenous world, is of less importance, since here the focus of the analyses is on an agent’s
subjective state space. This concept, however, does not belong to the set of the primitives of the theory. This
issue is closely related to the exogeneity vs endogeneity topic which we turn to discuss next.
5.2 Endogeneity of state space
In our view, the question of Machina (2003) [48, p 18]: “Do individuals making choices under uncertainty
face states of nature, or do they create them?” remains one of the most crucial and controversial in decision
theory. In Savage’s concept, the state space represents nature’s exogenous states, i.e., their emergence cannot
be influenced by agents’ decisions and actions; an agent just observes the states and is not an active part of the
decision situation.
In economics, as well as in the social sciences, however, there is increasing attention being payed to the
issue of creation of economic reality by the actions and decisions of economic agents; e.g., one considers the
notions of exogenous risk, performativity and reflexivity; cf. Danielsson and Shin (2003) [12], Danielson et
al (2009) [13], Callon (1998) [9], MacKenzie (2006) [50], and Soros (1998) [65]. There is also an interesting
movement in the direction of constructive decision theory, where decision-relevant states are not given but
“constructed by the DM (decision maker) in the course of deliberating about questions such as ‘How is choice
A different from choice B?’ and ‘In what circumstances will choice A turn out better than choice B?”’; see
Blume et al (2009) [7, p 1f].
Concerning the papers discussed in the article at hand, the works on uncertainty as ambiguity focus on the
non-knowledge of probabilities; here, the state space remains exogenous. However, the unawareness literature
makes an interesting and important move towards the conceptualisation of an endogenous state space: future
outcomes (“then”) are contingent on decisions made at present (“now”); the subjective state space is not directly
observable but can bederived from the only variable observable: an agent’s behaviour, or an agent’s preferences
(e.g., for flexibility).
However, an important question remains if, in the unawareness literature, we really deal with a truly en-
dogenous state space, as understood in the concepts of endogenous risk, performativity and reflexivity. There is
already raised some criticism in the literature, e.g. by Sagi (2006) [60], who is concerned about the static nature
of the theoretical construction in the unawareness approaches: “The decision maker chooses among menus, un-
certainty over her subjective states is assumed to resolve and then the decision maker selects from the menu.
However, there is no explicit modelling of ex-post choice and no role for consistency between realized tastes
and tastes inferred from ex-ante preferences”, see Sagi (2006) [60, p 307]. Wealso think that the representation
of true endogeneity — as a central determinant of non-knowledge — should be dynamic: uncertainty cannot
be resolved, as a situation constantly changes, so that the ex-post choice should not be modelled as a mechanic,
or empty (i.e., predetermined) decision. The other important dynamic aspect is the modelling of the evolution
of the state space itself (how it expands and changes); the genesis of a decision situation should be taken into
consideration. There are some interesting ideas aiming at these issues, e.g. by Hayashi (2012) [38], and Grant
and Quiggin (2007) [32]. The latter authors model the notion of discovery of the principally unknown space
states by decision-makers. We consider this issue to be crucial for making further progress in the unawareness
and non-knowledge literature.
REFERENCES
15
5.3 Social contingency
The idea of an endogenous space state (as well as the notions of endogenous risk, performativity and reflexivity)
goes beyond the subjective level of decision-making: future states are unknown because they are contingent
on thinking, deciding and acting of all interconnected economic agents. This is an issue which was neglected
in the unawareness papers presented here. However, there are interesting attempts to account for the social
construction of the (subjectively perceived) state space. Here we refer to the work on interactive unawareness
by Heifetz, Meier and Schipper (2006, 2008) [39,40], and on epistemic game theory; cf. Brandenburger (2008)
[8].
Finally, it may be noted that only a small number of authors proposing theoretical models of agents’ choice
behaviour under conditions of uncertainty are committed to making testable predictions that may be refuted
in principle. This state of affairs conflicts with logical positivism’s view that the falsification of hypotheses
by means of observation and/or experiment is the primary method for attaining meaningful progress in any
empirical scientific disciplines; see, e.g., Popper (2002) [55]. Ideally, future research in economics and decision
theory will address this problem more carefully.
References
[1] Akerlof G A 1970 The market for “Lemons”: quality uncertainty and the market mechanism
The Quarterly Journal of Economics 84 488–500
[2] Akerlof G A and Shiller R J 2009 Animal Spirits: How Human Psychology Drives the Economy, and Why
It Matters for Global Capitalism (Princeton, NJ: Princeton University Press)
[3] Beckert J 1996 What is sociological about economic sociology? Uncertainty and the embeddedness of
economic action Theory and Society 25 803–840
[4] Bernoulli D 1738 Specimen theoriae novae de mensura sortis
English translation: 1954 Exposition of a new theory on the measurement of risk Econometrica 22 23–36
[5] Bewley T F 1986 Knightian decision theory: part I (Cowles Foundation: discussion paper)
[6] Bewley T F 2002 Knightian decision theory: part I Decisions in Economics and Finance 25 79–110
[7] Blume L, Easley D and Halpern J Y 2009 Constructive decision theory (Institute for
Advanced Studies, Vienna: Economics Series) URL (cited on September 5, 2012):
www.ihs.ac.at/publications/eco/es-246.pdf
[8] Brandenburger A 2008 Epistemic game theory: an overview The New Palgrave Dictionary of
Economics Online eds S N Durlauf and L E Blume URL (cited on September 5, 2012):
www.dictionaryofeconomics.com/article?id=pde2008 E000257
[9] Callon M 1998 The Laws of the Market (Oxford: Blackwell)
[10] Colander D, F¨ollmer H, Haas A, Goldberg M D, Juselius K, Kirman A, Lux T and Sloth B 2009
The financial crisis and the systemic failure of academic economics (University of Copenhagen, Dept
of Economics: discussion paper No. 09-03, March 9, 2009) URL (cited on August 28, 2012):
http://ssrn.com/abstract=1355882
[11] Crocco M 2002 The concept of degrees of uncertainty in Keynes, Shackle, and Davidson
Nova Economia 12 11–28
REFERENCES
16
[12] Danielsson J and Shin H S 2003 Endogenous risk Modern Risk Management: A History ed P Field
(London: Risk Books) 297–314
[13] Danielsson J, Shin H S and Zigrand J-P 2009 Risk appetite and endogenous risk (Financial
Markets Group (LSE): FMG Discussion Papers DP647) URL (cited on September 5, 2012):
www.princeton.edu/∼hsshin/www/riskappetite.pdf
[14] Davidson P 1991 Is probability theory relevant for uncertainty? A post Keynesian perspective
Journal of Economic Perspectives 5129–143
[15] Dekel E, Lipman B L and Rustichini A 1998 Standard state-space models preclude unawareness
Econometrica 66 159–173
[16] Dekel E, Lipman B L and Rustichini A 2001 Representing preferences with a unique subjective state
space Econometrica 69 891–934
[17] Dequech D 2006 The new institutional economics and the theory of behaviour under uncertainty
Journal of Economic Behavior and Organization 59 109–131
[18] Dewey J 1915 The logic of judgments of practise The Journal of Philosophy, Psychology and Scientific Methods 12 505–523
[19] Diebold F X, Doherty N A and Herring R J 2010 The Known, the Unknown, and the Unknowable in
Financial Risk Management: Measurement and Theory Advancing Practice (Princeton, NJ: Princeton
University Press)
[20] The Economist 2007 A new fashion in modeling: what to do when you don’t know everything URL (cited
on August 28, 2012): www.economist.com/node/10172461
[21] Ellsberg D 1961 Risk, ambiguity, and the Savageaxioms The Quarterly Journal of Economics 75 643–669
[22] Epstein L G and Wang T 1994 Intertemporal asset pricing under Knightian uncertainty
Econometrica 62 283–322
[23] Epstein L G, Marinacci M and Seo K 2007 Coarse contingencies and ambiguity
Theoretical Economics 2355–394
[24] Epstein L G and Schneider M 2010 Ambiguity and asset markets
Annual Reviews of Financial Economics 2315–346
[25] Esposito E 2007 Die Fiktion der wahrscheinlichen Realit¨
at (Frankfurt am Main: Suhrkamp)
[26] Esposito E 2010 Die Zukunft der Futures: Die Zeit des Geldes in Finanzwelt und Gesellschaft (Heidelberg:
Carl-Auer-Systeme)
[27] de Finetti B 1937 La pr´evision: ses lois logiques, ses sources subjectives Annales de l’Institut Henri
Poincar´
e71–68
[28] von F¨orster H 1993 KybernEthik (Berlin: Merve)
[29] Gilboa I 2009 Theory of Decision under Uncertainty (Cambridge: Cambridge University Press)
[30] Gilboa I and Schmeidler D 1989 Maxmin expected utility with non-unique prior
Journal of Mathematical Economics 18 141–153
[31] Gilboa I, Postlewaite A W and Schmeidler D 2008 Probability and uncertainty in economic modeling
Journal of Economic Perspectives 22 173–188
REFERENCES
17
[32] Grant S and Quiggin J 2007 Awareness and discovery (Houston, Rice University: working paper)
[33] Hansen L P, Sargent T J and Tallarini T D 1999 Robust permanent income and pricing
Review of Economic Studies 66 873–907
[34] Hansen L P and Sargent T J 2001 Acknowledging misspecification in macroeconomic theory
Review of Economic Dynamics 4519–535
[35] Hansen L P and Sargent T J 2003 Robust control of forward-looking models
Journal of Monetary Economics 50 581–604
[36] Hawking S W and Ellis G F R 1973 The Large Scale Structure of Space-Time (Cambridge: Cambridge
University Press)
[37] Hayek F A 1945 The use of knowledge in society The American Economic Review 35 519–530
[38] Hayashi T 2012 Expanding state space and extension of beliefs Theory and Decision 73 591–604
[39] Heifetz A, Meier M and Schipper B C 2006 Interactive unawareness
Journal of Economic Theory 130 78–94
[40] Heifetz A, Meier M and Schipper B C 2008 A canonical model for interactive unawareness
Games and Economic Behavior 62 304–324
[41] IPCC Fourth Assessment Report: Climate Change 2007 URL (cited on September 2, 2012):
www.ipcc.ch/publications and data/ar4/wg3/en/ch2s2-3.html
[42] Keynes J M 1921 A Treatise on Probability (London: Macmillan)
[43] Keynes J M 1937 The general theory of employment The Quarterly Journal of Economics 51 209–233
[44] Keynes J M, Bonar J and Lloyd I H 1926 Francis Ysidro Edgeworth 1845–1926
The Economic Journal 36 140–158
[45] Knight F H 1921 Risk, Uncertainty and Profit (Boston: Houghton Mifflin)
[46] Kreps D M 1979 A representation theorem for “preference for flexibility” Econometrica 47 565–577
[47] Kreps D M 1992 Static choice and unforeseen contingencies Economic Analysis of Markets and Games:
Essays in Honor of Frank Hahn ed P Dasgupta, D Gale, O Hart and E Maskin (Cambridge, MA: MIT
Press) 259–281
[48] Machina M J 2003 States of the world and the state of decision theory The Economics of Risk ed D Meyer
(Kalamazoo, MI: W E Upjohn Institute for Employment Research) 17–49
[49] McGoey L 2012 Strategic unknowns: towards a sociology of ignorance Economy and Society 41 1–16
[50] Mackenzie D 2006 An Engine, Not a Camera: How Financial Models Shape Markets (Cambridge, MA:
MIT Press)
[51] Nash S J 2003 On pragmatic philosophy and Knightian uncertainty
Review of Social Economy 61 251–272
[52] Nehring K 1999 Preference for flexibility in a Savage framework Econometrica 67 101–119
[53] von Neumann J and Morgenstern O 1944 Theory of Games and Economic Behavior (Princeton, NJ:
Princeton University Press)
REFERENCES
18
[54] Nishimura K G and Ozaki H 2004 Search and Knightian uncertainty
Journal of Economic Theory 119 299–333
[55] Popper K R 2002 Conjectures and Refutations: The Growth of Scientific Knowledge 2nd Edition (London:
Routledge)
[56] Proctor R N 2008 Agnotology: the missing term to describe the cultural production of ignorance (and its
study) Agnotology: The Making and Unmaking of Ignorance ed R N Proctor and L Schiebinger (Stanford,
CA: Stanford University Press) 1–29
[57] Ramsey F P 1931 Truth and probability The Foundations of Mathematics and Other Logical Essays (Lon-
don: Routledge and Kegan Paul) 156–198
[58] Rooney D, Hearn G and Ninan A (eds) 2008 Handbook on the Knowledge Economy (Cheltenham: Edward
Elgar Publishing)
[59] Rigotti L and Shannon C 2005 Uncertainty and risk in financial markets Econometrica 73 203–243
[60] Sagi J S 2006 What is an ’endogenous state space’? Economic Theory 27 305–320
[61] Savage L J 1954 The Foundations of Statistics (New York: Wiley)
[62] Shackle G L S 1949 Expectations in Economics (Cambridge: Cambridge University Press)
[63] Shackle G L S 1955 Uncertainty in Economics and Other Reflections (Cambridge: Cambridge University
Press)
[64] Shackle G L S 1959 Time and thought The British Journal for the Philosophy of Science 9285–298
[65] Soros G 1998 The Crisis Of Global Capitalism: Open Society Endangered (New York: Public Affairs)
[66] Spence M A 1973 Job market signaling The Quarterly Journal of Economics 87 355–374
[67] Stigler G J 1961 The economics of information Journal of Political Economy 69 213–225
[68] Stiglitz J E 1975 The theory of ‘screening’, education, and the distribution of income
The American Economic Review 65 283–300
[69] Stiglitz J E 2002 Information and the change in the paradigm in economics
The American Economic Review 92 460–501
[70] Svetlova E and Fiedler M 2011 Understanding crisis: on the meaning of uncertainty and probability The
Recession of 2008 — Competing Explanations eds O D Asenjo and C Marcuzzo (Camberley: Edward
Elgar Publishing) 42–62
[71] Taleb N N 2007 The Black Swan — The Impact of the Highly Improbable (London: Penguin)
[72] Taleb N N 2010 Why did the crisis of 2008 happen? (SSRN: discussion paper) URL (cited on August 28,
2012): http://papers.ssrn.com/sol3/papers.cfm?abstract id=1666042
