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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.

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