Ambiguity and the Bayesian Paradigm by Gilboa Marinacci

Ambiguity and the Bayesian Paradigm by Gilboa Marinacci

Gilboa and Marinacci's work explores the complexities of decision-making under uncertainty, focusing on the limitations of the Bayesian paradigm. The text delves into ambiguity aversion, contrasting traditional Bayesian beliefs with alternative models that account for uncertainty. It discusses various decision-theoretic frameworks, including Choquet expected utility and maxmin expected utility, providing insights into how ambiguity influences economic behavior. This comprehensive analysis is essential for economists and decision theorists seeking to understand the implications of ambiguity in their models.

Key Points

  • Explores the limitations of the Bayesian paradigm in decision-making under uncertainty.
  • Discusses ambiguity aversion and its impact on economic behavior.
  • Analyzes various decision-theoretic frameworks like Choquet expected utility.
  • Contrasts traditional Bayesian beliefs with alternative models of uncertainty.
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Ambiguity and the Bayesian Paradigm
Itzhak Gilboa
y
and Massimo Marinacci
z
April 2011
Abstract
This is a survey of some of the r ecent decision-theoretic literature involving beliefs that cannot
be quanti…ed by a Bayesian prior. We discuss historical, philosophical, and axiomatic founda-
tions of the Bayesian model, as well as of several alternative models re cently propo sed. The
de…nition and comparison of ambiguity aversion and the updating of non-Bayesi an be liefs are
brie‡y discussed. Fi nally, several applications are mentioned to illustrate the way that ambiguity
(or “Kni ghtian uncertainty”) can change the way we think about econom ic problems.
1 Introduction
1.1 Varying probability estimates
John and Lisa are o¤ered additional insurance against the risk of a heart disease. They would like
to know the probability of developing such a disease over the next ten years. Th e happy couple
shares some key medical parameters: they are 70 years old, smoke, and never had a blood pressure
problem. A few tests show that both have a total cholesterol level of 310 mg/dL, with HDL-C (good
cholesterol) of 45 mg/dL, and that their systolic blood pressure is 130. Googling “heart disease risk
calculator”, they …nd several sites that allow them to calculate their risk. The results (May 2010)
are:
John Lisa
Mayo Clinic 25% 11%
National Cholesterol Education Program 27% 21%
American Heart Association 25% 11%
Medical College of Wisconsin 53% 27%
University of Maryland Heart Center 50% 27%
We thank Giulia Brancaccio, Simone Cerreia-Vioglio, Fabio Maccheroni, Andrew Postlewaite, Xiangyu Qu, and
David Schmeidler for comments on earlier drafts of this survey. We are also grateful to many members of the “decision
theory forum” for additional comment s and references. Gilboa gratefully acknowledges the …nancial sup po rt of the
Israel Science Foundation (grant 396/10) and of the Euro pean Reseach Council (advanced grant 269754) , and Marinacci
that of the European Reseach Council (advanced grant BRSCDP-TEA). An abridged version of this survey wi ll appear
in “Advances in Economics and Econometrics: Theory and Applicati ons”, Tenth World Congress of the Econometric
Society (D. Acemoglu, M. Arellano, and E. Dekel, eds.), Cambridge Uni versi ty Press.
y
HEC, Paris , and Tel-Aviv University, tzachigilboa@gmail.com
z
Università Bocconi, massimo.ma rinacci@unibocconi.it
1
The di¤erent calculators don’t completely agree on the probability in question. Partly, the reason
is that no two human bodies are identical, and therefore simple relative frequencies cannot be used
as a de…nition of “the probability of a heart disease”. Rather, the probabilities above are computed
by more sophisticated techniques such as logistic regression and variants thereof. These techniques
allow researchers to assess probabilities for di¤erent individuals depending on their characteristics.
But the resulting numbers are not perfectly objective: the researchers have to choose the variables
(predictors), the database, as well as the estimation technique itself. Consequently, the estimated
probability is not unique. The estimates vary substantially: the highest for John is 100% higher
than the lowest, whereas for Lisa the ratio is 5:2. Moreover, gender di¤erently a¤ects the estimated
numbers across risk calculators. Clearly, di¤erent probability estimates in the range of 25%-50% and
11%-27% may result in di¤erent decisions.
Next consider Mary, who contemplates an investment in a beach resort. The pro…tability of this
venture depends on global warming: the resort will not be very successful if, due to climate changes,
the beach b ec omes much rainier than it is now, or if the o ce ans’level increases signi…cantly. Mary
wonders what the probabilities of these eventualities are.
For this problem the estimation of probabilities seems conceptually h arder. We have but one
globe to base our predictions on. The history of global warmings in the past is only indirectly
attested to, and the number of past case s is not very large. Worse still, current conditions di¤er from
past ones in signi…cant ways, especially in the much-discussed human-generated conditions. Finally,
it is n ot obvious that one can assume causal independence across di¤erent warmings. Due to all
these di¢ culties, it is perhaps not very surprising that the estimates of the distribution of average
temperature several years hence vary considerably across experts.
1.2 Does rationality necessitate probability?
Since the mid-20th century, economic theory is dominated by the Bayesian paradigm, which holds
that any source of u nce rtainty can and should be quanti…ed probabilistically.
1
According to this view,
John and Lisa should have well-de…ned probabilities that they will develop a heart disease within
the next ten years, as should Mary for the temperature distribution anywhere on the globe …ve years
hence. But where should John, Lisa, or Mary get these probabilities from? If they are to consult
experts, they will typically obtain di¤erent estimates. Which experts are they to believe? Should
they compute an average of the experts’estimates, and, if so, how much weight should each expert
have in this average?
The standard line of reasoning of the Bayesian approach is th at, in the absence of objective
1
As Cyert and DeGroot (1974) write on p. 524 “To the Bayesian, all uncertainty can be represented by probability
distributions.”
2
probabilities, the decision maker (DM, for short) should have her own, subjective probabilities, and
that these probabilities should guide her decisions. Moreover, the remarkable axiomatic derivations of
the Bayesian approach (culminating in Savage, 1954), show that axioms that appear very compelling
necessitate that the DM behave as if she maximized expected utility relative to a certain probability
measure, which is interpreted as her subjective probability. Thus, the axiomatic foundations basically
say, “Even if you don’t know what the probabilities are, you should better adopt some probabilities
and make decisions in accordance with them, as this is the only way to satisfy the axioms.”
There is a heated debate regarding the claim that rationality necessitates Bayesian beliefs. Knight
(1921) and Keynes (1921, 1937) argued that not all sources of uncertainty can be probabilistically
quanti…ed. Knight suggested to distinguish between “risk”, referring to situations de scribed by known
or calculable p robabilities, and “uncertainty”, where probabilities are neither given nor computable.
Keynes (1937) wrote,
“By ‘uncertain’knowledge, let me explain, I do not mean merely to distinguish what
is known for certain from what is on ly probable. The game of roulette is not subject, in
this sense, to uncertainty ... The sense in which I am using the term is that in which the
prospect of a European war is uncertain, or the price of copp e r and the rate of interest
twenty years hence ... About these matters the re is no scienti…c basis on which to form
any calculable probability whatever. We simply do not know.”
Gilboa, Postlewaite, and Schmeidler (2008, 2009, 2010) argue that the axiomatic foundations of
the Bayesian approach are not as compelling as they seem, and that it may be irrational to follow
this approach. In a nutshell, their argument is that the Bayesian approach is limited because of
its inability to express ignorance: it requires that the agent express beliefs whenever asked, without
being allowed to say “I don’t know”. Such an agent may provide arbitrary answers, which are likely
to violate the axioms, or adopt a single p robability and provide answers based on it. But such a
choice would be arbitrary, and therefore a poor candidate for a rational mode of behavior.
The notion that rational individuals would select probabilities to serve as their subjective belief
might be particularly odd in the context of a market. Consider John and Lisa’s problem again.
Suppose that they receive a phone call from an agent who o¤ers them an insurance policy. They
…nd the policy too expensive. In the process of negotiation, the insurance agent quotes the Medical
College of Wisconsin’s site and estimates the probability of John and Lisa developing a heart disease
within ten years at 53% and 27%, respectively. John and Lisa, by contrast, looked up the site of the
Mayo Clinic and concluded that these probabilities are only 25% an d 11%. They decide not to buy
the insurance, and the conversation ends by John, Lisa, and the agent politely agreeing to disagree.
After John and Lisa hang up, they look up a few other sites and realize that there are estimates that
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FAQs of Ambiguity and the Bayesian Paradigm by Gilboa Marinacci

What is the main focus of Gilboa and Marinacci's work?
Gilboa and Marinacci's work primarily focuses on the limitations of the Bayesian paradigm in decision-making under uncertainty. They explore how traditional Bayesian beliefs may not adequately capture the complexities of real-world decision-making, especially in the presence of ambiguity. The authors delve into alternative models that account for ambiguity aversion, providing a richer understanding of how individuals make choices when faced with uncertain outcomes.
How does ambiguity aversion influence economic behavior?
Ambiguity aversion significantly influences economic behavior by affecting how individuals perceive and react to uncertain situations. In their analysis, Gilboa and Marinacci highlight that individuals often prefer to avoid ambiguous situations, leading them to make conservative choices. This aversion can result in a reluctance to engage in certain financial markets or investment opportunities, as individuals may opt for safer, more predictable options instead of those with uncertain outcomes.
What are the key decision-theoretic frameworks discussed in the document?
The document discusses several key decision-theoretic frameworks, including Choquet expected utility and maxmin expected utility. Choquet expected utility allows for non-additive probabilities, providing a way to model preferences under ambiguity. Maxmin expected utility, on the other hand, emphasizes the worst-case scenario approach, where decision-makers focus on minimizing potential losses. Both frameworks offer insights into how ambiguity affects decision-making processes in economic contexts.
What implications does the document suggest for traditional Bayesian models?
The document suggests that traditional Bayesian models may not fully capture the nuances of decision-making under ambiguity. By highlighting the limitations of the Bayesian approach, Gilboa and Marinacci argue for the necessity of incorporating alternative models that account for ambiguity aversion. This shift in perspective can lead to more accurate predictions and better understanding of economic behavior, particularly in markets characterized by uncertainty.

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