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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