Making decision under mixture of ignorance and risk
Phan H. Giang
Department of Health Administration & Policy
Example: Iraq WMD program
Decision under risk and ignorance
Decomposition of uncertainty into risk and ignorance
 Separate risk component from ignorance component in uncertainty
 Method of conditionalization:
• Risk is conditional on resolution of Ignorance or
• Ignorance is conditional on resolution of Risk.
 Important: the order between risk and ignorance is not reversible.
Decision under uncertainty becomes combination of decision under
risk and decision under ignorance.
“As we know, there are known knowns; there are things we
know we know. We also know there are known unknowns;
that is to say we know there are some things we do not know.
But there are also unknown unknowns - the ones we don’t
know we don’t know.” (Donald Rumsfeld 02/2002)
Technical definition
Ignorance is uncertainty pushed to the extreme, it represents
 the inability to form coherent probability and
 the inability to define the space of alternatives
Signs of Ignorance
• lack of knowledge about the factors influence the issues of
interest
• lack of reliable data, opponents actively practice deceit
• theories and opinions promoted by experts are contradictory with
each other
Reactions to ignorance
 to acknowledge our ignorance and explicitly deal with it or
 to ignore the voids in our knowledge and fill them with educated
guesses, analytic assumptions
Working assumption: Uncertainty in the real world is some
mixture of ignorance and risk.
Decision under risk: well understood
Variations and modifications of Expected utility theory or Prospect theory
Rational decision under ignorance: much less understood
Hurwicz - Arrow theorem: in comparison of acts under ignorance only
extreme outcomes (the best and the worst outcomes) matter. If both
extreme outcomes of an act is better than the counterpart values of another
act then the former is preferable to the later.
Example: investment choice
An investor is considering at the end of 2014 a one-year investment instrument that matures
on 1 January 2016. The return on the investment depends on two uncertain variables. The
first source of uncertainty is the prospect of a political settlement in country A (e.g.
Afghanistan) and the second source of uncertainty is the prospect of 2015 coffee crop in
country B (e.g. Brazil). The 2015 coffee harvest, denoted by C, can be either bumper (b) or
normal (n) or poor (p). On the one hand, from extensive historical data, the probability
distribution of Brazilian coffee crop in 2015 is estimated to be (0.46, 0.2, 0.34) where the
numbers are the chances of having bumper, normal and poor crop respectively. On the other
hand, the political settlement variable, denoted by S, is modeled with two possible values:
peaceful settlement among fighting factions in 2015 (s) or lack thereof (~s). The experts
whose advice the investor seeks on the political settlement question, offer contradictory
opinions and cannot come to any agreement. This underlies the fact that nobody knows the
true driving forces behind a political settlement in that region of the world.
Settlement /Crop
s
b
-3%
n
2%
p
8%
~s
5%
1.5%
-4%
1. Hurwicz α-rule: a set of outcomes is indifferent to the a α- combination
of the best and the worst outcomes. Decision rule based on α-rule is
complete and satisfies Hurwicz-Arrow theorem but it suffers from a
troubling property.
2. Median rule: an act (set of outcomes) is indifferent to the median
outcome does not satisfy Hurwicz-Arrow theorem.
Many results for decision under ignorance is unintuitive for those who are
used to think in terms of probability.
Rational decision under ignorance: issues
• Hurwicz-Arrow theorem does not specify a complete order on the act
space i.e., it is silent in the case that the best outcome of an act is better
than the best outcome of another but the worst outcome of the former is
worse than the worst outcome of the latter.
• α-rule specifies a complete order in the act space but suffers from a
problem called “dynamic inconsistency”. I.e. the same act under
ignorance is evaluated to different values under different mental
representations without change in outcomes or information.
Popular uncertainty models viewed as risk-ignorance mixture
Rational decision under ignorance: tau-anchor utility theory
Many popular uncertainty models considered in literature and used in
practical applications can be formulated as mixture of risk and
ignorance
 Sets of probability measures
 Non-additive convex capacity measure
 Dempster-Shaffer belief function theory
 Possibility theory
Theorem (Giang 2015) A decision rule that satisfies Hurwicz-Arrow
conditions and dynamically consistent must have the following form
max 𝐴 𝑖𝑓 max 𝐴 ≤ τ
CE(A)= τ 𝑖𝑓 min 𝐴 ≤ τ ≤ max(𝐴)
min 𝐴 𝑖𝑓 τ ≤ min 𝐴
where CE(A) is the certainty equivalence for a set of outcomes A and τ is a
real value that characterizes tolerance to ignorance of the decision maker.
Conclusion and findings
We develop a theoretical framework that can handle both risk and ignorance, allows faithful
and realistic representation of knowledge, avoids unjustified assumptions.
 Findings
• Decision maker’s tolerance to ignorance affects the choice
• The order in which the risk and ignorance are resolved is important and can affect the
value of decision.
 Practical implications
• You can change the value of your uncertain situation by manipulating the order in
which risk and ignorance are resolved.
• Method to estimate a person’s tolerance to ignorance and uncertainty from her choice
behavior.