Making decision under mixture of ignorance and risk
Phan H. Giang
Department of Health Administration & Policy, George Mason University
Example: Iraq WMD program
Decomposition of uncertainty into risk and ignorance
Separate risk component from ignorance component in uncertainty
Conditionalization method: Risk is conditional on resolution of Ignorance
or Ignorance is conditional on resolution of Risk.
Conditionalization between risk and ignorance is a directed operation.
That is one can not reverse the direction of conditionalization.
Important property: risk and ignorance are 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, represents inability
to form coherent probability and inability to define the space of
alternatives.
Ignorance can arise when
lack of knowledge about the factors that influence the issues of
interest
lack of reliable data, opponents actively use deceiving practice
theories and opinions adopted by experts are contradictory with
Reactions
each
other to ignorance
to acknowledge our ignorance and explicitly deal with it or
to ignore the holes in our knowledge and fill the void with educated
guesses, analytic assumptions
Working assumption
Uncertainty in the real world is mixture of ignorance and risk
(probability).
Problem
how to make decision under ignorance-risk uncertainty?
Popular uncertainty models viewed as risk-ignorance mixture
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 or convex capacity
• Dempster-Shaffer belief function
• Possibility theory
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.
Many results for decision under ignorance is unintuitive for those who is
thinking in terms of probability.
Hurwicz \alpha-rule: an act is indifferent to the a combination of the best
and the worst outcomes. Decision rule based on \alpha rule satisfies
Hurwicz-Arrow theorem.
Median rule: an act (set of outcomes) is indifferent to the median outcome
does not satisfy Hurwicz-Arrow theorem.
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.
\alpha 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.
Rational decision under ignorance: tau-anchor utility theory
Characterization of decision rule that satisfies Hurwicz-Arrow conditions
and dynamically consistent
\tau =