Confidence in Beliefs and Rational Decision Making · The higher the stakes involved in a decision,...
Transcript of Confidence in Beliefs and Rational Decision Making · The higher the stakes involved in a decision,...
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Confidence in Beliefs and Rational DecisionMaking
Brian [email protected]
www.hec.fr/hill
GREGHEC, CNRS & HEC Paris
2016(Presentation prepared for an
interdisciplinary audience)
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Flooding or drought defence (2016)
Floods Droughts
Floods Droughts
Flood Defence 100 0Drought Defence 0 100
Flooding or drought defence (1946)
Floods Droughts
Floods DroughtsFlood Defence 100 0Drought Defence 0 100
§ Which decision would you prefer to take?
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Flooding or drought defence (2016)
Floods Droughts
Floods Droughts
Flood Defence 100 0Drought Defence 0 100
Flooding or drought defence (1946)
Floods Droughts
Floods DroughtsFlood Defence 100 0Drought Defence 0 100
§ Which decision would you prefer to take?
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Flooding or drought defence (2016)
Floods Droughts
Floods Droughts
Flood Defence 100 0Drought Defence 0 100
Flooding or drought defence (1946)
Floods Droughts
Floods DroughtsFlood Defence 100 0Drought Defence 0 100
§ Which decision would you prefer to take?
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Ellsberg (1961)
Red or Black
§ Unknown urn: 100 balls, each red or black.§ Known urn: 100 balls, 50 red, 50 black.
Unknown urn Known urnRed Black Red Black
I $ 100 $ 0
$ 0 $ 0
II $ 0 $ 100
$ 0 $ 0
III
$ 0 $ 0
$ 100 $ 0IV
$ 0 $ 0
$ 0 $ 100
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Ambiguity
§ Ellsberg behaviour is inconsistent with Bayesianism.
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BayesianismA Quick Introduction
What is it? Three tenets:
Belief Beliefs can be represented by probabilities.Decision Decision Makers maximise expected utility.
Subjective Expected Utility (SEU)
Belief Formation Update by conditionalisation.
Expected utility
Eppupf qq “ÿ
sPS
upf psqq.ppsq
§ u: utility function, representing desires or tastes
§ p: probability measure, representing beliefs
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BayesianismA Quick Introduction
What does it do? Purportedly:
Descriptive it accurately captures peoples’ choice behaviour.Economic Modelling it is a tractable approximate model of
peoples’ choice behaviour.Normative it is the canon of rationality for belief and decision
making.Prescriptive it provides a tractable framework for taking
complex decisions (properly).
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BayesianismA Quick Introduction
Why? Purportedly:
1. it corresponds to a reasonable (pre-theoretical) intuition2. it has acceptable choice-theoretical consequences3. it is conceptually clear about the roles of different mental
attitudes4. it has a certain tractability for eg. decision analysis
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BayesianismA Quick Introduction
Why? Purportedly:
1. it corresponds to a reasonable (pre-theoretical) intuition2. it has acceptable choice-theoretical consequences3. it is conceptually clear about the roles of different mental
attitudes4. it has a certain tractability for eg. decision analysis
§ Dutch Book Arguments, Representation Theorems
i.e. results of the following form:
Properties of preferences/ choice
ô D p,u s.t. choice maximisesEppupf qq
And the p,u are appropriately unique.7 / 50
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BayesianismA Quick Introduction
Why? Purportedly:
1. it corresponds to a reasonable (pre-theoretical) intuition2. it has acceptable choice-theoretical consequences3. it is conceptually clear about the roles of different mental
attitudes4. it has a certain tractability for eg. decision analysis
§ u: utility function, representing desires or tastes
§ p: probability measure, representing beliefs
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Ellsberg (1961)Red or Black
§ Unknown urn: 100 balls, each red or black.§ Known urn: 100 balls, 50 red, 50 black.
Unknown urn Known urnRed Black Red Black
I $ 100 $ 0
$ 0 $ 0
II $ 0 $ 100
$ 0 $ 0
III
$ 0 $ 0
$ 100 $ 0IV
$ 0 $ 0
$ 0 $ 100
Which would you prefer (or would you be indifferent) between:
§ I and III?
§ I and II?
§ III and IV?8 / 50
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Ellsberg (1961)Red or Black
§ Unknown urn: 100 balls, each red or black.§ Known urn: 100 balls, 50 red, 50 black.
Unknown urn Known urnRed Black Red Black
I $ 100 $ 0
$ 0 $ 0
II $ 0 $ 100
$ 0 $ 0
III
$ 0 $ 0
$ 100 $ 0IV
$ 0 $ 0
$ 0 $ 100
Which would you prefer (or would you be indifferent) between:
§ I and III? ù ppRedK q ą ppRedUq
§ I and II?
§ III and IV?8 / 50
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Ellsberg (1961)Red or Black
§ Unknown urn: 100 balls, each red or black.§ Known urn: 100 balls, 50 red, 50 black.
Unknown urn Known urnRed Black Red Black
I $ 100 $ 0
$ 0 $ 0
II $ 0 $ 100
$ 0 $ 0
III
$ 0 $ 0
$ 100 $ 0IV
$ 0 $ 0
$ 0 $ 100
Which would you prefer (or would you be indifferent) between:
§ I and III? ù ppRedK q ą ppRedUq
§ I and II? ù ppRedK q “ ppBlackK q “ 0.5
§ III and IV? ù ppRedUq “ ppBlackUq “ 0.5
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Severe Uncertainty
§ Ellsberg behaviour is inconsistent with Bayesianism
§ Much work on finding models that:
§ are descriptively accurate and / or§ are sufficiently tractable for use in economic models
How about models that help us decide in these difficultdecisions?
AimTo formulate and defend:
§ Normatively valid models to give guidance for rationaldecision making under uncertainty.
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Severe Uncertainty
§ Ellsberg behaviour is inconsistent with Bayesianism§ Much work on finding models that:
§ are descriptively accurate and / or§ are sufficiently tractable for use in economic models
How about models that help us decide in these difficultdecisions?
AimTo formulate and defend:
§ Normatively valid models to give guidance for rationaldecision making under uncertainty.
9 / 50
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Severe Uncertainty
§ Ellsberg behaviour is inconsistent with Bayesianism§ Much work on finding models that:
§ are descriptively accurate and / or§ are sufficiently tractable for use in economic models
How about models that help us decide in these difficultdecisions?
AimTo formulate and defend:
§ Normatively valid models to give guidance for rationaldecision making under uncertainty.
9 / 50
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Proposal in a nutshellThe concept . . .
to express the proper state of belief, not one number buttwo are requisite, the first depending on the inferredprobability, the second on the amount of knowledge onwhich that probability is based. (Peirce, 1878, p179)
there may be another respect in which some kind ofquantitative comparison between arguments is possible.This comparison turns upon a balance, not between thefavourable and the unfavourable evidence, but between theabsolute amounts of relevant knowledge and of relevantignorance respectively. (Keynes, 1921, p71)
Beyond the degree to which one endorses a particular proposition
. . . there is the degree to which one is confident in this endorsement.
If the former is one’s beliefs, the latter is one’s
confidence in one’s beliefs
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Proposal in a nutshellThe concept . . .
to express the proper state of belief, not one number buttwo are requisite, the first depending on the inferredprobability, the second on the amount of knowledge onwhich that probability is based. (Peirce, 1878, p179)
Beyond the degree to which one endorses a particular proposition
. . . there is the degree to which one is confident in this endorsement.
If the former is one’s beliefs, the latter is one’s
confidence in one’s beliefs
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Proposal in a nutshellThe concept . . .
to express the proper state of belief, not one number buttwo are requisite, the first depending on the inferredprobability, the second on the amount of knowledge onwhich that probability is based. (Peirce, 1878, p179)
Beyond the degree to which one endorses a particular proposition
. . . there is the degree to which one is confident in this endorsement.
If the former is one’s beliefs, the latter is one’s
confidence in one’s beliefs
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Confidence and DecisionThere appear to be many significant decisions where
confidence in beliefs do, and should, play a role.
The action which follows upon an opinion depends asmuch upon the amount of confidence in that opinion as itdoes upon the favorableness of the opinion itself. (Knight,1921, p226-227)
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Confidence and DecisionThere appear to be many significant decisions where
confidence in beliefs do, and should, play a role.
The action which follows upon an opinion depends asmuch upon the amount of confidence in that opinion as itdoes upon the favorableness of the opinion itself. (Knight,1921, p226-227)
that course of action is preferable which involves leastrisk, and about the results of which we have the mostcomplete knowledge. . . . the coefficients of weight and riskas well as that of probability are relevant to our conclusion[as to the preferable course of action]. (Keynes, 1921,p315)
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Confidence and DecisionThere appear to be many significant decisions where
confidence in beliefs do, and should, play a role.
The action which follows upon an opinion depends asmuch upon the amount of confidence in that opinion as itdoes upon the favorableness of the opinion itself. (Knight,1921, p226-227)
that course of action is preferable which involves leastrisk, and about the results of which we have the mostcomplete knowledge. . . . the coefficients of weight and riskas well as that of probability are relevant to our conclusion[as to the preferable course of action]. . . . the only difficultyin this being the lack of any principle for the calculation ofthe degree of their influence. (Keynes, 1921, p315)
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Confidence and DecisionThere appear to be many significant decisions where
confidence in beliefs do, and should, play a role.
The action which follows upon an opinion depends asmuch upon the amount of confidence in that opinion as itdoes upon the favorableness of the opinion itself. (Knight,1921, p226-227)
But what role?
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Proposal in a nutshell. . . and the intuition
§ would we like decisions about climate change policy to betaken on the basis of “best hunch” estimates?
§ and what about wagers between us?
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Proposal in a nutshell. . . and the intuition
§ would we like decisions about climate change policy to betaken on the basis of “best hunch” estimates?
§ and what about wagers between us?
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Proposal in a nutshell. . . and the intuition
Maxim
The higher the stakes involved in a decision, the moreconfidence is needed in a belief for it to play a role in the
decision.
AimTo formulate and defend:
§ Normatively valid models to give guidance for rationaldecision making under uncertainty.
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Proposal in a nutshell. . . and the intuition
Maxim
The higher the stakes involved in a decision, the moreconfidence is needed in a belief for it to play a role in the
decision.
AimTo formulate and defend:
§ Normatively valid models to give guidance for rationaldecision making under uncertainty.
12 / 50
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Desiderata
The criteria evoked in favour of Bayesianism:
1. it corresponds to a reasonable (pre-theoretical) intuition2. it has acceptable choice-theoretical consequences3. it is conceptually clear about the roles of different mental
attitudes4. it has a certain tractability for eg. decision analysis
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Desiderata
The criteria evoked in favour of Bayesianism:
1. it corresponds to a reasonable (pre-theoretical) intuition2. it has acceptable choice-theoretical consequences3. it is conceptually clear about the roles of different mental
attitudes4. it has a certain tractability for eg. decision analysis
And of course:
5. it can fruitfully deal with “severe uncertainty” situationssuch as those above.
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Plan
1. develop a theory based on these claims§ model of confidence in beliefs§ family of decision rules incorporating confidence
2. defence and consequences of the theory:§ conceptual and choice-theoretic properties§ consequences and applications
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Modelling confidence
Idea (First attempt)
§ Represent beliefs by a set of probability measures (à laLevi (1986); Gilboa and Schmeidler (1989); Walley (1991);Bewley (2002) and . . . ).
ppAq ď 0.7
ppBq ď 0.2
ppCq ě 0.3
ppBq ď 0.2
∆pSq
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Modelling confidence
Interpretation
§ confident that the probability of B is less than 0.2
ppAq ď 0.7
ppBq ď 0.2
ppCq ě 0.3
ppBq ď 0.2
∆pSq
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Modelling confidence
Interpretation
§ unsure whether the probability of A is less than 0.7
ppAq ď 0.7
ppBq ď 0.2
ppCq ě 0.3
ppBq ď 0.2
∆pSq
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Modelling confidence
Problem
§ confidence is represented as “binary”.§ in reality, it comes in degrees.
ppAq ď 0.7
ppBq ď 0.2
ppCq ě 0.3
ppBq ď 0.2
∆pSq
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Modelling confidence
Idea
(First attempt)
§ Represent beliefs by a nested family of sets of measures
ppAq ď 0.7
ppBq ď 0.2
ppCq ě 0.3
ppBq ď 0.2
∆pSq
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Modelling confidence
Idea
(First attempt)
§ Represent beliefs by a nested family of sets of measures
ppAq ď 0.7
ppBq ď 0.2
ppCq ě 0.3
ppBq ď 0.2
∆pSq
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Modelling confidence
Interpretation
§ more confident in ppCq ě 0.3 than ppBq ď 0.2
ppAq ď 0.7
ppBq ď 0.2
ppCq ě 0.3
ppBq ď 0.2
∆pSq
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Modelling confidence
Definition
A confidence ranking Ξ is a nested family of closed subsets of∆pSq.
It is convex if every C P Ξ is convex.It is continuous if, for every C P Ξ, C “
Ť
C1ĹC C1 “Ş
C1ĽC C1.It is centered if it contains a singleton set.
C.f.§ Belief revision, conditionals, non-monotonic logics [though over ∆pSq]
. . .§ Gärdenfors and Sahlin (1982) . . .
For more details, see “Confidence and Decision” 16 / 50
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Modelling confidence
Definition
A confidence ranking Ξ is a nested family of closed subsets of∆pSq.
It is convex if every C P Ξ is convex.It is continuous if, for every C P Ξ, C “
Ť
C1ĹC C1 “Ş
C1ĽC C1.It is centered if it contains a singleton set.
C.f.§ Belief revision, conditionals, non-monotonic logics [though over ∆pSq]
. . .§ Gärdenfors and Sahlin (1982) . . .
For more details, see “Confidence and Decision” 16 / 50
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The role of confidence in choice
Maxim
The higher the stakes involved in a decision, the moreconfidence is needed in a belief for it to play a role in the
decision.
Decision
Stakes Confidence level
f
stakespf q Cf P Ξ
A cautiousness coefficient for a confidence ranking Ξ is asurjective function D : AÑ Ξ
§ the higher the stakes, the larger Dpf q
“Confidence and Decision”; “Incomplete Preferences and Confidence”; “Confidence as a Source ofDeferrral” 17 / 50
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The role of confidence in choice
Maxim
The higher the stakes involved in a decision, the moreconfidence is needed in a belief for it to play a role in the
decision.
Decision
Stakes Confidence level
f
stakespf q Cf P Ξ
A cautiousness coefficient for a confidence ranking Ξ is asurjective function D : AÑ Ξ
§ the higher the stakes, the larger Dpf q
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The role of confidence in choice
Maxim
The higher the stakes involved in a decision, the moreconfidence is needed in a belief for it to play a role in the
decision.
Decision
Stakes
Confidence level
f //
stakespf q
Cf P Ξ
A cautiousness coefficient for a confidence ranking Ξ is asurjective function D : AÑ Ξ
§ the higher the stakes, the larger Dpf q
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The role of confidence in choice
Maxim
The higher the stakes involved in a decision, the moreconfidence is needed in a belief for it to play a role in the
decision.
Decision
Stakes
Confidence level
f //D //
stakespf q
Dpf q P Ξ
A cautiousness coefficient for a confidence ranking Ξ is asurjective function D : AÑ Ξ
§ the higher the stakes, the larger Dpf q
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The role of confidence in choice
Maxim
The higher the stakes involved in a decision, the moreconfidence is needed in a belief for it to play a role in the
decision.
Decision
Stakes
Confidence level
f D //
stakespf q
Dpf q P Ξ
A cautiousness coefficient for a confidence ranking Ξ is asurjective function D : AÑ Ξ
§ the higher the stakes, the larger Dpf q“Confidence and Decision”; “Incomplete Preferences and Confidence”; “Confidence as a Source of
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A family of decision models
For each model in the family:Ingredients:
§ utility function u§ confidence ranking Ξ
§ cautiousness coefficient D
General form:
preferences concerning f are a function of
upf psqq and Dpf q
according to decision rule I
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Confidence and choiceExamples
Decision rules using a (closed) set of probabilities C:§ unanimity rule
f ĺ g iff Eppupf qq ď Eppupgqq for all p P C§ maxmin expected utility
f ĺ g iff minpPC Eppupf qq ď minpPC Eppupgqq
Ñ Belief functions / convex capacities
§ Hurwicz or α-maxmin ruleαminpPC Eppupf qq ` p1´ αqmaxpPC Eppupf qq
§ E-admissibility§ etc.
On stakes
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Plan
1. develop a theory based on these claims§ model of confidence in beliefs§ family of decision rules incorporating confidence
2. defence and consequences of the theory:§ conceptual and choice-theoretic properties§ consequences and applications
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Desiderata
1. it corresponds to a reasonable (pre-theoretical) intuition2. it has acceptable choice-theoretical consequences3. it is conceptually clear about the roles of different mental
attitudes4. it has a certain tractability for eg. decision analysis
And of course:
5. it can fruitfully deal with “severe uncertainty” situationssuch as those above.
Careful Preferences Incomplete Preferences
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Desiderata
1. it corresponds to a reasonable (pre-theoretical) intuition2. it has acceptable choice-theoretical consequences3. it is conceptually clear about the roles of different mental
attitudes4. it has a certain tractability for eg. decision analysis
And of course:
5. it can fruitfully deal with “severe uncertainty” situationssuch as those above.
Careful Preferences Incomplete Preferences
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Conceptual propertiesCareful preferences
§ “Maxmin EU” decision rule
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Conceptual propertiesCareful preferences
Choose to maximise:
minpPDpf q
Eppupf qq
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Conceptual propertiesCareful preferences
Choose to maximise:
minpPDpf q
Eppupf qq
Under such a rule:
§ for higher stakes, one is effectively only relying on beliefs inwhich one has sufficient confidence.
§ behaviour is as “pessimistic” as one’s confidence: themore confident in appropriate beliefs or the lower thestakes, the less pessimistic.
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Conceptual propertiesCareful preferences
Choose to maximise:
minpPDpf q
Eppupf qq
Conclusion This yields the following advice:
Choose boldly if one has sufficient confidence; choosecautiously if not.
Comparison Few “non-EU” rules correspond so closely to plausiblemaxims of this sort.Comparison This rule is not as extreme as maxmin EU.
Ellsberg Can accommodate Ellsberg behaviour in the same way asthe “standard maxmin EU rule”.
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Conceptual propertiesCareful preferences
Choose to maximise:
minpPDpf q
Eppupf qq
Conclusion This yields the following advice:
Choose boldly if one has sufficient confidence; choosecautiously if not.
Comparison Few “non-EU” rules correspond so closely to plausiblemaxims of this sort.Comparison This rule is not as extreme as maxmin EU.
Ellsberg Can accommodate Ellsberg behaviour in the same way asthe “standard maxmin EU rule”.
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Conceptual propertiesIncomplete preferences
§ “Unanimity” decision rule
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Conceptual propertiesIncomplete preferences
f ĺ g if and only if:
Eppupf qq ď Eppupgqq for all p P Dppf ,gqq
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Conceptual propertiesIncomplete preferences
f ĺ g if and only if:
Eppupf qq ď Eppupgqq for all p P Dppf ,gqq
Under such a rule:
§ choices made at low stakes may be suspended (but notreversed) at higher stakes.
§ for higher stakes, one is decisive only if one is confidentenough in appropriate beliefs.
One Interpretation of no preference between f and g:
§ Deferral
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Conceptual propertiesIncomplete preferences
f ĺ g if and only if:
Eppupf qq ď Eppupgqq for all p P Dppf ,gqq
Under such a rule:
§ choices made at low stakes may be suspended (but notreversed) at higher stakes.
§ for higher stakes, one is decisive only if one is confidentenough in appropriate beliefs.
One Interpretation of no preference between f and g:
§ Deferral
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Conceptual propertiesIncomplete preferences
f ĺ g if and only if:
Eppupf qq ď Eppupgqq for all p P Dppf ,gqq
Conclusion This yields the following advice:
Defer when one’s confidence in relevant beliefs is insufficient tomatch the importance of the decision.
Comparison Few “incomplete preference” rules defended by invokingplausible maxims of this sort.Comparison It is not as extreme as the unanimity rule.
Ellsberg Can accommodate Ellsberg behaviour in the same way asthe “standard unanimity rule”.
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Conceptual propertiesIncomplete preferences
f ĺ g if and only if:
Eppupf qq ď Eppupgqq for all p P Dppf ,gqq
Conclusion This yields the following advice:
Defer when one’s confidence in relevant beliefs is insufficient tomatch the importance of the decision.
Comparison Few “incomplete preference” rules defended by invokingplausible maxims of this sort.Comparison It is not as extreme as the unanimity rule.
Ellsberg Can accommodate Ellsberg behaviour in the same way asthe “standard unanimity rule”.
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Desiderata
X it corresponds to a reasonable (pre-theoretical) intuition2. it has acceptable choice-theoretical consequences3. it is conceptually clear about the roles of different mental
attitudes4. it has a certain tractability for eg. decision analysis
And of course:
X it can fruitfully deal with “severe uncertainty” situationssuch as those above.
Dutch Books Representation Theorems
Neither in detail
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Desiderata
X it corresponds to a reasonable (pre-theoretical) intuition2. it has acceptable choice-theoretical consequences3. it is conceptually clear about the roles of different mental
attitudes4. it has a certain tractability for eg. decision analysis
And of course:
X it can fruitfully deal with “severe uncertainty” situationssuch as those above.
Dutch Books Representation Theorems
Neither in detail
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Representation TheoremsPreliminaries
A typical framework
S non-empty finite set of states∆pSq set of probability measures on S
C set of consequences
[Convex subset of a vectorspace]
A set of acts (functions S Ñ C)ĺ preference relation on A
Notation:
§ fαg: shorthand for αf ` p1´ αqg.
Special case (Anscombe-Aumann framework)C “ set of lotteries (probability distributions with finite support)over a set X of prizes.
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Representation TheoremsPreliminaries
A typical framework
S non-empty finite set of states∆pSq set of probability measures on S
C set of consequences [Convex subset of a vectorspace]
A set of acts (functions S Ñ C)ĺ preference relation on A
Notation:
§ fαg: shorthand for αf ` p1´ αqg.
Special case (Anscombe-Aumann framework)C “ set of lotteries (probability distributions with finite support)over a set X of prizes.
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Representation TheoremsPreliminaries
A typical framework
S non-empty finite set of states∆pSq set of probability measures on S
C set of consequences [Convex subset of a vectorspace]
A set of acts (functions S Ñ C)ĺ preference relation on A
Notation:
§ fαg: shorthand for αf ` p1´ αqg.
Special case (Anscombe-Aumann framework)C “ set of lotteries (probability distributions with finite support)over a set X of prizes.
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Careful preferencesAxioms
Expected utility (Anscombe and Aumann):
For all f ,g,h P A, α P p0,1q:Non triviality and weak order ĺ is non-trivial, reflexive,
transitive and complete.Independence f ĺ g iff fαh ĺ gαh.
Continuity tα P r0,1s| fαh ĺ gu and tα P r0,1s| fαh ľ gu areclosed.
Monotonicity if f psq ĺ gpsq for all s P S, then f ĺ g.
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Careful preferencesAxioms
Expected utility (Anscombe and Aumann):
For all f ,g,h P A, α P p0,1q:Non triviality and weak order ĺ is non-trivial, reflexive,
transitive and complete.Independence f ĺ g iff fαh ĺ gαh.
Continuity tα P r0,1s| fαh ĺ gu and tα P r0,1s| fαh ľ gu areclosed.
Monotonicity if f psq ĺ gpsq for all s P S, then f ĺ g.
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Careful preferencesAxioms
Standard maxmin EU model (Gilboa-Schmeidler):
For all f ,g,h P A, c P C, α P p0,1q:Non triviality and weak order ĺ is non-trivial, reflexive,
transitive and complete.C-Independence f ĺ g iff fαc ĺ gαc.
Continuity tα P r0,1s| fαh ĺ gu and tα P r0,1s| fαh ľ gu areclosed.
Monotonicity if f psq ĺ gpsq for all s P S, then f ĺ g.Uncertainty Aversion if f „ g then fαg ľ f .
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Careful preferencesAxioms
Standard maxmin EU model (Gilboa-Schmeidler):
For all f ,g,h P A, c P C, α P p0,1q:Non triviality and weak order ĺ is non-trivial, reflexive,
transitive and complete.C-Independence f ĺ g iff fαc ĺ gαc.
Continuity tα P r0,1s| fαh ĺ gu and tα P r0,1s| fαh ľ gu areclosed.
Monotonicity if f psq ĺ gpsq for all s P S, then f ĺ g.Uncertainty Aversion if f „ g then fαg ľ f .
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Careful preferencesAxioms
Confidence-based careful preference model:
For all f ,g,h P A, c,d P C, α P p0,1q:Non triviality and weak order ĺ is non-trivial, reflexive,
transitive and complete.S-Independence (i) if f ľ c, then fαd ľ cαd whenever the
stakes are lower.(ii) if f ĺ c, then fαd ĺ cαd whenever the stakes
are higher.Continuity tα P r0,1s| fαh ĺ gu and tα P r0,1s| fαh ľ gu are
closed.Monotonicity if f psq ĺ gpsq for all s P S, then f ĺ g.Uncertainty Aversion For all f ,g P A, α P p0,1q, if f „ g then
fαg ľ f .
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Careful preferencesAxioms
Confidence-based careful preference model:
For all f ,g,h P A, c,d P C, α P p0,1q:Non triviality and weak order ĺ is non-trivial, reflexive,
transitive and complete.S-Independence (i) if f ľ c, then fαd ľ cαd whenever the
stakes are lower.(ii) if f ĺ c, then fαd ĺ cαd whenever the stakes
are higher.Continuity tα P r0,1s| fαh ĺ gu and tα P r0,1s| fαh ľ gu are
closed.Monotonicity applies to acts of the same stakesUncertainty Aversion applies to acts of the same stakes
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Careful preferencesRepresentation theorem
Theorem
ĺ satisfies axioms above
ô there exist u : X Ñ <, Ξ and D : AÑ Ξ such that, for all f ,g P A,f ĺ g iff
minpPDpf q
Epupf psqq ď minpPDpgq
Epupf psqq
Furthermore u is unique up to positive affine transformation, and Ξand D are unique.
Enough
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Incomplete preferencesAxioms
Expected utility (Anscombe and Aumann):
For all f ,g,h P A, α P p0,1q:Non triviality and reflexivity ĺ is non-trivial and reflexive.Completeness f ĺ g or f ľ g.Transitivity if f ĺ g and g ĺ h, then f ĺ h.
Independence f ĺ g iff fαh ĺ gαh.Continuity tpα, βq P r0,1s2| fαh ĺ gβhu is closed.
Monotonicity if f psq ĺ gpsq for all s P S, then f ĺ g.
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Incomplete preferencesAxioms
Expected utility (Anscombe and Aumann):
For all f ,g,h P A, α P p0,1q:Non triviality and reflexivity ĺ is non-trivial and reflexive.Completeness f ĺ g or f ľ g.Transitivity if f ĺ g and g ĺ h, then f ĺ h.
Independence f ĺ g iff fαh ĺ gαh.Continuity tpα, βq P r0,1s2| fαh ĺ gβhu is closed.
Monotonicity if f psq ĺ gpsq for all s P S, then f ĺ g.
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Incomplete preferencesAxioms
Standard unanimity model (Bewley):
For all f ,g,h P A, α P p0,1q:Non triviality and reflexivity ĺ is non-trivial and reflexive.Completeness f ĺ g or f ľ g whenever f ,g are constant acts.Transitivity if f ĺ g and g ĺ h, then f ĺ h.
Independence f ĺ g iff fαh ĺ gαh.Continuity tpα, βq P r0,1s2| fαh ĺ gβhu is closed.
Monotonicity if f psq ĺ gpsq for all s P S, then f ĺ g.
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Incomplete preferencesAxioms
Confidence-based incomplete preference model:
For all f ,g,h P A, α P p0,1q:Non triviality and reflexivity ĺ is non-trivial and reflexive.Completeness f ĺ g or f ľ g whenever f ,g are constant acts.Transitivity if f ĺ g and g ĺ h, then f ĺ h.
Independence f ĺ g iff fαh ĺ gαh.Continuity tpα, βq P r0,1s2| fαh ĺ gβhu is closed.
Monotonicity if f psq ĺ gpsq for all s P S, then f ĺ g.
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Incomplete preferencesAxioms
Confidence-based incomplete preference model:
For all f ,g,h P A, α P p0,1q:
Non triviality and reflexivity ĺ is non-trivial and reflexive.Completeness f ĺ g or f ľ g whenever f ,g are constant acts.S-Transitivity if f ĺ g and g ĺ h when the stakes are higher
than for pf ,hq, then f ĺ h.Independence f ĺ g iff fαh ĺ gαh, whenever both preferences
are determinate.Continuity tpα, βq P r0,1s2| fαh ĺ gβhu is closed.
Monotonicity if f psq ĺ gpsq for all s P S, then f ĺ g.
Consistency when the stakes decrease, one cannot suspend(determinate) preferences.
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Incomplete preferencesAxioms
Confidence-based incomplete preference model:
For all f ,g,h P A, α P p0,1q:
Non triviality and reflexivity ĺ is non-trivial and reflexive.Completeness f ĺ g or f ľ g whenever f ,g are constant acts.S-Transitivity if f ĺ g and g ĺ h when the stakes are higher
than for pf ,hq, then f ĺ h.Independence f ĺ g iff fαh ĺ gαh, whenever both preferences
are determinate.Continuity tpα, βq P r0,1s2| fαh ĺ gβhu is closed.
Monotonicity if f psq ĺ gpsq for all s P S, then f ĺ g.Consistency when the stakes decrease, one cannot suspend
(determinate) preferences.
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Incomplete preferencesRepresentation theorem
Theorem
ĺ satisfies axioms above
ô there exists affine u : C Ñ <, Ξ and D : A2 Ñ Ξ such that, for allf ,g P A, f ĺ g iff
ÿ
sPS
Eppupf qq ďÿ
sPS
Eppupgqq @p P Dppf ,gqq
Furthermore u is unique up to positive affine transformation, and Ξand D are unique.
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General moral
Expected utility
Remove “precision”§
§
§
đ
“Set of priors” rule
Remove stakes-independence
§
§
§
đ
Confidence rule
ConclusionThe basic issue between the confidence family and fixed set ofpriors (aka imprecise probability) models:
§ is stakes-independence a rationality constraint?
Enough
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Dutch Books
The standard argument (approximately):
Conditions on bets ô betting quotients are proba-bilities
Where:
Bet on A with stakes S yields eS if A and e0 if not A.Betting quotient qpAq value such that you are indifferent
between buying and selling the bet at stakes S foreqpAqS.
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Dutch Books
The standard argument (approximately):
Conditions on bets ô betting quotients are proba-bilities
Assumptions:For any bets on events A with stakes S:
§ eqpAqS is the price at which you are indifferent betweenbuying and selling the bet
§ eqpAqS is the buying / selling price for all stakes S
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Dutch Books
The standard argument (approximately):
Conditions on bets ô betting quotients are proba-bilities
Assumptions:For any bets on events A with stakes S:
§ eqpAqS is the price at which you are indifferent betweenbuying and selling the bet
§ eqpAqS is the buying / selling price for all stakes S
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Dutch Books
Removing it gives:
Conditions on bets ô buying / selling prices areminimal / maximal probabili-ties of fixed set
Assumptions:For any bets on events A with stakes S:
§ you have a buying price e qSpAqS and a selling priceeqSpAqS
§ eqpAqS is the buying / selling price for all stakes S
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Dutch Books
Removing it gives:
Conditions on bets ô buying / selling prices areminimal / maximal probabili-ties of fixed set
Assumptions:For any bets on events A with stakes S:
§ you have a buying price e qSpAqS and a selling priceeqSpAqS
§ eqpAqS is the buying / selling price for all stakes S
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Dutch Books
Which gives:
Conditions on bets ô buying / selling prices areminimal / maximal probabili-ties of a confidence ranking
Assumptions:For any bets on events A with stakes S:
§ you have a buying price e qSpAqS and a selling priceeqSpAqS
§ quotients qSpAq, qSpAq may depend on stakes
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Desiderata
X it corresponds to a reasonable (pre-theoretical) intuitionX it has acceptable choice-theoretical consequences3. it is conceptually clear about the roles of different mental
attitudes4. it has a certain tractability for eg. decision analysis
And of course:
X it can fruitfully deal with “severe uncertainty” situationssuch as those above.
Ducth Books Representation Theorems
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Separation of beliefs and desiresThe model contains three elements:
§ Utility function
= Desires over outcomes
§ Confidence ranking
= Beliefs and confidence in beliefs
§ Cautiousness coefficient
= Attitude to choosing on thebasis of limited confidence
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Separation of beliefs and desiresThe model contains three elements:
§ Utility function = Desires over outcomes§ Confidence ranking
= Beliefs and confidence in beliefs
§ Cautiousness coefficient
= Attitude to choosing on thebasis of limited confidence
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Separation of beliefs and desiresThe model contains three elements:
§ Utility function = Desires over outcomes§ Confidence ranking = Beliefs and confidence in beliefs§ Cautiousness coefficient
= Attitude to choosing on thebasis of limited confidence
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Separation of beliefs and desiresThe model contains three elements:
§ Utility function = Desires over outcomes§ Confidence ranking = Beliefs and confidence in beliefs§ Cautiousness coefficient = Attitude to choosing on the
basis of limited confidence
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Separation of beliefs and desiresThe model contains three elements:
§ Utility function = Desires over outcomes§ Confidence ranking = Beliefs and confidence in beliefs§ Cautiousness coefficient = Attitude to choosing on the
basis of limited confidence
And there is a natural comparison of attitude to uncertainty
that corresponds precisely to differences in the cautiousnesscoefficient.
§ DM 1 is more averse to uncertainty than DM 2 if @f , constant c,f ľ1 c ñ f ľ2 c.
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Separation of beliefs and desiresThe model contains three elements:
§ Utility function = Desires over outcomes§ Confidence ranking = Beliefs and confidence in beliefs§ Cautiousness coefficient = Attitude to choosing on the
basis of limited confidence
And there is a natural comparison of attitude to uncertainty
that corresponds precisely to differences in the cautiousnesscoefficient.
For DMs with the same u and Ξ
§ 1 is more averse to uncertainty
ô D1pf q Ě D2pf q for all acts f .
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Separation of beliefs and desiresThe model contains three elements:
§ Utility function = Desires over outcomes§ Confidence ranking = Beliefs and confidence in beliefs§ Cautiousness coefficient = Attitude to choosing on the
basis of limited confidence
There is a natural comparison of decisiveness
that corresponds precisely to differences in the cautiousnesscoefficient
§ DM 1 is more indecisive than DM 2 if @f ,g f ľ1 g ñ f ľ2 g.
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Separation of beliefs and desiresThe model contains three elements:
§ Utility function = Desires over outcomes§ Confidence ranking = Beliefs and confidence in beliefs§ Cautiousness coefficient = Attitude to choosing on the
basis of limited confidence
There is a natural comparison of decisiveness
that corresponds precisely to differences in the cautiousnesscoefficient
For DMs with the same u and Ξ
1 is more indecisive
ô D1ppf ,gqq Ě D2ppf ,gqq for all pairs f and g.
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Separation of beliefs and desiresThe model contains three elements:
§ Utility function = Desires over outcomes§ Confidence ranking = Beliefs and confidence in beliefs§ Cautiousness coefficient = Attitude to choosing on the
basis of limited confidence
Conclusion There is a clean separation between beliefs anddesires (attitudes to outcomes and to choosing in the absenceof confidence).
Comparison All other “incomplete preference” rules we are aware ofdo not exhibit such a separation.
Comparison Maxmin EU, as well as many other “non-EU” models ofdecision making, do not exhibit such a separation.
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An example: Maxmin EU
minpPC Epupf psqq
For DMs with the same u, DM 1 is more averse to uncertaintyiff C1 Ě C2. (Ghirardato and Marinacci, 2002)
Flood / drought exampleGovernor 1’s set of priors, C1 Ě C2, Governor 2’s set.
§ Does Gov 2 have further information / beliefs?§ Or is he just less cautious?
The pointMaximin EU can’t decide the question . . . or even properlyrepresent the possibilities.
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An example: Maxmin EU
minpPC Epupf psqq
For DMs with the same u, DM 1 is more averse to uncertaintyiff C1 Ě C2. (Ghirardato and Marinacci, 2002)
Flood / drought exampleGovernor 1’s set of priors, C1 Ě C2, Governor 2’s set.
§ Does Gov 2 have further information / beliefs?§ Or is he just less cautious?
The pointMaximin EU can’t decide the question . . . or even properlyrepresent the possibilities.
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An example: Maxmin EU
minpPC Epupf psqq
For DMs with the same u, DM 1 is more averse to uncertaintyiff C1 Ě C2. (Ghirardato and Marinacci, 2002)
Flood / drought exampleGovernor 1’s set of priors, C1 Ě C2, Governor 2’s set.
§ Does Gov 2 have further information / beliefs?§ Or is he just less cautious?
The pointMaximin EU can’t decide the question . . . or even properlyrepresent the possibilities.
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Desiderata
X it corresponds to a reasonable (pre-theoretical) intuitionX it has acceptable choice-theoretical consequencesX it is conceptually clear about the roles of different mental
attitudes4. it has a certain tractability for eg. decision analysis
And of course:
X it can fruitfully deal with “severe uncertainty” situationssuch as those above.
Ducth Books Representation Theorems
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What does the decision maker / expert need toprovide?
The confidence ranking
is an ordinal second-order structure.
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What does the decision maker / expert need toprovide?
The confidence ranking is an ordinal second-order structure.
Needed ordinal confidence comparisons over probabilities
Eg. I am more confident that ppAq ě 0.5 than ppBq ď 0.3
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What does the decision maker / expert need toprovide?
The confidence ranking is an ordinal second-order structure.
Needed ordinal confidence comparisons over probabilities
Comparison Many recent “belief-like parameters” used in theliterature are cardinal.
Needed numerical confidence comparisons over probabilities
Eg. I am confident to degree x that ppAq ě 0.5 but only confident to degree ythat ppBq ď 0.3
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What does the decision maker / expert need toprovide?
The confidence ranking
is an ordinal second-order structure.
Needed ordinal confidence comparisons over probabilities
Comparison Many recent “belief-like parameters” used in theliterature
are cardinal.
Needed numerical confidence comparisons over probabilities
Conclusion The confidence framework is much moreparsimonious than other recent models . . .
and hence easier to apply to analysis and guidance of actualdecisions!
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In summary
1. it embodies plausible maxims of choice2. it has acceptable choice-theoretical consequences3. it involve a neat separation of beliefs and tastes4. it is involves an ordinal notion of confidence
And of course:
5. it can fruitfully deal with “severe uncertainty” situationssuch as those above.
MoreoverIt is the only model (I am aware of) with 3, 4 and 5.
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Plan
1. develop a theory based on these claims§ model of confidence in beliefs§ family of decision rules incorporating confidence
2. defence and consequences of the theory:§ conceptual and choice-theoretic properties§ consequences and applications
Climate Uncertainty Deferral
Skip
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Plan
1. develop a theory based on these claims§ model of confidence in beliefs§ family of decision rules incorporating confidence
2. defence and consequences of the theory:§ conceptual and choice-theoretic properties§ consequences and applications
Climate Uncertainty Deferral
Skip
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Climate uncertainty
Current state of scientific knowledge about the climate:reported in IPCC’s uncertainty language (Mastrandrea et al.,2010).
How can it be used to inform decision making on climatepolicy?
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IPCC uncertainty language
Two metrics for communicating degree of uncertainty:
Confidence in the validity of a finding, based on the type,amount, quality, and consistency of evidence (e.g., mechanisticunderstanding, theory, data, models, expert judgment) and thedegree of agreement.
It is expressed qualitatively (five qualifiers, ranging from verylow to very high).
Quantified measures of uncertainty in a finding expressedprobabilistically (based on statistical analysis of observations ormodel results, or expert judgment).
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IPCC uncertainty language
• For findings with low agreement and limited evidence,assign summary terms for your evaluation of evidenceand agreement.
• In any of these cases, the degree of certainty in findingsthat are conditional on other findings should be evaluatedand reported separately.
9) A level of confidence is expressed using five qualifiers:“very low,” “low,” “medium,” “high,” and “very high.” Itsynthesizes the author teams’ judgments about the validityof findings as determined through evaluation of evidenceand agreement. Figure 1 depicts summary statementsfor evidence and agreement and their relationship toconfidence. There is flexibility in this relationship; for a givenevidence and agreement statement, different confidencelevels could be assigned, but increasing levels of evidenceand degrees of agreement are correlated with increasingconfidence. Confidence cannot necessarily be assigned forall combinations of evidence and agreement in Figure 1(see Paragraph 8). Presentation of findings with “low”and “very low” confidence should be reserved for areasof major concern, and the reasons for their presentationshould be carefully explained. Confidence should notbe interpreted probabilistically, and it is distinct from“statistical confidence.” Additionally, a finding that includesa probabilistic measure of uncertainty does not requireexplicit mention of the level of confidence associated withthat finding if the level of confidence is “high” or “veryhigh.”
10) Likelihood, as defined in Table 1, provides calibratedlanguage for describing quantified uncertainty. It can beused to express a probabilistic estimate of the occurrenceof a single event or of an outcome (e.g., a climate parameter,observed trend, or projected change lying in a given
range). Likelihood may be based on statistical or modelinganalyses, elicitation of expert views, or other quantitativeanalyses. The categories defined in this table can beconsidered to have “fuzzy” boundaries. A statement thatan outcome is “likely” means that the probability of thisoutcome can range from ≥66% (fuzzy boundaries implied)to 100% probability. This implies that all alternativeoutcomes are “unlikely” (0-33% probability). When thereis sufficient information, it is preferable to specify the fullprobability distribution or a probability range (e.g., 90-95%) without using the terms in Table 1. “About as likelyas not” should not be used to express a lack of knowledge(see Paragraph 8 for that situation). Additionally, there isevidence that readers may adjust their interpretation ofthis likelihood language according to the magnitude ofperceived potential consequences.11
11) Characterize key findings regarding a variable (e.g., ameasured, simulated, or derived quantity or its change)using calibrated uncertainty language that conveys themost information to the reader, based on the criteria (A-F)below.12 These criteria provide guidance for selectingamong different alternatives for presenting uncertainty,recognizing that in all cases it is important to include atraceable account of relevant evidence and agreement inyour chapter text.
A) A variable is ambiguous, or the processes determiningit are poorly known or not amenable to measurement:Confidence should not be assigned; assign summaryterms for evidence and agreement (see Paragraph 8).Explain the governing factors, key indicators, and
3
ipcc guidance note
High agreementLimited evidence High agreementRobust evidence
Low agreementLimited evidence Low agreementRobust evidence
Evidence (type, amount, quality, consistency)
Agreement
Low agreementMedium evidence
High agreementMedium evidence
Medium agreementMedium evidenceMedium agreementLimited evidence Medium agreementRobust evidence
ConfidenceScale
High agreementLimited evidence
High agreementRobust evidence
Low agreementLimited evidence
Low agreementRobust evidence
Evidence (type, amount, quality, consistency)Ag
reem
ent
Low agreementMedium evidence
High agreementMedium evidence
Medium agreementMedium evidence
Medium agreementLimited evidence
Medium agreementRobust evidence
ConfidenceScale
Figure 1: A depiction of evidence and agreement statements and their relationship toconfidence. Confidence increases towards the top-right corner as suggested by theincreasing strength of shading. Generally, evidence is most robust when there are multiple,consistent independent lines of high-quality evidence.
Table 1. Likelihood Scale
Term* Likelihood of the Outcome
Virtually certain 99-100% probability
Very likely 90-100% probability
Likely 66-100% probability
About as likely as not 33 to 66% probability
Unlikely 0-33% probability
Very unlikely 0-10% probability
Exceptionally unlikely 0-1% probability
* Additional terms that were used in limited circumstances in the AR4 (extremely likely –95-100% probability, more likely than not – >50-100% probability, and extremelyunlikely – 0-5% probability) may also be used in the AR5 when appropriate.
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IPCC uncertainty language
Often, both metrics appear together:
‘In the Northern Hemisphere, 1983–2012 was likely thewarmest 30-year period of the last 1400 years (mediumconfidence).’
‘Multiple lines of evidence provide high confidence that an[Equilibrium Climate Sensitivity] value less than 1˝C isextremely unlikely.’
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IPCC uncertainty language
Confidence in the validity of a finding, based on the type,amount, quality, and consistency of evidence (e.g., mechanisticunderstanding, theory, data, models, expert judgment) and thedegree of agreement.
It is expressed qualitatively (five qualifiers, ranging from verylow to very high).
Quantified measures of uncertainty in a finding expressedprobabilistically (based on statistical analysis of observations ormodel results, or expert judgment).
so it is perhaps not surprising that . . .
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IPCC uncertainty language
Confidence in the validity of a finding, based on the type,amount, quality, and consistency of evidence (e.g., mechanisticunderstanding, theory, data, models, expert judgment) and thedegree of agreement.
It is expressed qualitatively (five qualifiers, ranging from verylow to very high).
Quantified measures of uncertainty in a finding expressedprobabilistically (based on statistical analysis of observations ormodel results, or expert judgment).
so it is perhaps not surprising that . . .
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IPCC uncertainty language
§ IPCC assessments can easily be translated intoconfidence rankings
§ They are difficult to connect to other recent representationsof uncertainty used in decision theory
In fact, the confidence framework:
§ is a pragmatic vindication of IPCC practices§ shows how they can be used to guide decision making§ provides practical recommendations for the future use of
the language
Deferral Conclusion
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IPCC uncertainty language
§ IPCC assessments can easily be translated intoconfidence rankings
§ They are difficult to connect to other recent representationsof uncertainty used in decision theory
In fact, the confidence framework:
§ is a pragmatic vindication of IPCC practices§ shows how they can be used to guide decision making§ provides practical recommendations for the future use of
the language
Deferral Conclusion
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Conclusion
The higher the stakes involved in a decision, the moreconfidence is needed in a belief for it to play a role in the
decision.
We have:
§ a maxim concerning the role of confidence in choice§ a formal model of confidence in beliefs and a family of
decision rules embodying the maxim§ these rules are unique in having attractive conceptual and
choice-theoretic properties§ and potentially interesting consequences for high-stakes
decision making.
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Thank you.
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Material drawn from:§ Confidence and Decision, Games and Economic Behavior, 82:
675–692, 2013.§ Incomplete Preferences and Confidence, Journal of Mathematical
Economics, 65: 83-103, 2016.§ Confidence as a Source of Deferral, HEC Paris Research Paper No.
ECO/SCD-2014-1060.§ Climate Change Assessments: Confidence, Probability and Decision,
Philosophy of Science, forthcoming (with R. Bradley, C. Helgeson).§ Confidence in Beliefs and Rational Decision Making, mimeo HEC Paris.
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Confidence in Beliefs and Rational DecisionMaking
Brian [email protected]
www.hec.fr/hill
GREGHEC, CNRS & HEC Paris
2016(Presentation prepared for an
interdisciplinary audience)
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References I
Bewley, T. F. (2002). Knightian decision theory. part i. Decisionsin Economics and Finance, 25:79–110.
Chateauneuf, A. and Faro, J. H. (2009). Ambiguity throughconfidence functions. J. Math. Econ., 45:535–558.
Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms.Quart. J. Econ., 75:643–669.
Gärdenfors, P. and Sahlin, N.-E. (1982). Unreliableprobabilities, risk taking, and decision making. Synthese,53(3):361–386. Reproduced in Decision, Probability andUtility: Selected Readings, Gärdenfors, P. and Sahlin, N.-E.(eds), CUP, 1988.
Ghirardato, P. and Marinacci, M. (2002). Ambiguity madeprecise: A comparative foundation. J. Econ. Theory,102(2):251 – 289.
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References II
Gilboa, I. and Schmeidler, D. (1989). Maxmin expected utilitywith non-unique prior. J. Math. Econ., 18(2):141–153.
Keynes, J. M. (1921). A Treatise on Probability. DoverPublications.
Klibanoff, P., Marinacci, M., and Mukerji, S. (2005). A smoothmodel of decision making under ambiguity. Econometrica,73(6):1849–1892.
Knight, F. H. (1921). Risk, Uncertainty, and Profit. HoughtonMifflin, Boston and New York.
Levi, I. (1986). Hard Choices. Decision making underunresolved conflict. Cambridge University Press, Cambridge.
Maccheroni, F., Marinacci, M., and Rustichini, A. (2006).Ambiguity aversion, robustness, and the variationalrepresentation of preferences. Econometrica, 74:1447–1498.
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References IIIMastrandrea, M. D., Field, C. B., Stocker, T. F., Edenhofer, O.,
Ebi, K. L., Frame, D. J., Held, H., Kriegler, E., Mach, K. J.,Matschoss, P. R., Plattner, G.-K., Yohe, G. W., and Zwiers, F.W. . . (2010). Guidance note for lead authors of the IPCC fifthassessment report on consistent treatment of uncertainties.Technical report, Intergovernmental Panel on ClimateChange (IPCC). Available at ăhttp://www.ipcc.chą.
Peirce, C. S. (1878). Philosophical Writings of Peirce. Dover,New York.
Savage, L. J. (1954). The Foundations of Statistics. Dover,New York. 2nd edn 1971.
Schmeidler, D. (1989). Subjective probability and expectedutility without additivity. Econometrica, 57(3):571–587.
Walley, P. (1991). Statistical reasoning with impreciseprobabilities. Chapman and Hall, London.
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Belief functions / capacities
Capacity An increasing function ν : 2S Ñ r0,1s s.t. νpHq “ 0,νpSq “ 1.
§ A (Dempster-Shafer) belief function is a (convex) capacity.
Convex: νpEq ` νpF q ď νpE Y F q ` νpE X F q
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Belief functions / capacities
Capacity An increasing function ν : 2S Ñ r0,1s s.t. νpHq “ 0,νpSq “ 1.
§ A (Dempster-Shafer) belief function is a (convex) capacity.
DecisionThe appropriate decision rule for capacities is maximisation ofthe Choquet integral,
ş
upf psqqdν (Schmeidler, 1989).
ż
upf psqqdν “ż 8
0ν pts P S : upf psqq ě tuq dt
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Belief functions / capacities
Capacity An increasing function ν : 2S Ñ r0,1s s.t. νpHq “ 0,νpSq “ 1.
§ A (Dempster-Shafer) belief function is a (convex) capacity.
DecisionThe appropriate decision rule for capacities is maximisation ofthe Choquet integral,
ş
upf psqqdν (Schmeidler, 1989).
For convex capacities,ż
upf psqqdν “ minpPCν
Epuppf qq
Cν “
!
p P ∆pSq : ppEq ě νpEq @E P 2S)
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Belief functions / capacities
Capacity An increasing function ν : 2S Ñ r0,1s s.t. νpHq “ 0,νpSq “ 1.
§ A (Dempster-Shafer) belief function is a (convex) capacity.
Decision (in brief)The decision rule for convex capacities is (equivalent to)maxmin expected utility.
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Confidence, stakes and choiceExamples
Stakes involved in the choice:
§ of an act;
§ from a menu;§ in a context . . .
Eg. utility of worst consequence of the act
Back
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Confidence, stakes and choiceExamples
Stakes involved in the choice:
§ of an act;§ from a menu;
§ in a context . . .
Eg. utility of worst consequence of any act on the menu
Back
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Confidence, stakes and choiceExamples
Stakes involved in the choice:
§ of an act;§ from a menu;§ in a context . . .
Back
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Violations and abstentionsBlack White
f 15 -100 0 0g 1.5 M -1 Mf n 15 ˆn -10 ˆn
Bets on colour of next ball drawn from urn where:
§ only black and white balls§ observed 15 draws (with replacement): 9 black, 6 white.
Back
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Violations and abstentionsBlack White
f 15 -100 0 0g 1.5 M -1 Mf n 15 ˆn -10 ˆn
Bets on colour of next ball drawn from urn where:
§ only black and white balls§ observed 15 draws (with replacement): 9 black, 6 white.
f vs. 0 / f n`1 vs. f n / g vs. 0?
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Violations and abstentionsBlack White
f 15 -100 0 0g 1.5 M -1 Mf n 15 ˆn -10 ˆn
Bets on colour of next ball drawn from urn where:
§ only black and white balls§ observed 15 draws (with replacement): 9 black, 6 white.
f vs. 0 / f n`1 vs. f n / g vs. 0?
Standard Independence f ľ 0 ô g ľ 0Standard Transitivity f n`1 ľ f n @n ñ g ľ 0
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Violations and abstentionsBlack White
f 15 -100 0 0g 1.5 M -1 Mf n 15 ˆn -10 ˆn
Bets on colour of next ball drawn from urn where:
§ only black and white balls§ observed 15 draws (with replacement): 9 black, 6 white.
f vs. 0 / f n`1 vs. f n / g vs. 0?
Standard Independence f ľ 0 ô g ľ 0Pure Independence allows f ľ 0 and g ń 0
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Violations and abstentionsBlack White
f 15 -100 0 0g 1.5 M -1 Mf n 15 ˆn -10 ˆn
Bets on colour of next ball drawn from urn where:
§ only black and white balls§ observed 15 draws (with replacement): 9 black, 6 white.
f vs. 0 / f n`1 vs. f n / g vs. 0?
Standard Transitivity f n`1 ľ f n @n ñ g ľ 0Stakes Transitivity allows f n`1 ľ f n @n and g ń 0
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Violations and abstentionsBlack White
f 15 -100 0 0g 1.5 M -1 Mf n 15 ˆn -10 ˆn
Bets on colour of next ball drawn from urn where:
§ only black and white balls§ observed 15 draws (with replacement): 9 black, 6 white.
However§ Pure Independence ô Standard Independence§ Stakes Transitivity ô Standard Transitivity
whenever preferences are determinate.
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Choice theoretic propertiesS-independence
Blue Redf 10 000 -1 Mg 10 M -1 000 M0 0 0
Bets on colour of next ball drawn from urn where:
§ 1 M balls, each of which is either red or blue
§ Sure that at least 990000 blue, and at least 1 red.
§ Experts say: at most 10 red.
Story back Main
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Choice theoretic propertiesS-independence
Blue Redf 10 000 -1 Mg 10 M -1 000 M0 0 0
Bets on colour of next ball drawn from urn where:
§ 1 M balls, each of which is either red or blue
§ Sure that at least 990000 blue, and at least 1 red.
§ Experts say: at most 10 red.
f vs. 0 / g vs. 0?
Story back Main
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Choice theoretic propertiesS-independence
Blue Redf 10 000 -1 Mg 10 M -1 000 M0 0 0
Bets on colour of next ball drawn from urn where:
§ 1 M balls, each of which is either red or blue
§ Sure that at least 990000 blue, and at least 1 red.
§ Experts say: at most 10 red.
f vs. 0 / g vs. 0?
C-independence f ľ 0 ô g ľ 0.
S-independence g ľ 0 ñ f ľ 0, but not vice versa.
Story back Main
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Choice theoretic propertiesS-independence
Blue Redf 10 000 -1 Mg 10 M -1 000 M0 0 0
Bets on colour of next ball drawn from urn where:
§ 1 M balls, each of which is either red or blue
§ Sure that at least 990000 blue, and at least 1 red.
§ Experts say: at most 10 red.
f vs. 0 / g vs. 0?
C-independence f ľ 0 ô g ľ 0.S-independence g ľ 0 ñ f ľ 0, but not vice versa.
Story back Main
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Why confidence?Public decision making
The governor must decide whether to allow a factory project
§ fumes from the factory could affect district farming area.§ probabilities controversial, but he retains estimate of 10´5.§ with this probability, project retained.
The governor must decide whether to allow a GM crops project
§ probability of infecting non-GM area same as probability offumes arriving there.
§ consequences are larger by a factor of a thousand, ingovernor’s opinion.
§ Yet it is not prima facie unreasonable to turn down theproject!
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Why confidence?Public decision making
The governor must decide whether to allow a factory project
§ fumes from the factory could affect district farming area.§ probabilities controversial, but he retains estimate of 10´5.§ with this probability, project retained.
The governor must decide whether to allow a GM crops project
§ probability of infecting non-GM area same as probability offumes arriving there.
§ consequences are larger by a factor of a thousand, ingovernor’s opinion.
§ Yet it is not prima facie unreasonable to turn down theproject!
back
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Why confidence?Public decision making
The governor must decide whether to allow a factory project
§ fumes from the factory could affect district farming area.§ probabilities controversial, but he retains estimate of 10´5.§ with this probability, project retained.
The governor must decide whether to allow a GM crops project
§ probability of infecting non-GM area same as probability offumes arriving there.
§ consequences are larger by a factor of a thousand, ingovernor’s opinion.
§ Yet it is not prima facie unreasonable to turn down theproject!
back
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Why confidence?Public decision making
Stereotyped versionUrn with 106 balls; at least 990000 blue and at least 1 red.Advisers’ estimate: at most 10 are red.
Colour of ball drawn from urnBlue Red
f 10 000 -1 Mg 10 M -1 000 Mp0 0 0
f : factory; g: GM crops.
Preferences: f ą p0 and g ă p0.
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Second-order models
“Belief” represen-tation
Rule
Smooth Ambi-guity (Klibanoffet al., 2005)
second-order prob-ability µ P ∆p∆pSqq
ş
∆pSqφ pEpupf qqµppq
Variationalpreferences(Maccheroniet al., 2006)
c : ∆pSq Ñ r0,8s minpP∆pSq pEpupf q ` cppqq
Chateauneufand Faro (2009)
ϕ : ∆pSq Ñ r0,1s minpP∆pSq1
ϕpf qEpupf q
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