Operationalizing Individual Fairness in Harsanyi

1
Operationalizing Individual Fairness in
Harsanyis Utilitarianism

• Stefan Trautmann
• June 26, 2006

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outline

• Harsanyis theorem and criticism based on
  fairness
• Solution to criticisms all-inclusive inclusive
  individual utilities ? lose predictive power
• Propose two-stage approach to include individual
  fairness preferences in utilitarian welfare
  evaluation

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Harsanyis theorem (1)

• Harsanyi (1955) uses cardinal utility from risky
  choices to derive social welfare function
• assumptions
• individual agents max EU
• social planner max EU
• 3. Pareto-principle (all agents indifferent
  implies society indifferent)

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Harsanyis theorem (2)
Ui individual vNM utilities of outcomes xi W
social welfare function
Theorem (Harsanyi 1955) Assumptions 1 - 3 imply
a social welfare function of utilitarian form
W?i Ui
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Harsanyis theorem (3)
individual agents max EU social planner max
EU Pareto-principle
modest assumptions
?
strong individualistic values only marginal
distribution of outcomes of agents
matters distribution between agents not
considered (Anscombe-Aumann Ass1)
W?i Ui
strong result
distribution of utility over agents does not
matter
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criticisms based on fairness (1)
A always gets positive utility, B nothing
lack of fairness consideration by social planner
under utilitarianism criticized by
counterexamples Diamond 1967, Broome 1991
Diamond (1967)
both A and B have fair chance
A B 1 0 1 0
A B 1 0 0 1
0.5
0.5
?
P
Q
?
?
0.5
0.5
EW1
EW1
entries are utilities
under utilitarianism
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criticisms based on fairness (2)
always equality
Broome (1991)
A B 1 1 0 0
A B 1 0 0 1
0.5
0.5
?
P
Q
?
?
0.5
0.5
always inequality
EW1
EW1
Pareto vs AA assumption 1 only one horse matters
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criticisms based on fairness (3)

• utilitarian social planners indifference not
  convincing in these allocation examples
• how to save Harsanyis argument?
• all-inclusive utility
• Luce Raiffa 1957, Broome 1984, 1991, Binmore
  1994

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all-inclusive utility
Uis include already all social
comparisons UA(xA, xB , xA- xB , EXA-EXB,..)
A B 1 0 0 1
0.5
Q
0.5
pro saves Harsanyis argument formally
fairness included at individual level con
deprives it from predictive power
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all-inclusive utility prediction
A B 1 1 0 0
A B 1 0 0 1
Broome example
0.5
0.5
?
P
Q
0.5
0.5
say we know SP indiff in Broome expl what can we
predict in new decision?
A B
A B 1 1 0 0
0.25
? ? ? ?
0.5
?
P
Q
0.5
0.75
but same outcomes x
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all-inclusive utility prediction (2)
A B 1 1 0 0
A B 1 0 0 1
expl 1 selfish agents utility depends only on
own outcome
0.5
0.5
?
P
Q
0.5
0.5
what do these utilities include?
A B
A B 1 1 0 0
0.25
do not change outcomes, only prob
1 0 0 1
0.5
?
P
Q
0.5
0.75
EW1
EW1
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all-inclusive utility prediction (3)
expl 2 utility depends on both own outcome and
expected outcome difference
A B 1 1 0 0
A B 1 0 0 1
0.5
0.5
?
P
Q
0.5
0.5
A B
A B 1 1 0 0
expected outcome diffs change for Q, so do
all-inc utilities
0.25
a b c d
0.5
?
P
Q
0.5
0.75
EW1
EW0.25(ab)0.75(cd)
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two-stage approach
all-inclusive utility can justify social
planners preferences, but little predictive
power solution two-stage approach to obtain
empirically meaningful all-inclusive
utilities stage 1 agents evaluate risky
outcomes without social comparison
self-interested vNM utilities (Sugden 2000) stage
2 take self-interested vNM utilities as inputs
in tractable models of individual fairness
(Fehr-Schmidt 1999, Trautmann 2006)
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two-stage approach stage 2 fairness models

• outcome Fehr-Schmidt (1999)
• UA( xA , xB ) xA - ?A max xB-xA, 0
• - ?A max xA-xB, 0
• with 0? ? lt1 and ? ? ?
• process Fehr-Schmidt (Trautmann 2006)
• UA(xA,XA,XB) xA - ?A max EXB - EXA, 0
• - ?A max EXA - EXB, 0
• with 0? ? lt1 and ? ? ?

outcome fairness
procedural fairness
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two-stage approach stage 2 fairness models
why these models?

• empirically relevant individual fairness prefs
  originating from experimental econ, successfully
  predict data
• can be assessed by observing choices between
  (random) allocations can estimate individual ?
  and ?
• operational and tractable allow quantitative
  welfare evaluation under utilitarianism

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illustration of two-stage approach Diamond (1)
A B 1 0 1 0
A B 1 0 0 1
interpret as self-interested vNM utilities
0.5
0.5
?
P
Q
0.5
0.5
apply outcome FS
A B 1- ? -? 1- ? -?
A B 1- ? -? -? 1- ?
0.5
0.5
?
P
Q
0.5
0.5
assume ?A ?B ? gt0 ?A ?B ? gt0
EW1-?-?
EW1-?-?
planners preference still unconvincing
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illustration of two-stage approach Diamond (2)
A B 1 0 1 0
A B 1 0 0 1
interpret as self-interested vNM utilities
0.5
0.5
?
P
Q
0.5
0.5
apply process FS
A B 1- ? -? 1- ? -?
A B 1 0 0 1
0.5
0.5
?
P
Q
0.5
0.5
EW1-?-?
EW1
here planners preference is convincing
utilitarianism is supported by process FS
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illustration of two-stage approach Broome (1)
A B 1 1 0 0
A B 1 0 0 1
interpret as self-interested vNM utilities
0.5
0.5
?
P
Q
0.5
0.5
apply outcome FS
A B 1 1 0 0
A B 1- ? -? -? 1- ?
0.5
0.5
P
?
Q
0.5
0.5
EW1-?-?
EW1
planners preference is convincing
utilitarianism is supported by outcome FS
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illustration of two-stage approach Broome (2)
A B 1 1 0 0
A B 1 0 0 1
interpret as self-interested vNM utilities
0.5
0.5
?
P
Q
0.5
0.5
A B 1 1 0 0
A B 1 0 0 1
apply process FS
0.5
0.5
?
P
Q
0.5
0.5
EW1
EW1
planners preference is unconvincing
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appraisal of utilitarianism two-stage approach
with different fairness models
Diamonds example
Broomes example
unconvincing
unconvincing
self-interested
unconvincing
convincing, supports Harsanyi
outcome FS
convincing, supports Harsanyi
unconvincing
process FS
? both outcome and process fairness play role in
supporting utilitarianism
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conclusion (1)

• fairness not adequately considered by
  utilitarian SP under Harsanyis utilitarianism
• all-inclusive utility saves Harsanyis argument
  but deprives it from predictive power
• proposed two stage approach to obtain
  all-inclusive utilities

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conclusion (2)

• stage 1 evaluate outcomes by self-interested vNM
  utilities
• stage 2 use those as inputs in parametric models
  of individual fairness
• meaningful all-inclusive utilities
• quantitative evaluation of social allocations
• empirically assessable fairness models can
  apply to more specific settings than the ones
  above
• makes utilitarianism refutable

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conclusion (3)

• used approach in discussion of criticisms of
  Harsanyis theorem
• both process and outcome fairness play a role in
  making utilitarianism convincing in both examples
• if we accept utilitarianism and the criticisms,
  we need more complete individual fairness model