operationalizing individual fairness in harsanyi’s utilitarianism stefan trautmann june 26, 2006
TRANSCRIPT
Operationalizing Individual Fairness in Harsanyi’s
Utilitarianism
Stefan Trautmann
June 26, 2006
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• Harsanyi’s 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
outline
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Harsanyi (1955) uses cardinal utility from risky choices to derive social welfare function
assumptions:
1. individual agents max EU
2. social planner max EU
3. Pareto-principle (all agents indifferent implies society indifferent)
Harsanyi’s theorem (1)
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Harsanyi’s 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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Harsanyi’s theorem (3)
W=i Ui
individual agents max EU
social planner max EU
Pareto-principle
modest assumptions
strong result
?
distribution of utility over agents does not matter
strong: individualistic valuesonly marginal distribution of outcomes of agents mattersdistribution between agents not considered (Anscombe-Aumann Ass1)
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criticisms based on fairness (1)
Diamond (1967)
P
A B1 0
1 00.5
0.5
Q
A B1 0
0 10.5
0.5
EW=1 EW=1
lack of fairness consideration by social planner under utilitarianism
criticized by counterexamples: Diamond 1967, Broome 1991
under utilitarianism
A always gets positive utility, B nothing
both A and B have fair chance?
entries are utilities
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criticisms based on fairness (2)
Broome (1991)
P
A B1 1
0 00.5
0.5
Q
A B1 0
0 10.5
0.5
EW=1 EW=1
always equality
always inequality
?
Pareto vs AA assumption 1: only one horse matters
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criticisms based on fairness (3)
utilitarian social planner’s indifference not convincing in these allocation examples
how to save Harsanyi’s argument?
all-inclusive utility
[Luce & Raiffa 1957, Broome 1984, 1991, Binmore 1994]
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all-inclusive utility
Q
A B1 0
0 10.5
0.5
Ui‘s include already all social comparisons:
UA(xA, xB , xA- xB , E[XA]-E[XB],..)
pro: saves Harsanyi’s argument formally: fairness included at individual level
con: deprives it from predictive power
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all-inclusive utility: prediction
P
A B1 1
0 00.5
0.5
Q
A B1 0
0 10.5
0.5
P
A B1 1
0 00.5
0.5
Q? ?
? ??
0.25
0.75
Broome example
A B
but same outcomes x
say we know SP indiff in Broome expl
what can we predict in new decision?
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all-inclusive utility : prediction (2)
P
A B1 1
0 00.5
0.5
Q
A B1 0
0 10.5
0.5
P
A B1 1
0 00.5
0.5
Q1 0
0 10.75
0.25
expl 1: selfish agents; utility depends only on own outcome
do not change outcomes, only prob EW=1 EW=1
A Bwhat do these utilities include?
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all-inclusive utility : prediction (3)
P
A B1 1
0 00.5
0.5
Q
A B1 0
0 10.5
0.5
P
A B1 1
0 00.5
0.5
Q0.75
0.25
expl 2: utility depends on both own outcome and expected outcome difference
?
expected outcome diffs change for Q, so do all-inc utilities EW=1 EW=0.25(a+b)+0.75(c+d)
a b
c d
A B
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two-stage approach
all-inclusive utility can justify social planner’s 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 <1 and
• process Fehr-Schmidt (Trautmann 2006)
UA(xA,XA,XB)= xA - A max{ E[XB] - E[XA], 0}
- A max{ E[XA] - E[XB], 0}
with 0 <1 and
outcome fairness
procedural fairness
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two-stage approach: stage 2 fairness 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
why these models?
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illustration of two-stage approach: Diamond (1)
P
A B1 0
1 00.5
0.5
Q
A B1 0
0 10.5
0.5
?interpret as self-interested vNM utilities
apply outcome FS
P
A B 1- -1- -0.5
0.5
Q0.5
0.5
A B 1- - - 1-
EW=1--assume A= B= >0 A = B = >0
EW=1--
planner’s preference still unconvincing
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illustration of two-stage approach: Diamond (2)
P
A B1 0
1 00.5
0.5
Q
A B1 0
0 10.5
0.5
?interpret as self-interested vNM utilities
apply process FS
P
A B 1- -1- -0.5
0.5
Q0.5
0.5
A B 1 0
0 1
EW=1-- EW=1
here planner’s preference is convincing: utilitarianism is supported by process FS
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illustration of two-stage approach: Broome (1)
P
A B1 1
0 00.5
0.5
Q
A B1 0
0 10.5
0.5
?interpret as self-interested vNM utilities
apply outcome FS
P
A B 1 1
0 0 0.5
0.5
EW=1
planner’s preference is convincing: utilitarianism is supported by outcome FS
Q0.5
0.5
A B 1- - - 1-
EW=1--
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illustration of two-stage approach: Broome (2)
P
A B1 1
0 00.5
0.5
Q
A B1 0
0 10.5
0.5
?interpret as self-interested vNM utilities
apply process FS
EW=1
planner’s preference is unconvincing
EW=1
P
A B1 1
0 00.5
0.5
Q
A B1 0
0 10.5
0.5
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appraisal of utilitarianism: two-stage approach with different fairness models
convincing, supports Harsanyi
unconvincingprocess FS
unconvincingconvincing, supports Harsanyi
outcome FS
unconvincingunconvincing self-interested
Diamond’s example
Broome’s example
both outcome and process fairness play role in supporting utilitarianism
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conclusion (1)
• fairness not adequately considered by utilitarian SP under Harsanyi’s utilitarianism
• all-inclusive utility saves Harsanyi’s 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 fairnessmeaningful all-inclusive utilities quantitative evaluation of social allocationsempirically 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 Harsanyi’s theorem both process and outcome fairness play a role in making utilitarianism convincing in both examplesif we accept utilitarianism and the criticisms, we need more complete individual fairness model