Operationalizing Individual
Fairness in Harsanyi’s
Utilitarianism
Stefan Trautmann
June 26, 2006
outline
• 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
2
Harsanyi’s theorem (1)
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)
3
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
4
Harsanyi’s theorem (3)
individual agents max EU
social planner max EU
Pareto-principle
W=i Ui
modest
assumptions
?
strong: individualistic values
only marginal distribution of
strong
result of agents matters
outcomes
distribution between agents
distribution of utility over
not considered
agents does not matter
(Anscombe-Aumann Ass1) 5
criticisms based on fairness (1)
A always gets
positive
lack of fairness consideration
by social
utility, B nothing
planner under utilitarianism
criticized
counterexamples: Diamond 1967,
Diamondby(1967)
both
A
Broome 1991
and B
AB
AB
0.5 1 0
0.5 1 0
have

P
Q
fair
0.5 1 0
0.5 0 1
chance
?

EW=1
under utilitarianism
EW=1
entries are
6
utilities
criticisms based on fairness (2)
always equality
Broome (1991)
0.5
P
0.5
AB
11
00
EW=1
 Q
?
0.5
0.5
AB
10
01
always
inequality
EW=1
Pareto vs AA assumption 1: only one
horse matters
7
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]
8
all-inclusive utility
0.5
Q
0.5
AB
10
01
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
9
all-inclusive utility: prediction
0.5
P
0.5
AB
11
0.5

Q
00
0.5
say we know
SP indiff in
Broome expl
what can we
predict in new
decision?
0.5
P
0.5
AB
10
Broome
example
01
AB
11
00
?
0.25
AB
??
0.75
??
Q
but same outcomes x
10
all-inclusive utility : prediction (2)
0.5
P
0.5
AB
11
00

0.5
Q
0.5
AB
10
01
expl 1: selfish
agents; utility
depends only on
own outcome
what do these utilities include?
AB
AB
0.25
0.5 1 1
do not
10

P
Q
change
0.5 0 0
0.75 0 1
outcomes,
EW=1
EW=1
only prob
11
all-inclusive utility : prediction (3)
0.5
P
0.5
AB
11
00

0.5
Q
0.5
expl 2: utility
depends on both
own outcome and
expected outcome
difference
AB
10
01
AB
expected
0.5 1 1
outcome
P
s
diff change
0.5 0 0
for Q, so do
EW=1
all-inc utilities
?
0.25
AB
ab
0.75
cd
Q
EW=0.25(a+b)+0.75(c+d)
12
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)
13
two-stage approach: stage 2 fairness models
• outcome Fehr-Schmidt (1999)
UA( xA , xB )= xA - A max{ xB-xA, 0}
outcome
fairness
- 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}
procedural
fairness
14
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
15
illustration of two-stage approach: Diamond (1)
0.5
P
0.5
AB
10
10
? Q
0.5
0.5
AB
10
01
interpret as
self-interested
vNM utilities
A B
A B
apply
0.5 1-  0.5 1-  outcome

P
Q


FS
0.5
0.5
1-  - 1- assume
EW=1--
EW=1--


A= B= 
>0 A = B =
planner’s preference still unconvincing
16
 >0
illustration of two-stage approach: Diamond (2)
0.5
P
0.5
AB
10
10
? Q
0.5
0.5
AB
10
01
interpret as
self-interested
vNM utilities
A B
A B
apply
0.5 1-  0.5 1 0
process

P
Q

FS
0.5
0.5 0 1
1-  EW=1--
EW=1

here planner’s preference is convincing:
17
utilitarianism is supported by process FS
illustration of two-stage approach: Broome (1)
0.5
P
0.5
AB
11
? Q
00
0.5
01
interpret as
self-interested
vNM utilities
A B
apply
0.5
0.5 1-  outcome
P
Q

FS
0.5
0.5
- 1EW=1--
EW=1

planner’s preference is convincing:
18
utilitarianism is supported by outcome FS

AB
11
00
0.5
AB
10
illustration of two-stage approach: Broome (2)
0.5
P
0.5
0.5
P
0.5
AB
11
? Q
00
AB
11
00
EW=1
0.5
0.5

0.5
Q
0.5
AB
10
01
AB
10
01
interpret as
self-interested
vNM utilities
apply
process
FS
EW=1
planner’s preference is unconvincing
19
appraisal of utilitarianism: two-stage approach
with different fairness models
Broome’s
example
Diamond’s
example
selfunconvincing
unconvincing
interested
outcome FS convincing,
unconvincing
supports Harsanyi
process FS
unconvincing
convincing,
supports Harsanyi
 both outcome and process fairness play
role in supporting utilitarianism
20
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 allinclusive utilities:
21
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
22
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 examples
if we accept utilitarianism and the
criticisms, we need more complete individual
fairness model
23