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