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Operationalizing Individual Fairness in Harsanyi’s Utilitarianism

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

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Operationalizing Individual Fairness in Harsanyi’s Utilitarianism

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  1. Operationalizing Individual Fairness in Harsanyi’s Utilitarianism Stefan Trautmann June 26, 2006

  2. 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

  3. Harsanyi’s 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)

  4. 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

  5. Harsanyi’s 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

  6. 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 EW=1 EW=1 entries are utilities under utilitarianism

  7. 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 ? always inequality 0.5 0.5 EW=1 EW=1 Pareto vs AA assumption 1: only one horse matters

  8. 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]

  9. all-inclusive utility Ui‘s include already all social comparisons: UA(xA, xB , xA- xB , E[XA]-E[XB],..) A B 1 0 0 1 0.5 Q 0.5 pro: saves Harsanyi’s argument formally: fairness included at individual level con: deprives it from predictive power

  10. 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 0.75 0.5 but same outcomes x

  11. 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 do not change outcomes, only prob 0.25 1 0 0 1 0.5  P Q 0.75 0.5 EW=1 EW=1

  12. 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 expected outcome diffs change for Q, so do all-inc utilities A B 1 1 0 0 0.25 a b c d 0.5 ? P Q 0.75 0.5 EW=1 EW=0.25(a+b)+0.75(c+d)

  13. 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)

  14. two-stage approach: stage 2 fairness models • outcome Fehr-Schmidt (1999) • UA( xA , xB )= xA - Amax{ xB-xA, 0} • - Amax{ xA-xB, 0} • with 0  <1 and    • process Fehr-Schmidt (Trautmann 2006) • UA(xA,XA,XB)= xA - Amax{ E[XB] - E[XA], 0} • - Amax{ E[XA] - E[XB], 0} • with 0  <1 and    outcome fairness procedural fairness

  15. 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

  16. 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 A B 1-  - 1-  - A B 1-  - - 1-  apply outcome FS 0.5 0.5  P Q 0.5 0.5 assume A= B=  >0 A =B =  >0 EW=1-- EW=1-- planner’s preference still unconvincing

  17. 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 A B 1-  - 1-  - A B 1 0 01 apply process FS 0.5 0.5  P Q 0.5 0.5 EW=1-- EW=1 here planner’s preference is convincing: utilitarianism is supported by process FS

  18. 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 A B 1 1 0 0 A B 1-  - - 1-  apply outcome FS 0.5 0.5  P Q 0.5 0.5 EW=1-- EW=1 planner’s preference is convincing: utilitarianism is supported by outcome FS

  19. 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 EW=1 EW=1 planner’s preference is unconvincing

  20. appraisal of utilitarianism: two-stage approach with different fairness models Broome’s example Diamond’s example self-interested unconvincing unconvincing outcome FS convincing, supports Harsanyi unconvincing process FS unconvincing convincing, supports Harsanyi  both outcome and process fairness play role in supporting utilitarianism

  21. 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:

  22. 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

  23. 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

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