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

2. 2 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

3. 3 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)

4. 4 Harsanyi’s theorem (2) U i : individual vNM utilities of outcomes x i W : social welfare function Theorem (Harsanyi 1955): Assumptions 1 - 3 imply a social welfare function of utilitarian form W=  i U i

5. 5 Harsanyi’s theorem (3) W=  i U i individual agents max EU social planner max EU Pareto-principle modest assumptions strong result ? distribution of utility over agents does not matter strong: individualistic values only marginal distribution of outcomes of agents matters distribution between agents not considered (Anscombe-Aumann Ass1)

6. 6 criticisms based on fairness (1) Diamond (1967) P A B 1 0 0.5 Q A B 1 0 0 1 0.5  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

7. 7 criticisms based on fairness (2) Broome (1991) P A B 1 0 0.5 Q A B 1 0 0 1 0.5  EW=1  always equality always inequality ? Pareto vs AA assumption 1: only one horse matters

8. 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. 9 all-inclusive utility Q A B 1 0 0 1 0.5 U i ‘s include already all social comparisons: U A ( x A, x B, x A - x B, E[X A ]-E[X B ],.. ) pro: saves Harsanyi’s argument formally: fairness included at individual level con: deprives it from predictive power

10. 10 all-inclusive utility: prediction P A B 1 0 0.5 Q A B 1 0 0 1 0.5  P A B 1 0 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?

11. 11 all-inclusive utility : prediction (2) P A B 1 0 0.5 Q A B 1 0 0 1 0.5  P A B 1 0 0.5 Q 1 0 0 1 0.75 0.25 expl 1: selfish agents; utility depends only on own outcome  do not change outcomes, only prob EW=1 A B what do these utilities include?

12. 12 all-inclusive utility : prediction (3) P A B 1 0 0.5 Q A B 1 0 0 1 0.5  P A B 1 0 0.5 Q 0.75 0.25 expl 2: utility depends on both own outcome and expected outcome difference ? expected outcome diff s 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

13. 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. 14 two-stage approach: stage 2 fairness models outcome Fehr-Schmidt (1999) U A ( x A, x B )= x A -  A max{ x B -x A, 0} -  A max{ x A -x B, 0} with 0   <1 and    process Fehr-Schmidt (Trautmann 2006) U A (x A,X A,X B )= x A -  A max{ E[X B ] - E[X A ], 0} -  A max{ E[X A ] - E[X B ], 0} with 0   <1 and    outcome fairness procedural fairness

15. 15 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?

16. 16 illustration of two-stage approach: Diamond (1) P A B 1 0 0.5 Q A B 1 0 0 1 0.5 ? interpret as self-interested vNM utilities apply outcome FS P A B 1-  -  1-  -  0.5 Q A B 1-  -  -  1-  EW=1-  -  assume  A =  B =  >0  A =  B =  >0 EW=1-  -   planner’s preference still unconvincing

17. 17 illustration of two-stage approach: Diamond (2) P A B 1 0 0.5 Q A B 1 0 0 1 0.5 ? interpret as self-interested vNM utilities apply process FS P A B 1-  -  1-  -  0.5 Q A B 1 0 0 1 EW=1-  -  EW=1  here planner’s preference is convincing: utilitarianism is supported by process FS

18. 18 illustration of two-stage approach: Broome (1) P A B 1 0 0.5 Q A B 1 0 0 1 0.5 ? interpret as self-interested vNM utilities apply outcome FS P A B 1 1 0 0.5 EW=1  planner’s preference is convincing: utilitarianism is supported by outcome FS Q 0.5 A B 1-  -  -  1-  EW=1-  - 

19. 19 illustration of two-stage approach: Broome (2) P A B 1 0 0.5 Q A B 1 0 0 1 0.5 ? interpret as self-interested vNM utilities apply process FS EW=1 planner’s preference is unconvincing EW=1 P A B 1 0 0.5 Q A B 1 0 0 1 0.5 

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

21. 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. 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. 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