Social Aggregation Without Manipulation: Understanding Doctrinal Paradox

1. Aggregation of Binary Evaluations without Manipulations Dvir Falik Elad Dokow

2. “Doctrinal paradox” • Majority rule is not consistent!

3. “Doctrinal paradox” Assume that for solving this paradox the society decide only on p and q.

4. “Doctrinal paradox” Judge 1 can declare 0 on p and manipulate the result of the third column .

5. Linear classification

6. “Condorcet paradox” (1785) a>b>c c>a>b b>c>a • Majority rule is not consistent! • Arrow Theorem: There is no function which is IIA paretian and not dictatorial.

7. Example: My opinion 101 100 001 110 011 010 Social aggregator Facility location

8. Example: My opinion 101 100 001 110 011 010 Social aggregator Full Manipulation

9. Example: My opinion 101 100 001 110 011 010 Social aggregator Full Manipulation Partial Manipulation

10. Example: My opinion 101 100 001 110 011 010 Social aggregator Full Manipulation Hamming manipulation Partial Manipulation

11. Gibbard Satterhwaite theorem: Social choice function: Social welfare function: GS theorem: For any , there is no Social choice function which is onto A, and not manipulatable.

12. Example:GS theorem My opinion: c>a>b Social aggregator 101 100 c a 001 110 b 011 010

13. The model • A finite, non-empty set of issues K={1,…,k} • A vector is an evaluation. • The evaluations in are called feasible, the others are infeasible. • In our example, (1,1,0) is feasible ; but (1,1,1) is infeasible.

14. A societyis a finite set . • A profileof feasible evaluations is an matrix all of whose rows lie in X. • An aggregator for N over X is a mapping .

15. Different definitions of Manipulation Manipulation: An aggregator f is manipulatable if there exists a judge i, anopinion , an evaluation , coordinate j, and a profile such that: Partial partial

16. Different definitions of Manipulation Manipulation:An aggregator f is manipulatable if there exists a judge i, anopinion , an evaluation , coordinate j, and a profile such that: Full full And: We denote by and say that c is between a and b if . We denote by the set .

17. Different definitions of Manipulation Manipulation:An aggregator f is manipulatable if there exists a judge i, anopinion , an evaluation , coordinate j, and a profile such that: Full full

18. Different definitions of Manipulation • Any other definition of manipulation should be between the partial and the full manipulation. • If is not partial manipulable then f is not full manipulable .

19. Hamming Manipulation • Hamming distance: Hamming manipulation: An aggregator f is Hamming manipulatable if there exists a judge i, anopinion , an evaluation , and a profile such that:

20. Partial Manipulation Theorem (Nehiring and Puppe, 2002): Social aggregator f is not partial manipulatable if and only if f is IIA and monotonic. Theorem (Nehiring and Puppe, 2002): Every Social aggregator which is IIA, paretian and monotonic is dictatorial if and only if X is Totally Blocked.

21. Partial Manipulation Corollary (Nehiring and Puppe, 2002): Every Social aggregator which is not partial manipulable and paretian is dictatorial if and only if X is Totally Blocked.

22. IIA • An aggregator is independent of irrelevant alternatives(IIA) if for every and any two profiles and satisfying for all , we have

23. Paretian • An aggregator is Paretianif we have whenever the profile is such that for all .

24. Monotonic • An aggregator is IIA andMonotonicif for every coordinate j, if then for every we have .

25. Monotonic • An aggregator is IIA andMonotonicif for every coordinate j, if then for every we have .

26. Dictatorial • An aggregator is dictatorialif there exists an individual such that for every profile.

27. Almost dictator Almost dictator function: Fact: For any set is not Hamming/strong manipulatable. Question: what are the conditions on such that there exists an anonymous, Hamming\strong non-manipulatable social function?

28. Majority Nearest Neighbor Let be the majority function (|N| is odd) on each column. Let be an IIA and Monotonic function. Let be a function with the following property: there isn’t any between and . Let be a function with the following property: for every , . The sets of those function will be denoted byEasy to notice that

29. Nearest Neighbor Proposition: For any set is not full manipulatable.Furthermore, if is annonymous, then is annonymous. Proof: /0 /0 /0 /0

30. Nearest Neighbor Proof :

31. Nearest Neighbor Proof:

32. Hamming Nearest Neighbor Proposition: For any set 1. If then judge i can’t manipulate by choosing instead of . 2. If then judge i can’t manipulate by choosing instead of .

33. Hamming Nearest Neighbor Proof of part 1: Let ,

34. Hamming Nearest Neighbor Conclusions: 1. An Hamming Nearest Neighbor function is not manipulatable on . 2. Manipulation can’t be too ‘far’.

35. MIPE-minimally infeasiblepartial evaluation • Let , a vector with entries for issues in J only is a J-evaluation. • A MIPE is a J-evaluationfor some which is infeasible, but such that every restriction of x to a proper subset of J is feasible.

36. Hamming Nearest Neighbor Proposition: For any set 2. If then judge i can’t manipulate by choosing instead of . Proof: Let

37. Hamming Nearest Neighbor Proposition: For any set 1. If then judge i can’t manipulate by choosing instead of . 2. If then judge i can’t manipulate by choosing instead of . What happens in intermediate cases?

38. Example

39. Example Weighted columns: My opinion: 1 0 1 0 1 1 1 0

40. Conjectures: Let: What are the conditions on X such that Conjecture: For every set such that and there exists a weighting of the columns, such that for every Conjecture: thank you!!!