1. 110/20/2015 Aggregation of Binary Evaluations without Manipulations Dvir Falik Elad Dokow

2. 210/20/2015 “Doctrinal paradox” Majority rule is not consistent! The defendant is guilty The defendant was sane at the time The defendant killed the victim 001 Judge 1 010 Judge 2 111 Judge 3 011 Majority

3. 310/20/2015 “Doctrinal paradox” Assume that for solving this paradox the society decide only on p and q. The defendant is guilty The defendant was sane at the time The defendant killed the victim 001 Judge 1 010 Judge 2 111 Judge 3 111 Majority

4. 410/20/2015 “Doctrinal paradox” Judge 1 can declare 0 on p and manipulate the result of the third column. The defendant is guilty The defendant was sane at the time The defendant killed the victim 000 Judge 1 010 Judge 2 111 Judge 3 010 Majority

5. Linear classification 510/20/2015

6. 6 “Condorcet paradox” (1785) Majority rule is not consistent! IS c>aIS b>cIS a>b 011 Judge 1 101 Judge 2 110 Judge 3 111 Majority Arrow Theorem: There is no function which is IIA paretian and not dictatorial. a>b>c c>a>b b>c>a

7. Example: 10/20/20157 100 001 011 101 110 010 My opinion Social aggregator Facility location

8. Example: 10/20/20158 100 001 011 101 110 010 My opinion Social aggregator Full Manipulation

9. Example: 10/20/20159 100 001 011 101 110 010 My opinion Social aggregator Full Manipulation Partial Manipulation

10. Example: 10/20/201510 100 001 011 101 110 010 My opinion Social aggregator Full Manipulation Partial ManipulationHamming manipulation

11. 10/20/2015 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. 11

12. Example: GS theorem 10/20/201512 100 001 011 101 110 010 My opinion: c>a>b Social aggregator a b c

13. 1310/20/2015 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. 1410/20/2015 A society is a finite set. A profile of feasible evaluations is an matrix all of whose rows lie in X. An aggregator for N over X is a mapping.

15. 1510/20/2015 Different definitions of Manipulation Manipulation: An aggregator f is manipulatable if there exists a judge i, an opinion, an evaluation, coordinate j, and a profile such that: partialPartial

16. 1610/20/2015 Different definitions of Manipulation Manipulation: An aggregator f is manipulatable if there exists a judge i, an opinion, 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. 1710/20/2015 Different definitions of Manipulation Manipulation: An aggregator f is manipulatable if there exists a judge i, an opinion, an evaluation, coordinate j, and a profile such that: full Full

18. 1810/20/2015 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. 1910/20/2015 Hamming Manipulation Hamming manipulation: An aggregator f is Hamming manipulatable if there exists a judge i, an opinion, an evaluation, and a profile such that: Hamming distance:

20. 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. 10/20/201520 Partial Manipulation

21. 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. 10/20/201521 Partial Manipulation

22. 2210/20/2015 IIA An aggregator is independent of irrelevant alternatives (IIA) if for every and any two profiles and satisfying for all, we have 321 Judge 1 Judge 2 Judge 3 aggregator

23. 2310/20/2015 Paretian An aggregator is Paretian if we have whenever the profile is such that for all. 321 1 Judge 1 1 Judge 2 1 Judge 3 1 aggregator

24. 2410/20/2015 Monotonic An aggregator is IIA and Monotonic if for every coordinate j, if then for every we have. 321 1 Judge 1 0 Judge 2 0 Judge 3 1 aggregator

25. 2510/20/2015 Monotonic An aggregator is IIA and Monotonic if for every coordinate j, if then for every we have. 321 1 Judge 1 1 Judge 2 0 Judge 3 1 aggregator

26. 2610/20/2015 Dictatorial An aggregator is dictatorial if there exists an individual such that for every profile.

27. Almost dictator function: Fact: For any set is not Hamming/strong manipulatable. 10/20/201527 Almost dictator Question: what are the conditions on such that there exists an anonymous, Hamming\strong non-manipulatable social function?

28. 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 by Easy to notice that 10/20/201528 Majority Nearest Neighbor

29. 10/20/201529 Nearest Neighbor Proof: Third column Second column First column 111 Judge I Judge 2 Judge 3 011 m /0 Proposition: For any set is not full manipulatable. Furthermore, if is annonymous, then is annonymous.

30. 10/20/201530 Nearest Neighbor Proof:

31. 10/20/201531 Nearest Neighbor Proof:

32. Proposition: For any set 10/20/201532 Hamming Nearest Neighbor 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. 10/20/201533 Hamming Nearest Neighbor Proof of part 1: Let,

34. Conclusions: 10/20/201534 Hamming Nearest Neighbor 1. An Hamming Nearest Neighbor function is not manipulatable on. 2. Manipulation can’t be too ‘far’.

35. 3510/20/2015 MIPE- minimally infeasible partial evaluation Let, a vector with entries for issues in J only is a J-evaluation. A MIPE is a J-evaluation for some which is infeasible, but such that every restriction of x to a proper subset of J is feasible.

36. Proposition: For any set 10/20/201536 Hamming Nearest Neighbor 2. If then judge i can’t manipulate by choosing instead of. Proof: Let

37. Proposition: For any set 10/20/201537 Hamming Nearest Neighbor 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. 3810/20/2015 Example (P or q)ssqp 0000 0010 0001 0011 0100 1101 1110 1111

39. 10/20/2015 Example (p or q)s 3 s4s4 q2q2 p2p2 0000 0010 0001 0011 0100 1101 1110 1111 Weighted columns: My opinion: 1 0 6 8 4 6 2 3 7 5 1 1 1 0 8 6 6 4 4 5 5 3 5 2 0011 0100 1101 0101Maj: 0011 0100 1111 0111

40. 10/20/2015 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: