Aggregation of Binary Evaluations without Manipulations Dvir Falik
Aggregation of Binary Evaluations without Manipulations Dvir Falik Elad Dokow 10/3/2020 1
“Doctrinal paradox” The defendant killed the victim The defendant was sane at is guilty the time Judge 1 Judge 2 Judge 3 1 0 1 1 Majority 1 1 0 0 1 0 • Majority rule is not consistent! 2 10/3/2020
“Doctrinal paradox” The defendant killed the victim The defendant was sane at is guilty the time Judge 1 Judge 2 Judge 3 1 0 1 1 Majority 1 1 3 Assume that for solving this paradox the society decide only on p and q. 0 0 1 1 10/3/2020
“Doctrinal paradox” The defendant killed the victim The defendant was sane at is guilty the time Judge 1 Judge 2 Judge 3 0 0 1 1 Majority 0 1 4 Judge 1 can declare 0 on p and manipulate the result of the third column. 0 0 10/3/2020
Linear classification 5 10/3/2020
“Condorcet paradox” (1785) IS a>b IS b>c IS c>a Judge 1 Judge 2 Judge 3 1 1 0 1 0 1 1 Majority 1 1 a>b>c c>a>b b>c>a 1 • Majority rule is not consistent! • Arrow Theorem: There is no function which is IIA paretian and not dictatorial. 6 10/3/2020
Gibbard Satterhwaite theorem: Social choice function: GS theorem: For any , there is no Social choice function which is onto A, and not manipulatable. Social welfare function: 7 10/3/2020
Example: My opinion 101 100 001 110 011 010 Social aggregator Facility location 8 10/3/2020
Example: My opinion 101 100 001 110 011 010 Social aggregator Full Manipulation 9 10/3/2020
Example: My opinion 101 100 001 110 011 010 Social aggregator Full Manipulation 10 Partial Manipulation 10/3/2020
Example: My opinion 101 100 001 110 011 010 Social aggregator Full Manipulation 11 Hamming manipulation Partial Manipulation 10/3/2020
Example: GS theorem My opinion: c>a>b Social aggregator 101 100 a c 001 110 011 12 b 010 10/3/2020
The model n n 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. 10/3/2020 13
• 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. 10/3/2020 14
Different definitions of Manipulation Partial Manipulation: An aggregator f is partial manipulatable if there exists a judge i, an opinion , an evaluation , coordinate j, and a profile such that: 10/3/2020 15
Different definitions of Manipulation Full Manipulation: An aggregator f is full manipulatable if there exists a judge i, an opinion , an evaluation , coordinate j, and a profile such that: And: We denote by that c is between a and b if the set. 10/3/2020 and say. We denote by 16
Different definitions of Manipulation Full Manipulation: An aggregator f is full manipulatable if there exists a judge i, an opinion , an evaluation , coordinate j, and a profile such that: 10/3/2020 17
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. 10/3/2020 18
Hamming Manipulation • Hamming distance: Hamming manipulation: An aggregator f is Hamming manipulatable if there exists a judge i, an opinion , an evaluation , and a profile such that: 10/3/2020 19
Partial Manipulation Theorem (Nehiring and Puppe, 2002): Social aggregator f is PMF (partial manipulation free) 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. 20 10/3/2020
Partial Manipulation Corollary (Nehiring and Puppe, 2002): Every Social aggregator which is PMF and paretian is dictatorial if and only if X is Totally Blocked. 21 10/3/2020
IIA • An aggregator is independent of irrelevant alternatives (IIA) if for every and any two profiles and satisfying for all , we have 1 2 3 Judge 1 Judge 2 Judge 3 aggregator 22 10/3/2020
Paretian • An aggregator we have such that is Paretian if whenever the profile is for all. 1 23 2 Judge 1 Judge 2 1 1 Judge 3 1 aggregator 1 3 10/3/2020
Monotonic • An aggregator if for every coordinate j, if every we have 1 24 is IIA and Monotonic then for. 2 Judge 1 Judge 2 1 0 Judge 3 0 aggregator 1 3 10/3/2020
Almost dictator function: Fact: For any set is not Hamming/full manipulatable. 25 10/3/2020
Close to PMF (C-PMF) An aggregator PMF function in which is C-PMF if there exist a s. t for every profile we have that. is an IIA and monotonic function. 26 10/3/2020
Question: what are the conditions on such that there exists a C-PMF, Hammingfull nonmanipulatable social aggregator? 27 10/3/2020
Nearest Neighbor 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 28 10/3/2020
Social welfare maximizer (SWM) One special function which is C-PMF and depend not only in the outcome of but on the whole profile is the SWM aggregator. 29 10/3/2020
Full Manipulation Free aggregator Theorem: For any set Furthermore, if is not full manipulatable. is annonymous, then is annonymous. Remark: This proposition doesn’t hold for in which. Theorem: For any set is not full manipulatable. 30 the SWM aggregator 10/3/2020
Hamming Nearest Neighbor Theorem 1: For any set If manipulate by choosing 31 then judge i can’t instead of. 10/3/2020
Hamming Nearest Neighbor Conclusions: 1. An Hamming Nearest Neighbor function is not manipulatable on. 2. Manipulation can’t be too ‘far’. 32 10/3/2020
Hamming Nearest Neighbor Theorem 2: For any set If manipulate by choosing 33 then judge i can’t instead of. 10/3/2020
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. 10/3/2020 34
Hamming Nearest Neighbor What happens in intermediate cases? 35 10/3/2020
Example 36 p q s (P or q)s 0 0 0 1 1 0 0 1 0 1 1 1 10/3/2020
Example Weighted columns: p 2 q 2 s 4 (p or q)s 3 8 6 0 0 8 0 1 0 0 6 4 1 0 0 0 4 6 1 1 0 0 4 2 0 0 1 0 5 3 My opinion: 1 0 1 1 5 7 0 1 1 1 3 5 1 1 1 0 6 5 2 1 1 0 0 1 0 1 1 1 0 Maj: 1 0 1 1 0 0 1 0 1 1 Maj: 1 1 1 0 10/3/2020
Lines, Cycles Joint work with: Michal Feldman, Reshef Mair, Ilan Nehama. Main Theorem: An onto aggregator f on the line is HMF if and only if it is monotonic and 1 -SSI. Main Theorem: For sufficiently large cycles, any onto HMF aggregator is 1 -dictatorial. 38 10/3/2020
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