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

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 Judge Judge Majority

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 Judge Judge Majority

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 Judge Judge Majority

Linear classification 510/20/2015

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

Example: 10/20/ My opinion Social aggregator Facility location

Example: 10/20/ My opinion Social aggregator Full Manipulation

Example: 10/20/ My opinion Social aggregator Full Manipulation Partial Manipulation

Example: 10/20/ My opinion Social aggregator Full Manipulation Partial ManipulationHamming manipulation

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

Example: GS theorem 10/20/ My opinion: c>a>b Social aggregator a b c

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.

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.

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

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.

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

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.

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:

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/ 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. 10/20/ Partial Manipulation

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

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

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

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

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

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

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/ Majority Nearest Neighbor

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

10/20/ Nearest Neighbor Proof:

10/20/ Nearest Neighbor Proof:

Proposition: For any set 10/20/ 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.

10/20/ Hamming Nearest Neighbor Proof of part 1: Let,

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

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.

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

Proposition: For any set 10/20/ 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?

3810/20/2015 Example (P or q)ssqp

10/20/2015 Example (p or q)s 3 s4s4 q2q2 p2p Weighted columns: My opinion: Maj:

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: