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Choosing k from m Bezalel Peleg, Hans Peters Amsterdam, 19-03-2015 - - PowerPoint PPT Presentation

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Choosing k from m Bezalel Peleg, Hans Peters Amsterdam, 19-03-2015 Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 0 / 17 How it started One year, the department was asked by the Dean to suggest two people for slots

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Choosing k from m

Bezalel Peleg, Hans Peters Amsterdam, 19-03-2015

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 0 / 17

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How it started

“One year, the department was asked by the Dean to suggest two people for slots that were opening up in the Faculty of Natural Sciences and Mathematics. Four serious mathematicians were candidates; [...after the committee selection it turned out that...] not only [was] most of the department opposed to last night’s decision, but there [was] even a specific pair that most of the department prefers to the one chosen [...]” R.J. Aumann (2012) My scientific first-born. Special issue of International Journal of Game Theory in honor of Bezalel Peleg.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 1 / 17

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The central question

There are m candidates, from which a committee of size k has to be chosen: 1 ≤ k ≤ m − 1. There are n voters with linear preferences on the set of candidates. Is there a voting method such that no coalition of voters, by voting strategically, can guarantee a committee that all voters in the coalition prefer to the (or any) committee chosen by truthful voting?

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 2 / 17

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An example: n = 3, m = 4, k = 2

R1 R2 R3 a b c b c a c a b d d d R1 Q2 Q3 a c c b b b c a a d d d

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 3 / 17

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An example: n = 3, m = 4, k = 2

R1 R2 R3 a b c b c a c a b d d d R1 Q2 Q3 a c c b b b c a a d d d We apply Borda with weights 3, 2, 1, 0.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 3 / 17

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An example: n = 3, m = 4, k = 2

R1 R2 R3 a b c b c a c a b d d d R1 Q2 Q3 a c c b b b c a a d d d We apply Borda with weights 3, 2, 1, 0. In left profile: {(a, b), (b, a), (a, c), (c, a), (b, c), (c, b)}.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 3 / 17

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An example: n = 3, m = 4, k = 2

R1 R2 R3 a b c b c a c a b d d d R1 Q2 Q3 a c c b b b c a a d d d We apply Borda with weights 3, 2, 1, 0. In left profile: {(a, b), (b, a), (a, c), (c, a), (b, c), (c, b)}. Say (b, a) is chosen (according to some tie-breaking rule).

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 3 / 17

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An example: n = 3, m = 4, k = 2

R1 R2 R3 a b c b c a c a b d d d R1 Q2 Q3 a c c b b b c a a d d d We apply Borda with weights 3, 2, 1, 0. In left profile: {(a, b), (b, a), (a, c), (c, a), (b, c), (c, b)}. Say (b, a) is chosen (according to some tie-breaking rule). In right profile: {(b, c)}. (Second alternative: “chairman”)

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 3 / 17

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An example: n = 3, m = 4, k = 2

R1 R2 R3 a b c b c a c a b d d d R1 Q2 Q3 a c c b b b c a a d d d We apply Borda with weights 3, 2, 1, 0. In left profile: {(a, b), (b, a), (a, c), (c, a), (b, c), (c, b)}. Say (b, a) is chosen (according to some tie-breaking rule). In right profile: {(b, c)}. (Second alternative: “chairman”) Lexicographic preferences over sets: worst first chairman first In both cases, coalition {2, 3} “manipulates”.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 3 / 17

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An example: n = 3, m = 4, k = 2

R1 R2 R3 a b c b c a c a b d d d R1 Q2 Q3 a c c b b b c a a d d d We apply Borda with weights 3, 2, 1, 0. In left profile: {(a, b), (b, a), (a, c), (c, a), (b, c), (c, b)}. Say (b, a) is chosen (according to some tie-breaking rule). In right profile: {(b, c)}. (Second alternative: “chairman”) Lexicographic preferences over sets: worst first chairman first In both cases, coalition {2, 3} “manipulates”. Similarly for other choices in left profile.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 3 / 17

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The same example, now with FEP

Each alternative gets weight one. We eliminate alternatives and preferences at the same time, from bottom up. For instance: R1 R2 R3 a b c b c a c a b d d d

  • El. d, R1 →

R2 R3 b c c a a b

  • El. a, R2 → (b, c)

This way we get {(a, b), (b, a), (a, c), (c, a), (b, c), (c, b)}, say (b, a) is chosen.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 4 / 17

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The same example, now with FEP

Each alternative gets weight one. We eliminate alternatives and preferences at the same time, from bottom up. For instance: R1 R2 R3 a b c b c a c a b d d d

  • El. d, R1 →

R2 R3 b c c a a b

  • El. a, R2 → (b, c)

This way we get {(a, b), (b, a), (a, c), (c, a), (b, c), (c, b)}, say (b, a) is chosen.For the profile R1 Q2 Q3 a c c b b b c a a d d d we get {(a, b), (c, b), (b, c)}.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 4 / 17

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The same example, now with FEP

Each alternative gets weight one. We eliminate alternatives and preferences at the same time, from bottom up. For instance: R1 R2 R3 a b c b c a c a b d d d

  • El. d, R1 →

R2 R3 b c c a a b

  • El. a, R2 → (b, c)

This way we get {(a, b), (b, a), (a, c), (c, a), (b, c), (c, b)}, say (b, a) is chosen.For the profile R1 Q2 Q3 a c c b b b c a a d d d we get {(a, b), (c, b), (b, c)}. These are not all preferred to (b, a) for voters 2 and 3: these voters cannot guarantee something better.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 4 / 17

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Outlook of the paper and presentation

We focus on FEP, Feasible Elimination Procedures.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 5 / 17

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Outlook of the paper and presentation

We focus on FEP, Feasible Elimination Procedures. Background: FEP were introduced as an “escape” from the Gibbard-Satterthwaite result (Peleg, 1978).

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 5 / 17

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Outlook of the paper and presentation

We focus on FEP, Feasible Elimination Procedures. Background: FEP were introduced as an “escape” from the Gibbard-Satterthwaite result (Peleg, 1978). We show how FEP can be used to choose k from m.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 5 / 17

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Outlook of the paper and presentation

We focus on FEP, Feasible Elimination Procedures. Background: FEP were introduced as an “escape” from the Gibbard-Satterthwaite result (Peleg, 1978). We show how FEP can be used to choose k from m. We consider computation: equivalent to finding maximal matchings in bipartite graphs.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 5 / 17

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Outlook of the paper and presentation

We focus on FEP, Feasible Elimination Procedures. Background: FEP were introduced as an “escape” from the Gibbard-Satterthwaite result (Peleg, 1978). We show how FEP can be used to choose k from m. We consider computation: equivalent to finding maximal matchings in bipartite graphs. We have an axiomatic characterization for the case k = 1 (not in this presentation).

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 5 / 17

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Basic model and preliminaries

A is the finite set of alternatives, |A| = m ≥ 2.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 6 / 17

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Basic model and preliminaries

A is the finite set of alternatives, |A| = m ≥ 2. N is the finite set of voters, |N| = n ≥ 2.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 6 / 17

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Basic model and preliminaries

A is the finite set of alternatives, |A| = m ≥ 2. N is the finite set of voters, |N| = n ≥ 2. L is the set of preferences ( = linear orderings) on A.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 6 / 17

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Basic model and preliminaries

A is the finite set of alternatives, |A| = m ≥ 2. N is the finite set of voters, |N| = n ≥ 2. L is the set of preferences ( = linear orderings) on A. A social choice function is a map F : LN → A.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 6 / 17

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Basic model and preliminaries

A is the finite set of alternatives, |A| = m ≥ 2. N is the finite set of voters, |N| = n ≥ 2. L is the set of preferences ( = linear orderings) on A. A social choice function is a map F : LN → A. A pair (F, RN) with RN ∈ LN is a(n ordinal) voting game.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 6 / 17

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Basic model and preliminaries

A is the finite set of alternatives, |A| = m ≥ 2. N is the finite set of voters, |N| = n ≥ 2. L is the set of preferences ( = linear orderings) on A. A social choice function is a map F : LN → A. A pair (F, RN) with RN ∈ LN is a(n ordinal) voting game. F is non-manipulable (or strategy-proof) if RN is a Nash equilibrium in (F, RN) for every RN ∈ LN.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 6 / 17

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Basic model and preliminaries

A is the finite set of alternatives, |A| = m ≥ 2. N is the finite set of voters, |N| = n ≥ 2. L is the set of preferences ( = linear orderings) on A. A social choice function is a map F : LN → A. A pair (F, RN) with RN ∈ LN is a(n ordinal) voting game. F is non-manipulable (or strategy-proof) if RN is a Nash equilibrium in (F, RN) for every RN ∈ LN. THEOREM (Gibbard, 1973; Satterthwaite, 1975). Let F be non-manipulable with at least three alternatives in its range. Then F is dictatorial.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 6 / 17

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Social choice function F is exactly and strongly consistent (ESC) if for every RN ∈ LN there is a strong Nash equilibrium QN of (F, RN) such that F(QN) = F(RN). (Peleg, 1978)

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 7 / 17

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Social choice function F is exactly and strongly consistent (ESC) if for every RN ∈ LN there is a strong Nash equilibrium QN of (F, RN) such that F(QN) = F(RN). (Peleg, 1978) In other words, for an ESC social choice function there is for every profile of true preferences a strong Nash equilibrium of the voting game that results in the sincere (truthful) outcome. In order to obtain ESC social choice functions, feasible elimination procedures play a crucial role.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 7 / 17

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Feasible elimination procedures

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 8 / 17

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Feasible elimination procedures

Let RN be a profile of preferences.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 8 / 17

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Feasible elimination procedures

Let RN be a profile of preferences. Assign weights β(x) ∈ N to the alternatives x ∈ A such that

  • x∈A β(x) = n + 1.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 8 / 17

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Feasible elimination procedures

Let RN be a profile of preferences. Assign weights β(x) ∈ N to the alternatives x ∈ A such that

  • x∈A β(x) = n + 1.

Find an alternative x that occurs at bottom for at least β(x) many voters.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 8 / 17

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Feasible elimination procedures

Let RN be a profile of preferences. Assign weights β(x) ∈ N to the alternatives x ∈ A such that

  • x∈A β(x) = n + 1.

Find an alternative x that occurs at bottom for at least β(x) many voters. Pick exactly β(x) voters who have x at bottom and eliminate their preferences from the profile.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 8 / 17

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Feasible elimination procedures

Let RN be a profile of preferences. Assign weights β(x) ∈ N to the alternatives x ∈ A such that

  • x∈A β(x) = n + 1.

Find an alternative x that occurs at bottom for at least β(x) many voters. Pick exactly β(x) voters who have x at bottom and eliminate their preferences from the profile. Eliminate x everywhere from the profile.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 8 / 17

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Feasible elimination procedures

Let RN be a profile of preferences. Assign weights β(x) ∈ N to the alternatives x ∈ A such that

  • x∈A β(x) = n + 1.

Find an alternative x that occurs at bottom for at least β(x) many voters. Pick exactly β(x) voters who have x at bottom and eliminate their preferences from the profile. Eliminate x everywhere from the profile. Repeat the procedure for the remaining profile with β(x) voters less and without x.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 8 / 17

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Feasible elimination procedures

Let RN be a profile of preferences. Assign weights β(x) ∈ N to the alternatives x ∈ A such that

  • x∈A β(x) = n + 1.

Find an alternative x that occurs at bottom for at least β(x) many voters. Pick exactly β(x) voters who have x at bottom and eliminate their preferences from the profile. Eliminate x everywhere from the profile. Repeat the procedure for the remaining profile with β(x) voters less and without x. Continue doing this until all but one alternatives have been eliminated.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 8 / 17

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Feasible elimination procedures

Let RN be a profile of preferences. Assign weights β(x) ∈ N to the alternatives x ∈ A such that

  • x∈A β(x) = n + 1.

Find an alternative x that occurs at bottom for at least β(x) many voters. Pick exactly β(x) voters who have x at bottom and eliminate their preferences from the profile. Eliminate x everywhere from the profile. Repeat the procedure for the remaining profile with β(x) voters less and without x. Continue doing this until all but one alternatives have been eliminated. The remaining alternative is the outcome of the procedure.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 8 / 17

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Feasible elimination procedures

Let RN be a profile of preferences. Assign weights β(x) ∈ N to the alternatives x ∈ A such that

  • x∈A β(x) = n + 1.

Find an alternative x that occurs at bottom for at least β(x) many voters. Pick exactly β(x) voters who have x at bottom and eliminate their preferences from the profile. Eliminate x everywhere from the profile. Repeat the procedure for the remaining profile with β(x) voters less and without x. Continue doing this until all but one alternatives have been eliminated. The remaining alternative is the outcome of the procedure. The resulting alternative(s) is (are) called RN-maximal.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 8 / 17

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Feasible elimination procedures

Let RN be a profile of preferences. Assign weights β(x) ∈ N to the alternatives x ∈ A such that

  • x∈A β(x) = n + 1.

Find an alternative x that occurs at bottom for at least β(x) many voters. Pick exactly β(x) voters who have x at bottom and eliminate their preferences from the profile. Eliminate x everywhere from the profile. Repeat the procedure for the remaining profile with β(x) voters less and without x. Continue doing this until all but one alternatives have been eliminated. The remaining alternative is the outcome of the procedure. The resulting alternative(s) is (are) called RN-maximal. Note that all this depends on the exogenously chosen weights.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 8 / 17

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An example

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 9 / 17

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An example

A = {a, b, c}, N = {1, . . . , 5}, β(a) = β(b) = β(c) = 2.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 9 / 17

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An example

A = {a, b, c}, N = {1, . . . , 5}, β(a) = β(b) = β(c) = 2. R1 R2 R3 R4 R5 b c a c a c b b a c a a c b b

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 9 / 17

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An example

A = {a, b, c}, N = {1, . . . , 5}, β(a) = β(b) = β(c) = 2. R1 R2 R3 R4 R5 b c a c a c b b a c a a c b b There are two FEPs:

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 9 / 17

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An example

A = {a, b, c}, N = {1, . . . , 5}, β(a) = β(b) = β(c) = 2. R1 R2 R3 R4 R5 b c a c a c b b a c a a c b b There are two FEPs: (a, {1, 2}; b, {4, 5}; c)

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 9 / 17

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An example

A = {a, b, c}, N = {1, . . . , 5}, β(a) = β(b) = β(c) = 2. R1 R2 R3 R4 R5 b c a c a c b b a c a a c b b There are two FEPs: (a, {1, 2}; b, {4, 5}; c) (b, {4, 5}; a, {1, 2}; c).

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 9 / 17

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An example

A = {a, b, c}, N = {1, . . . , 5}, β(a) = β(b) = β(c) = 2. R1 R2 R3 R4 R5 b c a c a c b b a c a a c b b There are two FEPs: (a, {1, 2}; b, {4, 5}; c) (b, {4, 5}; a, {1, 2}; c). Hence c is the only RN-maximal alternative.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 9 / 17

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Relevance of FEP

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 10 / 17

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Relevance of FEP

Voter i ∈ N is a vetoer for social choice function F is there is an alternative x ∈ A and a preference for voter i such that by reporting this preference i can guarantee that x is not chosen.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 10 / 17

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Relevance of FEP

Voter i ∈ N is a vetoer for social choice function F is there is an alternative x ∈ A and a preference for voter i such that by reporting this preference i can guarantee that x is not chosen. THEOREM (Peleg and Peters 2010) Let F be anonymous without

  • vetoers. Then F is ESC if and only if there is a weight function β

such that F assigns an RN-maximal alternative to every profile RN.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 10 / 17

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Relevance of FEP

Voter i ∈ N is a vetoer for social choice function F is there is an alternative x ∈ A and a preference for voter i such that by reporting this preference i can guarantee that x is not chosen. THEOREM (Peleg and Peters 2010) Let F be anonymous without

  • vetoers. Then F is ESC if and only if there is a weight function β

such that F assigns an RN-maximal alternative to every profile RN. Remark: note that β(x) ≥ 2 for any β in this Theorem and any x ∈ A.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 10 / 17

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Relevance of FEP

Voter i ∈ N is a vetoer for social choice function F is there is an alternative x ∈ A and a preference for voter i such that by reporting this preference i can guarantee that x is not chosen. THEOREM (Peleg and Peters 2010) Let F be anonymous without

  • vetoers. Then F is ESC if and only if there is a weight function β

such that F assigns an RN-maximal alternative to every profile RN. Remark: note that β(x) ≥ 2 for any β in this Theorem and any x ∈ A. Goes back to results by Peleg (1978), Holzman (1986), and others.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 10 / 17

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Choosing k from m

The idea is to use FEPs to choose committees of k candidates from in total m candidates. For instance, for k = 2: R1 R2 R3 R4 R5 b c a c a c b b a c a a c b b

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 11 / 17

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Choosing k from m

The idea is to use FEPs to choose committees of k candidates from in total m candidates. For instance, for k = 2: R1 R2 R3 R4 R5 b c a c a c b b a c a a c b b (a, {1, 2}; b, {4, 5}; c) results in (b, c).

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 11 / 17

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Choosing k from m

The idea is to use FEPs to choose committees of k candidates from in total m candidates. For instance, for k = 2: R1 R2 R3 R4 R5 b c a c a c b b a c a a c b b (a, {1, 2}; b, {4, 5}; c) results in (b, c). (b, {4, 5}; a, {1, 2}; c) results in (a, c).

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 11 / 17

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In the paper we consider mainly two preference extensions. Lexicographic worst Lexicographic comparison starting from worst alternative. Example: m = 5, k = 3. Preference abcde. Then (d, c, a) is preferred

  • ver (b, a, e) and over (b, d, c). (Order of alternatives is irrelevant.)

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 12 / 17

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In the paper we consider mainly two preference extensions. Lexicographic worst Lexicographic comparison starting from worst alternative. Example: m = 5, k = 3. Preference abcde. Then (d, c, a) is preferred

  • ver (b, a, e) and over (b, d, c). (Order of alternatives is irrelevant.)

Lexicographic from top Lexicographic comparison starting from the right. Example: m = 5, k = 3. Preference abcde. Then (e, a, b) is preferred

  • ver (b, a, c) and over (e, c, b), but not over (e, d, a). (Order of

alternatives can matter.)

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 12 / 17

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Main result

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 13 / 17

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Main result

Theorem

Suppose we choose a committee of k members from a set of m alternatives according to a feasible elimination procedure (1 ≤ k ≤ m − 1). Assume the lexicographic worst or lexicographic from top preference

  • extension. Then there is no coalition who can guarantee (by reporting

some preference profile) a committee that is strictly preferred by all members of the coalition to the sincere one (any committee selected from the set of sincere committees).

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 13 / 17

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This result does not hold for other methods, e.g., scoring rules like Borda’s, STV, etc.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 14 / 17

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This result does not hold for other methods, e.g., scoring rules like Borda’s, STV, etc. The result does not hold for (e.g.) the lexicographic best preference extension.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 14 / 17

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This result does not hold for other methods, e.g., scoring rules like Borda’s, STV, etc. The result does not hold for (e.g.) the lexicographic best preference extension. The procedure cannot always be made neutral (all weights equal). But if the number of voters is relatively large then this does not matter too much, e.g., m = 10, n = 1000: take nine weights equal to 100 and one weight equal to 101.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 14 / 17

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Computation

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 15 / 17

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Computation

Given are profile RN, weights β(x), committee (a1, . . . , ak).

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 15 / 17

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Computation

Given are profile RN, weights β(x), committee (a1, . . . , ak). To check: can (a1, . . . , ak) result from FEP?

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 15 / 17

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Computation

Given are profile RN, weights β(x), committee (a1, . . . , ak). To check: can (a1, . . . , ak) result from FEP? Make bipartite graph with n left hand nodes (voters) and β(x) right hand nodes per alternative x = ak.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 15 / 17

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Computation

Given are profile RN, weights β(x), committee (a1, . . . , ak). To check: can (a1, . . . , ak) result from FEP? Make bipartite graph with n left hand nodes (voters) and β(x) right hand nodes per alternative x = ak. For x not in the committee, draw edges between i ∈ N on the left and all β(x) on the right if i prefers all committee members to x.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 15 / 17

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SLIDE 66

Computation

Given are profile RN, weights β(x), committee (a1, . . . , ak). To check: can (a1, . . . , ak) result from FEP? Make bipartite graph with n left hand nodes (voters) and β(x) right hand nodes per alternative x = ak. For x not in the committee, draw edges between i ∈ N on the left and all β(x) on the right if i prefers all committee members to x. For x = aℓ in the committee, draw edges between i ∈ N on the left and all β(x) on the right if i prefers all committee members aj, j > ℓ, to x.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 15 / 17

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SLIDE 67

Computation

Given are profile RN, weights β(x), committee (a1, . . . , ak). To check: can (a1, . . . , ak) result from FEP? Make bipartite graph with n left hand nodes (voters) and β(x) right hand nodes per alternative x = ak. For x not in the committee, draw edges between i ∈ N on the left and all β(x) on the right if i prefers all committee members to x. For x = aℓ in the committee, draw edges between i ∈ N on the left and all β(x) on the right if i prefers all committee members aj, j > ℓ, to x. LEMMA: (a1, . . . , ak) can result from FEP if and only if this graph has a maximal matching.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 15 / 17

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SLIDE 68

Computation

Given are profile RN, weights β(x), committee (a1, . . . , ak). To check: can (a1, . . . , ak) result from FEP? Make bipartite graph with n left hand nodes (voters) and β(x) right hand nodes per alternative x = ak. For x not in the committee, draw edges between i ∈ N on the left and all β(x) on the right if i prefers all committee members to x. For x = aℓ in the committee, draw edges between i ∈ N on the left and all β(x) on the right if i prefers all committee members aj, j > ℓ, to x. LEMMA: (a1, . . . , ak) can result from FEP if and only if this graph has a maximal matching. This can be checked in polynomial time (Hopcroft and Karp, 1973). Repeating this procedure m(m − 1) · · · (m − k + 1) times is still polynomial.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 15 / 17

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SLIDE 69

Concluding remarks

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 16 / 17

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SLIDE 70

Concluding remarks

There are many other, non-strategic approaches to choosing committees in the literature. For instance, approaches based on pairwise majority relations (Condorcet winners/losers). See the recent thesis of Eric Kamwa (Essais sur les modes de scrutins et la s´ election des comit´ es, Caen, 2014).

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 16 / 17

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SLIDE 71

Concluding remarks

There are many other, non-strategic approaches to choosing committees in the literature. For instance, approaches based on pairwise majority relations (Condorcet winners/losers). See the recent thesis of Eric Kamwa (Essais sur les modes de scrutins et la s´ election des comit´ es, Caen, 2014). In the paper we argue that methods based on pairwise majority do not have the core property and, hence, are sensitive to manipulation by coalitions.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 16 / 17

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SLIDE 72

Concluding remarks

There are many other, non-strategic approaches to choosing committees in the literature. For instance, approaches based on pairwise majority relations (Condorcet winners/losers). See the recent thesis of Eric Kamwa (Essais sur les modes de scrutins et la s´ election des comit´ es, Caen, 2014). In the paper we argue that methods based on pairwise majority do not have the core property and, hence, are sensitive to manipulation by coalitions. In the paper most of our results are framed in terms of cores of effectivity functions. A method has the core property of it assigns committees belonging to the core of the associated effectivity function.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 16 / 17

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SLIDE 73

Concluding remarks

There are many other, non-strategic approaches to choosing committees in the literature. For instance, approaches based on pairwise majority relations (Condorcet winners/losers). See the recent thesis of Eric Kamwa (Essais sur les modes de scrutins et la s´ election des comit´ es, Caen, 2014). In the paper we argue that methods based on pairwise majority do not have the core property and, hence, are sensitive to manipulation by coalitions. In the paper most of our results are framed in terms of cores of effectivity functions. A method has the core property of it assigns committees belonging to the core of the associated effectivity function. The (an) axiomatization for k > 1 is still open.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 16 / 17

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SLIDE 74

Concluding remarks

There are many other, non-strategic approaches to choosing committees in the literature. For instance, approaches based on pairwise majority relations (Condorcet winners/losers). See the recent thesis of Eric Kamwa (Essais sur les modes de scrutins et la s´ election des comit´ es, Caen, 2014). In the paper we argue that methods based on pairwise majority do not have the core property and, hence, are sensitive to manipulation by coalitions. In the paper most of our results are framed in terms of cores of effectivity functions. A method has the core property of it assigns committees belonging to the core of the associated effectivity function. The (an) axiomatization for k > 1 is still open. There are (open) issues concerning neutrality, other preference extensions.

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 16 / 17

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SLIDE 75

THE END

Bezalel Peleg, Hans Peters Choosing k from m Amsterdam, 19-03-2015 17 / 17