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Example 1: Probabilistic Models for Approval Voting and Majority Rule A: 50 votes A 40 AB 2 B: 32 votes B 20 AC 8 C 20 BC 10

Example 1: Probabilistic Models for Approval Voting and Majority Rule A: 50 votes A 40 AB 2 B: 32 votes B 20 AC 8 C: 38 votes C 20 BC 10 A is the Approval Voting Winner! Is there a Majority Winner? Who is it?

Sorry! Majority Winner not defined for Approval Voting Majority Winner • Candidate who is ranked ahead of any other candidate by more than 50% • Candidate who beats any other candidate in pairwise competition Majority Winner is Counterfactual

Example 1: Probabilistic Models for Approval Voting and Majority Rule A 40 AB 2 A beats B 48 times B 20 AC 8 B beats A 30 times C 20 BC 10 A is majority preferred to B

Example 1: Probabilistic Models for Approval Voting and Majority Rule A 40 AB 2 A beats C 42 times B 20 AC 8 C beats A 30 times C 20 BC 10 A is majority preferred to B A is majority preferred to C

Example 1: Probabilistic Models for Approval Voting and Majority Rule A 40 AB 2 B beats C 22 times B 20 AC 8 C beats B 28 times C 20 BC 10 A is majority preferred to B A C A is majority preferred to C B C is majority preferred to B

Example 1: Probabilistic Models for Approval Voting and Majority Rule ABC 8 A 40 AB 2 ACB 32 ABC 2 B 20 AC 8 BCA 20 ACB 8 C 20 BC 10 CBA 20 BCA 5 CBA 5

Example 1: Probabilistic Models for Approval Voting and Majority Rule ABC 8 A 40 AB 2 ACB 32 ABC 2 B 20 AC 8 BCA 20 ACB 8 C 20 BC 10 CBA 20 BCA 5 CBA 5 A is majority tied with B A is majority tied with C C C is majority preferred to B B

Majority Winner may be Model Dependent First computation: Topset Voting Model (Regenwetter, 1997, MSS) (Niederee & Heyer, 1997, Luce volume) Second computation: Size-Independent Model (Falmagne & Regenwetter, 1996, JMP) (Doignon & Regenwetter, 1997, JMP) (Regenwetter & Grofman, 1998a,b; SCW, MS) (Regenwetter & Doignon, 1998, JMP) (Regenwetter, Marley & Joe, 1998, AJP)

Order by Majority Order Majority Order AV scores Topset Model SIM Model TIMS E1 TIMS E2 MAA1 MAA2 A25 A72 IEEE

Order by Majority Order Majority Order AV scores Topset Model SIM Model b c b Same as TIMS E1 c b c or AV order a a a

Order by Majority Order Majority Order AV scores Topset Model SIM Model b c b Same as TIMS E1 c b c or AV order a a a c b c Same as TIMS E2 b c b or AV order a a a

Order by Majority Order Majority Order AV scores Topset Model SIM Model b c b Same as TIMS E1 c b c or AV order a a a c b c Same as TIMS E2 b c b or AV order a a a c a c Same as MAA1 a c a or AV order b b b

Order by Majority Order Majority Order AV scores Topset Model SIM Model b c b Same as TIMS E1 c b c or AV order a a a c b c Same as TIMS E2 b c b or AV order a a a c a c Same as MAA1 a c a or AV order b b b b Same as Same as MAA2 c AV order AV order a

Order by Majority Order Majority Order AV scores Topset Model SIM Model b c b Same as TIMS E1 c b c or AV order a a a c b c Same as TIMS E2 b c b or AV order a a a c a c Same as MAA1 a c a or AV order b b b b Same as Same as MAA2 c AV order AV order a b Same as Same as A25 c AV order AV order a

Order by Majority Order Majority Order AV scores Topset Model SIM Model b c b Same as TIMS E1 c b c or AV order a a a c b c Same as TIMS E2 b c b or AV order a a a c a c Same as MAA1 a c a or AV order b b b b Same as Same as MAA2 c AV order AV order a b Same as Same as A25 c AV order AV order a c Same as Same as A72 a AV order AV order b

Order by Majority Order Majority Order AV scores Topset Model SIM Model b c b Same as TIMS E1 c b c or AV order a a a c b c Same as TIMS E2 b c b or AV order a a a c a c Same as MAA1 a c a or AV order b b b b Same as Same as MAA2 c AV order AV order a b Same as Same as A25 c AV order AV order a c Same as Same as A72 a AV order AV order b a Cycle a a Same as IEEE b c b or , AV order c one of b c

Preliminary Conclusions: Majority Preference Relation is model dependent should be treated in an inference framework may or may not be robust

A General Concept of Majority Rule Linear Orders “complete rankings” Weak Orders “rankings with possible ties” Semiorders “rankings with (fixed) threshold” Interval Orders “rankings with (variable) threshold” Partial Orders asymmetric, transitive Asymmetric Binary Relations

> B Real Representation of Weak Orders a b 7 7 | | c 3 | | ∈ ⇔ > ( , ) ( ) ( ) a b B u a u b d 1

Variable Preferences: Probability Distribution on Binary Relations Variable Utilities: Jointly Distributed Family of Utility Random Variables (Random Utilities) (parametric or nonparametric)

Random Utility Representations Semiorders Interval Orders > ∈ ⎛ ⎞ L U | ( , ) i j B ⎜ ⎟ i j = ⎜ ⎟ ( ) and P B P ⎜ ⎟ ≤ ∉ L U | ( , ) i j B ⎝ ⎠ i j ω = ω + ε U L With ( ) ( ) i i ∀ ω

A General Definition of Majority Rule Given a probabilit y distributi on → : [ 0 , 1 ] B P $ ( ) B P B on any set B of binary relations, o a is strictly majority preferred t b if and only if ∑ ∑ > ( ) ( ' ) P B P B ∈ ∈ ( , ) ( , ) ' a b B b a B

A General Definition of Majority Rule Given a probabilit y distributi on → : B [ 0 , 1 ] P $ ( ) B P B on any set B of binary relations, o a is strictly majority preferred t b if and only if ∑ ∑ > ( ) ( ' ) P B P B ∈ ∈ ( a , b ) B ( b , a ) B ' For Utility Functions or Random Utility Models choose a Random Utility Representation and obtain a consistent Definition

Examples: majority preferred to i j ⇔ ( ) ( ) > > > Proportion ( ) ( ) Proportion ( ) ( ) u i u j u j u i majority preferred to i j ⇔ > + > > + U U U U ( 54 ) ( 54 ) P P i j j i

Weak Utility Model Weak Stochastic Transitivity Transitivity of Majority Preferences

Remember: No Cycles in 7 Approval Voting Data Sets (1 analysis ambiguous) Let’s analyze National Survey Data! 1968, 1980, 1992, 1996 ANES Feeling Thermometer Ratings translated into Weak Orders or Semiorders

.03 H 1968 NES W .04 Weak Order N 0 Probabilities H H W W N N H .02 W N .32 H W N H N W .03 .08 W H N .02 N W .06 H N W H N W N .27 H W H N .05 .01 W H .07

-.04 H 1968 NES .03 W Weak Order N -.05 Net Probabilities H H W W N N H -.25 W .26 N H W N H N W -.05 .05 W H N 0 N W Majority -.26 H N W H N W N .25 H W H N .05 -.03 W H .04

-.02 H 1968 NES Threshold W .03 Semiorder N of 10 -.09 Net Probabilities H H W W N N H W H .23 N -.19 H N 0 W -.01 N 0 W H N W H N W -.10 .10 0 W H N N W W .01 H 0 N W Majority 0 H -.23 N N W H H N W N .19 H W H N -.03 .09 W H .02

0 H 1968 NES Threshold W .02 -.04 Semiorder N of 54 Net Probabilities H H W W N N H 0 W H N H 0 N - .01 W N .12 W H N W -.10 H N W -.19 0 .19 W H N N W .10 W -.12 H .01 0 N W Majority H N N W H H N W N 0 H W H N .04 -.02 W H 0

ANES Strict Majority Social Welfare Orders Year Threshold SWO Nixon 1968 0, …, 96 Humphrey Wallace

ANES Strict Majority Social Welfare Orders Year Threshold SWO Clinton 1992 0, …, 99 Bush Perot

However: There is no Theory-Free Majority Preference Relation

ANES Strict Majority Social Welfare Orders SWO Threshold Carter Year Reagan 0, …, 29 Anderson 1980 Reagan 30, …, 99 Carter Anderson

ANES Strict Majority Social Welfare Orders SWO Threshold Year Clinton 0, …, 49 Dole 85, …, 99 Perot 1996 Dole 50, …,84 Clinton Perot

Preliminary Conclusions: Majority Preference Relation is model dependent We did not see any indication of cycles!

Borda Scoring rule: • 1 st ranked candidate gets 2 points, • 2 nd ranked candidate gets 1 point, • 3 rd ranked candidate gets 0 point. In general, the i th ranked among n candidates gets n-i points.

Scoring rule: • 1 st ranked candidate gets x points, • 2 nd ranked candidate gets y < x points, • 3 rd ranked candidate gets z < y points. In general, the i th ranked among n candidates gets f(n-i) many points with f increasing.

Plurality Scoring rule: 1 st ranked candidate gets 1 point, • • other candidates get 0 points.

How about a General Concept of Scoring Rules? Let’s generalize the concept of Ranks from Linear Orders to Arbitrary Finite Binary Relations

beyond linear orders Generalizing ranks (?)

In-degree, Out-degree and Differential of an object In-degree (c) = 1 Out-degree (c) = 2 (3) ∆ (c) = Differential (c) = In-degree (c) - Out-degree (c) = - 1 n+1+ ∆ (c) Rank (c) = 2

Generalizing ranks beyond linear orders n+1+ ∆ (c) Rank (c) = 2

Some properties of generalized rank • Average generalized rank is n+1 2 • Minimal possible rank is 1 • Maximal possible generalized rank is n

Borda Scoring rule: (for n=3 candidates) • 1 st ranked candidate gets 2 points, • candidate with rank = 1.5 gets 1.5 points, • 2 nd ranked candidate gets 1 point, • candidate with rank = 2.5 gets 0.5 points, • 3 rd ranked candidate gets 0 point. In general, the i th ranked among n candidates gets n-i points.

Borda scores derived from semiorder probabilities R R R .04 Borda ( R ) = C C .05 .11 A A R R R C R R C R 2*(.1+.11+.04) + C A C A A A R R R C R R 1.5*(.07+.01+.01+.05)+ A A R R .05 .01 .1 A C C A .01 .02 C C 1*(.04+.12+.02+.05) + R R A A C 0.12 R A C R .14 .07 C R A R .5*(.03+.01+.02+.1) + A A C .02 .00 R C C R 0*(.04+.08+.07) = A C .01 R R A A = 1.02 R C .04 R R .07 R A C A R R C R R .08 A .03 n+1+ ∆ (c) C Rank (c) = Semiorder R R .04 1980 NES 1980 NES 2 Threshold=10

Borda scores derived from semiorder probabilities R R .04 C C .05 .11 Borda (R) = A A R R R C R C 1.02 C A C A A A R R C R A A R .05 .01 .1 A Borda (A) = C C A .01 .02 0.92 C R A 0.12 C R A C .14 .07 C R A Borda (C) = A C 1.07 .02 .00 C R A C .01 R A A C .04 R .07 R A C A R C R .08 A .03 C Semiorder R .04 1980 NES Threshold=10

Plurality Scoring rule: (for n candidates) 1 st ranked candidate gets 1 point, • • other candidates get 0 points. Note: I f no (single) candidate has rank equal to 1, a given ballot is effectively ignored

Plurality scores derived from semiorder probabilities R R R .04 Plurality ( R )= C C .05 .11 A A 1*(.1+.11+.04) = R R R C R C R C A C A A A R R = 0.25 R C C R A A R .05 .01 .1 A C C A .01 Plurality ( A )= .02 C R A 0.12 C = 0.11 R A C C .14 .07 C R A A Plurality ( C )= C .02 .00 C R A A C C .01 = .26 R A A C .04 R .07 R A C A A R C R .08 A A .03 C Semiorder R .04 1980 NES Threshold=10

Empirical example: NES thermometer scores Social ordering depends on: - model of preferences [translation of raw data into binary relations] - social choice function [Majority, Borda, Plurality, others] - data

Empirical example: 1968 NES Various scoring rules Threshold=0, 1, 2, …, 97 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 3 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 4 0 4 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 5 0 5 1 5 2 5 3 5 4 5 5 5 6 5 7 5 8 5 9 6 0 6 1 6 2 6 3 6 4 6 5 6 6 6 7 6 8 6 9 7 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 7 8 7 9 8 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 8 9 9 0 9 1 9 2 9 3 9 4 9 5 9 6 Plur Plur w\sh Out-degree Borda N In-degree A-pl w\sh H Antipl W HWN (H=W)>N WHN W>(H=N) Candidates: H , N, W WNH (W=N)>H NWH Data: thermometer scores {1, …, 97} N>(H=W) NHW (H=N)>W HNW Model: semiorders with threshold: 0 … 97 H>(W=N) Scoring rules: Plurality, Antiplurality (with or without sharing), Borda, In-degree, Out-degree

ANES Strict Majority Social Welfare Orders Year Threshold SWO Nixon 1968 0, …, 96 Humphrey Wallace

Empirical example: 1980 NES Various scoring rules Threshold=0, 1, 2, …, 100 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 3 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 4 0 4 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 5 0 5 1 5 2 5 3 5 4 5 5 5 6 5 7 5 8 5 9 6 0 6 1 6 2 6 3 6 4 6 5 6 6 6 7 6 8 6 9 7 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 7 8 7 9 8 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 8 9 9 0 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 Plur Plur w\sh Out-degree Borda In-degree A-pl w\sh Antipl CAR (C=A)>R ACR Candidates: A, C, R A>(C=R) ARC (A=R)>C RAC Data: thermometer scores {1, …, 100} R>(C=A) RCA (C=R)>A Model: semiorders with threshold: 0 … 100 CRA C>(A=R) Scoring rules: Plurality, Antiplurality (with or without sharing), Borda, In-degree, Out-degree

ANES Strict Majority Social Welfare Orders SWO Threshold Carter Year Reagan 0, …, 29 Anderson 1980 Reagan 30, …, 99 Carter Anderson

Empirical example: 1992 NES Candidates: B, C, P Data: thermometer scores {1, …, 100} Model: semiorders with threshold: 0 … 100 Scoring rules: Plurality, Antiplurality (with or without sharing), Borda, In-degree, Out-degree

ANES Strict Majority Social Welfare Orders Year Threshold SWO Clinton 1992 0, …, 99 Bush Perot

Empirical example: 1996 NES Candidates: C, D, P Data: thermometer scores {1, …, 100} Model: semiorders with threshold: 0 … 100 Scoring rules: Plurality, Antiplurality (with or without sharing), Borda, In-degree, Out-degree