α . Hene, the distane between two andidates

nolongerdependsontheirvaleneparameters. Thequotient

a(c)a(c ^′ )/a(c, c ^′ )

an be interpreted as a measure of orrelation between the andidates. The

more strongly the eletorate of two andidates overlap, the smaller will be

the distane between them.

Obtaining the Spatial Representation

The key for obtainingaspatial representation of the alternatives is

Multidi-mensionalSaling and PrinipalComponent Analysis. Let

y c

^be

k

^points ⁱⁿ

R ^k

^entered ^at ^zero. ^A ^priori, ^one ^an ^assume

k

^to^be ^equal ^to^the ^number

Unfortunately, this formula was misprinted in Laslier (2006). This is the orreted

version.

of available alternatives, sine

k

^points ⁱⁿ ^a ^Eulidean ^spae ^of ^a ^higher

di-mensionan alwaysbeembeddedin

R ^k

^. ^The^inertia ^of

y = (y) c∈C

^is ^dened

as the sum of the squareddistanes,

I(y) = X

c,c ^′ ∈ C

|| y c − y c ^′ || ² .

Let

y ^E

^denote^the^projetion^of^the^system ^of^points

y

^on^a^linear^subspae

E

R ^k

I (y ^E )

^is^the^inertia^explained^by

E

^and ^the^ratio

I(y ^E )/I(y)

^measures

the qualityof the representation of

y

^by

y ^E

^. ^The ^goal^is ^now^to^nd^a

lower-dimensional representation of

y

^suh ^that ^the ^explained^inertia ^is^maximal.

Let

D

^be^the

(k × k)

^symmetri^matrix^of^squared^distanes,^i.e.

D 12

^is^the

squared distane between alternative 1 and 2 alulated aording to (2.3).

The representation isbased ona matrix

Γ

^derived ^from

D

^, ^alled ^matrix ^of

inertia, whih is dened as

Γ = (D · J + J · D − J · D · J − D)/2

^(2.4)

where

J

^is^a

(k × k)

^matrix ^with ^all^entries ^equal^to ^1. ^Let

µ 1 , ..., µ k

^be ^the

eigenvalues of

Γ

^, ^ordered ^from ^largest ^to ^smallest, ^and ^let

ν 1 , ..., ν k

^be ^the

orresponding normed eigenvetors. The best projetion of the initialpoint

system into a linear,

i

-dimensional subspae

E _i

^of

R ^k

^ontains ^the ^enter,

P

c y c

^, ^and ^is ^spanned ^by ^the ^rst

i

eigenvetors. Under the speiations above,theperentageofexplainedinertiainthe

i

-dimensionalrepresentation is given by

The quality of representation for eah single point an be measured by the

osine of the projetion, whih is given by

r(c, i) = || y ^E ⁱ ||

|| y || .

^(2.5)

That is, the loser

r(c, i)

^is ^to^1, ^the ^better ^andidate

c

^isrepresented.

As mentioned above, the model leaves unspeied the poliy saliene

parameter

α

^, ^and ^needs ^to^be^alibrated ^by ^nding ^a^proper^value ^of

α

^, ⁱⁿ

the following sense. First,

α

^needs ^to ^be ^suh ^that ^the ^estimated ^distanes

are non-negative. From Equation2.3, we obtainalowerbound on

α

^. ^F^or^all

pairwise omparisonsof alternatives, it must be that 11

2.6 Appendix: Auray of the Spatial

Repre-sentations

The spatial representation relies on the eigenvetors and eigenvalues of the

matrixofinertiagivenin(2.4),whihisderivedfromtheestimateddistanes.

The positions of the parties along the rst axis of the representation are

given by

√ µ 1 ∗ ν 1

^, ^(see ^Laslier, ¹⁹⁹⁶⁾ ^where

µ 1

^is ^the ^largest ^eigenvalue

and

ν 1

^is ^the orresponding eigenvetor. Only the positive eigenvalues and their orresponding eigenvetors an be used for the spatial representation

of the parties, in the sense that the dimensionality of the representation is

onstrainedby thenumberofpositiveeigenvalues. The smallerweset

α

^,^the

morelikelyitisthatweobtainnegativeeigenvaluesandhene,themorelikely

it is that weare not able to nd agoodrepresentation inthe

k

-dimensional spae. An aurate Eulidean representation is possible for larger values of

α

^. ^However, ^inreased ^auray ^of ^the representing point system omes at a ost as the fration of explained inertia by the rst eigenvalues dereases

with larger values of

α

^. ^Hene, ^a ^reliable, low-dimensional representation requires a `proper' alibrationof

α

Party Vote, Messel

Let us start with the ase of the Party Vote in Messel. If

α = 3.69

^, ^whih

is the (rounded) lower bound derived from (2.3) in order for all estimated

Duetothemisprintmentionedpreviously,theversionofthisformulagivenin Laslier

(2006) does not allow for a omputation of the threshold for

α

^. ^This ^is ^the ^orreted

version.

dimensional representation, Messel,Party Vote.

Parties SPD CDU GRE FDP Left APP FAM REP NPD

SPD 0.00 0.21 0.01 0.40 0.03 1.72 2.97 1.32 0.29

CDU 0.21 0.00 0.16 0.03 0.09 0.62 1.40 0.74 0.03

GRE 0.01 0.16 0.00 0.33 0.01 1.67 2.81 1.18 0.22

FDP 0.40 0.03 0.33 0.00 0.21 0.39 1.02 0.44 0.00

Left 0.03 0.09 0.01 0.21 0.00 1.49 2.41 1.12 0.20

APP 1.72 0.62 1.67 0.39 1.49 0.00 0.20 0.00 0.42

FAM 2.97 1.40 2.81 1.02 2.41 0.20 0.00 0.17 1.29

REP 1.32 0.74 1.18 0.44 1.12 0.00 0.17 0.00 3.39

NPD 0.29 0.03 0.22 0.00 0.20 0.42 1.29 3.39 0.00

SPD= Soial Demorati Party, CDU= Christian Demorati Union, GRE=

Greens, FDP= Liberal Demorati Party, Left= The Left, APP= Animal Pr

ote-tionParty, FAM= FamilyParty,REP= Republians, NPD=NationalDemorati

Party.

distanes to be positive, the rst seven eigenvalues are real and positive.

The distane aording to equation (2.3) between REP and NPD is 0.0058.

Given the orresponding set of eigenvalues and vetors, the distane in the

seven-dimensional representation would be 0.11536. Although we an then

explain about 65

%

^of ^the ^inertia ^with ^a ^projetion ^on ^three dimensions, the pointsystem thatweprojetisaninaurateimageoftheoriginaldistanes.

In order to provide anaurate piture with enough explanatory power,

we hose

α = 4

^. ^The ^rst ^eight eigenvalues are then real and positive and the eight-dimensionalrepresentation seems reliable. Inorder toquantify the

auray ofthe representation,we onsider perentual dierenes asfollows.

Let

d 1

^be ^the ^distane ^between ^two ^parties ^aording ^to ^equation ^(2.3) ^and

let

d 2

^be ^the ^distane ^we ^observe ⁱⁿ ^the eight-dimensional representation.

The perentual dierene isgiven by

(d 2 − d 1 )/d 1

With our hoie of

α = 4

^and ^an eight-dimensional representation, the perentual dierene between the two distanes is never larger than 3.39%

Cosines 1 dim 2dim 3 dim 4 dim

SPD 66.83 67.45 80.09 97.44

CDU 40.59 75.58 76.25 78.98

GRE 70.42 73.28 77.65 81.43

FDP 20.47 69.81 72.63 83.91

Left 58.16 74.89 81.29 89.97

APP 15.89 29.61 73.20 73.20

FAM 2.19 30.97 53.20 64.56

REP 76.88 84.17 88.51 88.55

NPD 91.87 97.42 98.03 98.03

SPD= Soial Demorati Party, CDU= Christian Demorati Union, GRE=

Greens, FDP= Liberal Demorati Party, Left= The Left, APP= Animal Pr

ote-tionParty, FAM= FamilyParty,REP= Republians, NPD=NationalDemorati

Party.

(see Table 2.5). In fat, for most of the pairwise omparisons, it is stritly

smaller than 1%. Further, the inertia explained by the lower-dimensional

projetion that we represent is relatively high. With three dimensions, we

explain about 63% of the inertia.

The osines of the projetion omputed

aording to(2.5) indiatethat most of the partiesare well-represented (see

Table 2.6). The only exeption is the Family Party, whih would require

more dimensions.

Party Vote, Konstanz

WeproeedanalogouslyforthePartyVoteinKonstanz. Equation(2.3)

deliv-ersaboundof

α ≥ 3.71

^for^all^model^distanes^to^be^positive. ^With

α = 3.85

the rst ten eigenvalues are real and positive. The largest perentual

dier-ene between the model distane and the ten-dimensional representation is

Onedimension: 34.38%. Twodimensions: 51.20%Three dimensions: 62.89%. Four

dimensions: 73.34%.

then 3.05%, for the CDU and the Violet Party (see Table 2.7). All other

dierenes are stritly smaller, hene the overall ten-dimensional

represen-tation is an aurate piture of the model distanes. As mentioned in the

text, with three dimensionswe onlyexplain about 56% ofthe inertia.

The

osinesof projetionforthe Greens,the PirateParty,the AnimalProtetion

Party and the ödp are rather low (see Table 2.8). The Pirate Party and the

ödp require even more than ve dimensions.

Table 2.7: Perentual dierene of the distanes between the model and the

ten-dimensional representation, Konstanz, Party Vote.

Parties GRE SPD CDU FDP PIR Left APP ödp VIO REP NPD

GRE 0.00 0.20 0.30 0.21 0.45 0.12 0.61 1.06 2.01 0.51 1.03

SPD 0.20 0.00 0.02 0.00 1.16 0.60 1.44 2.02 2.97 1.13 1.78

CDU 0.30 0.02 0.00 0.01 1.29 0.57 1.64 2.37 3.05 2.15 2.41

FDP 0.21 0.00 0.01 0.00 1.20 0.49 1.45 1.95 2.49 1.83 2.53

PIR 0.45 1.16 1.29 1.20 0.00 0.12 0.01 0.12 0.57 0.04 0.27

Left 0.12 0.60 0.57 0.49 0.12 0.00 0.21 0.44 1.23 0.29 0.80

APP 0.61 1.44 1.64 1.45 0.01 0.21 0.00 0.07 0.48 0.01 0.13

ödp 1.06 2.02 2.37 1.95 0.12 0.44 0.07 0.00 0.20 0.02 0.02

VIO 2.01 2.97 3.05 2.49 0.57 1.23 0.48 0.20 0.00 0.37 0.09

REP 0.51 1.13 2.15 1.83 0.04 0.29 0.01 0.02 0.37 0.00 1.48

NPD 1.03 1.78 2.41 2.53 0.27 0.80 0.13 0.02 0.09 1.48 0.00

GRE= Greens, SPD= Soial Demorati Party, CDU= Christian Demorati

Union, FDP= Liberal Demorati Party, PIR= The Pirates, Left= The Left,

APP= Animal Protetion Party, ödp= Eologi Demorati Party, VIO=The

Vi-olets, REP= Republians, NPD=National Demorati Party.

Candidate Vote, Messel

Foralldistanestobepositivethe poliysalieneparameterhastobelarger

thanorequalto3.84. AnalogouslytothedisussionforPartyVoteinMessel,

Onedimension: 23.58%. Twodimensions: 43.95%Three dimensions: 56.23%. Four

dimensions: 65.67%.

Cosines 1 dim 2dim 3 dim 4 dim

GRE 55.48 61.70 61.88 85.83

SPD 66.18 68.97 73.07 93.77

CDU 29.49 79.14 81.63 81.97

FDP 23.14 77.54 80.71 81.63

PIR 1.39 19.78 41.87 56.78

Left 9.21 55.21 75.42 76.96

APP 17.08 30.28 60.50 60.89

ödp 2.77 18.66 69.74 69.98

VIO 41.15 63.01 74.79 74.82

REP 87.36 94.47 95.20 96.03

NPD 94.13 94.40 95.65 95.68

GRE= Greens, SPD= Soial Demorati Party, CDU= Christian Demorati

Union, FDP= Liberal Demorati Party, PIR= The Pirates, Left= The Left,

APP= Animal Protetion Party, ödp= Eologi Demorati Party, VIO=The

Vi-olets, REP= Republians, NPD=National Demorati Party.

weareledtoadoptahighervalue. With

α = 4.45

^,^the^rst^seven eigenvalues of the analysis are real and positive. The piture of the seven-dimensional

representationisanaurateimageofthedistanes alulatedbythe model.

Thediereneformostof theomparisonsisbelow1% (seeTable2.9). With

three dimensions, we explain about 69% of the inertia.

Further, all eight

andidates are well-represented with only three dimensions, as indiated by

the osines of the projetion (see Table 2.10).

Candidate Vote, Konstanz

The threshold valuefor

α

^suh ^that ^all^model^distanes ^are ^positive^is ^equal

to 2.55. With

α = 3.35

^, ^the ^rst ^four eigenvalues are real and positive.

As indiated by Table 2.11, inthis ase four dimensions are already enough

Onedimension: 32.43%. Twodimensions: 54.18%Three dimensions: 68.76%. Four

dimensions: 80.59%.

dimensional representation, Messel,Candidate Vote.

Parties SPD CDU GRE FDP Left FRV REP NPD

SPD 0.00 0.11 0.01 0.23 0.45 1.42 2.44 0.48

CDU 0.11 0.00 0.07 0.03 0.05 0.70 2.07 0.26

GRE 0.01 0.07 0.00 0.18 0.37 1.39 2.20 0.39

FDP 0.23 0.03 0.18 0.00 0.01 0.54 1.60 0.10

Left 0.45 0.05 0.37 0.01 0.00 0.41 1.31 0.03

FRV 1.42 0.70 1.39 0.54 0.41 0.00 0.40 0.13

REP 2.44 2.07 2.20 1.60 1.31 0.40 0.00 4.43

NPD 0.48 0.26 0.39 0.10 0.03 0.13 4.43 0.00

SPD= Soial Demorati Party, CDU= Christian Demorati Union, GRE=

Greens, FDP= Liberal Demorati Party, Left= The Left, FRV= Free Voters,

REP= Republians, NPD= NationalDemorati Party.

to nd aurate representation of the modeldistanes. Further, with three

dimensions,weobtainanexplainedinertiaof91%.

Allandidatesare

well-represented with only three dimensions, as indiated by the osines of the

projetion (see Table 2.12).

Onedimension: 44.34%. Twodimensions: 77.74%Three dimensions: 90.75%. Four

dimensions: 100.00%.

Table 2.10: Cosinesof projetions, Messel,Candidates.

Cosines 1 dim 2dim 3 dim 4 dim

SPD 65.78 66.89 84.09 95.06

CDU 46.44 70.77 76.08 76.91

GRE 71.72 72.61 72.82 96.24

FDP 23.25 77.79 80.33 81.57

Left 44.03 81.82 81.94 81.94

FRV 18.49 18.76 87.28 97.51

REP 70.91 87.13 88.72 88.83

NPD 79.60 90.08 95.47 97.89

SPD= Soial Demorati Party, CDU= Christian Demorati Union, GRE=

Greens, FDP= Liberal Demorati Party, Left= The Left, FRV= Free Voters,

REP= Republians, NPD= NationalDemorati Party.

Figure2.3: Three-dimensionalprojetion oftheseven-dimensionalandidate

rep-resentation, Messel.

(a) Axes1and 2 (b) Axes1and 3

Figure 2.4: Three-dimensional projetion of the four-dimensional andidate

rep-resentation, Konstanz.

(a) Axes1and 2 (b) Axes1and 3

Table 2.11: Perentual dierenes of the distanes between the model and the

four-dimensional representation, Konstanz, Candidate Vote.

Parties GRE SPD CDU FDP Left NPD

GRE 0.00 0.37 2.05 0.22 0.70 0.33

SPD 0.37 0.00 1.02 0.30 0.66 0.04

CDU 2.05 1.02 0.00 2.82 0.97 0.89

FDP 0.22 0.30 2.82 0.00 0.02 0.92

Left 0.70 0.66 0.97 0.02 0.00 2.06

NPD 0.33 0.04 0.89 0.92 2.06 0.00

GRE= Greens, SPD= Soial Demorati Party, CDU= Christian Demorati

Union, FDP= Liberal Demorati Party, Left= The Left, NPD=National

Demo-rati Party.

Table 2.12: Cosinesof projetions, Konstanz,Candidates.

Cosines 1 dim 2 dim 3 dim 4dim

GRE 32.48 64.46 97.41 100.00

SPD 8.78 82.11 99.20 100.00

CDU 89.73 90.56 91.05 100.00

FDP 72.11 88.07 88.59 100.00

Left 95.86 96.10 96.54 100.00

NPD 39.08 99.65 99.76 100.00

GRE= Greens, SPD= Soial Demorati Party, CDU= Christian Demorati

Union, FDP= Liberal Demorati Party, Left= The Left, NPD=National

Demo-rati Party.

Preferene Aggregation Under

Unertainty

3.1 Introdution

Attheveryheartofeverydividedmajorityproblemliesaoordination

prob-lem. Anabsolutemajorityofvotersisuniedintheirdislikeofonepartiular

andidate they jointly onsider to be inferior. However, their support is

di-vided between two(or more)otherandidates. Theyhavetoombineeorts

and onentrate their supportbehind one andidate. Otherwise, their worst

option, baked up by a weak majority him/herself, will emerge as the

win-ner of the eletion. The reognition of these kindof problems follows a long

tradition in the study of eletoral systems and an at least be dated bak

to de Borda (1781). He used a divided majority problem to illustrate the

drawbaks of the voting method the Frenh Aademy of Sienes employed

for the seletion proess of new board members at that time. The method

was Plurality Voting (or PV) and de Bordaexpressed his onerns that the

method an selet new board members a majority onsiders to be inferior,

even if everybody votes sinerely (inaordane with their preferene).

Inthe meantime, real world instanes ofdivided majorityproblems have

been observed, espeially in eletions based on or involving PV. `One man,

one vote' systems make it partiularly easy for suh `paradoxial' results

to appear. One infamous example ourred during the Frenh presidential

eletion in2002, where right-extremistJean-Marie LePen, the leader of the

extreme Right party Front National, took seond plae in the rst round of

the eletion although 82% of the voters in the seond round voted against

him. Le-Pen did not win, but him entering the seond round aused

in-ternational mass media overage shedding bad light onthe Frenh politial

landsape.

Another example is the 1998 governor eletion in Minnesota

where former professional wrestler Jesse `The Body' Ventura won although

64% of the voters preferred any of the twoother majorandidates.

The Frenh president is eleted via a plurality vote run-o system. If a andidate

reeivesanabsolutemajorityofvotesintherstround,he/sheiseleted. Ifnoandidate

is ableto reeivean absolutemajorityof votes, the twoandidateswith the mostvotes

meetin aseond round. Inthe rstround, JaquesChirareeived19.88%of thevotes,

Jean Marie Le Pen 16.86%, and Linoel Jospin 16.18%. Le Pen, together with Chira

enteredtheruno. Chirawontheseondround withmorethan82%ofthevotes.

The very fat that a andidate an win an eletion despite being

on-sidered inferior by an absolutemajority poses a major and grave threat. It

an be argued that suh results ompromise the wishes of the voters. They

derail the demorati proess by undermining the legitimay of appointed

leaders. The ulprit inase of PV eletionsis easilyidentied. The method

respets alleged maxima only and disregards any other type of preferene

information. Any eletion in whih the ombined vote share of andidates

promoting similar poliies exeeds that of the winner raises suspiion and

doubt about PV aurately reeting voters' wishes. Eetive attempts in

replaing the `standard' eletoralsystem must addressthis ore diulty.

Multi-votesmethodspossessthepotentialtosuessfullyreplaePV.

Ap-proval Voting or AV and the Borda Count or BC an be onsidered to be

among the most promisingof suh methods both in terms of sienti work

devoted totheir study and the many positive(desirable) properties they

ex-hibit. Both methods give voters additional opportunities to express their

wishes and desires. AV wasrst analytially desribed by Brams and

Fish-burn (1978). It merely requires voters to reveal whih andidates they nd

`aeptable', i.e.,eah voter onlyneeds tomarkthe namesof the andidates

he approves of. Arguments have been put forward inthe literature that the

methodprovides anaurate reetion of voters' wishes and is not

vulnera-bletovotermanipulation(seeBramsandFishburn,1978;Fishburn,1978a,b;

Brams andFishburn,2005; Wolitzky,2009). In ontrast, BC requiresvoters

toprovideaompleterankingof andidates. Therst mathematialtreatise

on BC dates bak to de Borda (1781), however, the history of the method

an be traed bak to as early as the 15th entury in the work of Niolas

Cusanus.

Asshown bySaari(1994),BCminimizestheamountofparadoxes

withinthe set ofpositionalvotingmethods(inludingAV and PV).Further,

the BC-winner(thealternativewinningBCeletionswhenvotersusesinere

strategies) is the alternativeranked highestonaverage in the eletorate.

Reformresolution proposing a hange in the eletoral system in favor of

AV and BC must take intoaount that both methods have identied

de-2

Seee.g.MLean(1990)foranearlyhistorialaountofBordaandCondoret

prin-iples.

ienies too. The main theoretial ritiques against AV in the literature is

the multipliity of potential outomes for the same set of preferenes (see

Saari and Van Newenhizen, 1988b,a). AV and BC respet neither the

Con-doret nor the majority riterion: Condoret-Winners and alternatives that

are preferredto any otheralternativeby anabsolutemajority donot

nees-sarily win.

In terms of strategi inentives, BC is often ritiized as being

easy to manipulate and vulnerable to strategi misrepresentation of

prefer-enes. It is relatively easy for voters to prevent an undesirable outome by

insinerely raising (`ompromising') or lowering (`burying') the positions of

alternatives. Eletoralsystemreformsmustguaranteethatthesystemavoids

aws of the past system and that it does not introdue new problems of a

dierent kind.

In order to fully understand the impliations of suh alternative

meth-odsfor soiety,it is essentialto olletempirial data ontheir performane.

Field experiments have been a rst step in this diretion providing us with

invaluabledata and with lear evidene ontheir atual feasibility(see

Alós-Ferrer and Grani¢,2012b;Baujardand Igersheim,2009;Laslierand Vander

Straeten, 2008). Thereare,however, anumberoflimitationsassoiatedwith

eld experiments. First, due to legal onerns (ensuring voter anonymity in

atual eletions) aswell asmethodologialonsiderations, one isnot able to

fully identify the partiipants' atual preferenes. Seond, self-seletion

bi-ases may our, whihannot fully be aounted for. Hene, without

know-ing either the preferenes of the partiipants or whether the samples are

truly unbiased, ertain properties of voting methods annot be tested. As

a omplement to eld experimentation, the ontrolled laboratory

environ-ment typial of experimental eonomis allows usto indue any preferenes

overandidatesby modelingthe partiipants'expetedpayoonditionalon

the eletionoutomes. Hene, we an study theoretial properties of voting

methodsin the laboratory that annotbe tested inthe eld.

This paper reports on the results of series of experimental (laboratory)

4-alternative eletions. The experimental design is based on Forsythe,

My-3

ACondoret-Winner(Loser)isanalternativeableto beat(isbeatenby)everyother

alternativeinapairwiseomparison.

erson, Rietz, and Weber (1993) and Forsythe, Rietz, Myerson, and Weber

(1996) andthe indued preferene proleorresponds toanadapted divided

majority problem. We assess and ompare the Condoret-Eieny (how

often the Condoret-Winner is eleted) of three voting rules: AV, BC, and

PV. In addition,we independently vary the underlyinginformational

stru-ture fromafull-informationframeworkwherepartiipantsarefullyinformed

about the preferene struture in the whole eletorate to an

inomplete-information framework where partiipants only know their own payos and

the eletion histories and analyze its impat on the Condoret-Eieny of

the voting methods.

One feature of our design is the interation of the

partiipantsoverseveral roundsexpliitlyenablingthemtoaquire

familiar-ity with the voting method and allowing for learningeets. This repeated

environment iswell suited for analyzing the partiipants' adaptivebehavior

and the relevane of strategi onsiderations as it almost begs for strategi

behavior onpart of the voters.

We ontribute to the existing literature in the following ways. First, we

provide evidenethat multi-votes systems inrease the frequeny of

suess-ful oordination in divided majority settings in omparison to PV. A

de-parture from the full-informationframework to a more realisti

inomplete-informationframeworkaninreasethesuperiorityofthesemethodsoverPV.

Information imperfetions, in our ase inomplete-information,in

ombina-tion with the relatively low information provided by a PV ballot message,

an hange eletion results dramatially. Coordination on the

Condoret-Winner breaks down and almost never ours with inomplete-information

under PV.At the sametime, the underlyinginformationstruture has little

to no eet on the eletion outomes generated under AV and BC. Our

re-sults suggest that, in general, the responsiveness towards the informational

struture may serve as another important dimension to evaluate the merits

of voting methods.

Seond, the analysis of the individual voting behavior reveals interesting

For atheoretial aount on how information imperfetions an inuene the

Con-doret Eieny of voting methods see, e.g., Laslier (2009); Bouton and Castanheira

(2012).

patterns. In line with, e.g. Ballester and Rey-Biel (2009), we observe that

unertainty redues the amountofstrategivotingand inreases sinerity to

a great extent under PV. Interestingly, this is also true for AV, but not for

BC. We have lear evidene that, althoughAV and BC are roughly equally

Im Dokument Essays on Approval Voting and Choice-Induced Attitude Change (Seite 87-158)

α . Hene, the distane between two andidates

a(c)a(c ′ )/a(c, c ′ )

y c

k

R k

k

k

R k

y = (y) c∈C

I(y) = X

c,c ′ ∈ C

|| y c − y c ′ || 2 .

y E

y

E

R k

I (y E )

E

I(y E )/I(y)

y

y E

y

D

(k × k)

D 12

Γ

D

Γ = (D · J + J · D − J · D · J − D)/2

J

(k × k)

µ 1 , ..., µ k

Γ

ν 1 , ..., ν k

i

E i

R k

P

c y c

i

i

r(c, i) = || y E i ||

|| y || .

r(c, i)

c

α

α

α

α

√ µ 1 ∗ ν 1

µ 1

ν 1

α

k

α

α

α

α = 3.69

α

%

α = 4

d 1

d 2

(d 2 − d 1 )/d 1

α = 4

α ≥ 3.71

α = 3.85

α = 4.45

α

α = 3.35

a(c)a(c ^′ )/a(c, c ^′ )

R ^k

R ^k

c,c ^′ ∈ C

|| y c − y c ^′ || ² .

y ^E

R ^k

I (y ^E )

I(y ^E )/I(y)

y ^E

E _i

R ^k

r(c, i) = || y ^E ⁱ ||