Probabilities of electoral outcomes: from three-candidate to four-candidate elections

Abstract

The main purpose of this paper is to compute the theoretical likelihood of some electoral outcomes under the impartial anonymous culture in four-candidate elections by using the last versions of software like LattE or Normaliz. By comparison with the three-candidate case, our results allow to analyze the impact of the number of candidates on the occurrence of these voting outcomes.

Appendix: Computation of \( \Pr \left( {E_{3} ,\infty } \right) \)

Let $\left( {E_{3} , n} \right)$ be the set of all voting situations, of size $n$, in which $E_{3}$ occurs. Since $a$ must be first or second in the first stage of the sincere vote, by symmetry, we can write:

$$ \left| {\left( {E_{3} , n} \right)} \right| = 3\left( {\left| {\left( {G,n} \right)} \right| + \left| {\left( {H,n} \right)} \right|} \right), $$

(11)

where $\left( {G,n} \right)$ is the set of the voting situations in $\left( {E_{3} , n} \right)$ for which $a$ is first and $b$ is second in the first stage, and $\left( {H,n} \right)$ the set of the voting situations in $\left( {E_{3} , n} \right)$ for which $a$ is second and $b$ is first in the first stage. Considering all possibilities for the choice of the candidate who wins after manipulation and the candidate who goes with him to the second stage, we obtain:

$$ \left( {G,n} \right) = \left( {G^{bc} ,n} \right) \cup \left( {G^{cb} ,n} \right) \cup \left( {G^{bd} ,n} \right) \cup \left( {G^{db} ,n} \right) \cup \left( {G^{cd} ,n} \right) \cup \left( {G^{dc} ,n} \right) $$

(12)

$$ \left( {H,n} \right) = \left( {H^{bc} ,n} \right) \cup \left( {H^{cb} ,n} \right) \cup \left( {H^{bd} ,n} \right) \cup \left( {H^{db} ,n} \right) \cup \left( {H^{cd} ,n} \right) \cup \left( {H^{dc} ,n} \right). $$

(13)

Here, for $\alpha$, $\beta$ in $\left\{ {b,c,d} \right\}$ and $\alpha \ne \beta$, the notation $\left( {G^{\alpha \beta } ,n} \right)$ (resp. $\left( {H^{\alpha \beta } ,n} \right)$) denotes the subset of $\left( {G,n} \right)$ (resp. $\left( {H,n} \right)$) of voting situations where $\alpha$ and $\beta$ go to the second stage after manipulation (in favor of $\alpha$) and $\alpha$ beats $\beta$ by a majority of votes. For simplicity, in what follows, these subsets will be denoted by $G^{\alpha \beta }$ (resp. $H^{\alpha \beta }$).

Using Proposition 2 and deleting the redundant inequalities, it follows that the voting situations $x$ in $G^{bc}$, $G^{cb},$ and $G^{cd}$ are characterized by the following parametric linear systems:

$$ \begin{array}{*{20}c} {(S_{n}^{bc} )\, \left\{ \begin{aligned} n_{1} + \ldots + n_{24} = n \hfill \\ n_{1} + \cdots + n_{24} = n \hfill \\ P_{x} \left( {a, b} \right) - P_{x} \left( {b, a} \right) > 0 \hfill \\ P_{x} \left( {a, c} \right) - P_{x} \left( {c, a} \right) > 0 \hfill \\ P_{x} \left( {a, d} \right) - P_{x} \left( {d, a} \right) > 0 \hfill \\ F_{x} \left( a \right) - F_{x} \left( b \right) > 0 \hfill \\ F_{x} \left( b \right) - F_{x} \left( c \right) > 0 \hfill \\ F_{x} \left( b \right) - F_{x} \left( d \right) > 0 \hfill \\ P_{x} \left( {b,a} \right) - F_{x} \left( a \right) > 0 \hfill \\ P_{x} \left( {b,a} \right) + F_{x}^{ab} \left( c \right) - 2F_{x} \left( a \right) > 0 \hfill \\ P_{x} \left( {b,c} \right) - P_{x} \left( {c,b} \right) > 0 \hfill \\ \end{aligned} \right.} & {(S_{n}^{cb} )\, \left\{ \begin{aligned} n_{1} + \cdots + n_{24} = n \hfill \\ n_{i} \ge 0, i = 1, \ldots , 24 \hfill \\ P_{x} \left( {a, b} \right) - P_{x} \left( {b, a} \right) > 0 \hfill \\ P_{x} \left( {a, c} \right) - P_{x} \left( {c, a} \right) > 0 \hfill \\ P_{x} \left( {a, d} \right) - P_{x} \left( {d, a} \right) > 0 \hfill \\ F_{x} \left( a \right) - F_{x} \left( b \right) > 0 \hfill \\ F_{x} \left( b \right) - F_{x} \left( c \right) > 0 \hfill \\ F_{x} \left( b \right) - F_{x} \left( d \right) > 0 \hfill \\ P_{x} \left( {c,a} \right) - F_{x} \left( a \right) > 0 \hfill \\ P_{x} \left( {c,a} \right) + F_{x}^{ac} \left( b \right) - 2F_{x} \left( a \right) > 0 \hfill \\ P_{x} \left( {c,b} \right) - P_{x} \left( {b,{\text{c}}} \right) > 0 \hfill \\ \end{aligned} \right.} & {(S_{n}^{cd} ) \, \left\{ \begin{aligned} n_{1} + \cdots + n_{24} = n \hfill \\ n_{i} \ge 0, i = 1, \ldots , 24 \hfill \\ P_{x} \left( {a, b} \right) - P_{x} \left( {b, a} \right) > 0P_{x} \left( {a, b} \right) - P_{x} \left( {b, a} \right) > 0 \hfill \\ P_{x} \left( {a, c} \right) - P_{x} \left( {c, a} \right) > 0 \hfill \\ P_{x} \left( {a, d} \right) - P_{x} \left( {d, a} \right) > 0 \hfill \\ F_{x} \left( a \right) - F_{x} \left( b \right) > 0 \hfill \\ F_{x} \left( b \right) - F_{x} \left( c \right) > 0 \hfill \\ F_{x} \left( b \right) - F_{x} \left( d \right) > 0 \hfill \\ P_{x} \left( {c,a} \right) - F_{x} \left( a \right) > 0 \hfill \\ P_{x} \left( {c,a} \right) + F_{x}^{ac} \left( d \right) - 2F_{x} \left( a \right) > 0 \hfill \\ P_{x} \left( {c,d} \right) - P_{x} \left( {d,{\text{c}}} \right) > 0 \hfill \\ \end{aligned} \right.} \\ \end{array}. $$

By symmetry between candidates $c$ and $d$, the systems characterizing $G^{bd}$, $G^{db},$ and $G^{dc}$ are obtained by permuting $c$ and $d$ in $S_{n}^{bc} , S_{n}^{cb} {\text{and }}S_{n}^{cd},$ respectively; therefore, we have $\left| {G^{bd} } \right| = \left| {G^{bc} } \right|$, $\left| {G^{db} } \right| = \left| {G^{cb} } \right|,$ and $\left| {G^{dc} } \right| = \left| {G^{cd} } \right|$.

Now, we use (12) and we apply the inclusion–exclusion principle to calculate $\left| {\left( {G,n} \right)} \right|$. For the 15 pairwise intersections, it is obvious that $G^{bc} \cap G^{cb}$, $G^{bd} \cap G^{db},$ and $G^{cd} \cap G^{dc}$ are empty, and that by symmetry, we have $\left| {G^{cb} \cap G^{bd} } \right| = \left| {G^{bc} \cap G^{db} } \right|$, $\left| {G^{bd} \cap G^{dc} } \right| = \left| {G^{bc} \cap G^{cd} } \right|$, $\left| {G^{bd} \cap G^{cd} } \right| = \left| {G^{bc} \cap G^{dc} } \right|$, $\left| {G^{db} \cap G^{dc} } \right| = \left| {G^{cb} \cap G^{cd} } \right|$, and $\left| {G^{db} \cap G^{cd} } \right| = \left| {G^{cb} \cap G^{dc} } \right|$. Of the 20 triple intersections, the only ones that are (possibly) non-empty are the 8 that are obtained by choosing one and only one element in each of the three sets $\left\{ {G^{bc} , G^{cb} } \right\}$, $\left\{ {G^{bd} , G^{db} } \right\}$ and $\left\{ {G^{cd} , G^{dc} } \right\}$; and by symmetry we have $\left| {G^{bc} \cap G^{bd} \cap G^{cd} } \right| = \left| {G^{bc} \cap G^{bd} \cap G^{dc} } \right|$, $\left| {G^{bc} \cap G^{db} \cap G^{cd} } \right| = \left| {G^{cb} \cap G^{bd} \cap G^{dc} } \right|$, $\left| {G^{cb} \cap G^{bd} \cap G^{cd} } \right| = \left| {G^{bc} \cap G^{db} \cap G^{dc} } \right|,$ and $\left| {G^{cb} \cap G^{db} \cap G^{cd} } \right| = \left| {G^{cb} \cap G^{db} \cap G^{dc} } \right|$. Finally, all intersections of 4, 5, or 6 subsets $G^{\alpha \beta }$ ($\alpha$, $\beta$ in $\left\{ {b,c,d} \right\}$ and $\alpha \ne \beta$) are empty, because each of them is included in (at least) one of the three empty intersections, $G^{bc} \cap G^{cb}$, $G^{bd} \cap G^{db},$ and $G^{cd} \cap G^{dc}$ (to form an intersection of 4, 5 or 6 subsets $G^{\alpha \beta }$, it is necessary to choose the two elements of at least one of the sets $\left\{ {G^{bc} , G^{cb} } \right\}$, $\left\{ {G^{bd} , G^{db} } \right\}$ and $\left\{ {G^{cd} , G^{dc} } \right\}$).

We can now write the formula giving the cardinality of $\left( {G,n} \right){:}$

$$ \begin{aligned} \left| {\left( {G,n} \right)} \right| = 2\left( {\left| {G^{bc} } \right| + \left| {G^{cb} } \right| + \left| {G^{cd} } \right|} \right) - \left( {\left| {G^{bc} \cap G^{bd} } \right| + \left| {G^{cb} \cap G^{db} } \right|} \right) + 2\left| {G^{bc} \cap G^{db} } \right| + 2\left| {G^{bc} \cap G^{cd} } \right| + 2\left| {G^{cb} \cap G^{cd} } \right| + 2\left| {G^{cb} \cap G^{dc} } \right|) + 2\left| {G^{bc} \cap G^{dc} } \right| \hfill \\ + 2(\left| {G^{bc} \cap G^{bd} \cap G^{cd} } \right| + \left| {G^{bc} \cap G^{db} \cap G^{dc} } \right| + \left| {G^{cb} \cap G^{db} \cap G^{dc} } \right| + \left| {G^{cb} \cap G^{bd} \cap G^{dc} } \right| \hfill \\ \end{aligned}. $$

(14)

To obtain ${ Pr }\left( {G,\infty } \right)$, we replace each cardinality that appears in the second member of (14) by the volume of the associated polytope (for example, the polytope associated with $G^{bc}$ is the one described by the system $S_{1}^{bc}$), and then, we divide by the volume associated with the total number of voting situations (i.e., by$1/23!$). Using the method based on LattE and Lrs (and Normaliz for the triple intersections), we get the following results:

	Volume of the associated polytope
$G^{bc}$	21579965102214833661822341895472578264296147661301820748996857636891534517195601│1696627485451304216801706693769459989090225696942760631389397188608000000000000000
$G^{cb}$	5573287256158164544822279100674154863906571658254932679572826363232135917│5381633372644158916218281473636071209286170854780673282015232000000000000000
$G^{cd}$	690763588675926208892200201275635242424317217116771810711456400097168422586192409│1085841590688834698753092284012454393017744446043366804089214200709120000000000000000
$G^{bc} \cap G^{bd}$	4374958709307366498551275282550829945168341340158005803124163906777355886621592418407│15783821593121799418221188750549477369596125747406423643972960256000000000000000000000000
$G^{cb} \cap G^{db}$	371668307604755922004052912809417026311491652333420360381433590516107912320558408681833214907876615097308537557287│149865786232755160714603713492567144588895472965955187951797051990988320128219435626412335975849000960000000000000000
$G^{bc} \cap G^{db}$	73728795097691910608032675802979649504177582112609999559155361106901059950083499782664269443828800493654344962886188347693│319769506827021125451226886147064465003853770355663104931071786212847273139053736326218431673677463018274816000000000000000000
$G^{bc} \cap G^{cd}$	1128678697638588406020837480381418976300873421867855410386970684729753714450967582951899081133195953970779591091254147647624909│5730269562340218568085985799755395212869059564773482840364806408934223134651842954965834295592300137287484702720000000000000000000
$G^{bc} \cap G^{dc}$	1791208843746302469037817356603077744950568899273425055696702056165779778059081619821156647740788159470371527351823357│7196555054896902817515270321913074283158760611825168125445294436607259132557097298780320373560269026099200000000000000000
$G^{cb} \cap G^{cd}$	156090914075338229952814649806280933859900502452413621807273601730707932075993177383819│104173222514603876160259845753626550639334429932882396050221537689600000000000000000000000
$G^{cb} \cap G^{dc}$	16987196522687599537363832144822196085155735807228425387516053556339294374740663419980423405997963504376939242487320875885987│116944276782453440165020118362355004344266521730071078374791967529269859890853937856445597869230615046683361280000000000000000000
$G^{bc} \cap G^{bd} \cap G^{cd}$	674324033423332239489570724345486219235182962634052751321547924919297782590615439333218137279830648518311866993975500318502409742149903│800337852832421947317129795594046017972207282304842286498696861526387919674178698272333760160140951092350184062976000000000000000000000000
$G^{bc} \cap G^{db} \cap G^{dc}$	66224084947732991948095057625409755209795263748146831963545633222617962161248153298557211890052296338881958196326711071188575733861237│1026074170297976855534781789224877694611821446449338754679320110452101015342843422855427405148736019370814126161920000000000000000000000000
$G^{cb} \cap G^{db} \cap G^{dc}$	172499040886591309105147664280974842140465399609634373368829431585429271796091241844107326681316968683271609632632217239661825101449│3374674704134010572259781563482056846842725283481549285924564370689149907098236927927278529330469282801631887360000000000000000000000000
$G^{cb} \cap G^{bd} \cap G^{dc}$	6433925339577046922056349489616669597071348937361564870673006196214723723│147708854420427693761988302384320276840921234854209315143680000000000000000000

After calculation, we obtain:

$$ { Pr }\left( {G,\infty } \right) = \frac{52683297709949532142119507583496364663740732115091118336072352908066132417}{13487448260227442240858216769041163302467817484765364224000000000000000000000}. $$

To compute ${ Pr }\left( {H, \infty } \right)$, we use (13) and we proceed in exactly the same way as for ${ Pr }\left( {G,\infty } \right)$. Using Proposition 2 and deleting the redundant inequalities, we obtain the systems characterizing the voting situations $x$ in $H^{bc}$, $H^{cb},$ and $H^{cd}$:

$$ \begin{array}{*{20}c} {(T_{n}^{bc} )\,\left\{ \begin{aligned} n_{1} + \vdots + n_{24} = n \hfill \\ n_{i} \ge 0, i = 1, \ldots , 24 \hfill \\ P_{x} \left( {a, b} \right) - P_{x} \left( {b, a} \right) > 0 \hfill \\ P_{x} \left( {a, c} \right) - P_{x} \left( {c, a} \right) > 0 \hfill \\ P_{x} \left( {a, d} \right) - P_{x} \left( {d, a} \right) > 0 \hfill \\ F_{x} \left( b \right) - F_{x} \left( a \right) > 0 \hfill \\ F_{x} \left( a \right) - F_{x} \left( c \right) > 0 \hfill \\ F_{x} \left( a \right) - F_{x} \left( d \right) > 0 \hfill \\ P_{x} \left( {b,a} \right) - F_{x} \left( a \right) > 0 \hfill \\ P_{x} \left( {b,a} \right) + F_{x}^{ab} \left( c \right) - 2F_{x} \left( a \right) > 0 \hfill \\ P_{x} \left( {b,c} \right) - P_{x} \left( {c,b} \right) > 0 \hfill \\ \end{aligned} \right.} & {(T_{n}^{cb} )\,\left\{ \begin{aligned} n_{1} + \cdots + n_{24} = n \hfill \\ n_{i} \ge 0, i = 1, \ldots , 24 \hfill \\ P_{x} \left( {a, b} \right) - P_{x} \left( {b, a} \right) > 0 \hfill \\ P_{x} \left( {a, c} \right) - P_{x} \left( {c, a} \right) > 0 \hfill \\ P_{x} \left( {a, d} \right) - P_{x} \left( {d, a} \right) > 0 \hfill \\ F_{x} \left( b \right) - F_{x} \left( a \right) > 0 \hfill \\ F_{x} \left( a \right) - F_{x} \left( c \right) > 0 \hfill \\ F_{x} \left( a \right) - F_{x} \left( d \right) > 0 \hfill \\ P_{x} \left( {c,a} \right) - F_{x} \left( a \right) > 0 \hfill \\ P_{x} \left( {c,a} \right) + F_{x}^{ac} \left( b \right) - 2F_{x} \left( a \right) > 0 \hfill \\ P_{x} \left( {c,b} \right) - P_{x} \left( {b,{\text{c}}} \right) > 0 \hfill \\ \end{aligned} \right.} & {(T_{n}^{cd} )\,\left\{ \begin{aligned} n_{1} + \cdots + n_{24} = n \hfill \\ n_{i} \ge 0, i = 1, \ldots , 24 \hfill \\ P_{x} \left( {a, b} \right) - P_{x} \left( {b, a} \right) > 0 \hfill \\ P_{x} \left( {a, c} \right) - P_{x} \left( {c, a} \right) > 0 \hfill \\ P_{x} \left( {a, d} \right) - P_{x} \left( {d, a} \right) > 0 \hfill \\ F_{x} \left( b \right) - F_{x} \left( a \right) > 0 \hfill \\ F_{x} \left( a \right) - F_{x} \left( c \right) > 0 \hfill \\ F_{x} \left( a \right) - F_{x} \left( d \right) > 0 \hfill \\ P_{x} \left( {c,a} \right) - F_{x} \left( a \right) > 0 \hfill \\ P_{x} \left( {c,a} \right) - F_{x} \left( b \right) > 0 \hfill \\ P_{x} \left( {c,a} \right) + F_{x}^{ac} \left( d \right) - 2F_{x} \left( a \right) > 0 \hfill \\ P_{x} \left( {c,a} \right) + F_{x}^{ac} \left( d \right) - 2F_{x} \left( b \right) > 0 \hfill \\ P_{x} \left( {c,d} \right) - P_{x} \left( {d,{\text{c}}} \right) > 0 \hfill \\ \end{aligned} \right.} \\ \end{array}. $$

Here again, $c$ and $d$ being symmetrical, we obtain the same symmetries as before. Therefore, the formula describing the cardinality of $\left( {H,n} \right)$ is exactly the same as (14), except that $\left| {\left( {G,n} \right)} \right|$ is replaced with $\left| {\left( {H,n} \right)} \right|$ and each $G^{\alpha \beta }$ ($\alpha$, $\beta$ in $\left\{ {b,c,d} \right\}$ and $\alpha \ne \beta$) is replaced with $H^{\alpha \beta }$. By applying the method based on LattE and Lrs (and Normaliz for the triple intersections), we calculate the volumes of all the polytopes associated with the cardinalities involved in the second member of this formula. We then get:

	Volume of the associated polytope
$H^{bc}$	1122570228285484416840414329038859444597│22032517669125135387328512000000000000000
$H^{cb}$	4815102613831086008681845331774767357│2754064708640641923416064000000000000000
$H^{cd}$	4859814977353382934768278656966765458531281829079037659495505447570096549131│2636152533698047281461986677988445611214432454458846070397992960000000000000000
$H^{bc} \cap H^{bd}$	6833482604299574922319228350257359109218343485126709565221│228793803992037731981659487629131907072000000000000000000000
$H^{cb} \cap H^{db}$	5049881685008345434475345496190114508046016393│10408519742515465059176289612595200000000000000000
$H^{bc} \cap H^{db}$	675827524940449071804022414564519275245666212472817163│831977469061955389811512540956047966208000000000000000000
$H^{bc} \cap H^{cd}$	54527549003500598877896693999359561653981068895014160097048506522520901040183861295097633826231569261│4610787019086802067117459185013926303116690035637510458825406244812151254034364026060800000000000000000
$H^{bc} \cap H^{dc}$	8308091188337554623944084932295328721407928052921637390327492939829616693208962854674889│6253787586569434428002409935673694836855808351567447350530718940594490572800000000000000000
$H^{cb} \cap H^{cd}$	536467444833359046580624594631977457827077563033371231007512557540128954421668859753948816150778793│1129430486744758491847310206009682124024272495502035679704440095241071735752097792000000000000000000000
$H^{cb} \cap H^{dc}$	8829802817815659461561114981596342439514389717111976228094675237975903263366333445667│24510239414342286607887164160978619779956136984391328044408069530058752000000000000000000
$H^{bc} \cap H^{bd} \cap H^{cd}$	6526360468678669861037589052034470126857779241331314839826230041617566152568440550895761885023785390756518058117863413│7216599736503166408006324991787941691613794336591761487594800388300067534635371463393316550373015552000000000000000000000
$H^{bc} \cap H^{db} \cap H^{dc}$	987760979695491729713131336886222849186173537468343019201090980213224186369431218746360706439232516885140014895617│3299771255831351809787985821576562273257336230723256281479103972702362841625684253952133767888896000000000000000000000
$H^{cb} \cap H^{db} \cap H^{dc}$	5404398077528054472521504460835385332865736755704757514533585222718698855027095804933932537271634529787736605385657│38183067388905642370403835935385934876263462098369108399972488826984484310240060652874690742714368000000000000000000000
$H^{cb} \cap H^{bd} \cap H^{dc}$	141767031443658485490372638618892830741501772123401878931601845278076851988166862281062607360239016458736050692837│763661347778112847408076718707718697525269241967382167999449776539689686204801213057493814854287360000000000000000000

After calculation, we obtain:

$$ { Pr }\left( {H,\infty } \right) = \frac{86196191235167272312652407600525591350591906553065213523273899326331417319}{9968983496689848612808247177117381571389256401783095296000000000000000000000}. $$

Finally, going to the limit in (11), we have:

$$ { Pr }\left( {E_{3} ,\infty } \right) = 3\left( {\left| {\left( {G,\infty } \right)} \right| + \left| {\left( {H,\infty } \right)} \right|} \right) = \frac{1087728064806496337719968633307455328929405251956556660146836615246691931}{28884683852842846824715253851562078123198903658479616000000000000000000000}. $$