Crowdsourcing  Applications  of  Voting  Theory  

Daniel  Hughart  5/17/2013  

   

   

     

Most  large  scale  marketing  campaigns  which  involve  consumer  participation  through  voting  makes  use  of  plurality  voting.    In  this  work,  it  is  questioned  whether  firms  may  have  incentive  to  utilize  alternative  voting  systems.    Through  an  analysis  of  voting  criteria,  as  well  as  a  series  of  voting  systems  themselves,  it  is  suggested  that  though  there  are  no  necessarily  superior  voting  systems  there  are  likely  enough  benefits  to  alternative  systems  to  encourage  their  use  over  

plurality  voting.  

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Contents  Introduction  ....................................................................................................................................  3  

Assumptions  ...............................................................................................................................  6  

Voting  Criteria  ..............................................................................................................................  10  

Condorcet  .................................................................................................................................  11  

Smith  .........................................................................................................................................  14  

Condorcet  Loser  ........................................................................................................................  15  

Majority  ....................................................................................................................................  16  

Independence  of  Irrelevant  Alternatives  ..................................................................................  17  

Consistency  ...............................................................................................................................  21  

Participation  ..............................................................................................................................  21  

Favorite  Betrayal  .......................................................................................................................  23  

Monotonicity  ............................................................................................................................  26  

Pareto  Efficiency  .......................................................................................................................  28  

Arrow’s  Impossibility  Theorem  .....................................................................................................  29  

Voting  Systems  .............................................................................................................................  31  

Plurality  .....................................................................................................................................  34  

Approval  ....................................................................................................................................  38  

Range  ........................................................................................................................................  41  

Borda  Count  ..............................................................................................................................  43  

Approval  Preference  Hybrids  ....................................................................................................  46  

Random  Ballot  ..........................................................................................................................  49  

Conclusions  ...................................................................................................................................  50  

Glossary  ........................................................................................................................................  53  

Bibliography  ..................................................................................................................................  54  

     

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Introduction    

The ever-increasing interconnectivity of the information age has provided firms,

organizations which trade goods and services to consumers, with greater access to a previously

underutilized source of labor and research, the general public. The act of disseminating tasks to

the large, undefined networks of people has recently been defined as crowdsourcing. An

important subset of crowdsourcing practices, crowd voting, involves a firm collecting

information through some form of a vote. The applications of crowd voting, including product

reviews and cheap market research among others, are manifold. Crowd voting has seen a lot of

use in recent years as a marketing mechanism. In 2011, as Toyota first launched multiple lines

of their Prius model, they asked the public to decide on the proper plural term for Prius.1 In their

annual “Crash the Super bowl” campaign, Doritos uses crowd voting to select a winner from

their crowd sourced advertisements to air during the Super Bowl.2 These and a number of other

uses of crowd voting poll the public about something they may find interesting, though have no

stake in.

This thesis is more concerned with instances of crowd voting where consumers are

directly affected, if minimally, by the outcome of the vote. This often occurs through the

potential release of a new product. If a consumer enjoys a product, they derive some utility by

the availability of that product. Utility, in the economic meaning of the word, is a quantified but

incomparable measure of an individual’s satisfaction. There have been a number of campaigns

which employ voting in order to select an alternative for production. In early 2012, Samuel

Adams used social networking to have its consumers vote in sequence to determine each aspect

of a beer they would release for the South by Southwest festival. Later in the same year, the

                                                                                                               1  (Toyota)  2  (Frito-­‐Lay)  

4    

Australian division of Domino’s pizza held a series of votes to determine the crust, sauce, and

toppings of a pizza that they then added to their menu.3 On a weekly basis, NBA.com uses

voting to determine what game will air the following Tuesday on NBAtv.4 On multiple

occasions and in a variety of countries around the world, Pepsi-co brands Lay’s and Mountain

Dew have released a small set of new flavors, and had consumers vote to keep a single flavor in

production. Dubbed “Do us a Flavor,”5 and “DEWmocracy”6 respectively, these promotions are

quintessential examples of crowd voting in which both the firms and voters have distinct

interests in the winners. This is because both are single “elections” held to determine a single

winner from a set of alternatives that the voter has ample opportunity to be familiar with. Due to

the food centric nature of most of these instances of crowd voting, I will occasionally use the

term flavor to denote an alternative in such a promotion. Each of these crowd voting marketing

examples use plurality voting, as do most such campaigns.

My discussion will focus around this very specific form of crowd voting, which I will

refer to as a promotional election. I define a promotional election as a firm surrendering some

production decision to the public at large through an actively publicized voting mechanism. This

term is used to capture both the promotional and electoral elements of these marketing

campaigns.

The term promotional is used in both the sense of advertising a brand, as well as

encouraging a positive customer relationship and goodwill through interaction. Ideally,

promotional elections make voters feel as though their input is valued by the brand, and build

excitement about that brand. For example, Mountain Dew’s marketing director has asserted that

                                                                                                               3  (DominosAustralia)  4  (NBATVfannight)  5  (Frito-­‐Lay  North  America)  6  (Mountain  Dew)  

5    

“[DEWmocracy] contributes to our growth. ... The Dew fan is excited about engaging with new

offerings from Dew. But it also attracts new people into the Dew fan base that say, 'hey, this is

something really interesting, let me give it a try.”7 There is some evidence that this is the case.

As an interactive social media marketing campaign, DEWmocracy not only involved and

inspired loyalty in the brand’s 726 thousand Facebook fans and 19 thousand Twitter followers8,

but helped the brand increase sales volume in a shrinking market.9 Many other promotional

elections have been similarly successful. The firm’s choice of voting system has a distinct

effect on these promotional gains, particularly customer loyalty. The more voters perceive a

sense of efficacy, the more goodwill and loyalty the campaign can potentially generate.

The term election refers to the social choice format of the campaign, using voters to make

some decision for the firm. The vote itself conveys valuable information about the participating

consumers. Though the magnitude of promotional gains are likely much larger, the ability of

promotional elections to double as market research is also valuable. However, if the process of

crowd voting is too explicit about its market research function, it runs the risk of diminishing the

consumer’s sense of involvement, and thus the promotional gains.

My thesis is concerned with the question of whether or not firms utilizing a promotional

election campaign have incentive to use some voting system other than plurality voting. In the

context of political elections the shortcomings of plurality voting have been evident as early as

the late 18th century: “If there are more than two candidates, and none of them obtains more than

half the votes, this method can in fact lead to error."10 There is a wealth of research and

discussion about the merits and shortcomings of plurality voting, as well as other voting systems

                                                                                                               7  (Zmuda)  8  (Mountain  Dew)  9  (Zmuda)  10  (Marquis  de  Condorcet  113)  

6    

within the closely related fields of voting theory and social choice theory. Most theorists agree

that plurality voting is far from optimal in a democratic context. The context of promotional

elections is notably different than that of political elections. Where a social planner selecting a

voting system for a political election is concerned with the subjective judgments of fairness, a

“social planner” for promotional elections, the firm running the campaign, has much more

specific, quantifiable goals. Their goals are the maximization of each of the benefits of

promotional elections through the selection of a voting system. These benefits come primarily in

the form of promotional gains, valuable voter preference information, and sales of the selected

product itself. My discussion in this work centers around an effort to assess the effects different

voting systems have on the maximization sales of the product itself. However, as different

voting systems have radically different effects on the promotional and information gains of

promotional elections, those effects are taken in to account and mentioned as well. By taking

voting theory out of the political context and placing it within the context of crowd voting I seek

to determine whether firms, like democracies, have incentive to use voting systems other than

plurality voting

Assumptions  

In my discussion of promotional elections, I make a number of simplifying assumptions

about costs, voter behavior, and firm’s motives. With regards to costs, I assume that they are

effectively equivalent for each of the alternatives being proposed. I also assume that price will

not change significantly as a result of increased output costs during or after the election. Though

these are assumptions, they are rather reasonable when put into the context of firms that have

already used promotional elections. The cost differences between different flavors of pizza,

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beer, soda, or potato chips should be minimal, particularly as the firm selects the alternative for

public consideration. As these promotional elections for food items are produced by international

giants, the simplifying assumption that increasing production does not significantly change

marginal cost is not egregious. As an extension of these two assumptions, let us assume that the

products for consideration by the consumer are priced identically. This has been the case for

both Lay’s and Mountain Dew.

Let us also assume that firms value the condition of anonymity amongst their voters.

Simply put, this condition requires that the voting rules treat each voter equally, such that the

identity of the voters is not required to produce a result. The appeal of anonymity is its intuitive

sense of fairness. Each individual has the same power as any other individual. In the context of

a promotion which focuses around empowering the consumer, this is highly desirable. Though

there are some potential benefits of removing anonymity, such as weighting voters of target

demographics, let us assume that those are outweighed by the promotional and participation

benefits of equal and fair elections.

There are also some assumptions to be made about the consumer-voter that participates in

this type of election. The first assumption is that they are familiar with all of the alternatives in

the election. Being familiar with each alternative, every voter develops both cardinal and ordinal

preferences between the alternatives. Their ordinal preferences are simple ranking of the

alternatives in order of their preference. Cardinal preferences express magnitude, for example

through a voter’s numeric estimation of how much they like each alternative on some arbitrary

scale. Though this information reflects the voter’s relative preference size from their

perspective, it is useless to a firm. As the numeric values attached to preferences are arbitrary,

there is no universal metric that allows comparison across either voters or alternatives. Because

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of this, cardinal reports of preferences provide firms no additional information beyond ordinal

rankings. Cardinal preference information in and of itself is somewhat valuable. If cardinal

preferences are collected through some metric, for example number of votes cast, they are

comparable across voters. The potential value of cardinal preference information comes from its

implications of willingness to buy. Let us assume that there is some positive relationship

between a voter’s cardinal preferences and cardinal willingness to buy.

Let us additionally assume that consumers know their own behavior well enough to

estimate how much they would be willing to buy of each flavor if it were released. Such an

estimate ideally incorporates both the estimated magnitude and frequency of their purchases.

This estimation is a cardinal measure of willingness to buy. Cardinal information is extremely

desirable as it translates into the best estimate a vote could provide about number of units sold.

In order for cardinal information be useful, it must be compared against some metric. As

opposed to preference, such a metric exists for willingness to buy. An expression of the

estimated volume of their purchases over a given time period could be compared between voters.

However, if voters are estimating their willingness to buy on some arbitrary scale, this

expression only provides ordinal information to the firm. Reducing willingness to buy to a

binary statistic removes the estimation and requires very simple comparable information from

voters. A voter will either be willing to buy some amount of a product if it is released, or none

of it. For some promotional elections, like NBA.com’s Fan Night, willingness to buy is solely

binary; an individual cannot watch the live airing of a sport broadcast more than once.

In addition to the binary of willingness to buy and the cardinal information regarding

magnitude of preference and willingness to buy, a voter has an ordinal preference ranking of

each of the alternatives. This preference ranking orders each of the flavors based on which one

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is preferred. Let us assume that voters who are interested enough to participate are interested

enough to purchase at least their highest ordinal preference. Though this may seem intuitively

reasonable, a number of promotional elections involve giveaway drawings to encourage

participation. This practice provides incentive for consumers to vote even if they have no

interest in the outcome. For discussion of voting systems, let us assume that the number of

voters who have no intent to purchase any of the products is negligible. Let us also assume that

voters are rational consumers, and if they are willing to buy any single flavor, they will also be

willing to buy any flavor higher on their ordinal preference ranking. Similarly, when one

alternative is preferred in the ordinal sense to another, the voter’s cardinal willingness to buy

should be greater than or equal to the less preferred alternative. Let us also assume that the

election results cannot harm any voters. If some flavor that a voter does not care for is selected,

they can simply choose not to buy it, unlike political elections. Finally, for the sake of

discussion, it is assumed that the voter understands the voting rules used in the promotional

election to the extent that they are presented.

A profit maximizing firm should have an interest in maximizing the willingness to buy in

both its binary and cardinal representations. The former maximizes the size of the customer base

for the new product, while the latter maximizes the number of units sold overall. The voting

system that is most desirable is the one that is most effective at teasing information about

willingness to buy out of the voter, without jeopardizing the promotional aspects of the

campaign. To some undetermined extent, accounting for voter’s ordinal preferences plays a role

in the promotional aspect of the campaign. In order to compare voting systems, or social choice

functions, the mathematic criteria upon which they are compared in social choice theory is

assessed from within the context of a promotional election. These criteria are generally used to

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assess social choice functions, which have a more specific meaning than voting systems. A

social choice function is a method of taking complete preference information from all individual

voters and translating that into a ordering of preferences that is representative of the entire voting

body. As opposed to voting systems, which are often only required to select a single alternative,

a social choice function must determine social preference rankings for all alternatives. From an

understanding of which conditions are valuable in the context of promotional elections, the

desirability of certain voting systems becomes more apparent. This helps narrows the search for

improvement over plurality voting to a few specific systems, whose relative merit and

shortcomings are assessed.

Voting  Criteria  

Discussion of voting criteria must take place within a context that determines what

preferences are possible for voters to have. For the purposes of this discussion, promotional

elections will take place under the assumption of unrestricted domain. This is also known as

“unrestricted scope. This requires that an acceptable [social choice function] be able to process

any (logically) coherent set of individual preference rankings of any number of choice

alternatives.” This also implies that voters’ individual preferences must be

transitive.11Transitivity is a small assumption in the context of promotional elections. There is

no quality about different flavors of food items which would cause a rational individual to

circularly prefer x Pi y, y Pi z, and z Pi x. The notation Pi stands between two alternatives to

indicate a preference of individual i. For example, x Pi y should be read, x is preferred by i to y.

In addition to other terms and mathematic notation, this information is also available in the

                                                                                                               11  (MacKay  7)  

11    

glossary provided at the end of the thesis. With the assumption of transitivity, unrestricted

domain simply means that the voter can rank the alternatives in an ordinal fashion in any order,

and that the social choice function must be able to process that information.

Comparisons of voting systems are based in their compliance to a variety of voting

criteria. At the root of these criteria are some very simple questions. What makes a fair winner?

What should be counted to determine a winner? Voting criteria offer some answers to these

questions by describing conditions which ensure some desirable aspects of voting system,

including what characteristics a winner must have, limits on what can influence a winner, and

what factors should influence a rational voter. The primary difference between political

elections and the promotional elections is the addition of a desire to collect information about a

voter’s willingness to buy the products that they are voting about. The important distinction is

that while a voter’s preferences are ordinal information, their willingness to buy is cardinal

information. There is very little translation between the two. Beyond their first preference, it is

unclear and likely to vary largely between voters how many, if any, of their other preferences

they would be willing to purchase. Nonetheless, some conclusions can be made about how the

satisfaction of certain criteria will affect willingness to buy maximization, or promotional

elections more broadly.

Condorcet  

One of the oldest voting criteria is the Condorcet criterion, which mandates that if a

candidate would win in direct comparison against every other candidate that they win the

election. The Condorcet criterion is satisfied if a voting system will always succeed in selecting

a Condorcet winner. Such a winner is defined as “when there is an alternative that would defeat

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all the other alternatives in a pairwise comparison”12. To illustrate, imagine a three voter

election between three alternatives as follows:

Voter 1 Voter 2 Voter 3

x y z

y z y

z x x

In pairwise comparison, yPx 2:1 and yPz 2:1, therefore y is the Condorcet winner. In absence of

the subscript i, the notation P denotes social preference. In this situation, y is social preferred to

both x and z by the mechanism of pairwise comparison. In order to satisfy the Condorcet

criteria, a voting system must select such a winner in every situation where such a winner exists.

There are three variations of the Condorcet criterion. The weak Condorcet criterion describes

the situation above, mandating the selection of an alternative that is preferred by strict simple

majorities to each other alternative if such an alternative exists. The other two criteria dictate

how a system should select in ties. These are situations where a set of alternatives exist that are

preferred by strict simple majorities to all alternatives outside of the set, but are indifferent to

other members of the set. To illustrate this, imagine a fourth voter whose individual preferences

are identical to those of Voter 3. In pairwise comparison, yPx 3:1 and zPx 3:1, however yIz 2:2.

Functioning similarly to P, I simply denotes social indifference. This set of y and z, known as

the core13, is addressed by the Condorcet criterion and the strong Condorcet criterion. The

Condorcet criterion mandates that the selected alternative(s), or winner(s) of the election be a

subset of the core, while the strong mandates that they be identical to the core. The strong

criterion implies satisfaction of both other criteria and the Condorcet criterion implies

                                                                                                               12  (Nurmi  38)  13  (Nurmi  19)  

13    

satisfaction of its weak variant.14 For purposes of our discussion, references to the Condorcet

criteria will be to the Condorcet criterion, its middling variant. This is because promotional

elections are generally single-winner, which renders adherence to the strong Condorcet criterion

impossible when the core contains more than one alternative.

Satisfaction of this criterion has a certain innate appeal to a sense of fairness. This has

some promotional implications. In order to nurture the sentiment of consumer participation, it is

in a firm’s interest to appear and be as fair as possible when selecting a response to consumers.

However, pairwise comparisons are not the only method that appeals to fairness and in certain

situations the Condorcet winner may not seem like the intuitively fair winner.15

In terms of maximizing either binary or cardinal willingness to buy, satisfaction of the

Condorcet criteria means almost nothing. Satisfaction of the Condorcet criterion relies solely

upon the ordinal preferences of voters. Assuming that voters are willing to purchase their most

preferred alternative, and may be willing to purchase some varied number of their next most

preferred alternatives, it is easy to imagine a situation where in satisfying the Condorcet criteria,

a voting system fails to maximize willingness to buy. Consider the following preferences and

(unreported) estimated number of units bought per week. The binary aspect of willingness to

buy is denoted by the lack of such a rating for those voters that would not buy.

Voter 1 Voter 2 Voter 3

x(10) y(5) z(7)

y(3) z(4) y

z(2) x x

                                                                                                               14  (Fishburn,  The  Theory  of  Social  Choice  146)  15  (Nurmi  39)  

14    

The Condorcet criterion would mandate the selection of y; however z is the alternative that

maximizes both binary and cardinal willingness to buy. In fact, y is the alternative which

minimizes cardinal willingness to buy! With little information other information about voters

willingness to buy, very little can be assumed about how many voters would be willing to buy

the Condorcet winner, and how often they would buy. Satisfaction of the Condorcet criterion is

far from necessary from the perspective of maximizing willingness to buy.

Smith  

The Condorcet criteria are implied by the stronger Smith criterion. The Smith criterion

refers to a set called the Smith set. As proposed by Peter Fishburn, the Smith set is made up of

alternatives which defeat all candidates not within the Smith set in pairwise comparisons. If the

Smith set is non-empty, the choice set must be a subset of the Smith set. The smith set is named

for John H. Smith16, who proposed the following expanded Condorcet criterion: “If the set of

candidates can be divided into T and U so that each candidate in T has a majority over each

candidate in U, then each candidate in T finishes ahead of each candidate in U.”17 For this

discussion, Fishburn’s Smith set is discussed over Smith’s Condorcet criteria, as the latter is

unnecessarily strict for promotional election, as preference relations between alternatives not in

the choice set have minimal value in this situation. The Smith criterion implies the Condorcet

criteria, as in the case of a Condorcet winner, the Smith set will be made up of solely the

Condorcet winner. As the Smith criterion is stronger than the Condorcet criterion, the

implications for promotional elections are marginally more significant at best. The gains from

compliance with the Smith criterion are entirely through an increased sense of fairness. The

                                                                                                               16  (Fishburn,  Condorcet  Social  Choice  Functions  478)  17  (Smith  1038)  

15    

implications of satisfaction on willingness to buy are completely uncertain. If the number of

alternatives m is small, the Smith criterion demands nothing that the Condorcet criterion does

not. If the size of the Smith set is 1, that alternative is a Condorcet winner, if it is 2 alternatives

large it is identical to the core, and makes the same demands as the Condorcet criterion. It is

only when  𝑚 ≥ 4, such that a 3 alternative Smith set is possible and that the demands of the

Smith criterion expand beyond those of the Condorcet.

Condorcet  Loser  

The specific case of a three alternative Smith set where 𝑚 = 4 effectively eliminates the

single alternative not in the Smith set from contention to be selected. This situation describes a

Condorcet loser, a single alternative which loses in majority pairwise comparisons with every

other alternative. Implied by satisfaction of the Condorcet criterion, the Condorcet loser

criterion mandates that such an alternative cannot be selected.18 This criterion has also been

referred to as the inverse Condorcet condition.19 This criterion, despite being much weaker, also

has very uncertain implications for willingness to buy. It is possible that a Condorcet loser is the

alternative which would maximize willingness to buy. Imagine the following sets of preference

and willingness:

Voter 1 Voter 2 Voter 3 Voter 4 Voter 5

x(!) x(!) y(!) z(!) w(!)

y w z y y

z y w w z

                                                                                                               18  (Straffin  23)  19  (Richelson  465)  

16    

w z x x x

For this set of preferences, x is a Condorcet loser as it is defeated in pairwise comparisons

against every other alternative, yPx 3:2, zPx 3:2, wPx 3:2. Despite this, two individuals would

be willing to purchase a, while only one would purchase any of the other alternatives. Similarly

to the Condorcet and Smith criteria, the only sure gains from satisfying the Condorcet loser

criteria come from ensuring some undesirable alternative is not selected because of the nature of

the system. Satisfaction of the Condorcet loser criterion implies no specific information about

either voter’s binary or cardinal willingness to buy.

Majority  

In addition to the Condorcet loser criterion, the Condorcet criterion also implies the

satisfaction of a far weaker and simpler criterion, the majority criterion.20 This criterion simply

mandates that if a single alternative is preferred by a simple majority of the voters, it must be

selected.21 As this criterion is concerned solely with the first preferences of voters, there are

some conclusions that can be drawn about binary willingness to buy. This is because binary

willingness to buy is implied by the first preference of the voter. A voting system that passes the

majority criterion ensures that if a majority winner is available, at least half of the voting body

will be willing to buy that alternative at some level. However, with the extreme uncertainty of

information about willingness to buy, it is possible that this majority winner could be the worst

possible choice to maximize binary willingness to buy. Consider the following, where each

exclamation point indicates a binary willingness to buy for a single voter.

                                                                                                               20  (Nurmi  62)  21  (FairVote)  

17    

3 Voters 1 Voter 1 Voter

x(!!!) y(!) z(!)

y(!!)

z(!)

z(!)

x

y(!)

x

In this situation, though x is the majority winner, only the 3 voters who prefer x would buy. z

would also yield 3 buyers, and 4 voters would buy y. Nonetheless, so long as individuals are

willing to purchase their highest preference, satisfaction of the majority criteria is desirable. If

there is a majority winner, that winner will be purchased by half the voters in a worst case

scenario. The majority criterion also carries an enormous appeal to fairness. With its appeal to

the simple democratic value of majority rule, selecting a majority winner if such a winner exists

seems intuitive. The large concern about majority rule in democracies which inspired much of

the U.S. constitution is the tyranny of the majority. Such a tyranny describes a majority which

places its own interests above those of individuals or minority groups. As it does not hurt an

individual to see a flavor of potato chip be put into production, for example, the repercussion of

such a tyranny are minimal, if at all existent. The implication of the majority criterion is the

strongest connection the Condorcet criterion has to the maximization of binary willingness to

buy.

Independence  of  Irrelevant  Alternatives  

There are a number of other criteria for evaluating social choice functions that do not

share any sort of implication relationship with the Condorcet criterion. The independence of

irrelevant alternatives (IIA) criterion is based on the idea that a social choice function should

make a choice about a preference ranking between a set of alternatives based solely on the

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preference orderings of individuals. For that set, {x,y} for example, a social choice function

must not rely on any alternatives outside of the set, the “irrelevant alternatives”. In more positive

terms, the social preference ordering between x and y must be based on nothing other than the

individual ordinal preference relation between x and y.22 More formally, “If PROF is a profile

of individual preferences over some set of alternatives that includes x and y, if [social choice

function F] 𝐹(𝑃𝑅𝑂𝐹, {𝑥,𝑦}) = 𝑥  𝑃  𝑦 and if R’ is another preference profile such that each

person’s preference between x and y is the same in R’ as in R then  𝐹(𝑃𝑅𝑂𝐹’, {𝑥,𝑦}) =  𝑥  𝑃  𝑦.”23

The profile of individual preferences PROF is a set that contains all voters’ individual

preferences. Arrow’s IIA criterion seems intuitively very reasonable, and has a great appeal to

fairness.

IIA is often confused with a criterion that shares the name. This other independence of

irrelevant alternatives, also known and henceforth referred to as Sen’s property α, states:

∀𝑥:  𝑥 ∈ 𝑆! ⊂ 𝑆! → [𝑥 ∈ 𝐶 𝑆! → 𝑥 ∈ 𝐶 𝑆! ].24 This condition demands that if x is selected out

of the set S2, and S1 is a subset of S2, x must be selected from S1 as well. This confusion was

made by Arrow himself, when providing an illustrative example for his IIA condition. In his

example, a hypothetical candidate dies during an election and he compares the results of a

hypothetical Borda count before and after deleting that candidate.25 Both of these examples are

defenses of α, rather than IIA. Arrow acknowledged as much as his mistake, by clarifying the

difference between the two. α “refers to variations in the set of opportunities, mine to variations

in the preference orderings… The two uses are easy to confuse (I did so myself in Social Choice

                                                                                                               22  (MacKay  9)  23  (Ordeshook  60)  24  (Sen  17)  25  (Arrow  26-­‐27)  

19    

and Individual Values at one point).”26 Despite its intuitive appeal similar to that of IIA,

condition α has dramatically different implications.

It is nearly impossible for a socially desirable voting system to pass α. α cannot be

passed by any system that when reduced to two alternatives resorts to majority judgment.

Imagine the following set of voter preferences, the simple voting paradox of cyclical social

preference:

Voter 1 Voter 2 Voter 3

x y z

y z x

z x y

In this way, social preferences are xPyPzPx. Suppose a voting system somehow differentiates x

as the winner. Majority judgment would dictate the selection of z if y were eliminated, as zPx

2:1. Similarly, if y is the winner, the elimination of z mandates selection of x, and removing x

from a system which selects y would require the selection of z. In this way, it is impossible for a

voting system with unrestricted domain to satisfy α if it reduces to majority judgment between

two alternatives. Even when voters have multiple scoring options for their vote, there is a

strategic incentive to vote the maximum score for their preferred candidate and the minimum for

their less preferred candidate. This obvious strategy maximizes the voters’ chance of their

seeing their preferred alternative selected, which is a desirable outcome.

IIA is not without its criticism as well. As it is incompatible with the Condorcet criteria

(viz Arrow’s impossibility theorem), a number of alternatives have been suggested in order to

decrease the strength of IIA and work around the impossibility theorem. Among these is the

Independence of Smith Dominated Alternatives. This independence criterion, which resembles                                                                                                                26  Arrow,  Kenneth  J.  The  Functions  of  Social  Choice  Theory,  As  cited  in  (Mackie  127)  

20    

condition α and is compatible with the Condorcet criteria, mandates that a voting system not

select a different winner if alternatives not in the Smith set are eliminated.27 Such a condition is

much weaker than IIA, yet it carries a similarly strong appeal to fairness. Additionally, some

criticism is aimed at the lack of an end normative justification for IIA. The content of IIA is

certainly normative in nature: it makes judgments for what information a choice should be made.

However, IIA makes no normative justifications about how limiting the inputs of a choice

function in such a way is socially desirable.28 When compared to a criterion with a normatively

justified end, IIA seems much less intuitively desirable. Within the context of maximizing

willingness to buy, IIA implies nothing. As the condition is only a prescription for the inputs of

a social welfare function, making no claims whatsoever about what alternative should win, there

is no translation to the binary or cardinal language of willingness to buy. A voting system could

possibly be perceived as more fair if it adheres to IIA. In the context of promotional elections, it

is easy to imagine how undesirable a violation of IIA could be. Imagine a promotion election in

which alternative x would be selected, defeating y, and z. If a voting system fails to pass IIA, a

group of voters deciding that they prefer z more than they originally did could somehow, all else

constant, cause y to be selected. The effects of such a violation could decrease the sentiment that

the election is fair, however its conceptual unfairness may be difficult to perceive. Take the

example of alternatives x, y, and z. In the initial reality where x is selected, it may not be

immediately apparent that some change in voter preferences for z could change the selection to

y. A voting system that is more subjectively “fair” may yield greater promotional gains than one

that is not. However, these gains, if they exist, are ambiguous in magnitude at best.

                                                                                                               27  (Shulze  295-­‐296)  28  (Brennan  and  Hamlin  107-­‐108)  

21    

Consistency  

Closely related to IIA, and similarly incompatible with the Condorcet criteria is the

consistency criterion. The consistency criterion states that if there are two independent groups

that, using the same social choice function, select the same alternative from identical sets of

alternatives, the combination of the two groups should select the same alternative. Put

mathematically, let N1 and N2 be two groups of voters such that  𝑁! ∩ 𝑁! = ∅ with preference

profiles PROF1 and PROF2 respectively, while 𝑁 = 𝑁! ∪ 𝑁! with preference profile PROF.

Additionally, let F be a social choice function for a given set of M alternatives with choice set C,

such that F(M, PROFx)=Cx: which is consistent if and only if whenever 𝐹 𝑀,𝑃𝑅𝑂𝐹! ∩

𝐹 𝑀,𝑃𝑅𝑂𝐹! ≠ ∅ → 𝐹 𝑀,𝑃𝑅𝑂𝐹 = 𝐹(𝑀,𝑃𝑅𝑂𝐹!) ∩ 𝐹(𝑀,𝑃𝑅𝑂𝐹!).29 A choice set is the set

of alternatives that can be selected as winner(s) from a social choice function. It is worth noting

that the language used implies that if either choice set for the separate groups has more than one

element and there is a nonzero intersection between the two, the choice set of the combined

groups must only contain that intersection. The consistency criterion has a certain intuitive

appeal to fairness, just as IIA does. However, the consistency criterion, in stark contrast to IIA,

has enormous implications for willingness to buy maximization. These implications are much

clearer in a discussion of the participation criterion, which is implied by the consistency

criterion.

Participation  

The participation criterion regards incentives of potential voters to participate in

elections. This, simply put, implies that a voter i cannot cause social choice function F to yield a                                                                                                                29  (Nurmi  94)  

22    

result less favorable to themselves, or lower on their individual preference orderings, by

participating and reporting their true preferences.30 The participation criterion is implied by the

consistency criterion. This is clear when testing the consistency criterion with the size of one

group, or N1= {i}. When this is the case, both the consistency and participation criteria mandate

the same thing; that the addition of a single voter i to some other set of voters N2 cannot result in

a less favorable outcome for i. The normative appeal of this criterion from a democratic fairness

standpoint is tremendous. Democracy should encourage participation of informed rational voters

rather than discourage it. Participation should be encouraged in promotional election marketing

campaigns as well.

Satisfaction of the participation criterion has enormous implications for firms seeking to

maximize the promotional gains of their campaign. If a system were to fail the participation

criteria, it risks discouraging consumers from voting. This prevents the consumer from

interacting with the brand through the marketing campaign and erodes at the goodwill the

campaign may have produced. Not only does a consumer’s participation and thus exposure

afford a firm the opportunity to gain a more loyal customer, it also gets another individual to

potentially be excited and talk about the promotion itself. Participation in the campaign grows

exponentially in this way. Higher turnout also has some implications for willingness to buy.

Even operating under the pessimistic assumption that participating in the campaign does not

increase a consumer’s willingness to buy, increased voter turnout will provide information about

a larger selection of the customer base as a whole. Though voting in a promotional election

provides far from perfect information about customers’ willingness to buy, it is superior to the

complete lack of information a firm receives from non-voters. As turnout increases, firms gain

this valuable information about a larger portion of their customer base. However, it is unlikely                                                                                                                30  (Moulin  55)  

23    

that the participating group of customers is representative of the customer base as a whole. The

variables that affect an individual’s participation in a promotional election are likely to also

affect their willingness to buy.

It is also possible that voters could react to a failure of the participation criterion by

voting strategically. If, by misrepresenting their preferences, a voter can influence the outcome

of the election in their favor, they have incentive to do so. This situation is also undesirable to

firms. The more a voting system encourages voters to vote strategically rather than honestly, the

more the market research benefits of promotional elections are diminished.

Additionally, there is a related condition, known as the twin’s welcome axiom.31 It states

that the addition of a voter with identical preferences to some voter i cannot result in a change in

social choice to an alternative lower on i’s personal preference ordering. This has similar

implications to voter participation and its appeal to marketing campaigns. Campaigns can reach

more people largely through word of mouth between consumers, and it is only to a firms benefit

if consumers have incentive to encourage their like-minded friends to participate and expose

themselves to the campaign. A voting system that fails the twins-welcome axiom would have

similar failures as a system that fails the participation criterion. The failure would discourage

voter turnout, which has severe negative implications for scale of willingness to buy information

as well as the marketing aspects of promotional elections.

Favorite  Betrayal  

                                                                                                               31  (Moulin  53)  

24    

Voter participation may also be affected by the favorite betrayal condition, which simply

states that “voters should have no incentive to vote someone else over their favorite.”32 This

condition is best illustrated by its violation in plurality voting where it is often in a voter’s

interest to “betray” their favorite alternative when two less desirable candidates are perceived as

likely to win. A voter can maximize their expected utility from the election outcome by

strategically voting as if a lesser-evil alternative were higher on their preference rankings in

order to avoid some greater-evil. If a voting system is to fail this criterion, it could cause

disillusion among voters, which has some chance of decreasing voter turnout. However, this

criterion is likely to have little to no effect on binary willingness to buy. If a voter is willing to

buy their lesser-evil alternative, they are accurately conveying a binary willingness to buy.

There are information losses about ordinal preferences, and as a result, cardinal willingness to

buy. If a voter is willing to buy the lesser-evil alternative and they vote strategically, they are

failing to report their favorite, and thus their highest cardinal willingness to buy. If a voter is not

willing to purchase the lesser-evil alternative however, they have no incentive to betray their

favorite, as a voter in a marketing election is completely unaffected by the outcome of

alternatives they are not willing to buy. As a result, there is no incentive for such a voter to

convey their preferences strategically. There is a chance that that would lead to the voter not

participating, however this will only occur if the magnitude of the cost is greater than the

expected value of participation for that voter. Even if the voter believes the odds of their favorite

winning are exceptionally low, with the internet, the cost in time to the voter is next to nothing.

Take for example, NBA.com’s fan night vote, which requires no login and no site redirect from

the front page of their website. As their website generates a large amount of traffic from

potential voters for non-voting related information, the cost of voting for a basketball enthusiast                                                                                                                32  (Ossipoff  and  Smith)  

25    

is as little as a handful of seconds. With the ubiquitous use of the internet for promotional

elections, the cost of voting is similarly low for other such marketing campaigns.

Though the favorite betrayal condition itself has few implications for the customer

generation of promotional votes, the stricter condition of non-strategic voting which implies it

does have relatively significant implications. The non-strategic voting condition is simply that a

voting system should be structured such that any individual i, or group of individuals has no

incentive to report anything other than their true preferences for all social preference ordering

profiles. This condition is too strict to be satisfied by any reasonable social choice mechanism.

In fact strategic voting cannot be completely removed for a voting system with a single winner

unless the system either ensures the loss of some alternative regardless of preferences, or the

system is dictatorial such that a single voter can determine the outcome of the election. This is

the Gibbard–Satterthwaite theorem.33 Though it is impossible to ensure that honest voting is the

only rational strategy in a non-dictatorial voting system, it is possible and highly desirable to

limit the incentives for strategic voting as much as possible. This can be done primarily through

limiting the information available to the voter. If no ‘polling’ information, live voting statistics,

or some similar source is available to voters, their only perceptions of social preference orderings

come from their own assumptions or community research. Joining or building a community to

discuss the brand is highly desirable to a firm from a promotional standpoint. If a voter makes

that effort, the information cost of their strategic vote is likely outweighed by the growth of an

active fan community. Without information about other voter’s preferences, it can be extremely

difficult or impossible for a voter to make a rational strategic vote. Given the lightness of the

subject matter, it is likely not worth the effort for a voter to display preferences strategically

when voting on a new chip flavor to see in production unless such a strategy is immediately                                                                                                                33  (Svensson)  

26    

apparent after a second’s reflection. Promotional elections have this advantage over political

elections; they are relatively inconsequential from most voters’ perspectives. There is little to no

available information about social preferences, and a voter would have to put the research effort

into estimating this themselves. Promotional elections lose this advantage if they make

information about the vote available as the vote occurs. Though this may bring promotional

gains through apparent transitivity, providing information about already tallied votes makes

strategic voting much easier and potentially obfuscates the true individual preferences of a

number of consumers. One possible method to dramatically reduce strategic voting is ambiguity

about the voting system used. Assuming that some non-dictatorial system is actually being used,

if voters are unaware how their votes will count, they are much less likely to be able to vote

strategically. Threadless, a website that crowdsources t-shirt designs and chooses some for

production through crowd voting, is ambiguous with how it translates the 1-5 user ratings into a

production decision.34 Use of such an ambiguous system for promotional elections would

unfortunately run the risk of jeopardizing the promotional gains. Voters might feel less involved,

and even less inclined to participate if they do not understand how their vote will count, and will

also have no assurance that they have any efficacy whatsoever.

Monotonicity  

Strategic voting plagues all specified, non-dictatorial social choice functions, however it

does not affect all of them evenly. In some voting systems, it may even harm an alternative to be

voted for. The monotonicity criteria represented here solely by, the mono-raise criterion

demands that: “a candidate x should not be harmed if x is raised on some ballots without

                                                                                                               34  (Threadless)  

27    

changing the orders of the other candidates.”35 The other criteria also mandate that some change

in individual preference orderings which add or move x towards or to the top of preference

orderings or delete entire profiles with x at the bottom of the preference ordering must not hurt x.

These criteria, particularly mono-raise, intuitively seem like a desirable trait of social welfare

functions, and its satisfaction has unambiguously positive implications for the maximization of

collective willingness to buy. If some individual i’s preferences change, such that alternative x is

raised in their preference ordering: the only possible change, if any change occurs, in willingness

to buy is a change for that alternative for that individual from a 0 to a 1. That is, the rise in

preference ranking may or may not result in a switch from not being willing to purchase a

product to being willing. Such a change should result in a nonnegative change for the social

willingness to buy, as well as in the social preference ordering, as dictated by mono-raise. The

contrary is perhaps a more powerful assertion. If mono-raise is violated by some choice function

F, for the individual preferences pref, contained within profile PROF between feasible

alternatives M: 𝑝𝑟𝑒𝑓! ∈ 𝑃𝑅𝑂𝐹 , 𝑝𝑟𝑒𝑓!! ∈ 𝑃𝑅𝑂𝐹!, and 𝑝𝑟𝑒𝑓! 𝑥 < 𝑝𝑟𝑒𝑓!!(𝑥) such that all

preference relations between non-x alternatives are identical between prefi and prefi’, the ranking

of x according to 𝐹(𝑀,𝑃𝑅𝑂𝐹) > the ranking of x according to F(M, PROF’). In terms of

number of individuals willing to buy, this violation implies that a potential increase in number of

buyers results in a decrease of social preference order rank, which runs directly counter to

maximization.

Monotonicity is one of the more desirable criteria, particularly for promotional elections.

Because of the market research applications of the votes themselves, it is in a firm’s interest to

assure that a voting system is used such that voting for some alternative can only help. This

                                                                                                               35  (Woodall  85)  

28    

becomes most clear when examining a theoretical violation of monotonicity, where voting for a

an alternative decreases that alternatives chances of winning. In this scenario, it becomes

strategic for voters to vote for alternatives they would not actually buy, which would yield wildly

incorrect market research information, and would likely lead to a poor choice in the context of

willingness to buy.

Pareto  Efficiency  

The condition of Pareto efficiency in the context of social choice is very weak, and

almost as a result has relatively significant implications for consumers’ willingness’ to purchase.

The condition is based on a principle of the same name that demands that “if (𝑥𝑃!𝑦) for all

persons i in a group then (𝑥𝑃𝑦) in the social preference order.”36 As the principle involves

placing an alternative above another in social preferences and mentions nothing of individual or

social preferences for other alternatives, it cannot make any conclusions about which alternative

should be chosen, but rather makes a statement about what alternatives cannot be chosen. The

condition itself is split into two, the strong and weak Pareto conditions. The weak Pareto

condition mandates that if (𝑥𝑃!𝑦) for all persons i, y cannot be selected by the social choice

function. The stronger variation expands the argument to assert

that ∀𝑖 𝑥𝑅!𝑦  𝑎𝑛𝑑   ∃𝑖 (𝑥𝑃!𝑦)37. Both of these conditions have a very fundamental appeal to

fairness through unanimity and have very certain positive effects on maximizing willingness to

buy. Eliminating an alternative y to which alternative x is unanimously preferred in accordance

with even the strong Pareto condition necessarily removes an alternative that is at the very best

equal in new consumers generated. This is true because if voter i is willing to purchase some

                                                                                                               36  (Ordeshook  61)  37  (Nurmi  81)  

29    

alternative y, they are necessarily also willing to purchase all alternatives preferred to y. A

voting system in this discussion is said to be Pareto efficient if it satisfies Pareto efficiency under

complete voter honesty. This is because in the presence of misinformed strategic voters, any

voting system may be inefficient in equilibrium.

Arrow’s  Impossibility  Theorem  

The Pareto condition is perhaps best known for its presence in voting paradoxes,

primarily Arrow’s. Arrow’s paradox, or impossibility theorem identifies a glaring issue with

establishing a social welfare function, which Arrow defines as the process or rule that produces

ordinal rankings of social preferences from individual ordinal rankings.38 This issue is the

incompatibility of a number of conditions or axioms that are all highly desirable aspects of a

social welfare function. The exact titles and content of these conditions varies from publication

to publication, as many of the criteria imply satisfaction of other close criteria, but they are given

as follows by Arrow in his 1963 work Social Choice and Individual Values:

Condition P: 𝐼𝑓  𝑥  𝑝!𝑦  𝑓𝑜𝑟  𝑎𝑙𝑙  𝑖, 𝑡ℎ𝑒𝑛  𝑥  𝑃  𝑦…

Condition 1’: All logically possible orderings of the alternative social states are

admissible…

Condition 2’: For a given pair of alternatives, x and y, let the individual preferences be

given. (By Condition 3, these suffice to determine the social ordering.) Suppose that x is

then raised in some or all of the individual preferences. Then if x was originally socially

preferred to y, it remains socially preferred to y after the change.39

                                                                                                               38  (Inada  396)  39  (Arrow  96-­‐7)  

30    

Condition 3: Let pref1 … , prefn and pref1‘ … , prefn’ be two sets of individual orderings

and let F(M) and F’(M) be the corresponding social choice functions . If, for all

individuals I and all x and y in a given environment M, x prefi y if and only if x prefi’ y,

then F(M) and F’(M) are the same (independence of irrelevant alternatives).40

Condition 4: The social welfare function is not to be imposed.

Condition 5: The social welfare function is not to be dictatorial (non-dictatorship).41

Arrow’s logic follows that as Condition P, the Pareto condition is implied by conditions 2’, 3 and

4, proof of the incompatibility of conditions 1’ (unrestricted domain), 3 (Independence of

Irrelevant Alternatives), P (Pareto efficiency) and 5 (Non-dictatorship) leads to the proof of the

inconsistency of the 5 numbered conditions. The remaining conditions 2 and 4 are monotonicity

and Citizen’s Sovereignty. Citizen’s Sovereignty is simply the condition that there may be no

pair of alternatives x and y, such that xRy, regardless of individual preferences.42 Arrow’s

impossibility theorem has enormous implications for the compatibility of voting criteria. IIA is

the primary cause of these implications. Arrow’s IIA condition shares remarkable similarity to

the Condorcet criterion in that explicitly avoids taking the precise position of alternatives in

individual preferences into account. Arrow specifically mentions the Condorcet criterion

claiming that it “implicitly accepts the view of what I have termed the independence of irrelevant

alternatives.”43 Despite their shared non-positional nature, the Condorcet criteria and IIA are

incompatible, despite what appears to be Arrow’s 1963 belief to the contrary.

Arrow’s independence [of irrelevant alternatives] condition is explicitly against the view

that the positions of an alternative will have a direct influence on the social ordering.

                                                                                                               40  (Arrow  27)  41  (Arrow  30)  42  (Arrow  28)  43  (Arrow  94-­‐95)  

31    

Applying the Condorcet conditions, one completely ignores the positions of an alternative

and finds the optimal alternatives by pairwise majority votings… The non-positionalist

view is inconsistent in itself, or more exactly stated: no voting function satisfied both

[IIA] and [weak Condorcet].44

This is proven by the existence of a theoretical set of individual preferences which are compared

to alternative situations where a single voter’s preferences for some non-selected alternative were

altered. For every situation in which a different alternative were to be selected, there is some

alternate situation with an altered individual preference that force a contradiction between IIA

and the Condorcet winner.45 Additionally, the Condorcet criterion is incompatible with the

consistency criterion. In order for a social choice function that uses individual ordinal

preferences as inputs to be consistent, it must a scoring function similar to the Borda count.

Such systems are incompatible with the Condorcet criterion.46

Voting  Systems  

Satisfaction of most voting criteria in and of themselves has minimal implications for

maximization of willingness to buy. Voting criteria are, on the whole, insufficient to assess

voting systems for their maximization of willingness to buy. This is because voting criteria are

rules regarding social choice functions. As the translation of individual preferences to social

preferences, social choice functions exclusively involve ordinal preference information.

Willingness to buy is cardinal information. Nonetheless, each criterion is somewhat valuable, if

for nothing other than perceived fairness. Some are particularly desirable in the context of

                                                                                                               44  (Gärdenfors  9)  45  (Gärdenfors  9-­‐10)  46  (Young  829)  

32    

promotional elections. The majority criterion ensures that if alternatives with very broad appeal

exist, one of them must be chosen. The participation criterion and monotonicity are essential for

voter efficacy, which encourages participation. Satisfaction of the favorite betrayal criterion

ensures a firm receives honest voting information about at least voter favorites. Finally, Pareto

efficiency is actually a requirement for willingness to buy maximization. Voting criteria are a

valuable tool to compare voting systems; however more detailed assessment of the voting rules

themselves is required to discuss willingness to buy maximization.

The incompatibility of large sets of voting criteria form a few clear divisions that allow

voting systems, or social choice functions to be sorted into groups that share similar

characteristics. The fundamental splits come in determining what information can be used as

inputs for the social choice function. The first such split designates a distinction between

positionalist voting functions and non-positionalist voting functions. The most common subset

of positionalist voting functions is representable voting functions. A representable voting

function “is a method where we assign numbers of the alternatives according to their positions in

the preference orders and then determine the social ordering from the sum of these numbers for

each alternative.”47 The most famous of such voting systems is the Borda count, the theory of

which predates Condorcet. The Borda count and most others like it fail the vast majority of the

aforementioned criteria. This is primarily because many of the criteria involve a strict

adherence to ordinal preference relations as the inputs of social choice functions. While

positional social choice functions use ordinal preferences as inputs, these inputs are translated by

a scoring function into scores. A winner is determined through these scores rather than the

ordinal rankings themselves. Though a representable voting system necessarily fails the

                                                                                                               47  (Gärdenfors  2)  

33    

Condorcet criteria, as well as Arrow’s IIA,48 it is possible for such a system to pass the

consistency and participation criteria. There are unique costs and benefits in the context of a

promotional election to the use of a positional voting system, which will be assessed in

conjunction with the later overview of the Borda count.

The chief division between alternative non-positionalist voting systems is their adherence

to either Condorcet’s criteria or Arrow’s IIA and consistency criteria. The two sets of systems

will be referred to as Condorcet and Arrovian systems. The set of Condorcet systems has the

potential to pass the Condorcet criteria, the Condorcet loser criterion, and the Smith criterion. To

select a Condorcet system, however is highly undesirable in the context of a marketing election.

The logic of a Condorcet winner simply does not map onto a promotional voting campaign very

well. Though an individual may have preference orderings between alternatives that they would

not be willing to buy, a function that seeks to maximize willingness to buy should be

independent of that information. Methods that satisfy the Condorcet criterion do so by only

taking ordinal preference relations into account. To reduce the inputs of a social choice function

to ordinal preference, relations would render any available information about the voter’s

willingness to buy useless in selection of the choice set. Even more undesirable, satisfaction of

the Condorcet criterion necessitates the failure of the Arrovian family of criteria, most

importantly the participation criterion.49 Satisfaction of the participation criterion is extremely

valuable to political elections, and is even more valuable to promotional elections. By

potentially discouraging participation through failure of the participation condition, a firm allows

their chosen voting system to not only disenfranchise voters but to discourage participation in

their marketing campaign. The impact of this failure on both the maximization of willingness to

                                                                                                               48  (Gärdenfors  9)  49  (Moulin  56)  

34    

buy, as well as the success of the voting campaign as a marketing campaign is distinctly

negative.

Mutually exclusive with Condorcet systems, let Arrovian systems, which necessarily fail

the Condorcet family of criteria, be defined as non-positional systems which pass at least one of

the Arrovian family of criteria, which include consistency, participation, IIA and the favorite

betrayal condition. This bundle of criteria is vastly more desirable than the Condorcet family of

criteria in the context of promotional elections. This is primarily because of the participation

criterion; however favorite betrayal and IIA have some desirable characteristics to prevent

disillusionment. Defined in such a way, the set of Arrovian systems is incredibly broad. It

includes plurality voting despite its failure to adhere to IIA or favorite betrayal. The runoff

variations of plurality voting cannot be considered Arrovian, as such variations fail to satisfy any

of the Arrovian criteria, failing consistency and participation in addition to plurality voting’s

failings. Runoff voting systems also fail the monotonicity criterion.50 It also includes the closely

related approval and range voting systems which both pass all four of the Arrrovian criteria

assessed.

Plurality  

Plurality voting is very simple, intuitive, and is popularly used across the world in

political elections. Quite simply, each voter selects one alternative, and the alternative with the

largest number of votes wins. It is the voting system that is primarily employed in promotional

elections, notably including Pepsico’s “DEWmocracy” and “Do us a Flavor” campaigns as well

as NBA.com’s Fan Night. Plurality voting passes the monotonicity, consistency and

                                                                                                               50  (Straffin  24)  

35    

participation criteria and is Pareto efficient, however fails IIA and the favorite betrayal condition.

Though the Condorcet criteria is incompatible with IIA and consistency, the majority criterion,

which is implied by the Condorcet criterion can be consistent with Arrovian criteria. Plurality

voting demonstrates this possibility, as it passes both the majority criterion and the consistency

criterion. Suppose the following set of voters and preferences for a 3 candidate, 7 voter

elections:

3 Voters 2 Voters 2 Voters

x y z

z z y

y x x

This example illustrates the failure of plurality voting to adhere to the Condorcet, Arrow’s IIA,

and the favorite betrayal criteria. If all voters were to vote honestly, x would be chosen, despite

the fact that z is a Condorcet winner, as zPx 4:3 and zPy 5:3. Additionally, the plurality winner x

is a Condorcet loser, as yPx 4:3 and zPx 4:3, a violation of the Condorcet loser criterion.

However, plurality voting does not fail the majority criterion. If any alternative were to receive a

simple majority of the votes, they would emerge victorious. Imagine some information is

revealed which makes the 2 voters who originally prefer z prefer y over z. This would cause y to

be selected by plurality voting, a violation of IIA. If, knowing the preferences of the other

voters, the voters who prefer z strategically change their votes to y, they can change the outcome

to the more desirable result y. This violation of the favorite betrayal condition is a large concern,

however due to an individual’s indifference between alternatives they would not buy; it is not

rational for anyone to switch their vote to any alternative they would not buy. Additionally, the

strategic voting relies on a voter’s knowledge of other voter’s opinions. For non-repeated

36    

elections like those used by Pepsi-co, this sort of information is unavailable unless a firm makes

current election information available. Because information about social preferences between

newly released soda flavors is not aggregated and made widely available in the way that

preferences between political candidates are, a firm can effectively eliminate strategic voting by

not publishing that information. This is not the case for NBA.com’s fan night. Because there are

a number of active communities that discuss basketball, a voter can gain a very good estimate of

social preferences. Additionally, the voting website publishes statistics of what the current

voting percentages are, which ensure voters have all the information they need to vote

strategically.

Plurality voting’s chief benefits in the context of promotional elections are its ease of use

and familiarity for most voters. Plurality voting is somewhat more desirable under the

assumptions that voters will only vote if there is at least one alternative they are willing to buy

and will only strategically vote for alternatives they would buy. If this is the case plurality

voting uses as inputs only information about alternatives voters are willing to buy.

Unfortunately, plurality voting fails to capture any information regarding cardinal

willingness to buy for different alternatives, however the ability of voters to vote multiple times

in promotional elections allows for the cardinal strength of their preferences to be expressed in a

relatively effective manner. This feature of promotional elections provides firms with

comparable, however imperfect, information about the strength of voter preferences. This

cardinal preference information maps roughly onto cardinal willingness to buy. In this way,

allowing voters to vote multiple times offers firms a rough estimate of cardinal willingness to

buy, however flawed.

37    

The general rule for promotional elections is that a voter is limited to a single vote per

time period, for example a day, for the duration of the election. This allows for a voter with

stronger preferences to express those and the daily limit helps sort these voters. By preventing a

voter from voting over and over again in a single day, a firm keeps the cost of voting each time

almost as high as the first vote cast. Though a voter may know exactly how to cast their vote,

they must make the decision to navigate to the appropriate website and cast their vote each time.

This allows firms to collect some measure of the cardinal information associated with

willingness to buy for most goods. NBA.com’s fan night is an exception to this, as the relevant

willingness to buy is binary. Each individual can only watch a single game once, but for a more

typical good like Mountain Dew, a strong cardinal preference difference could mean the

difference between a loyal customer who would buy some new flavor daily and an occasional

customer who may only purchase a few times a year. Though this is a likely explanation for

differences in voting behavior, there are other, much less desirable explanations of unequal

voting behavior. It could be the case that an unemployed voter with little to no disposable

income may have all the time in the world to vote as often as possible, while an loyal consumer

with sufficient disposable income simply may not have the time to invest in voting more than a

few times. Even with some metric to be compared by, cardinal preferences are difficult to

compare between individuals.

The, the choice of firms to allow multiple votes has a few additional benefits. Among

these is the increased consumer interaction and web traffic generated by allowing multiple votes.

Customers are encouraged by the ability to vote multiple times to develop a stronger relationship

with the brand. Keeping the lockout window relatively short maximizes the number of potential

page visits as a result of the promotional election from each individual voter. An alternative

38    

method of roughly capturing cardinal information is currently being used in the Canadian

iteration of DEWmocarcy. In addition to their single vote per day, voters can get additional

votes by redeeming codes found on mountain dew products.51 As the alternative flavors are

temporarily available to familiarize voters, voters are already expressing their cardinal

willingness to buy in how many of units of the temporary flavors they buy. Voters can express

high cardinal willingness to buy before the vote itself by purchasing a lot of the temporary

product. By affording these voters additional votes, a firm includes cardinal voting information

in their choice between alternatives to select. Plurality voting fails however, to capture all of the

information about willingness to buy, by ignoring the possibility of an individual being willing to

purchase more than a single alternative.

Approval  

In an exceptionally simple deviation from plurality voting, approval voting removes the

requirement of voting for a single alternative. “In an election among three or more candidates

for a single office, voters are not restricted to voting for just one candidate. Instead, each voter

can vote for, or ‘approve of,’ as many candidates as he or she wishes.”52 The alternative with the

largest number of these “approval votes” is selected as the winner. Approval voting, like

plurality voting satisfies the consistency and participation criteria, however does not similarly

fail the favorite betrayal condition. Approval voting is, like plurality voting, monotonic, a vote

for an alternative can only help that alternative. However unlike plurality voting, there is no cost

attached to voting for a favorite in the form of a missed opportunity to vote for the preferred

candidate of the most likely alternatives. As though the binary of approval is cardinal measure,

                                                                                                               51  (PepsiCo  Canada  ULC  2)  52  (Brams  and  Fishburn,  Approval  Voting  xi)  

39    

it is plagued by the same problem all cardinal measures share; there is no way to exactly

compare individual cardinal preferences, particularly for political elections. What constitutes

“approval” likely varies between individuals. For promotional elections however, the cardinal

values for utility of a given alternative being selected are either positive or zero. If a voter likes

some alternative enough to be willing to buy it, they will have some interest in that alternative

being selected; however, the voter is completely unaffected and has no negative interest in the

selection of some alternative they would not be willing to buy. Because of this, binary

willingness to buy is a natural cutoff point for voter approval. As a result, approval voting is

effective at teasing out the binary of willingness to buy, and mapping it onto the binary of

approval. The process of approval voting does not contain any element that could tease out the

magnitude aspect of willingness to buy. However, if a promotional election utilizing approval

voting is held in a similar format to current plurality promotional elections, allowing single

voters to cast multiple votes, the magnitude of willingness to buy could be estimated by that

behavior. In this manner, the quantity and content of approval votes could provide a complete

picture that captures both the magnitude and binary nature of willingness to buy respectively.

That approval information is then conveyed directly to the firms through the vote. This is not

without its problems. Using multiple approval votes as a mechanism to determine magnitude of

willingness to buy has the same shortcomings as the use of multiple plurality votes, it cannot be

determined if this information can accurately translate into magnitude of willingness to buy. In

addition, approval voting is subject to strategy that may cause votes to not accurately convey the

binary of willingness to buy. The obvious plurality voting strategy of voting for a lesser-evil to

prevent a greater-evil still exists in approval voting. If voters are concerned about a popular

greater-evil candidate, the same strategy of voting for a lesser-evil remains effective. Voters

40    

simply vote such that their approval cutoff lies between their lesser-evil and greater-evil choices,

approving of all alternatives above that cutoff. Because a voter is indifferent between the

outcomes of different alternatives that they would not buy, it is not rational for a voter to

strategically vote in this way between two alternatives they would not buy, despite some

preference between the two. This sort of strategic voting may however cause a voter to not

approve some alternatives they are actually willing to buy. Suppose individual i’s preference

ordering is {u,v,w,x,y,z}, and i is willing to purchase the set of alternatives {u,v,w,x}. If i has

some knowledge about the preferences of other individuals and believes the election to be

closely contested between v and w, it is in i’s interest to vote such that they approve only {u,v}

to maximize the chance that v is selected over w. This type of strategic voting is certainly a

concern for approval voting; however at its very worst, strategic plurality voting provides the

same information as honest plurality voting. If every voter either only approves of their favorite

or perceives their second preference as a threat to win over their favorite, they will simply vote

for their favorite. There is no incentive for an individual to vote for a single alternative that is

not their favorite. Because in this context voters have no incentive to vote for an alternative they

would not buy, any additional votes only provide a firm with more complete information about

binary willingness to buy. When compared to honest votes, a strategic vote in approval voting

loses some information about willingness to buy lower-preferred alternatives, while a strategic

vote within a plurality voting election loses information about willingness to buy an individual’s

highest-preferred alternative.

Another significant shortcoming of approval voting is its failure of the Pareto condition.

Imagine two well liked alternatives x and y such that every voter is willing to purchase both,

however    ∀  𝑖  𝑥  𝑝!  𝑦. If a promotional election was held with approval voting with these

41    

individual social preference, it would have no information to conclude that  𝑥𝑃𝑦, but would rather

lead to the assumption that  𝑥𝐼𝑦. Though a small concern for promotional elections due to their

relative unimportance, it is possible for misinformed voters to strategically vote such that  𝑦  𝑃  𝑥

appears to be the social choice.53 Even if this is the case, it is never strategically rational to not

vote for an individual’s favorite, and the information yielded by approval voting will provide a

more clear representation of voters’ binary willingness to buy.

Range  

Closely related to approval voting is range voting. The similarity is most obvious if

approval voting is described as a voter expressing cardinal preferences on each candidate within

a binary context. A voter can give a candidate a score of either 1 or 0 by “approving” or not.

Range voting removes the binary context and opens up a range for voters to score candidates on.

Though range voting can use any range, a 1-10 scale is not uncommon and will be used here as

representative for any discussion of particular ranking strategies. The scores are summed and the

alternative with the highest score is selected. Range voting passes the exact same set of criteria

that approval voting passes, and is subject to the same issue as approval voting. The difference

lays in the opportunity range voting offers voters to express cardinal information about their

preferences. This strength is also range voting’s weakness. In order to be able to compare the

valuable information about cardinal willingness to buy, the ballot must be structure such that

voters are scoring their candidates in accordance with how much they intend to purchase.

In approval voting, what constitutes “approval” may vary between individuals,

particularly for political elections. With a non-binary upon which to express cardinal preferences,

                                                                                                               53  (Brams  and  Fishburn,  Approval  Voting  140-­‐1)  

42    

the meaning behind each vote would be impossible to tease out without direct comparison to a

metric. If a ballot asks for arbitrary score estimates of preference or willingness to buy, rather

than an estimate of number of units purchased in some timer period, it will yield results that are

impossible to interpret. This issue is very significant within the context of promotional elections.

The binary nature of approval voting valuably encourages voters to express their binary

willingness to purchase. Range voting, because of its non-binary format is employed commonly

for online product reviews. If voters intuitively perceive a range voting ballot in this way, they

will likely vote according to the cardinal information associated with their preferences. Even if

they are not willing to purchase x or y, voters are likely have some opinion between x and y, and

that opinion is easy to naturally express through range voting. Beneficially, this allows voters to

express cardinal preferences. Unfortunately, in the absence of a comparable metric, this offers

little to no information to firms beyond ordinal preferences. Each individual has some cutoff

point, a score above which indicates willingness to buy, however this cutoff point is likely

unclear and different for every individual. In this way, range voting risks making information

about binary willingness to buy indistinguishable.

Additionally, the presence of strategy removes a lot of the benefit of a wider range of

options to express cardinal preference. To not rate at least one candidate the maximum is

irrational, and effectively lowers the strength of that individual’s vote. If a voter is generally

aware of the preferences of other individuals, an expected utility maximizing voter will vote

nearly identically to how they would in approval voting. They would vote the maximum score of

10 for the more preferred of two tightly contested candidates, referred to as lesser and greater

evil for consistency. So as not to decrease the chances of their more preferred candidates relative

to the lesser evil, a strategic voter would vote the maximum for all alternatives more preferred to

43    

the lesser evil. In order to minimize the chance of the greater evil winning, they would score that

alternative the minimum of 1. So as not to increase the chances of their less preferred

alternatives to the greater evil, each of those alternatives would receive a 1 as well. The

difference from approval voting would come solely in the ability of the strategic voter to

honestly score alternatives that they rank between their lesser and greater evil. As it does for all

voting systems, strategic range voting relies on information about the preferences of other

individuals participating in the system. As a result, the issue of strategic voting can be avoided if

the alternatives are all unfamiliar to the voter.

Borda  Count  

One of the oldest voting systems, which predates the Condorcet criteria, is the Borda

count. Proposed by, and named after Jean-Charles de Borda, the Borda count is the most famous

representable voting system. Describing his voting system for a three candidate election, Borda

states that “if we take a to be the degree of merit which each voter attributes to last place and a+b

the degree of merit attributed to second place, we can represent first place by a +2b.”54 This

voting system attributes point values to each of the positions in a ranking and selects the

alternative with the highest point total. It fails the Majority criteria and Condorcet criteria,

however passes the Condorcet loser criteria. It fails the Arrovian favorite betrayal and IIA

criteria, though passes the consistency and participation criteria. To illustrate how the Borda

count works in practice and conflict with Condorcet methods, consider the following:

3 Voters 2 Voters

x z

                                                                                                               54  (Borda)  

44    

z y

y x

The Condorcet winner is clearly x, as xPy 3:2 and xPz 3:2. The Borda count however, if a first

place ranking is valued at 3, a second at 2, and a third at 1, yields the following scores; x:

3(3)+2(1)=11, y: 3(1)+2(2)=7, z: 3(2)+2(3)=12. The Borda count winner is z. The chief issue

with the Borda count in the context of promotional elections is that the Borda count does not

offer any information regarding a voter’s willingness to buy. There is some cutoff within the

voter’s preference ordering above which they would be willing to buy, and the Borda count is

completely insensitive to this aspect of the voter’s information. This is because even binary

willingness to buy is cardinal information, which the Borda count only offers a mechanism to

express ordinal preferences. This can have horrible implications, such as in the following

extreme case. The score associated with each preference ranking is provided to the left of the

preferences.

Score Voter 1 Voter 2 Voter 3 Voter 4

4 x(!) b(!) z(!) b(!)

3 w w w w

2 z z x x

1 b x b z

Alternative Borda Score

x 9

y 10

45    

z 9

w 12

Even though no individual would be willing to purchase w, the borda count dictates its selection!

A possible solution is allowing truncated ballots, which allows voters to rank as many candidates

as they would like, while all unranked candidates receive the minimum possible score. Similar

to range voting, this runs the risk of voters intuitively voting on their preferences between the

alternatives rather than their utilities of each candidate’s selection. Additionally, this leaves a

wealth of possibility for strategic voting. In situations where a voter perceives their favorite two

alternatives as the likely, it is rational to cast a single vote for only the favorite alternative. This

strategic voting can similarly be minimized if individuals do not have a perception of each

other’s preferences. If a truncated Borda count with complete voter honesty was used and

voters voted on their willingness to buy rather than their preferences, the firm would receive a

very clear picture of the willingness to buy binary and ordinal information about willingness to

buy magnitude. The act of providing a rank for an alternative would ideally serve as an

indication of willingness to buy.

Though this is highly desirable, under the same assumptions about voters voting in

accordance with their willingness to buy, a range system would provide more complete

information, as it would allow voters to express cardinal information about willingness to buy.

Additionally, as the Borda count fails the favorite betrayal condition, a truncated Borda count

may solicit inaccurate information about voter favorites as a result of strategic voting, unlike

range voting. An issue with both systems would be ballot complexity. If a voter had to fill out a

complicated ballot, the promotional election could feel less like a promotion and more like

46    

market research, causing the voter loses the feeling of interactivity and excitement, diminishing

the possibility for valuable customer relations gains from the promotional campaign.

Approval  Preference  Hybrids  

The appeal that exists for the truncated Borda system lies in mix between ordinal

preferences and binary willingness to buy. Though the ballot provides valuable information

from a market research perspective, the voting rule itself makes minimal use of the binary

element. By not ranking an alternative, voters score them the lowest possible. By the nature of

the Borda count, such a system could select the alternative that the fewest number of people were

willing to buy if a passionate minority existed amongst an otherwise competitive field. A

truncated Borda system is not alone in combining binary and ordinal information. A number of

ordinal preference and approval hybrids share this feature.

One such voting system is simply titled preference approval voting (PAV). A PAV ballot

would require voters to list both their complete ordinal rankings as well as approval votes. If no,

or a single alternative receives a majority, or over fifty percent, of the approval votes, the

approval winner is selected. If however, multiple alternatives receive a majority of the approval

votes, the ordinal preference rankings become important. Through pairwise comparisons, it is

determined if there is a Condorcet winner amongst the alternatives receiving a majority of the

approval votes. If there a Condorcet winner that alternative is selected. However it may be that

there is no Condorcet winner which must be a result of a cyclical paradox. If this is the case,

than the approval winner from those alternatives is selected.55 PAV passes the monotonicity

criterion for both the binary approval vote as well as the ordinal ranking vote. PAV is also

                                                                                                               55  (Brams  and  Sanver,  Voting  Systems  That  Combine  Approval  and  Preference  7-­‐8)  

47    

Pareto efficient; however PAV fails every other voting criterion. Despite this, it has a number of

desirable characteristics. PAV would be ideal for a firm interested in striking some balance

between maximizing binary and cardinal willingness to buy. PAV ensures that a majority

approval winner will be selected, if available. It only fails the majority criterion as it allows for

multiple alternatives to receive a majority of the votes. If two or more alternatives do not receive

majorities, PAV functions identically to approval voting. As an alteration from approval voting,

PAV offers a mechanism to differentiate between majority winners of the approval vote. This

causes the system to potentially fail to maximize binary willingness to buy, however allows the

voter to express their ordinal preferences, distinguishing alternatives from one another.

Another, similar system is Fallback voting. Contrary to PAV, Fallback voters only report

their approval as well as their ordinal preferences between the alternatives they approve of. In

such system the highest rank candidates are considered alone, much like plurality voting. If any

alternative is a majority winner, that alternative is selected. If no candidate receives a majority,

the procedure continues. Now considering both the first and second preferences of voter, the

question of majority winners is posed again. If there are any majority winners, the one with the

most approval votes wins. If not, the top three alternatives are considered, and so on and so

forth. If no majority is reached before running out of lower approved preferences to consider,

the alternative with a plurality of approval votes is selected.56 Fallback voting, similarly to PAV,

passes the monotonicity criteria and is Pareto efficient. However, unlike PAV, as a result of the

different role ordinal preferences play, Fallback voting passes the participation criterion as well.

Because it begins by only considering voter’s highest ranked alternatives, Fallback voting

resembles plurality voting with a majority requirement. However, in the lack of a majority, more

and more information becomes available, and if an alternative receives a majority of approvals, it                                                                                                                56  (Brams  and  Sanver,  Voting  Systems  That  Combine  Approval  and  Preference  13-­‐15)  

48    

will be selected. Fallback voting, in the manner it resembles plurality voting runs the risk of

selecting a majority approved candidate that is not majority preferred between multiple approval

majority winners. Fallback voting’s strength lies in its ballot simplicity relative to PAV.

Finally, let LLull-Smith(L-S) voting be considered. In such a voting system, similarly to

PAV, approval votes and total ordinal preference rankings are provided by voters. If a majority

of voters approve of a single candidate, that candidate wins. If multiple or no alternatives

receive majority votes, the Smith set is used to narrow down the alternatives. The winner is the

alternative with the highest number of approval votes within the Smith set. L-S passes the same

criteria as PAV does. As a deviation from approval voting, L-S is relatively mild. In fact, the

sole effect of ordinal preferences is to ensure that a Smith dominated alternative cannot be

selected by the voting system.

Each of these hybrid voting systems has a somewhat ambiguous impact on willingness to

buy. If a majority of voters are willing to buy at least one alternative, each voting system will

select an alternative that receives a majority of approval votes. Each trades off the binary

willingness to buy of the approval winner for some measure of higher ordinal preference.

Valuing higher ordinal preferences could but will not necessarily result in a voting system that is

perceived as more fair, and reflects higher cardinal willingness to buy. There are distinct

differences between each voting system and the situations in which they would be optimal. L-S

voting makes minimal changes to approval voting, and attempts to eliminate universally bland

yet acceptable alternatives if another option exists. PAV similarly seeks ensure that an

alternative that has less overall approval, but is strongly preferred is selected. Fallback voting

resembles plurality voting more than either of the other systems, and ensures that any alternative

that would be selected by a majority with plurality voting would be selected. All hybrid

49    

approval preference systems, like all voting systems would suffer from strategic voting. Strategy

would likely be similar to that of approval voting. However, as each of these systems passes

monotonicity, strategic voting can only results in incomplete binary willingness to buy

information as opposed to inaccurate information.

Random  Ballot  

To eliminate strategic voting completely, it would need to be clear that there was a

dictator element to the choice function. Within the context of voting rules, the term dictator has

a specific definition. A voter is a dictator if any preferences they express are also expressed in

the social preferences, regardless of all other voter preferences. For a single winner promotional

election, the cost associated with strategic voting is likely not as large as the cost of introducing

dictator power to the choice function, even when just taking voter disillusionment into effect.

The main purpose of a promotional election is promotional, and in order for a voter to be

enthused as a result of that promotion, their vote should count.

If, however a hypothetical firm were intent on eliminating strategy, but still wanted some

element of voter interaction, they could employ a random ballot voting system. In such a

system, voters would fill out a ballot with their preference information normally; however the

winner is selected by randomly selecting a ballot and choosing as if that individual’s preference

were social preferences. A random ballot system would fail all criteria concerning how a winner

should be chosen, as well as the consistency criterion. However, it would pass IIA, participation,

and would cause voters to have no incentive to vote strategically whatsoever. Because voters

would have no incentive to vote strategically, such a system would ensure Pareto efficiency.

Such a system would also provide firms with detailed, accurate information about voter

50    

preferences, but would completely fail to maximize willingness to buy. Unless a firm is

primarily interested in gaining accurate information about voter preferences, such a random

ballot voting system would be undesirable.

Conclusions  

My goal in this thesis was to determine whether or not a firm managing a promotional

election campaign has incentive to deviate from the use of plurality voting. In order to compare

voting systems, the criteria upon with they are conventionally analyzed are put into the context

of promotional elections. Voting criteria are however rules for the translation of individual

ordinal preferences to a social ordinal preference ranking. As such, the satisfaction of voting

criteria has very limited application to the maximization of willingness to buy. In order to

determine how a voting system might affect the sales of the winning alternative, the voting

systems themselves had to be analyzed.

An ideal choice function for a promotional election is one that solely and completely

incorporates the inputs of the voters, so long as those inputs are the information most valuable to

firms. These desirable inputs are the binary and cardinal information about voter’s willingness

to buy a product. Quite simply, if a firm desires any particular type of information from a voting

system, the system must ask for that information in its ballots. Approval and range voting are

examples of a firm simple asking each voter for their binary and cardinal willingness to buy

respectively. In a world where all voters voted honestly, these systems would maximize their

respective willingness to buy. Voters would express their willingness to buy directly on the

ballot, and the voting system selects that alternative that maximizes collective willingness to buy.

The presence of strategic voting makes it difficult to draw specific conclusions about willingness

51    

to buy. Though it is impossible to claim that any voting system is necessarily an improvement

over plurality voting for willingness to buy maximization, there is also no evidence that plurality

voting is superior to other systems. There is however some other certain benefit of approval

voting relative to plurality voting. Ballot results convey consumer information much like market

research to firms. Approval voting necessarily accomplishes this more completely.

Approval voting would account for, and allow voters to express exclusively binary

information about willingness to buy. If a firm is interested in utilizing ordinal ranking

information, any hybrid approval preference system could be an improvement upon approval

voting. If, a firm is particularly concerned about cardinal rather than binary willingness to buy

and trusts that voters will give honest cardinal information relative to some comparable scale,

range voting would be superior still, providing perfect market research information, and ensuring

maximization of cardinal willingness to buy.

Through a comparison of voting systems and their adherence to criteria, it has been

demonstrated that there are a variety of potential incentives for firms to deviate from the use of

plurality voting. It is possible to imagine a set of preferences for any voting system which cause

that system to lead to willingness to buy maximization, however no system can guarantee

maximization for all voter preferences. As such, no specific conclusions can be reached about

willingness to buy maximization. There are a few conclusions that can be reached about the

collection of voter preferences. Approval voting, even under purely strategic voting, yields at

least as complete information about voter’s binary willingness to buy as plurality voting.

Despite this, the simplicity and Pareto efficiency of plurality voting may make it the best choice.

The strong appeal that exists for other systems leads to an even less clear picture of what system

should be used. If a firm intends to deviate from the use of plurality voting, their decision should

52    

rest largely on what type of willingness to buy they are interested in maximizing. Approval

voting lends itself towards maximization of binary willingness to buy, while providing a clear

improvement over the information that plurality voting provides. If a firm trusts their voters to

vote honestly rather than strategically, a range system would provide yet more complete

information, and would allow the maximization of cardinal willingness to buy. If a firm is not

concerned with necessarily maximizing either binary or cardinal willingness to buy, but finding

some balance between the two, a hybrid approval preference system or range voting may be even

more desirable. To some extent, the choice between voting systems boils down to a subjective

decision for a firm. There are a number of reasons to believe that some alternative system may

yield better results than plurality voting. A firm that is willing to take a risk on an alternative

voting system for a promotional election has relatively little to lose, and the potential to

maximize their gains in what is already a very effective marketing technique.

53    

Glossary  Variables  

Term   Definition  i   An  individual  P   Strict  social  preference.    xPy  means  x  is  strictly  

preferred  to  y  Pi   Strict  preferences  of  individual  voter  i  R   Social  preference  relation  that  implies  either  

preference  or  indifference.  If  xRy  either  xPy  or  xIy  is  true,  but  not  yPx  

Ri   Preference  relation  for  voter  i  I   Social  indifference  m   Number  of  feasible  alternatives  |M|=  m  M   The  set  of  feasible  alternatives  

u,v,w,x,y,z   Variable  alternatives  PROF   Profile  of  individual  preferences  prefi   Individual  preferences  of  voter  i  F   Social  choice  function  C   Choice  set  N   Set  of  voters  

Mathematic  terms  Term   Definition  {}   Set  brackets  ∀   For  all  

→      →   If,  then  ∈   Is  an  element  of  ⊂   Is  a  subset  of  |    |   Size  of  a  set  ∃   There  exists  ∪   Union  ∩   Intersection  ∅   Empty  set  

 

54    

Bibliography    Arrow,  Kenneth  J.  Social  Choice  and  Individual  Values.  New  York:  John  Wiley  &  Sons  Inc.,  1963.  

Borda,  Jean-­‐Charles.  "On  Elections  by  Ballot."  McLean,  Iain  and  Arnold  B.  Urken.  Classics  of  

Social  Choice.  Ann  Arbor:  University  of  Michigan  Press,  1995.  83-­‐90.  

Brams,  Steven  J.  and  M.  Remzi  Sanver.  March  2006.  Voting  Systems  That  Combine  Approval  and  

Preference.  16  May  2013  

<http://www.nyu.edu/gsas/dept/politics/faculty/brams/approval_preference.pdf>.  

Brams,  Steven  J.  and  Peter  C.  Fishburn.  Approval  Voting.  Boston:  Birkhäuser,  1983.  

Brennan,  Geoffrey  and  Alan  Hamlin.  Democratic  Devices  and  Desires.  Cambridge:  Cambridge  

University  Press,  2000.  

DominosAustralia.  Domino's  Australia  Social  Pizza.  15  March  2012.  10  2013  May  

<http://www.youtube.com/watch?v=mQx97f9gIvM&list=UUpdl_zwlPHT96OVYmlfJWrQ

&index=3&feature=plcp>.  

FairVote.  Fairvote.org  |  Single-­‐winner  Voting  Method  Comparison  Chart.  April  2009.  30  April  

2013  <http://www.fairvote.org/single-­‐winner-­‐voting-­‐method-­‐comparison-­‐chart#32>.  

Fishburn,  Peter  C.  "Condorcet  Social  Choice  Functions."  SIAM  Journal  on  Applied  Mathematics  

November  1977:  469-­‐489.  

—.  The  Theory  of  Social  Choice.  Princeton,  New  Jersey:  Princeton  University  Press,  1973.  

Frito-­‐Lay.  Doritos  Crash  the  Super  Bowl.  2012.  10  May  2013  

<https://apps.facebook.com/crashthesuperbowl/>.  

55    

Frito-­‐Lay  North  America.  Lay's  Potato  Chips,  Eva  Longoria  And  Chef  Michael  Symon  To  

Introduce  Three  Finalists  Vying  For  $1  Million  Or  More  In  Grand  Prize  Winnings  In  The  

Lay's  "Do  Us  A  Flavor"  Contest.  12  Febuary  2013.  10  May  2013  

<http://www.pepsico.com/PressRelease/Lays-­‐Potato-­‐Chips-­‐Eva-­‐Longoria-­‐And-­‐Chef-­‐

Michael-­‐Symon-­‐To-­‐Introduce-­‐Three-­‐Finalis02122013.html>.  

Gärdenfors,  Peter.  "Positionalist  Voting  Functions."  Theory  and  Decisions  September  1973:  1-­‐

24.  

Inada,  Ken-­‐Ichi.  Alternative  Incompatible  Conditions  for  a  Social  Welfare  Function.  Vol.  23.  The  

Econometric  Society,  October  1955.  

MacKay,  Alfred  F.  The  Paradox  of  Social  Choice:  A  Case  Study  in  the  Philosophy  of  Economics.  

New  Haven:  Yale  University  Press,  1980.  

Mackie,  Gerry.  Democracy  Defended.  Cambridge:  Cambridge  Universtiy  Press,  2003.  

Marquis  de  Condorcet,  M.  J.  A.  N.  de  Caritat.  "On  the  Constitutionand  the  Functions  of  

Provincial  Assemblies."  McLean,  Iain  and  Arnold  B.  Urken.  Classics  of  Social  Choice.  Ann  

Arbor:  The  University  of  Michigan  Press,  1995.  113-­‐145.  

Moulin,  Hervé.  "Condorcet's  Principle  Implies  the  No  Show  Paradox."  Journal  of  Economic  

Theory  June  1988:  53-­‐64.  

Mountain  Dew.  DEWmocracy  2  Media  Hub.  10  May  2013  

<http://www.dewmocracymediahub.com/>.  

NBATVfannight.  NBA  TV  Fan  Night  on  Twitter.  10  May  2013  

<https://twitter.com/NBATVfannight>.  

56    

Nurmi,  Hannu.  Comparing  Voting  Systems.  Dordrecht,  Holland:  D.  Reidel  Publishing  Company,  

1987.  

Ordeshook,  Peter  C.  Game  theory  and  political  theory.  Cambridge:  Cambridge  University  Press,  

1986.  

Ossipoff,  Mike  and  Warren  D.  Smith.  Survey  of  Voting  Methods  that  avoid  Favorite-­‐Betrayal.  

January  2007.  2  May  2013  <http://rangevoting.org/FBCsurvey.html>.  

PepsiCo  Canada  ULC.  "MOUNTAIN  DEW®  DEWMOCRACY  OFFICIAL  CONTEST  RULES."  12  May  

2013  <https://dew.appsrvs.com/pdf/Dewmocracy_ContestRules.pdf>.  

Richelson,  Jeffrey  T.  "Running  off  empty:  Run-­‐off  point  systems."  Public  Choice  35  (1980):  457-­‐

468.  

Sen,  Amartya  K.  Collective  Choice  and  Social  Welfare.  San  Francisco  ;  Edinburgh:  Holden-­‐Day  

Inc.;  Oliver  &  Boyd,  1970.  

Shulze,  Markus.  "A  new  monotonic,  clone-­‐independent,  reversal  symmetric,  and  condorcet-­‐

consistent  single-­‐winner  election  method."  Social  Choice  and  Welfare  Febuary  2011:  

267-­‐303.  

Smith,  John  T.  "Aggregation  of  Preferences  with  Variable  Electorate."  Econometrica  November  

1973:  1027-­‐1041.  

Straffin,  Philip  D.,  Jr.  Topics  in  the  Theory  of  Voting.  Boston:  Birkhäuser,  1980.  

Svensson,  Lars-­‐Gunnar.  "The  proof  of  the  Gibbard-­‐Satterthwaite  Theorem  revisited."  Working  

Paper.  1999.  

Threadless.  Frequently  Answered  Questions.  11  November  2010.  2  May  2013  

<http://support.threadless.com/ics/support/default.asp?deptID=15140>.  

57    

Toyota.  Toyota  Prius  Projects:  No.007.  January  2011.  10  May  2013  

<http://www.toyotapriusprojects.com/#/007>.  

Woodall,  Douglas  R.  "Monotonicity  and  Single-­‐Seat  Election  Rules."  Discrete  Applied  

Mathematics  1997:  81-­‐98.  

Young,  H.P.  "Social  Choice  Scoring  Functions."  SIAM  Journal  on  Applied  Mathematics  June  1975:  

824-­‐838.  

Zmuda,  Natalie.  New  Pepsi  'Dewmocracy'  Push  Threatens  to  Crowd  Out  Shops.  2  November  

2009.  10  May  2013  <http://adage.com/article/agency-­‐news/pepsi-­‐dewmocracy-­‐push-­‐

threatens-­‐crowd-­‐shops/140120/>.