Crowdsourcing Applications of Voting Theory

Transcription

1 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.

2 Contents Introduction... 3 Assumptions... 6 Voting Criteria Condorcet Smith Condorcet Loser Majority Independence of Irrelevant Alternatives Consistency Participation Favorite Betrayal Monotonicity Pareto Efficiency Arrow s Impossibility Theorem Voting Systems Plurality Approval Range Borda Count Approval Preference Hybrids Random Ballot Conclusions Glossary Bibliography

3 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) 3

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) 4

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 followers 8, 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 18 th 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) 5

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, 6

7 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 7

8 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 8

9 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 9

10 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. 11 Transitivity 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 P i y, y P i z, and z P i x. The notation P i stands between two alternatives to indicate a preference of individual i. For example, x P i 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) 10

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 11

12 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 core 13, 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) 12

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) 13

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. Smith 16, 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) 14

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) 15

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) 16

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 17

18 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 S 2, and S 1 is a subset of S 2, x must be selected from S 1 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) 18

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) 19

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 ) 28 (Brennan and Hamlin ) 20

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 N 1 and N 2 be two groups of voters such that N! N! = with preference profiles PROF 1 and PROF 2 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, PROF x )=C x : 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) 21

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 N 1 = {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 N 2 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) 22

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) 23

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) 24

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) 25

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) 26

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 pref i and pref i, 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) 27

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) 28