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Background and Motivation

Early criticism of manipulable voting systems equated manipulation with dishonesty. The eighteenth century mathematician Jean-Charles de Borda is said to have responded to criticism that his Borda count voting scheme was manipulable by saying, ``My scheme is only intended for honest men'' [2]. But later theoreticians dismissed Borda's assumption that people are honor-bound to vote for their most preferred candidate. In an 1876 pamphletgif written well before the development of game theory, C. L. Dodgson suggested that an election be thought of ``more as a game of skill than a real test of the wishes of the electors.'' He proposed that ``as my own opinion is that it is better for elections to be decided according to the wish of the majority than of those who happen to have most skill in the game, I think it desirable that all should know the rule by which this game may be won.'' Although Dodgson elaborated on the rules for winning certain types of games, little additional work was done in this area until 1953 when Farquharson set out to apply game theory to voting procedures [9].

Farquharson began by introducing the concept of a voting strategy. Farquharson described a voting strategy as a plan made prior to an election that prescribes the course of action a voter should take given any contingency that can arise. In a single round election, a strategy is simply the voter's plan for how to cast a single ballot. However, in a multiple round election such as one that involves choosing between two alternatives at a time, a strategy must include a plan for all pairs of alternatives that could possibly be presented. Farquharson also discussed sincere voting, in which voters always vote for their most preferred alternatives, and sophisticated voting in which voters select utility maximizing strategies that take into account the strategies of the other voters [9].

In this paper we follow Farquharson's terminology and distinguish sophisticated voting from strategic voting, using the following definitions:

Sincere vote:
A vote such that for any two alternatives under consideration, the voter has given greater or equal weight to the more preferred of those alternatives.
Strategic vote:
A vote that is rational, but not sincere. Often such a vote will be determined through decision-theoretic analysis, which takes into account the preferences of the other voters.
Sophisticated vote:
A vote determined optimal through game theoretic analysis, which takes into account the likely strategies of the other voters. Note that all sophisticated votes are also strategic. Sophisticated votes may be viewed as higher-order functions than simple strategic votes.

We will also refer to strategic or sophisticated votes as insincere votes.

The application of game theory to voting theory brought on a variety of new approaches to studying voting schemes and voter behavior. It led to further investigation of the conditions under which voting schemes might be immune to manipulation, and it led to the development of models designed to provide rational choice explanations for voter behavior.

Several voting theorists, including Farquharson and Dummett [8], conjectured that it would be impossible to find a voting scheme immune to manipulation. Indeed, in the early 1970s, Gibbard and Satterthwaite independently proved that all non-dictatorial voting schemes with at least three possible outcomes are manipulable [12,21]. Gibbard also demonstrated by example the existence of a ``mixed decision scheme'' that is neither manipulable nor dictatorial, and can allow more than two outcomes. In this scheme each voter submits a ballot containing a vote for a single alternative. One ballot is then selected at random, and the alternative specified in that ballot is declared the winner. This scheme is unsuitable for most purposes because it relies heavily on chance. Gibbard suggested that further work be done to explore decision schemes that do not leave ``too much to chance.'' This suggestion was taken up by Barbera [1], who showed that ``the only selection methods that can be both nondictatorial and nonmanipulable are those in which chance plays an extensive role.'' Gardenfors [11] also extended the work of Gibbard and Satterthwaite, focusing on voting schemes that do not necessarily select a single outcome. He showed that most of these schemes are manipulable, and those which are not tend to be very indecisive.

While these theorists were exploring the extent to which voting schemes are manipulable, other voting theorists sought to determine the optimal strategies for voters to use. McKelvey and Ordeshook developed ``A General Theory of the Calculus of Voting'' in which they derived decision rules that could be used by voters to determine their optimal strategies [15]. This article extended ``A Theory of the Calculus of Voting'' in which Riker and Ordeshook derived decision rules that could be used to determine whether it was rational for a particular voter to vote at all [20]. The theory was further generalized by Hoffman in ``A Model For Strategic Voting'' [13].

All of the decision models cited here are expected-utility models that require voters to consider their personal preferences as well as the probable preferences of the rest of the electorate. Information about the preferences of others allows voters to determine the relative probability of each candidate winning the election. When voters are able to determine such probabilities, they are said to be making decisions under risk. Without information about the preferences of others, voters are not able to determine the probabilities of the various contingencies. When this occurs voters are said to be making decisions under uncertainty. Merrill showed that the sincere strategy is always the optimal strategy when decisions are made under uncertainty in plurality, Borda, and approval voting elections [17]. This is likely the case for other voting schemes as well. Merrill's results suggest that voters are only able to manipulate voting systems when they are making decisions under risk. When voters have no information about the preferences of others and therefore must make their decisions under uncertainty, they cannot manipulate the voting system.

Others have also pointed out that the fact that a voting system is manipulable does not imply that it will actually be manipulated. As Gibbard explained:

... to call a voting scheme manipulable is not to say that, given the actual circumstances, someone really is in a position to manipulate it. It is merely to say that, given some possible circumstances, someone could manipulate it [12].
The circumstances required for manipulation are:
  1. a set of preferences for each voter such that at least one voter or group of voters can obtain a more preferred outcome by voting insincerely,
  2. at least one voter who has sufficient information about the preferences of other voters to determine that he or she can obtain a more preferred outcome by voting insincerely, and
  3. at least one voter who has sufficient knowledge about how to solve utility maximization problems to determine his or her utility maximizing strategy.

In an election where the required circumstances for manipulation are present, not all voters who could benefit from voting strategically will have sufficient knowledge and information to formulate a utility maximizing strategy. As we have discussed, the ability to manipulate is dependent on the individual voter's ability to gather information about the preferences of the other voters --- but there may be costs associated with gathering this information [3]. Depending on the type of election, money, education, access to media, time, and political alliances may be necessary resources for gaining accurate voter preference information. Because of the difficulty some voters may have in obtaining this information themselves, it is conceivable that voters might accept inaccurate estimates --- perhaps propagated by certain candidates or their supporters --- as the truth. Indeed, it has been reported that supporters of Pat Buchanan actively sought out poll takers during the 1992 and 1996 Republican Presidential primaries so that pre-election polls would show more support for Buchanan than actually existed [7]. In addition, as noted by Black [3], some voters ``may hold probability estimates that are inflated or clearly erroneous or the product of the simple mimicking of the views of a close friend or spouse.'' Moreover, the ways that poll data and other preference information flow through society and are interpreted by voters are not well understood [4].

Thus a significant problem with manipulable voting systems is that they allow voters with more accurate information about the preferences of others to use their votes more effectively than other voters, essentially granting these voters a weighted vote. This is contrary to the principles espoused by many democratic countries and organizations in which all people are given the same amount of say in the electoral process, regardless of their means. Those who subscribe to these principles often expect that by granting each individual one vote, they are granting all people equal voting power. However, this is not necessarily the case when some voters have the means to obtain information about the preferences of other voters which is unavailable to everyone. These voters may use this information to formulate optimal voting strategies which cannot be identified in the absence of such information.

Another problem with manipulable voting systems is that not all voters understand the formulation of utility maximizing functions. Thus even if all voters have equal information about the preferences of others, those who understand the formulation procedure have more power than those who do not. Riker [19] notes that when strategic voting occurs frequently, the meaning of social choices ``may consist simply of the tastes of some people (whether majority or not) who are skillful or lucky manipulators.''

Finally, manipulable voting systems frustrate attempts at analyzing and interpreting election results because analysts cannot distinguish between voters who voted sincerely and those who voted insincerely. Thus election results give little indication as to whether a winning alternative had strong support from those who voted for that alternative. As noted by Knight and Johnson [14], ``Social choice theorists demonstrate that any electoral outcome is at least partly an artifact of the aggregation mechanism through which it is produced. Therefore, electoral results always require interpretation and justification.'' Insincere voters add a degree of uncertainty to such interpretation.

In the next section we will develop a framework for a voting system in which all voters are given equal access to information about the preferences of others, voters may vote strategically without having to understand the formulation of utility maximizing strategies, and election results reveal the sincere preferences of the electorate in addition to the election outcome.

next up previous
Next: A Framework for Up: Towards an Information-Neutral Previous: Introduction

Lorrie Faith Cranor
Thu Apr 25 14:31:14 CDT 1996