Before determining what strategies should be permitted and how they should be formulated, we considered the strategies that rational voters might like to use. We assume that voters are rational if they take the actions they believe are most likely to lead to their most preferred outcome. We are not concerned with whether it is rational for them to prefer that outcome. As already discussed, several authors have suggested that voters may use expected-utility models involving their personal preferences and the probable preferences of the rest of the electorate to determine their rationally optimal strategies. In fact, Black [3] used the approach outlined by McKelvey and Ordeshook [15] to model the 1968 and 1972 Canadian Federal elections and found ``significant empirical support for the role of probabilities in the multicandidate calculus of voting.''

McKelvey and Ordeshook [15] deduced that in **N**-candidate
elections one can setup **N** difference equations that calculate the
expected gain in utility associated with a given voter voting for a
particular candidate rather than abstaining. Assuming that the voter
has established a set of utility values
specifying his or her personal utility for candidate **N** winning the
election, and assuming that the probability of a three-way tie is
negligible, the difference equations for a plurality election which
selects a single winner can be written in the form:

where is the expected utility associated with voting for
candidate **k**, is the expected utility associated with
abstaining, and (which we shall refer to as a * pivot
probability*) is the probability that the voter will be *
decisive* in creating or breaking a tie between **i** and **k**. Note
that the voter affects the election result if and only if he or she
casts a vote that creates or breaks a first-place tie. A rational
voter will vote for the candidate **k** that maximizes the equations.

Consider, for example, a three candidate plurality election. The voter can combine the difference equations for his or her two most preferred candidates (denoted and ) to form the equation:

Thus can be used to determine the candidate for which it is rationally optimal to vote:

The McKelvey and Ordeshook model is ambiguous about how the voter should vote if . When this occurs, the voter cannot expect to be better off voting for rather than or vice versa. For example, this situation occurs when the voter is indifferent between all candidates. In this case the voter might decide to abstain, make a random selection, adopt a bandwagon strategy (voting for the candidate most likely to win), or adopt an underdog strategy (voting for the candidate most likely to lose). Although unlikely, this situation can also occur when the voter is not indifferent between all candidates. In this case the voter may choose any of the above options or vote for as a show of support.

Hoffman [13] paraphrased the equation into a
* critical value* formula:

indicating that a voter should vote for if and if . We shall use Hoffman's formulation throughout the remainder of this paper.

In developing our declared-strategy voting system, we assume that rational voters will develop declared strategies consistent with the expected-utility model. Therefore, to accommodate these voters, our system need only accept strategies based on the expected-utility equations for the voting procedure being used (simple plurality, approval, etc.) and the number of candidates under consideration. In fact, a tie-breaking rule (for example, bandwagon strategy) plus a single strategy function in terms of utilities and pivot probability estimates should be sufficient for representing the declared strategy of any rational voter. This approach even accommodates those voters who wish to vote despite being indifferent between all candidates. These voters may submit equal utility values for all candidates and have their votes based solely on their tie-breaking rules. One situation where this might prove useful involves voters who are indifferent between the candidates running in their party's primary but want to vote for whichever candidate is most likely to win in order boost the winner's share of the vote and make their party appear more unified.

The expected-utility calculus can also be extended for elections that
use decision rules other than plurality and that select more than one
winner. For example, Merrill [16] presented a model in
which a voter's utility differences for a pair of candidates was
multiplied by the difference in votes that the voter awarded to each
member of that pair of candidates. This accommodates a voting system
such as Borda count in which the voter casts **N-1** votes for his or
her most preferred candidate, **N-2** votes for his or her second most
preferred candidate, and so on.

For the purposes of this paper, all voters in a given election are essentially using the same (rational) voting strategy with differing parameters; thus, voters need only submit the relevant parameters as part of their ballots. In this system we will ask voters to submit their utility values for each candidate on a scale of 0 to 10, as well as their tie-breaking rules. Depending on the probability calculation method used, we may also require the voters to submit a number that indicates their attitude toward risk. The other parameters---the probability values---can be calculated on a voter's behalf as explained in the next section. Thus, voters need not understand the process of formulating an expected-utility maximization function to use this system.

Note that we ask voters to submit their sincere utility values for all candidates. Our system will calculate the optimal decision-theoretic strategy based on these values, enabling voters to vote strategically while simultaneously revealing their sincere preferences. Thus it is generally advantageous for voters to submit their sincere utility values. However, as we will discuss later, voters who have the necessary knowledge to formulate sophisticated strategies might not find it to their advantage to reveal their sincere utility values (and will act accordingly).

Thu Apr 25 14:31:14 CDT 1996