Several authors have presented evidence that some voters 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 actual elections. In fact, Black [14] used the approach outlined by McKelvey and Ordeshook [69] 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 [69] deduced that in N-candidate
elections one can establish N difference equations that model 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 each candidate
winning the election, and assuming that the probability of a three-way
tie is negligible, the expected-gain equations for a plurality
election that selects a single winner can be written in the form
where 
is the expected utility associated with voting for
candidate i, 
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 first-place tie between i and
j. For elections that select M alternatives, the pivot
probability is the probability that the voter will be decisive in
creating or breaking an 
-place tie. Note that the voter affects
the election result if and only if he or she casts a vote that creates
or breaks an -place tie. A rational voter will vote for the
candidate i that maximizes the 
expected-gain
equations.
Consider, for example, a three-candidate plurality election. The
voter can combine the expected-gain equations for his or her two most
preferred candidates (denoted 
and 
) to form the
equation 
:
By plugging in the voter's utilities 
and the
pivot probabilities 
, can be
calculated and used as follows 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 
. For example, this situation occurs
when the voter is indifferent between all candidates, that is 
. When this occurs, the voter cannot expect to be
better off voting for rather than or vice versa. In
this case the voter may defer to a tie-breaking rule which
might, for example, specify abstention, a random selection, or a
bandwagon or underdog strategy. 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
(the voter's preferred candidate) as a show of support.
The expected-gain equations presented for plurality elections in
Equation 
may also be applied to cumulative vote
elections [55] and Borda count elections [70]
that select a single alternative. In cumulative vote elections the
strategic voter should distribute all of his or her votes to the
alternative i that maximizes . In Borda count
elections the strategic voter should rank the candidates according to
their values. The expected-gain equations may also be
used for single non-transferable vote elections (SNTV). In this case
they correctly result in rational voters abandoning their most
preferred candidates if they are either too far behind or sufficiently
far ahead that their victory is assured [34].
Approval voting requires a slightly different approach: strategic
voters should always vote for their first choice candidate, never vote
for their last choice candidate, and vote for any other candidate
if and only if [55]
The formulation of optimal strategies becomes somewhat more complicated when multiple alternatives are to be selected and voters must vote for L>1 alternatives, as in the limited vote method and plurality rule applied to multiple alternatives. Hoffman suggests that voters consider every possible group of alternatives that might be elected and assign a utility value to each group (this may be accomplished more easily by assigning utilities to each alternative and summing the utilities for each group member to determine the group utility). Each group can then be considered as a distinct alternative and the plurality expected-gain formula may be applied [55]. However, as the number of alternatives increases this approach can result in large numbers of expected-gain equations; for example, 210 equations must be considered when selecting six alternatives from among 10 choices. (This approach is by no means intractable; however, as the number of equations remains of polynomial order.) Another approach would be to apply the plurality expected-gain formula as with SNTV, but vote for the candidates that correspond to the top L expected gains. This approach assumes the alternatives are essentially independent.
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-gain
equations for the voting procedure being used (simple plurality,
approval, etc.) and the number of candidates under consideration. In
fact, we have found that a tie-breaking rule plus a single
strategy function in terms of utilities and pivot probability
estimates are 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, as we shall
see in the Section .
