The structure of the Electoral College based U.S. Presidential elections system suggests a certain approach to choosing campaign strategies by U.S. Presidential candidates, and problems associated with finding competitive strategies of the candidates are considered. Most of the problems are formulated as discrete mathematical programming ones or as those with mixed variables, whereas some of the problems are formulated as game ones. Approaches to solving all the considered problems with the use of both widely available and experimental software are proposed.
The U.S. Presidential elections system is unique and very logically designed although rather
complicated for understanding in depth [1]. From the author's viewpoint, this system has not
been studied to a degree allowing one to understand, in particular, how quantitative regularities
embedded in the system affect campaigns of U.S. Presidential candidates. Only a few publications
address some of these problems and propose certain approaches to their solving in particular
cases (however, mostly, when only two candidates really compete in the race) [2-5]. At the same
time, the Electoral College mechanism proposed by the Founding Fathers in the form similar to
that used in the Centurial Assembly system of the Roman Republic [6] immediately suggests a
manner in which U.S. Presidential candidates may design their campaigns. Namely, according to
the U.S. Constitution, each of 51 places (states and the District of Columbia (DC)) appoints a
particular number of the electors, and this number is subject to corrections every ten years [7].
The " winner-take-all" principle determines a manner in which a U.S. Presidential candidate who
receives a plurality of the popular vote in each of the states (except for Maine and Nebraska) and
in DC is awarded the whole number of the electoral votes which each such state or DC appoints
in the election [8]. To win a particular U.S. Presidential election in the Electoral College, the
successful candidate must receive a majority of the whole number of the electoral votes that are
in play in the election. Currently, such a number does not exceed 538, and the above-mentioned
majority can vary depending on this number.It is clear that winning a plurality of the popular vote in a particular state and DC implies
extensively campaigning there, which requires monetary and time resources. It is natural to
assume that these resources are limited for each U.S. Presidential candidate in each U.S. Presidential
election. So it is expedient to solve a problem of finding such combinations of states
and DC the winning of the electoral votes in which secures the winning of the election in the
Electoral College for a particular U.S. Presidential candidate while the total amounts of both
resources fall within the limits existing for the candidate. Such a problem is easily formulated as
a discrete optimization one of a particular kind, namely, as a Boolean knapsack problem (with
an additional constraint) [9,10], and its solution determines possible ways of designing campaigns
for U.S. Presidential candidates.
It turns out that this problem is not the only one which U.S. Presidential candidates' teams
could be interested to consider; however, even this problem appears in several modifications. In
particular, as long as winning a plurality of the popular vote in each state and DC can, generally,
be attained only with a certain probability, approaches to allocating the resources depending on
such probabilities for the states and DC are also expedient to consider.
The present article addresses the above-mentioned problems, along with others relevant to
them, and suggests mathematical models for a formalized analysis of all the considered problems,
as well as approaches to solving these problems. In all considerations throughout the article, for
the sake of simplicity, we use current values of some parameters of the U.S. Presidential elections
system. In particular, we assume that, for instance, the total number of the electoral votes that
are in play in the election under consideration equals 538, and no stipulations on this matter are
further made.
