Mathematical modeling of elections has progressed since the work of Anthony Downs. This research can be roughly divided into two types. The first asks, “What kind of policies will candidates offer?” and the second, “Will voters vote or not?”
The first type is called the “spatial theory of elections.” It places rationally-acting candidates into a hypothetical election under certain assumptions, and asks what policies will represent the maximum exercise of candidates' rationality. Recent research has built spatial theories addressing the two-step process of primary and general elections, and analyzed elections under the single non-transferable vote system, among other topics.
The second type of research tries to establish the conditions under which rationally-acting voters will vote or abstain. This work has so far identified the size of voting costs and of differentials in expected utility from different candidates' victories as key influences over voting participation.
There seem to be two trends ahead in the field. The first will be continued efforts to bring hypothesized election conditions closer to reality. The second will be the application of actual election data to various mathematical models in order to identify the best among them.
