Disclaimer. Don't rely on these old notes in lieu of reading the literature, but they can jog your memory. As a grad student long ago, my peers and I collaborated to write and exchange summaries of political science research. I posted them to a wiki-style website. "Wikisum" is now dead but archived here. I cannot vouch for these notes' accuracy, nor can I even say who wrote them. If you have more recent summaries to add to this collection, send them my way I guess. Sorry for the ads; they cover the costs of keeping this online.
Goldman. 2002. A causal responsibility approach to voting. In Democracy, ed. David Estlund.
People vote because they bear partial responsibility for the electoral outcome. Analysis of voting should not focus on expected outcomes, but rather on the (moral or) quasi-moral credit people earn by casting votes.
Simple counterfactual view of causation: x causes y if y does not occur without x (inadequate for analyzing voting).
This seems to exclude cases of overdetermination. Who pushes the car out of the snowbank when one too many people helps? In the simple counterfactual view, nobody does, if the loss of one pusher would still lead to the same result. There is no room for partial responsibility in this picture.
INUS view of causation: x causes y if it is an indispensable part of a sufficient condition of y. That is, if x is an Insufficient but Necessary part of a condition which is itself Unnecessary but Sufficient for the result (hence, "INUS"). So if Jones defeats Smith 60-40, and I vote for Jones, my vote is not a necessary part of the condition sufficient for Jones to win (her having a majority) because had I abstained, she would have still won with 59 votes. If the vote is 51-49, my vote is a necessary part of a sufficient condition, but there are other sufficient conditions that would cause Jones to win (overdetermining votes). It is still not clear what role these play. (Note that others generally interpret INUS differently: Even in the 60-40 case, my vote is a necessary part of the various sets of 51 votes which would have been sufficient for Jones to win; therefore, I had a causal role according to the INUS criterion even if my vote was not decisive.)
Vectoral causal systems are like tugs-of-war. Forces push and pull on the rope, whose movement represents the sum of the vectors (positive, negative and zero). There are thus contributing, counteracting and neutral forces determining the outcome. Elections are conventional vectoral causal systems: a vote for Jones is a positive vector in regard to her possible election. A vote against is a negative vector, and an abstention is neutral. You support Jones more by voting for her than abstaining, and much more than voting against her. This seems intuitive.
However, Goldman adds the idea that voters bear quasi-moral "blameworthiness" or "culpability" by abstaining or by voting for the wrong candidate. He has to define the "objectively best" candidate as the one who produces a set of outcomes higher on the preference-ordering of a majority of citizens than the set of outcomes produced by any other candidate. Comment: Goldman's "quasi-moral blameworthiness" model starts to look a little like Rousseau's General Will, in which voting against the GW is not dissenting, but choosing wrongly based on incorrect deliberation. It also seems to resemble the Beauty Contest game, in which players base their responses on guesses about the average response. Is this the idea of voting?
Don't think of simple models of causation in considering voting: their emphasis on decisiveness is misleading. Instead think of a tug-of-war, in which everybody's vector is a factor in the outcome.
Research on similar subjects