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.
Riker and Ordeshook. 1970. A theory of the calculus of voting. American Political Science Review 1970:25-41.
The authors begin where Downs (1957) left off: Rational choice appears to predict unrealistically low levels of turnout. They express this in equation 1, where R = the reward in utils of voting; B = the benefits of having your candidate win (compared to benefits of opponent); P = probability that your vote matters; and C = costs of voting:
Equation (1): R = (B*P) - C
In a population with 100 million voters, Downs assumed P would always be 1/100,000,000. Thus, any positive C would make R negative. The surprising result: Nobody will ever vote. Rather than ask why so many people don't vote, then, rational choice leads us to ask why so many people do.
The authors suggest a couple possible solutions to this paradox, although the "paradox of non-voting" has been the subjected of repeated scholarly inquiry. For more, see turnout.
The authors insert D into the equation. D represents your sense of civic duty, your satisfaction from voting, your desire to affirm your partisanship or efficacy, etc (see pg 28). Thus, we get a new equation, in which C and D become the most important figures (since B*P is effectively zero, at least so far):
Equation (7): R = B*P - C + D
Downs had also proposed a D term, although his D represented a voter's desire to support democracy as a system. For Downs, a voter might turn out merely to prevent the democratic collapse that would follow zero turnout. (Note that this argument collapses on itself; just as no vote has much chance of deciding the election result, no vote has much chance of deciding the fate of democracy.)
P can be much larger than Downs suggested. As the race gets closer to a tie, voters perceive that their vote has a much higher probability of affecting the outcome. Moreover, we tend to overestimate P (for more on the latter point, see Quattrone and Tversky 1988). Thus:
Equation (21): P = a bunch of calculus. The point: as your candidate's vote share approaches 50%, P increases.
The authors test their ideas using survey data from 1952, 1956, and 1960, showing that P, B, C, and D (at the individual level) could predict Y, the probability that a particular respondent (claims that he) voted.
See Table 3 (p 38). The main hypotheses are validated.
Note that a model driven by D and C doesn't explain why voters are strategic. ("Strategic voting" means that voters consider not only a candidate's positions, but also whether he is electable--thus, as an example, voters abandon the Green (or Libertarian) Party and vote for the Democrats (or Republicans) instead.) Strategic behavior implies that B (and therefore P) must matter.
An arcane and technical point that later critics have made: This analysis shows only that P, B, C, and D have marginal effects on Y (the probability of turnout). That is, it shows that increasing B, P, or D (or decreasing C) can change the probability of turnout, but it does not show that there is a particular (observed) combination of P, B, C, and D that will consistently lead us to predict that a particular voter will turn out.
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