Boscardin: Estimating Probability Nonoccurring Events your vote decisi - PDF document

Gelman, Boscardin: Estimating Probability Nonoccurring Events your vote decisive, under assumption that all the other actors vote by flipping game- theoretic analyses (such as that 1992), allow votes additional information, also im- assume that the (b) approaches 0 number of voters is large. ELECTION FORECASTS WILL Conditional and Marginal presidential vote two conditions without your vote, your state's election outcome must be tied or consider the case either vote not vote. voter who is considering switch- from the Republican candidate will, of higher probability being decisive.) the national election; that it tied, neither party an electoral vote jority. We following mathematical notation for the following known constants: ei assigned to et,tal = ei total number electoral votes; the following election outcomes that to be ni number of vi Democratic share the two-party vote your vote), �ifvi.5 v, = 0 otherwise, and EPi = Cjfi ejV, Democratic electoral vote in the states excluding i, an unambiguous and easily predictable, we assume that its 3 electoral are certain to is not a controver- e.g., Rosenstone 1983). One can minor parties separately estimating won by minor parties and setting the total votes in the states contested the Democrats you live in i, Pr(your vote matters) Pr(your in your Pr(your will be lyour on the right side a conditional probability that state a popular vote that state. We now to evaluate given any state-by- presidential election (i.e., the values ni and vi states). In practice, ni be fairly rately and higher turnout that closer contests. Because electoral we require a forecast (vl, . . . u~~), representing some knowledge before the presidential election question. Such a forecast input to our method and, like all statistical forecasts, must include uncertainty as as a separate forecasts all the states enough; it must conditional probability (e.g., the probability that strongly Republican decisive in that it the two turn. First, the large number of voters in any can with negligible error model the Democratic vi, as continuous variables. If ni is even, then the (1) is Pr(your vote is decisive in your state) = Pr(nivi = .5ni) = Pr(vi = .5) KZ f,% (.5)/ni, (2) the discrete approximation the continuous distribu- tion and the notation fvt for the probability density func- the continuous vi under the forecasting ni odd, then the (1) is Pr(nivi = .5(ni - 1)) = Pr(vi = .5 - .5/ni) KZ fvt (.5)/ni assuming that ni reasonably large. can compute the factor on the right side forecasting model determine the conditional the condition vi = .5 all practical vi = .5 and vi = .5 - .5/ni identical conditions): Pr(your state is Pr(EPi E (.5etOtal - ei, .5etotal)lvi arise because a vote that causes national election be tied is only half as decisive as a vote that changes the election outcome.) the other states must then be combined into a forecast national electoral right side (I), which are by (2) and (3), be computed posterior simula- analytic expressions where to estimate probabilities directly to estimate all or, in a Bayesian context, obtain a large Gelman, Boscardin: Estimating Probability Nonoccurring Events tion, voting behavior, not the size Our results and general obvious inter- est to candidates to allocate their campaign resources and states concerned attracting the atten- prospective presidents. In the election outcome by mobilizing porters to vote in a single state times the prob- that a single vote in that so state-by-state campaign efforts can be chosen probability, with the optimal decision varying the campaign and the election forecasts change. been discussed by Brams and Davis (1973, the election voters in a single state times the that a single vote In addition, our results rational choice theorists interested that the voter may influence other, nonpresidential contests Mathematical Discussion Comparison to Methods Not Based a state is on the order l/ni, and the probability that a state be decisive occurs is (crudely) proportional ei, which is roughly proportional ni (except in the smallest states). two factors be approximately constant, smallest states. for 1992 log-probability that a state decisive given that it tied versus the log-probability that it be tied. Most the dotted line indicating a probability -6.0 -5.5 log1 0 Probability That a State Decisive Given Tied Probability That the State a Log Scale. ., product of lo-'. 0 , I I I 'Same Plot the Model With 0 Set to (ie., Inversely Proportional models in the literature (see Sec. 1) assume that the standard deviation vi a state 116. Our model replicate this assumption be exactly 1; performed this computation investigate whether our would change measurably with such an assump- previous years: probability that a be decisive increases largest states, the extent anticipated the binomial-based because the forecasting model has several variance components, and the regional and national errors do not, of as sensitive to the parameter might fear. analysts thus homoscedastic regression-based Gelman Another possible modeling choice the compound bi- nomial: modeling an expected vote outcome ui linear model as this article and then nivi - Bin(ni, ui). though this class models seems reasonable, we adopt it because US. so large that the binomial variability negligible com- pared to forecast uncertainty 1992, turnout all states greater than 160,000, ,/(.5) (.5)/l6OlOOO = .00125, as compared