While the expected-utility model includes a straightforward decision rule, it does not include a method for calculating pivot probabilities. Indeed this calculation appears to be anything but straightforward. Several approaches to this calculation from the literature are presented and analyzed here. These approaches all become more computationally complex as the number of election alternatives increases. We introduce a novel approach that does not increase in complexity with the number alternatives.
Before discussing methods for calculating pivot probabilities, it is useful to examine the definition of pivot probability in more detail. As already mentioned, a voter's pivot probability for any two candidates is the probability that he or she will be decisive in making or breaking an -place tie between those two candidates. If we assume that for each possible election outcome we can determine the probability associated with the occurrence of that outcome, the pivot probability is the sum of the probabilities associated with each of the possible election outcomes that involve an -place tie between those two candidates.
Figure: A three-dimensional view of the barycentric coordinate system
The election outcome space for a three-candidate election can be represented visually as a barycentric coordinate system -- an equilateral triangle on the three-dimensional plane , where , , and represent the proportions of votes for candidates 1, 2, and 3 respectively. Each of the triangle corners represents one candidate. The closer an outcome point is to a particular corner, the more votes the alternative represented by that corner received. Thus, an outcome point in a corner of the triangle represents a ``shut-out'' in which one alternative received all the votes, while an outcome point in the geometrical center of the triangle represents a three-way tie. As shown in Figure , the line segments that bisect the triangle represent two-way winning ties. This model can be extended arbitrarily. For an election with four alternatives the barycentric triangle becomes a solid tetrahedron. An election with N alternatives can be represented by a simplex polyhedron of N - 1 dimensions.
Because finding the candidate that maximizes the expected-gain equations (Equation 1) depends on the ratio of the pivot probabilities rather than the actual probabilities themselves, we do not need to find a method for calculating the absolute probability of reaching each point in the outcome space; rather, it is sufficient to find a method of calculating relative probabilities. Thus the methods we examine here do not necessarily result in the probabilities over the entire outcome space summing to 1.
Figure: Example contour plot
When comparing probability calculation methods and analyzing the suitable of each for use in the expected-utility model of voting, it is helpful to be able to visualize the behavior of the expected-utility model when different probability calculation methods are used. Hoffman  includes contour plots in his paper to help illustrate this behavior. To allow visual comparison of our new method with Hoffman's we wrote a computer program to generate contour plots such as the one shown in Figure . The curves on these plots represent predicted outcome-proportions for a voter's first choice, , in a three-candidate election. Predicted outcome-proportions for a voter's second choice, , are shown along the x-axis. , a voter's utility rating for is shown along the y-axis. Given a predicted outcome at point , a voter should vote for for if and only if the point lies above the curve for outcome ; otherwise the voter should vote for . Consider, for example a predicted outcome of P = (.15, .4, .45). Voters who rate should vote for ; voters who rate should vote for . The plots shown in this chapter all assume normalized ratings such that .
We now consider several methods for calculating pivot probabilities.
One approach to calculating pivot probabilities involves making a prediction about the likely outcome of the election, plotting that outcome as a point on a barycentric coordinate system, and computing the relative distances between that outcome point and the outcome lines for each of the two-way ties. Black  used this method in a model of a three-candidate plurality election in which each voter was assumed to be able to make a reasonable prediction about the election outcome based on the results of previous elections, opinion polls, or other data. In batched DSV we can determine the voters' sincere strategies from their utility information, and aggregate these strategies to determine a first-round predicted outcome. Subsequent rounds can use the results of the previous round as a predicted outcome. In ballot-by-ballot DSV, the interim tally of ballots counted so far can serve as the predicted outcome point.
Figure: Distances used to calculate Black's probability estimates
Black assumes that the pivot probability for any pair of candidates is proportional to 1-d, where d is the Euclidean distance between the line segment representing an -place tie between those candidates and the predicted outcome point A, as shown in Figure . For this distance is equivalent to the length of the perpendicular line segment from the predicted outcome point to the first-place tie line .
The distance calculation for can be expressed as:
where and . The other pivot probabilities can be calculated in a similar manner.
We developed and experimented with a variation on Black's method involves calculating the distance d between the predicted outcome and a given point on a two way tie line. Using this technique, the pivot probability for any two candidates can be found by summing the differences 1-d for every point on the two-way tie line for those candidates. This method makes more intuitive sense than Black's method if one considers what the pivot probabilities really represent.
Figure: Contour plot for 3-candidate plurality election using our simple distance pivot probabilities
One problem with both Black's method and our variation is that they assume that the probability of a tie gets linearly larger the farther away a predicted outcome point is from a two-way tie line. This uniform probability distribution, while simple, does not seem consistent with empirical evidence. For example, as Figure illustrates, these methods rarely select the strategy of voting for a second-choice candidate unless the second-choice candidate has a utility rating at least 80% as high as the first-choice candidate. In addition, these methods do not take into consideration the certainty of the predicted outcome point.
Hoffman  offers another approach that allows for the modeling of voting schemes other than simple plurality, and assumes a Gaussian distribution of outcomes rather than a uniform distribution. Although Hoffman introduces his approach by describing a three-candidate election, the approach may be applied to elections involving any number of candidates and is most easily understood by considering a two-candidate election first. A two-candidate election with B voters can be modeled as a line segment. This line segment can be divided into B smaller segments, each one vote wide, as shown in Figure . Each of the smaller line segments represents a possible election outcome. If there are an odd number of voters, there will be a line segment T at the center of the line that represents a tie. If we randomly sample the voters, we can predict an election outcome and represent it somewhere along our line at segment P.
Figure: Gaussian geometry for a 2-candidate election
Given P, we are interested in determining the probability that the election outcome will be a tie. As Figure illustrates, we can construct a normal distribution curve around the center point of P, given by the formula
where D is the distance from each point to (the center of) P, and is a measure of the uncertainty of the prediction. Hoffman does not specify how should be calculated, but the ``standard error of the estimate'' seems to be an appropriate measure of uncertainty. As we shall see, is an important factor and can affect the outcome of a DSV election dramatically. When using the standard error to calculate in a ballot-by-ballot DSV system, we multiply it by the finite-population correction factor yielding:
where and represent the proportion of votes for each candidate thus far, B represents the total voter population, b represents the number of votes counted thus far, and represents the area under the standard normal curve for confidence level . Thus as the election progresses, b will increase and will decrease. The fact that uncertainty decreases as the election progresses is consistent with the fact that the amount of information increases.
The above method of calculating uncertainty is useful for ballot-by-ballot DSV, in which the predicted outcome involves a sample that increases in size until it equals the entire population; however, in batch DSV, the predicted outcome point is based on polling the entire population -- thus our prediction derives no uncertainty from sampling error. In batch DSV the uncertainty of the prediction is based on not knowing how many voters will find that their optimal strategies are not their sincere strategies.
The simplest way to account for uncertainty in batch DSV would be to select an arbitrary value for . For example, has the property that as long as a is predicted to receive at least 5% of the vote, voters will not vote for unless is at least 10% as large as (on a 10 point scale, a voter with must have in order to vote for ). We used this approach in our simulations, described in Chapter . We also tried conducting batch DSV simulations with the number of rounds equal to the number of voters and adjusting according to Equation , letting b equal the number of rounds run so far. However this approach generally resulted in the same election outcome as was produced using a constant equal to the value produced by Equation when b = B.
Another approach to calculating would be to develop a formula based on some aggregation of the utilities submitted by the voters -- perhaps taking into account the percentage of voters for whom it might never be optimal to voter for . Such a calculation is also likely to be somewhat arbitrary, however. Still another approach would be to assign random values to within a reasonable range (say .001 - .5). Finally, voters could select their own uncertainty values based on their attitudes toward risk, formulated perhaps as a function of the ``maturity'' of the election: number of rounds, position in the voting sequence, etc.
Regardless of what method is used to calculate , the probability that the election outcome will involve a first-place tie between candidates 1 and 2 can be computed as follows:
where U = 1/B (the length of a one-vote wide line segment). That is, the probability of a tie is proportional to the area under the segment of the normal curve that lies above the tie segment T.
Figure: Hoffman's geometry for a 3-candidate election
Hoffman illustrates his approach for a three-candidate election. As shown in Figure , Hoffman specifies the region as the portion of the outcome triangle in which candidate i loses to candidate j by one vote (or less in systems that allow fractional votes). He defines the pivot probability as ``the probability that the election result lies in the region .'' Thus can be expressed as:
where D is the distance from the predicted outcome point to a point in the region . Because is only one vote wide, can be approximated by using Simpson's rule or another numerical integration technique to integrate over the outcome points for which . Irrespective of the number of candidates N, first-place ties can occur only when . Values of and greater than 1/2 would be impossible because ; and values of and less than 1/N would indicate that at least one other candidate has a higher proportion of the vote. Values of and between 1/N and 1/3 may or may not indicate first-place ties, depending on the outcome proportions for the other candidates. However, when , and must be in a first-place tie. Thus, for a three-candidate election, can be approximated by:
where x represents the proportion of votes for candidates i and j at a given outcome point. Note that when N > 2, the standard error of the estimate becomes quite difficult to calculate. We use the approximation
to calculate the standard error of the estimate for N > 2.
A four-candidate election is modeled as a solid tetrahedron. However, finding the outcome region in which first-place ties occur is not as easy in four dimensions as it is in two and three. As discussed above, first-place ties may occur in regions where , depending on the proportion of votes received by each of the other candidates. With four candidates, the outcome region in which candidates i and j tie for first place is a kite-shaped section of the plane. The probability of an outcome occurring in this region can be approximated by:
where x represents the proportion of votes for candidates i and j, and y represents the proportion of votes for candidate k at a given outcome point.
Hoffman's approach is appealing because it does not assume a uniform probability distribution and because it takes into account the uncertainty associated with the predicted outcome. Indeed, this uncertainty proves to be a significant factor in calculating a voter's optimal strategy. Figure shows the contours we have calculated for values of .02 and .05. The figure illustrates that the more certain the prediction, the more willing voters should be to vote for rather than . For example, given a predicted outcome P of (.15, .4, .45) the optimal strategy when is to vote for if . However under the same circumstances but with , it is optimal to vote for if .
Figure: Contour plot for 3-candidate plurality election, Hoffman's method pivot probabilities
A drawback to Hoffman's approach is that the pivot probability calculations increase in complexity with the number of candidates. For elections involving many candidates, these calculations can be quite difficult and time consuming. The reason for the increasing complexity is that Hoffman's method requires the projected outcome for all candidates to be considered in every pivot probability calculation. However, we have discovered that the only projected outcomes crucial to the pivot probability calculation are the outcomes for the candidate predicted to win and for the two candidates whose pivot probability is being computed. Furthermore, a simplifying assumption allows us to calculate pivot probabilities for N candidate elections using the predicted outcomes for just two candidates.
Figure: Predicted outcomes for candidates i and j
To calculate using our 2-D Gaussian method, we being by plotting the predicted outcomes for i and j along an outcome line and constructing normal distribution curves around each outcome point, as shown in Figure . In ballot-by-ballot DSV it is important that the values used to construct the normal curves decrease as the election progresses. We can simplify our calculation while still ensuring that decreases by substituting an average value for the numerator in the standard error approximation presented in Equation :
Thus, as a ballot-by-ballot DSV election progresses, the normal curves will become increasingly narrower. However all the curves calculated for a given ballot will have the same width.
The shaded area in Figure , where the curves intersect, represents the probability that the final outcome will involve a tie between i and j. But not all ties are winning ties. If our election involved only three candidates we could declare the pivot probability to be the section of the intersection area representing outcomes where i and j each receive between 1/3 and 1/2 of the votes. However, elections involving four or more candidates cannot be represented this simply. To solve this problem we plot the outcome point for the predicted winner on our outcome line and construct a normal curve around it. As illustrated in Figure , the pivot probability is the intersection of all three curves.
Figure: Geometry for calculating pivot probability using 2-D Gaussian method
Note, that when either i or j is the predicted winner, only two curves need be plotted. Also note, that because the curves all use the same value of , the intersection of three curves is equal to the intersection of the left-most and right-most curve. Thus, to compute we need only determine the area of the intersection of the curve for the predicted winner and either i or j, whichever is predicted to receive fewer votes.
To compute the area of the intersection we must first determine the point at which the two normal curves intersect. The point of intersection I can be calculated:
where is the predicted outcome for the predicted winner and is the predicted outcome for either i or j, whichever is predicted to receive fewer votes. The pivot probability is then calculated:
where . Because there is no region in which ties are guaranteed to occur, we assume the curve is infinite in both directions. However, only half the area need be computed, as the curve is symmetrical about the intersection point.
The above integral can be approximated using Simpson's rule or another numerical integration technique. However, as decreases, Simpson's rule requires increasingly more steps to converge. A more efficient approach is to use a table of areas under the standard normal curve or a computer function that returns the values in such a table. In this case a change of variable must be performed to remove from the exponent. The pivot probability is then calculated:
In our simulations we use the erf function from the C math library. However, the erf function is not intended for use with the large upper bound values that occur when gets very small. When our computations using the erf function return 0, we approximate the area as
Figure: Contour plot for 3-candidate plurality election, 2-D Gaussian method pivot probabilities
Our 2-D Gaussian method results in contour plots (shown in Figure quite similar to those produced using Hoffman's method. The most obvious difference between the results of these two methods can be seen in the effects of varying . An value approximately 0.4 times smaller than the one used in Hoffman's method must be used in the 2-D Gaussian method to produce similar results.
Another difference can be observed by comparing the upper right-hand corners of Figures and . When a voter's second-choice is predicted to win by a large margin the 2-D Gaussian method produces pivot probabilities that will result in a vote for the voter's first choice, regardless of . Under the same circumstances Hoffman's method produces pivot probabilities that may result in a vote for the voter's second choice, depending on . In this situation, the voter's first choice has practically no chance of winning and the voter's second choice has practically no chance of loosing. Therefore, the voter cannot influence the outcome with his or her vote, and thus, we believe, should vote sincerely. Hoffman's method and the 2-D Gaussian method indicate different strategies in this situation due to the fact that for a three-candidate race Hoffman's method indicates a higher probability of a winning tie between third-place and first-place candidates than between third-place and second-place candidates, while our method indicates that the probability of a winning tie is the same for these two pairs of candidates. Because the number of voters who must change their votes to achieve a winning tie in each of these two situations is identical, we believe our method is a more accurate model of pivot probabilities involving third-place candidates.