Standard-Score Voting System

The Standard-Score Voting System was invented by by Samuel Merrill III for single-winner contests. It could be useful also if voters use points to raise and lower several budgets.

9.4. The Standard-Score Voting System

The characterization in the preceding section of potentially unique optimal strategies as a set of extreme points suggests that we might be able to construct a minimal voting system by specifying as permissible set a region of Euclidean space R^k, all of whose points are extreme. In fact, a sphere is such a set, and the subset of it defined in Appendix F specifies a permissible region consisting entirely of extreme points that are potentially uniquely optimal.⁴ Furthermore, the voting system defined by this permissible region is a simple modification of the cardinal-measure voting system, but one in which the system does not induce the voter to strategically avoid all but the extremes of the scale.

This voting system — although its invention was motivated by the foregoing considerations — can be described much more simply. The procedure, to be called standard-score voting system, is defined as follows:

Rule 1. The balloting method is identical to that for cardinal-measure voting: rate each candidate on a fixed scale, 0 to 100.

Rule 2. The decision rule is different: for each voter separately, replace her ratings r_i by their statistical standard scores, i.e.,

Z_i = (r_i - μ) / σ, (9.1)

where ( and ( are the mean and standard deviation of the voter's ratings. For each candidate, sum the standard scores over all voters. The candidate with the largest total wins.

Recall that for each voter, her set of standard scores has mean zero and standard deviation 1. Thus, the standard scores are independent of the scale and position used by the voter (the use of a fixed interval such as 0 to 100 is only a convenience).

As was shown in section 5.4, under cardinal-measure voting, the optimal strategy for a utility maximizer is to dichotomize the candidates using the extremes of the scale as ratings. By contrast, under standard-score voting, the voter is rewarded for reporting ratings intermediate between her extremes, since they reduce the standard deviation in the denominator of the standard score. The standard scores of such a voter are expanded relative to those of a voter using only the extremes of the scale, enhancing her effect on the outcome. The expected utilities in table 9.2, computed for equiprobable outcomes from formula (5.2)⁵, illustrate these effects.

For the voter with utilities for the four candidates given in table 9.2, the optimal strategy (100, 100, 0, 0) under cardinal-measure voting yields a higher expected utility (4,000) than does the vote (100, 70, 50, 0), which exactly reflects her utilities. The latter gives an expected utility of 3,533. By contrast, under standard-score voting, the straightforward strategy (100, 70, 50, 0) yields as expected utility of 97.1, higher than the 80 offered by the strategy (100, 100, 0, 0).⁶ If outcome probabilities are not equally likely (i.e., if the t_ij are not all the same), the optimal strategies still employ the full scale but may not be exactly in proportion to utilities.

Table 9.2. Example of Voter Profile for Cardinal-Measure and Standard-Score Voting

			Cardinal- measure		Standard- score
Candidate	Utility	Z_i	Strategy 1	Strategy 2	Strategy 3	Strategy 4
A	100	1.24	100	100	100	100
B	70	0.41	70	100	70	100
C	50	-0.14	50	0	50	0
D	0	-1.51	0	0	0	0
Expected	utility		3,533	4,000	97.1	80.0

Coleman (1982) derives a procedure equivalent to the standard-score system. He solves directly the problem of choosing a transformation of votes-as-cast into votes-as-counted so that the voter will maximize expected utility by casting votes that are a linear transformation of her utilities. If Z_i denotes the standard score as defined by (9.1), the Coleman value for votes-as-counted is

(Z_i + μ/σ)/√6,

i.e., a linear transformation of the standard-score value. The standard-score system, because of its decision rule, should be recommended only for a mathematically knowledgeable electorate.

Footnotes

4 The hollow sphere is not convex, but that is not a necessary condition for a voting system to be minimal.

5 For equiprobable outcomes with four candidate, t_ij = 1/3 for i not equal j. Note that comparison of expected utilities across voting systems is meaningless.

6 In particular, in a decision under uncertainty, standard-score voting is a preference revealing procedure that avoids the compensatory weighting of votes needed when, say, the Clarke tax method (see Straffin 1980, section 3.5) is applied to voters with nonhomogeneous resources.

9.5 Conclusions.

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A modification of cardinal-measure voting, this procedure specifies that the votes-as-counted are the statistical standard scores of the votes-as-cast. Unlike cardinal-measure voting, standard-score voting does not reduce to approval voting, but rather it offers optimal intermediate strategies that closely reflect the voters' utilities.

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An interactive ballot that shows the voter her standardized scores makes this decision rule easier to understand. PoliticalSim^™ uses this rule, with such a ballot, to let players select the election rule for a game.