## 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. The following blockquote is from pages 101-102 in Making Multi Candidate Elections more Democratic, by Samuel Merrill III, ©1988; ## 9.4. The Standard-Score Voting SystemThe 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. Rule 2. Z
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) 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). 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.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## 9.5 Conclusions....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. ... An interactive ballot that shows the voter her standardized scores makes this decision rule easier to understand. PoliticalSim |