I can't find any online OCRed copy of this content, so I'm transcribing a bunch of stuff.
Citation:
- Weber, Robert J. (September 1978). "Comparison of Public Choice Systems". Cowles Foundation Discussion Papers. Cowles Foundation for Research in Economics. No. 498.
Actually this contains three previous unpublished papers, which is why it's sometimes listed as "1977". Example citations for the individual papers:
- Weber, R.J. (1977a). ‘Comparison of Voting Systems.’ New Haven: Cowles Foundation discussion paper # 498A.
- Weber, R. J. 1977. Comparison of Voting Systems. New Haven: Cowles Foundation discussion paper No. 498
- Weber, Robert James. "Comparison of voting systems.” Mimeographed, 1977.
- Weber, Robert James. “Multiply-Weighted Voting Systems.” Mimeographed, 1977.
- Weber, R.J. (1977b). 'Multiply-weighted Voting Systems.' Presented at MSSB Conference on Game Theory and Political Science
- Weber, Robert James. “Reproducing Voting Systems.” Mimeographed, 1977.
- Introduction 6
- Voting Systems 7
- Random societies 10
- Two-candidate elections 12
- Optimal voting strategies 13
- The effectiveness of a voting system 16
- The standard voting system 17
- The Borda voting system 19
- The approval voting system 20
- Three-candidate elections 21
- The case for approval voting 23
- Related work 26
- Appendix: Social utility and random societies 27
- Definitions 34
- Essentially equivalent voting systems 38
- Basic results 40
- Average-dependent voting systems 41
- Computing the effectiveness of an average-dependent voting system 44
- The effectiveness of the best three-candidate average-dependent voting system 48
- Midpoint-optimal voting systems 50
- The most effective four-candidate midpoint-optimal voting system 53
- The effectiveness of the best four-candidate midpoint-optimal voting system 54
- Summary 55
- Voting Systems 59
- The Effectiveness of a Voting System 60
- Reproducing Voting Systems 64
- The Effectiveness of a Reproducing Voting System 67
- The Vote-for-k Voting System 69
- The Vote-for-or-against-k Voting System 71
- The Borda Voting System 73
- Summary 74
- The Effectiveness of Several Voting Systems table 76
- Appendix: Individual Utility Functions, and a Characterization of the Uniform Probabiliy Distribution 77
Page numbers are relative to the PDF that contains all three.
p. 5:
p. 6:
An important criterion is that a voting system should be likely to lead to the election of a candidate who represents the desires of the voting public. In this paper, we develop a measure of the effectiveness of a voting system in meeting this goal.
p. 7:
The effectiveness of a voting system is defined as a comparison between the expected social utility of the socially-optimal candidate and the expected social utility of the candidate actually elected under the system.
p. 11:
The social utility of a candidate is the sum of his utilities to the voters.
If there are n voters in a random society, the social utility of any given candidate is the sum of n independent uniformly-distributed random variables. By the Central Limit Theorem, this sum will be asymptotically normal, with mean
$\frac{n}{2}$ and variance$\frac{n}{12}$ . In particular, consider the candidate of greatest social utility. His expected social utility is asymptotic to
$$ \frac{n}{2}+\sqrt{\frac{n}{12}}\cdot\mathrm{Norm}_\mathrm{max}(m), $$ where
$\mathrm{Norm}_\mathrm{max}(m)$ is the expected value of the maximum of m independent unit normal random variables (see Table 3). This provides an (asymptotic) upper bound on the expected social utility of the candidate elected under any particular voting system.
p. 12:
Since a randomly-selected candidate has expected social utility
$\frac{n}{2}$ , this result suggests defining the "effectiveness" of a two-candidate voting system as the ratio between the coefficients of$\sqrt{n}$ in this formula and in the formula for the expected social utility of the socially-optimal candidate. Then the effectiveness of every two-candidate voting system will be$\sqrt{\frac{2}{3}} = 0.8165$ , since$\mathrm{Norm}_\mathrm{max}(2) = 0.56419 = \frac{1}{\sqrt{\pi}}$ .
p. 16:
We define the effectiveness of a voting system for an m-candidate election as the asymptotic (for large societies) difference between the expected social utilities of the elected candidate and a typical candidate, normalized by the maximum possible difference. Formally, let
$E_\mathrm{maximal}(m,n)$ be the expected social utility of the best of m candidates in an n-voter random society, and for a given voting system let$E_\mathrm{elected}(m,n)$ be the expected social utility of the candidate elected under this system when all of the voters follow optimal voting strategies. The effectiveness of this voting system for m-candidate elections is $$ \lim_{n \to \infty} \frac {E_\mathrm{elected}(m,n)-\frac{n}{2}}{E_\mathrm{maximal}(m,n) - \frac{n}{2}}. $$This definition at last provides a method for comparing voting systems, in terms of the extent to which they tend to the election of a candidate representative of the preferences of the voters.
p. 22:
Table 4
The expected social utility of the elected candidate, under three voting systems.
Number of voters Standard voting system Borda voting system Approval voting system 2 1.2500 1.2917 1.2917 3 1.8333 1.8750 1.8646 4 2.3889 2.4236 2.4213 5 2.9167 2.9765 2.9726 10 5.5975 5.6706 5.6719 15 8.2245 8.3206 8.3245 20 10.8328 10.9472 10.9531 25 13.4328 13.5588 13.5662 30 16.0190 16.1597 16.1684
p. 31:
p. 36
Assume that the utilities of the candidates to a specific voter are
$u_1,\dots,u_m$ , and let$\overline u = \frac{\sum u_c}{m}$ be the average of these utilities.
p. 37:
Under the approval voting system, it is optimal to vote for all candidates of greater-than-average utility (that is, take
$w_c=1$ if$u_c-\overline u>0$ , and$w_c=0$ otherwise.)
The expected social utility of a randomly-selected candidate is
$\frac{n}{2}$ .
p. 38:
$$ \mathrm{Effectiveness}=\lim_{n \to \infty} \frac {E_\mathrm{elected}(m,n)-\frac{n}{2}}{E_\mathrm{maximal}(m,n) - \frac{n}{2}}. $$
p. 40:
it follows that every two-candidate voting system has effectiveness
$\sqrt{\frac{2}{3}}=81.65%$ .
p.50:
would require allowing twelve different types of voter responses.
p.57:
p.59
The approval voting system, which has attracted recent interest
We will not study this system directly, but will consider the vote-for-or-against-k voting system with weight sets
p. 61:
Select a fixed voter k, and let
$E^{(k)}_\mathrm{elected}(m,n)$ be the expected utility of the elected candidate to this voter. (The expectation is taken over the nm random utilities which determine the preferences, and therefore the actions, of the voters.)Define $E^{(k)}\mathrm{maximal}(m,n)$ to be the expected value of the maximum of m quantities, each the average of n independent random variables drawn from voter k's (uniform) utility distribution. This is an upper bound for $E^{(k)}\mathrm{elected}(m,n)$.
p. 62:
Let $ E^{(k)}\mathrm{random}(m)$ be the expected utility to voter k of a randomly selected candidate. Clearly, we hope that any reasonable voting system would out-perform random selection. We define the effectiveness of an m-candidate voting system to be: $$ \lim{n \to \infty} \frac {E^{(k)}\mathrm{elected}(m,n) - E^{(k)}\mathrm{random}(m)}{E^{(k)}\mathrm{maximal}(m,n) - E^{(k)}\mathrm{random}(m)}. $$
p. 63:
In this case,
$E_\mathrm{random}(m)=1/2$ .
The effectiveness of the m-candidate standard voting system was shown in [3] to be
$\sqrt{3m}/(m+1)$ .
We have viewed effectiveness through the eyes of a single voter. A mathematically equivalent approach considers the utilities of the voters as being drawn from a common distribution, and defines the social utility of a candidate as the sum of his utilities to the n voters.
$E^{(k)}_\mathrm{elected}(m,n)$
p. 64:
Since all two-candidate voting systems are essentially equivalent, this implies that no two-candidate voting system can be more than
$\sqrt{\frac{2}{3}}=81.65%$ effective
p. 70
Therefore, the effectiveness of the m-candidate vote-for-k voting system is $$ \frac{1}{(m+1)}\left[\frac{3mk(m-k)}{m-1}\right]^{1/2}. $$
It is interesting to observe that the vote-for-k and vote-for-(n-k) voting systems are equally effective.
(Should be
The effectiveness of these "vote-for-half" systems is $$ \begin{array}{ll} \frac{\sqrt 3}{2}\frac{m}{m+1}\left(\frac{m}{m-1}\right)^{1/2} & \text{if }m\text{ is even, and} \ \frac{\sqrt 3}{2}\left(\frac{m}{m+1}\right)^{1/2} & \text{if }m\text{ is odd.} \end{array} $$
p. 71
Consider an m-candidate election in which this system is employed, where
$k<m/2$ .
Actually should be
a voter is equally likely to be of either type.
p. 72:
Therefore, the effectiveness of the m-candidate vote-for-or-against-k voting system is $$ \omega\left[\frac{12k(m-k)}{m(m-1)}\right]^{1/2}= \begin{cases} \frac{5m^2+1-6mk}{4(m+1)}\left[\frac{3k}{m(m-1)(m-k)}\right]^{1/2}, & \text{if }m\text{ is odd,} \ \frac{5m^2-6mk}{4(m+1)}\left[\frac{3k}{m(m-1)(m-k)}\right]^{1/2}, & \text{if }m\text{ is even.} \end{cases} $$
When m is even, and
$k=m/2$ , the vote-for-or-against-k system coincides with the vote-for-k system.
p. 74:
Therefore, the effectiveness of the m-candidate Borda voting system is $$ \left[\frac{m}{m+1}\right]^{1/2} $$
p. 76:
The Effectiveness of Several Voting Systems
Number of Candidates Standard Vote-for-half Best Vote-for-or-against-k Borda 2 81.65% 81.65% 81.65% 81.65% 3 75.00 75.00 87.50 86.60 4 69.28 80.00 80.83 89.44 5 64.55 79.06 86.96 91.29 6 60.61 81.32 86.25 92.58 $\vdots$ 10 49.79 82.99 88.09 95.35 $\vdots$ $m \to \infty$ 0.00 86.60 92.25 100.00
[There are closed-form expressions for each of these, so the entire table can be re-generated relatively easily.]