## Numerical experiments with Apportionment methods

By Dan Bishop with some contributions by Warren D. Smith

H.Peyton Young argued in his book [M.L. Balinski & H. Peyton Young: Fair Representation: Meeting the Ideal of One Person, One Vote (2nd edition), Brookings Institution Press 2001] and (more accessibly) in his essay here that the best apportionment method was Webster's.

We now do computer simulations that reach the same conclusion as Young. The simulations assumed:

• There are 50 states and 435 seats.
• A state with population p is given max(m, r(p/q)) seats, where r is the appropriate rounding function for that apportionment method, m is the minimum number of seats a state can have (in our simulations we always take either m=0 or m=1) and q is chosen so that the total number of seats comes out to 435.
• State population is a uniformly-distributed, or exponentially-distributed (there were two kinds of simulations) positive random variable.
• Bias is defined in three ways:
1. Pearson's [centered] correlation coefficient between the states' populations and log(seats/population).
2. Spearman's [rank-based centered] correlation coefficient between the states' populations and their seats/population ratios. A Spearman bias of +1 means that a high-population state will always be better-represented than a low-population state, and a bias of -1 means vice versa.
3. Pearson's [centered] correlation coefficient between the states' populations and their seats/population ratios.
(Whether Spearman is log-based or ratio-based makes no difference since Spearman is simply the Pearson correlation of the integer rank vectors and hence is invariant under monotonic function transformations of either data vector.)
• 10,000 simulations were performed for each method.
• Each bias-measurement datapoint given as mean±std.error.

### Definitions of the apportionment methods (for noninteger 2x)

ADAMS' METHOD: r(x) = ⌈x⌉    [most favorable to small-pop states]

DEAN'S METHOD: r(x) = if x > hmean then ⌈x⌉ else ⌊x⌋ where hmean=2/(1/⌈x⌉ + 1/⌊x⌋)

HUNTINGTON-HILL'S METHOD: r(x) = if x > gmean then ⌈x⌉ else ⌊x⌋ where gmean=√(⌈x⌉⌊x⌋)

NEW METHOD: r(x) = ⌈x-0.495⌉; or more generally the magic constant "0.495" can be replaced by one chosen to yield the best retrospective performance (least "bias") over past history (0.495 is optimal in a certain theoretical model with the parameters used in the contemporary USA; I discuss that elsewhere.)

WEBSTER'S METHOD: r(x) = ⌊x+½⌋; equivalently r(x) = if x > amean then ⌈x⌉ else ⌊x⌋ where amean=(⌈x⌉+⌊x⌋)/2

JEFFERSON'S METHOD: r(x) = ⌊x⌋    [most favorable to large-pop states]

We write "WebsterM" (where M=0 or M=1) for the Webster method with minimum number of seats m=0 or 1, etc. For Hill, Dean, and Adams it does not matter whether m=0 or m=1 because these methods "naturally" enforce the one-seat minimum.

### Uniform Distributions for State-Populations:

Apportionment Method Log-based Pearson bias Spearman bias Ratio-based Pearson bias
Dean0=Dean1 -0.343019 ± 0.00120 -0.231386 ± 0.00161 -0.319941 ± 0.00101
HuntingtonHill0=HH1 -0.303224 ± 0.00139 -0.173667 ± 0.00166 -0.294303 ± 0.00114
New0(0.495) +0.036963 ± 0.00171 +0.135006 ± 0.00219
Webster0 +0.040412 ± 0.00172 +0.141884 ± 0.00217
Webster1 -0.263111 ± 0.00160 -0.119540 ± 0.00168 -0.261334 ± 0.00139
Jefferson0 +0.651469 ± 0.00101 +0.584829 ± 0.00059
Jefferson1 -0.018986 ± 0.00271 +0.354741 ± 0.00162 -0.099855 ± 0.00275

### Exponential Distributions for State-Populations:

Apportionment Method Log-based Pearson bias Spearman bias Ratio-based Pearson bias Probability of "quota violation"
Dean0=Dean1 -0.281804 ± 0.00067 -0.345293 ± 0.00155 -0.233027 ± 0.00061 9.2%
HuntingtonHill0=HH1 -0.258760 ± 0.00074 -0.287029 ± 0.00161 -0.222645 ± 0.00061 4.4%
New0(0.495) +0.076093 ± 0.00173 +0.129916 ± 0.00131 (0.13 ± 0.012)%
Webster0 +0.079859 ± 0.00174 +0.138316 ± 0.00129 (0.16 ± 0.013)%
Webster1 -0.232864 ± 0.00083 -0.224715 ± 0.00167 -0.210107 ± 0.00065 (2.0 ± 0.05)%
Jefferson0 +0.772881 ± 0.00075 +0.483876 ± 0.00054 98%
Jefferson1 -0.083986 ± 0.00143 +0.229443 ± 0.00179 -0.139785 ± 0.00122 78%

Notation: a "quota violation" is said to occur if some state gets X seats, as opposed to its ideal fractional number Y, where |X-Y|>1. Every "divisor method" is known to suffer quota violations in some circumstances, although the Hamilton/Vinton method (which is not a divisor method) can boast that it never violates quota. The best "classic divisor method" in terms of suffering quota violations the most rarely, evidently is Webster.

If we employ 9380 seats instead of 435 (with m=1; and with either exponential or uniform; both work; note 9380 is the largest House size currently constitutionally permitted – 1 person per 30,000 residents), then:

• Webster1 is still the least-biased (using Spearman bias).
• Webster1, Hill, and Dean all are an order of magnitude less biased than they were with only 435 seats; also, they all have an order of magnitude (or more) smaller chance of having a quota violation. But that's not the case with Jefferson0, Jefferson1 and Adams, whose biases increase or stay about the same, and whose chances of quota-violation remain high (over 97%).

In view of these facts we advocate Webster, in general, as the best of these 5 divisor-based methods, albeit the new method seems even better.

The order of small-state favoritism is Adams>Dean>Hill>Webster>Jefferson, and Webster's method is (at least in the large #seats limit) the least biased. These results are identical to Young's, despite our different definitions of bias. Also, Walter F. Willcox, in 1916, did the first similar numerical experiments (which was very laborious since that was before computers!) which also ended up supporting Webster's method and not Huntington-Hill. This conclusion was unfortunately not agreed to by the National Academy of Sciences, which ignored numerical experiments and preferred Huntington-Hill based on misleading purely theoretical considerations.

The moral is that this is really an experimental question that is not soluble purely by theorizing. That is because the best quality measures and most realistic population distributions are beyond the reach of theorem proving – but fully within the reach of computer simulation.

### Remaining to do???

Our new apportionment method appears to do even better than Webster (see table above, "new" is now included). Should discuss that. (And 0.495 does not appear to be the best value; can do even better?)

It would be good also to add the Hamilton-Vinton and Balinski-Young apportionment methods to our computer simulations above and to redo them all with millions, not merely 10000, Monte-Carlo experiments. Also of interest might be "subset" versions of Webster, e.g. where you do Webster separately on the different subsets of states that fall into different population ranges.

Joe Malkevitch "AMS feature" article on apportionment schemes

Our own article on apportionment schemes