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What happens when Webster's method can't narrow down to a working divisor?
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The Politicus
I've been poking around and without trying, managed to bump into some scenarios where Webster's method gives inconclusive results. The fact I managed it more than once without trying implies to me that this is a highly probable situation, though maybe only with small or rounded numbers.
For example, with 20 seats available, and states with populations of: 10000, 5000, 9000, 1000, 3000, 2000, 8000 (total: 38,000)
| State | Population | Standard Q | Modified Q | Rounded Q (2,000.00000000) | Final (2,000.00000000) | Rounded Q (2,000.00000001) | Final (2,000.00000001) | |
|---|---|---|---|---|---|---|---|---|
| A | 10000 | 5.263157895 | 5 | 5 | 5 | 5 | 5 | |
| B | 5000 | 2.631578947 | 2.5 | 3 | 3 | 2 | 2 | |
| C | 9000 | 4.736842105 | 4.5 | 5 | 5 | 4 | 4 | |
| D | 1000 | 0.526315789 | 0.5 | 1 | 1 | 0 | 1 | |
| E | 3000 | 1.578947368 | 1.5 | 2 | 2 | 1 | 1 | |
| F | 2000 | 1.052631579 | 1 | 1 | 1 | 1 | 1 | |
| G | 8000 | 4.210526316 | 4 | 4 | 4 | 4 | 4 | |
| total | 38000 | 20 | 19 | 17 | 21 (too high) | 17 | 18 (too low) |
I can't tell what it is about these numbers but the tiniest fraction of a change to the divisor will always swing between the result being too high or too low, with the a perfect result being impossible to hit.
I'm wondering primarily if there's a lesser-known official next-step for dealing with this situation as written in Webster's original algorithm, or a common follow-up or modified step for dealing with this situation that someone else has come up with?
Otherwise, is there a way of telling in advance if the numbers will work or not?
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