Voting power (3): The American swing states

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As we all know, American Presidential elections are done with a two-stage process: first the public votes, and then the Electoral College votes. It is the Electoral College that actually votes for the President; but they vote (in their respective states) in accordance with the plurality determined by the public vote. This unusual system was devised by the Founding Fathers as a compromise between mob rule and autocracy, of which both they were determined to guard against. The Electoral College is not now an independent body: in all states but two all electoral college votes are given to the winner in that state. This means that the Electoral College may "amplify" the public vote; or it may return a vote which differs from the public vote, in that a candidate may receive a majority of public votes, and yet still lose the Electoral College vote. This means that there are periodic calls for the Electoral College to be disbanded, but in reality that seems unlikely. And in fact as far back as 1834 the then President, Andrew Jackson, was demanding its disbanding: a President, according to Jackson, should be a "man of the people" and hence elected by the people, rather than by an elite "College". This is one of the few instances where Jackson didn't get his way.

The initial idea of the Electoral College was that voters in their respective states would vote for Electors who would best represent their interests in a Presidential vote: these Electors were supposed to be wise and understanding men who could be relied on to vote in a principled manner. Article ii, Section 1 of the USA Constitution describes how this was to be done. When it became clear that electors were not in fact acting impartially, but only at the behest of the voters, some of the Founding Fathers were horrified. And like so many political institutions the world over, the Electoral College does not now live up to its original expectations, but is also too entrenched in the political process to be removed.

The purpose of this post is to determine the voting power of the "swing states", in which most of a Presidential campaign is conducted. It has been estimated that something like 75% of Americans are ignored in a campaign; this might be true, but that's just plain politics. For example California (with 55 Electoral College cotes) is so likely to return a Democrat candidate that it may be considered a "safe state" (at least, for the Democrats); it would be a waste of time for a candidate to spend too much time there. Instead, a candidate should stump in Florida, for example, which is considered a swing state, and may go either way: we have seen how close votes in Florida can be.

For discussion about measuring voting power using power indices check out the previous two blog posts.

The American Electoral College

According to the excellent site 270 to win and their very useful election histories, we can determine which states have voted "the same" for any election post 1964. Taking 2000 as a reasonable starting point, we have the following results. Some states have voted the same in every election from 2000 onwards; others have not.

Safe Democrat Safe Republican Swing
California 55 Alabama 9 Colorado 9
Connecticut 7 Alaska 3 Florida 29
Delaware 3 Arizona 11 Idaho 4
DC 3 Arkansas 6 Indiana 11
Hawaii 4 Georgia 16 Iowa 6
Illinois 20 Kansas 6 Michigan 16
Maine 3 Kentucky 8 Nevada 6
Maryland 10 Louisiana 8 New Hampshire 4
Massachusetts 11 Mississippi 6 New Mexico 5
Minnesota 10 Missouri 10 North Carolina 15
New Jersey 14 Montana 3 Ohio 18
New York 29 Nebraska 4 Pennsylvania 20
Oregon 7 North Dakota 3 Virginia 13
Rhode Island 4 Oklahoma 7 Wisconsin 10
Vermont 3 South Carolina 9 Maine CD 2 1
Washington 12 South Dakota 3 Nebraska CD 2 1
Tennessee 11
Texas 38
Utah 6
West Virginia 5
Wyoming 3
195 175 168

From the table, we see that since 2000, we can count on 195 "safe" Electoral College votes for the Democrats, and 175 "safe" Electoral College votes for the Republicans. Thus of the 168 undecided votes, for a Democrat win the party must obtain at least 75 votes, and for a Republican win, the party needs to amass 95 votes.

Note that according to the site, of the votes in Maine and Nebraska, all but one are considered safe - remember that these are the only two states to apportion votes by Congressional district. Of Maine's 4 Electoral College votes, 3 are safe Democrat and one is a swing vote; for Nebraska, 4 of its votes are safe Republican, and 1 is a swing vote.

All this means is that a Democrat candidate should be campaigning considering the power given by

\[ [75; 9,29,4,11,6,16,6,4,5,15,18,20,13,10,1,1] \]

and a Republican candidate will be working with

\[ [95; 9,29,4,11,6,16,6,4,5,15,18,20,13,10,1,1] \]

A Democrat campaign

So let's imagine a Democrat candidate who wishes to maximize the efforts of the campaign by concentrating more on states with the greatest power to influence the election.

In [1]: q = 75; w = [9,29,4,11,6,16,6,4,5,15,18,20,13,10,1,1]

In [2]: b = banzhaf(q,w); bn = [sy.Float(x/sum(b)) for x in b]; [sy.N(x,4) for x
in bn]
Out[2]: [0.05192, 0.1867, 0.02271, 0.06426, 0.03478, 0.09467, 0.03478, 0.02271,
0.02870, 0.08801, 0.1060, 0.1196, 0.07515, 0.05800, 0.005994, 0.005994]

In [3]: s = shapley(q,w); [sy.N(x/sum(s),4) for x in s]
out[3]: [0.05102, 0.1902, 0.02188, 0.06375, 0.03367, 0.09531, 0.03367, 0.02188,
0.02770, 0.08833, 0.1073, 0.1217, 0.07506, 0.05723, 0.005662, 0.005662]

The values are not the same, but they are in fact quite close, and in this case they are comparable to the numbers of Electoral votes in each state. To compare values, it will be most efficient to set up a Dataframe using Python's data analysis library pandas. We shall also convert the Banzhaf and Shapley-Shubik values from sympy floats int ordinary python floats.

In [4]: import pandas as pd
In [5]: bf = [float(x) for x in bn]
In [6]: sf = [float(x) for x in s]

In [5]: d = {"States":states, "EC Votes":ec_votes, "Banzhaf indices":bf, "Shapley-Shubik indices:sf}

In [6]: swings = pd.DataFrame(d)

In [7]: swings.sort_values(by = "EC Votes", ascending = False)
In [8]: ssings.sort_values(by = "Banzhaf indices", ascending = False)
In [9]: swings.sort_values(by = "Shapley-Shubik indices", ascending = False)

We won't show the results of the last three expressions, but they all give rise to the same ordering.

We can still get some information by not looking so much at the values of the power indices, but their relative values to the number of Electoral votes. To do this we need a new column which normalizes the Electoral votes so that their sum is unity:

In [10]: swings["Normalized EC Votes"] = swings["EC Votes"]/168.0
In [11]: swings["Ratio B to N"] = swings["Banzhaf indices"]/swings["Normalized EC Votes"]
In [12]: swings["Ratio S to N"] = swings_d["Shapley-Shubik indices"]/swings["Normalized EC Votes"]
In [13]: swings.sort_values(by = "EC Votes", ascending = False)

The following table shows the result.

States EC Votes Banzhaf indices Shapley-Shubik indices EC Votes Normalized Ratio B to N Ratio S to N
Florida 29 0.186702 0.190164 0.172619 1.081585 1.101640
Pennsylvania 20 0.119575 0.121732 0.119048 1.004430 1.022552
Ohio 18 0.106034 0.107289 0.107143 0.989655 1.001360
Michigan 16 0.094671 0.095309 0.095238 0.994049 1.000743
North Carolina 15 0.088014 0.088330 0.089286 0.985754 0.989293
Virginia 13 0.075149 0.075057 0.077381 0.971155 0.969966
Indiana 11 0.064261 0.063752 0.065476 0.981447 0.973660
Wisconsin 10 0.058004 0.057227 0.059524 0.974471 0.961422
Colorado 9 0.051922 0.051017 0.053571 0.969215 0.952318
Iowa 6 0.034777 0.033670 0.035714 0.973770 0.942774
Nevada 6 0.034777 0.033670 0.035714 0.973770 0.942774
New Mexico 5 0.028695 0.027704 0.029762 0.964169 0.930862
Idaho 4 0.022714 0.021877 0.023810 0.953972 0.918823
New Hampshire 4 0.022714 0.021877 0.023810 0.953972 0.918823
Maine CD 2 1 0.005994 0.005662 0.005952 1.007058 0.951282
Nebraska CD 2 1 0.005994 0.005662 0.005952 1.007058 0.951282

We can thus infer that a Democrat candidate should indeed campaign most vigorously in the states with the largest number of Electoral votes. This might seem to be obvious, but as we have shown in previous posts, there is not always a correlation between voting weight and voting power, and that a voter with a low weight might end up having considerable power.

A Republican candidate

Going through all of the above, but with a quota of 95, produces in the end the following:

States EC Votes Banzhaf indices Shapley-Shubik indices EC Votes Normalized Ratio B to N Ratio S to N
Florida 29 0.186024 0.190086 0.172619 1.077658 1.101190
Pennsylvania 20 0.119789 0.121871 0.119048 1.006230 1.023718
Ohio 18 0.106258 0.107258 0.107143 0.991741 1.001075
Michigan 16 0.094453 0.095156 0.095238 0.991756 0.999140
North Carolina 15 0.088106 0.088410 0.089286 0.986789 0.990194
Virginia 13 0.075362 0.074940 0.077381 0.973906 0.968460
Indiana 11 0.064064 0.063568 0.065476 0.978439 0.970862
Wisconsin 10 0.058073 0.057394 0.059524 0.975628 0.964219
Colorado 9 0.052133 0.051209 0.053571 0.973140 0.955892
Iowa 6 0.034692 0.033612 0.035714 0.971363 0.941142
Nevada 6 0.034692 0.033612 0.035714 0.971363 0.941142
New Mexico 5 0.028776 0.027715 0.029762 0.966885 0.931235
Idaho 4 0.022912 0.021963 0.023810 0.962300 0.922436
New Hampshire 4 0.022912 0.021963 0.023810 0.962300 0.922436
Maine CD 2 1 0.005877 0.005621 0.005952 0.987357 0.944289
Nebraska CD 1 0.005877 0.005621 0.005952 0.987357 0.944289

and we see a similar result as for the Democrat version, an obvious difference though being that Michigan has decreased its relative power, at least as measured using the Shapley-Shubik index. \