The Electoral College, for better or for worse, is the method of selecting the President of the United States of America. Each state is apportioned a certain number of electors, based on its representation in the United States Congress. Thus, for Alaska (the oft-cited example), the state receives 3 electors: 2 for its Senators and 1 for its sole Representative. California receives 54 electors: 52 for its ample House representation, and its token 2 Senators as well.

There are many arguments that claim this is unfair to California. The House has been limited to 435 members since 1911. If we were to divide those members evenly across our approximately 280 million people, it would give one representative for every 730,000 people. States like Wyoming and Alaska do not have this many people living in them, and yet still receive one full representative. In addition, every state has two Senators regardless of population. Thus California's 54 electors is only 18 times as powerful as Alaska's 3 electors, even though they have nearly 50 times the population of Alaska. People use this as an example to prove that the Electoral College is actually more fair, giving a voice to people in Alaska who would otherwise go unheard.

Yet there is a flipside to this argument, and it rests in the Banzhaf Power Index (BPI). Conceived by John H. Banzhaf III, it relies on game theory and the power of critical players to prove that, in fact, states with more electoral votes have more weight and power than smaller states in the Electoral College.

Essentially, a BPI is a set of data that analyzes a voting group for their power within the group on achieving their ends. A simple example would be a company owned equally by 2 people, Bob and Jim. The *power set* of this voting group would be

{51: 50, 50}

Where 51 is the *quota* the group needs for a motion to pass (i.e. a majority vote) and the two 50s represent each voter's power. Generally these are ordered from highest to lowest, but in this case, both Bob and Jim have equal votes - 50%. In this case, both Bob and Jim must agree on the motion in order to achieve the 51% quota required to pass. Both of them have equal weight and equal power. Their BPI would look like this:

{Bob: .5; Jim: .5}

Note that a BPI always adds up to 1. In total, there is 100% of power to be distributed.

A more complicated example would be a company owned disproportionately by three people. Let's say Bob sells 40% of his stock to his friend Dan. Now Bob owns 30% of the stock, Dan owns 20%, and Jim still has his 50%. Let's also say the group agrees that in order to pass a bill, they must have 75% agreement. The power set is now

{75: 50, 30, 20}

With this arrangement, we can see a problem. If Bob and Dan want to do something, they must get Jim's approval to reach the 75% quota. And even if Dan refuses to go along with an idea of Bob and Jim, they will still have enough votes above the quota (80%) to pass the motion. To analyze specifically how much power each *player* has in this scenario, we generate a Banzhaf set of all of the possible voting block for this company:

Voters Weight Outcome
{} 0 L
{D} 20 L
{B} 30 L
{J} 50 L
{D,B} 50 L
{D,J} 70 L
{B,J} 80 W
{D,B,J} 100 W

The next step is to identify *critical players*. A critical player is a person who, if removed from a winning bloc of voters, turns it into a losing block. In the scenario set out above, Jim is a critical player twice, and Bob is a critical player once. Dan is never a critical player. The power index is

{Jim: .67; Bob: .33; Dan: 0}

This system obviously favors Jim. Interestingly, if you converted back to the 51% majority that the company had required before, the system again changes.

Voters Weight Outcome
{} 0 L
{D} 20 L
{B} 30 L
{J} 50 L
{D,B} 50 L
{D,J} 70 W
{B,J} 80 W
{D,B,J} 100 W

Now Dan is a critical player once, Bob is a critical player once, and Jim is a critical player three times. The power index becomes

{Jim: .6; Bob: .2; Dan: .2}

Here both Jim and Bob lose power at the expense of Dan. Whether or not this system is desirable is moot - the point is that with certain requirements, the power of voters change, often dynamically.

#### Application to the Electoral College

Ok, now that we've had our boring vote theory tutorial of the day, let's take a look at the BPI for the Electoral College. First things first, calculating all of those critical players and their values is simply ridiculous. There are **2^n possible outcomes for a set of n voters** - so with 51 voters (the states plus Washington, D.C.) that totals well into the quadrillions. Instead, let's use a computer! There is an excellent Banzhaf power generator at *http://www.warwick.ac.uk/~ecaae/ipgenf.html* which can generate BPI totals for any potential voting blocs - including our EC. (It's even preset for just such a calculation!)

So all of the data is inputted, as well as the quota (270 for the Electoral College) and then the numbers are calculated. Here is the Banzhaf Power Index for the 51 electors in the US Electoral College:

ST EV BPI
CA 54 0.1114
NY 33 0.0620
TX 32 0.0600
FL 25 0.0463
PA 23 0.0424
IL 22 0.0405
OH 21 0.0386
MI 18 0.0330
NJ 15 0.0274
NC 14 0.0256
VA 13 0.0237
GA 13 0.0237
IN 12 0.0219
MA 12 0.0219
WA 11 0.0200
TN 11 0.0200
WI 11 0.0200
MO 11 0.0200
MN 10 0.0182
MD 10 0.0182
LA 9 0.0164
AL 9 0.0164
OK 8 0.0145
CT 8 0.0145
CO 8 0.0145
SC 8 0.0145
AZ 8 0.0145
KY 8 0.0145
MS 7 0.0127
IA 7 0.0127
OR 7 0.0127
AR 6 0.0109
KS 6 0.0109
NE 5 0.0091
WV 5 0.0091
UT 5 0.0091
NM 5 0.0091
RI 4 0.0072
ID 4 0.0072
HI 4 0.0072
NH 4 0.0072
ME 4 0.0072
NV 4 0.0072
SD 3 0.0054
MT 3 0.0054
AK 3 0.0054
VT 3 0.0054
WY 3 0.0054
DE 3 0.0054
ND 3 0.0054
DC 3 0.0054

As you can see, California has a better thann 11% chance of being the dealbreaker for a President getting elected, while the 7 states and D.C. with only 3 electoral votes pose a mere .54*%* (that's 54 out of 10000, folks) chance of actually being the critical vote in an electoral college. In short, California is over 20 times more likely to push the election in favor of one candidate than South Dakota is. Whether or not this is fair, of course, depends on the populations in each of these two states.

Let's return to the voting theory, shall we?

#### Adding Populations to our BPI

Now in our earlier examples, our voting blocs were controlled by a single person. That is, if Bob voted yes on a proposal, then all of Bob's votes went towards that proposal. In state elections, we have a winner takes all plurality vote. If there are three candidates A, B, and C, with candidate A taking 40% of the vote, candidate B taking 35% of the vote, and candidate C taking 25% of the vote, candidate A receives all of a state's electoral votes, despite not having a majority in the state. (There are a few exceptions here, primarily involving Maine and Nebraska splitting their electoral votes among their voting districts, but we will discount this for the theoretical moment.)

So, let's make up a new example. Picture an imaginary country - let's call it Acirema. It is made up of 5 states: Aksala, Amabala, Nogero, Saxet, and Ohadi. Its electoral system is comparable to the US - each state is doled out electors, and the majority winner of the electoral college wins the election. Now Aksala has 12 electoral votes, and a population of 601. Amabala has 8 electoral votes, and a population of 361. Nogero has 6 electoral votes, and a population of 301. Both Saxet and Odahi have the minimum number of electors, 3. While Saxet has a population of 101, Odahi has a population of only 21.

So in total we have 32 electoral votes, and thus a candidate needs 17 votes to capture the Presidency. Here are the BPIs for the 5 states.

STATE EV BPI
Aksala 12 0.384
Amabala 8 0.231
Nogero 6 0.231
Saxet 3 0.077
Ohadi 3 0.077

This is pretty kosher with what we've learned so far. But now let's examine things from a population standpoint. And for that, we're gonna need some combinatorics all up in this business. (For those you with a strong statistical background, you can go read The Onion, but no talking!)

First, we must find out how often a person's vote will be the critical vote in a given population of **N+1** people. (Our critical voter is the extra 1, and we're assuming that N is even. Bear with me.) The formula for figuring out the number of different ways that you can choose **p** items from a population **n** is

n! / p! * (n-p)!

which in our case, p is equal to N/2 (exactly half of the population, minus our critical voter) and n is equal to - well, N. So our forumula is

N! / (N/2)! * (N/2)!

And we know that all of the possible outcomes of the voting in the population is equal to 2^(N+1) from our formula earlier. So the probability that there will be a critical voter in our population is

N! / (N/2)! * (N/2)! * (2^(N+1))

Which we can calculate. Now to make things easier for you, there is an approximation known as Stirling's approximation, which states that a good rough estimate for the factorial of a number m is

e^(-m)*(m^m)*sqrt(2*pi*m)

If we set N/2 equal to our m in the formula and substitute it into our original calculation, we get

e^(-2m)*(2m^2m)*sqrt(4*pi*m) / e^(-m)*(m^m)*sqrt(2*pi*m) * e^(-m)*(m^m)*sqrt(2*pi*m) * 2^m

Which can be simplified to

e^(-2m)*(2m^2m)*sqrt(4*pi*m) / e^(-2m)* 2m^2m * sqrt(4*pi*pi*m*m)

Which can be simplified to

sqrt(4 * pi * m / 4 * pi * pi * m * m)

Which simplifies to

sqrt(1/pi* m)

And plugging our N/2 back in for m, we get

sqrt(2 / (pi * N))

Which works out to be about 4/5 * sqrt(N). So, interestingly enough, all of this goes to prove that the power of the individual voter in a population is inversely proportional to the square root of the population. So in order to find out the power of individual voters, all we have to do is divide the BPIs of each state by the square root of its population. Note this is not the actual voting power of each voter, but merely a scale comparing the power of one voter in one state to another voter in another state.

STATE EV BPI POPULATION RATIO NBPI
Aksala 12 0.384 601 .0156 2.03
Amabala 8 0.231 361 .0122 1.58
Nogero 6 0.231 301 .0133 1.73
Saxet 3 0.077 101 .0077 1.00
Ohadi 3 0.077 21 .0169 2.19

What's this? The state with the smallest population has the most powerful voters! At the far right is a Normalized BPI - the smallest value has been set equal to 1 and the other ratios have been adjusted proportionally. Thus in Ohadi a vote there is 2.19 times as powerful as one in Saxet - even though they have the same number of electoral votes! And although Aksala is almost 5 times as likely to determine the winner of the election as Ohadi is, its voter's powers are still not as powerful as the 21 lonely citizens of Ohadi.

The reasons this is true are fairly logical. Imagine if Ohadi had a population of only one person - say, our old friend Bob. Since Ohadi has 7% of the electoral power, Bob's single vote also has 7% of the electoral power. But for every person who joins Ohadi, the power does not merely get diluted by a linear percentage, but exponentially, because the combinations of votes increase that way.

Now, as it has been pointed out, deriving all of the possible coalitions for the 50 states would be time-consuming beyond all belief. The number of voting blocs in a group with *n* members is *2^n*. For 50 states, this reaches the quadrillions! Instead, data is generated by creating random voting blocs that exceed the majority requirement for selecting the President (270 at the moment) and then determining critical players within that bloc. This is repeated billions of times to create a general overview of how much power each state has. The results, as culled from *http://www.cs.unc.edu/~livingst/Banzhaf/* (thank you, Mark Livingston!), are as follows:

STATE POPULATION ELECTORS POWER INDICES
CA 29760021 54 3.344
NY 17990455 33 2.394
TX 16986510 32 2.384
FL 12937926 25 2.108
PA 11881643 23 2.018
IL 11430602 22 1.965
OH 10847115 21 1.923
MI 9295297 18 1.775
NC 6628637 14 1.629
NJ 7730188 15 1.617
VA 6187358 13 1.564
GA 6478216 13 1.529
IN 5544159 12 1.524
WA 4866692 11 1.490
TX 4877185 11 1.489
WI 4891769 11 1.486
MA 6016425 12 1.463
MO 5117073 11 1.453
MN 4375099 10 1.428
MD 4781468 10 1.366
OK 3145585 08 1.346
AL 4040587 09 1.337
WY 453588 03 1.327
CT 3287116 08 1.317
CO 3294394 08 1.315
LA 4219973 09 1.308
MS 2573216 07 1.302
SC 3486703 08 1.278
IA 2776755 07 1.253
AZ 3665228 08 1.247
KY 3685296 08 1.243
OR 2842321 07 1.239
NM 1515069 05 1.211
AK 550043 03 1.205
VT 562758 03 1.192
RI 1003464 04 1.190
ID 1006749 04 1.188
NE 1578385 05 1.186
AR 2350725 06 1.167
DC 606900 03 1.148
KS 2477574 06 1.137
UT 1722850 05 1.135
HI 1108229 04 1.132
NH 1109252 04 1.132
ND 638800 03 1.118
WV 1793477 05 1.113
DE 666168 03 1.095
NV 1201833 04 1.087
ME 1227928 04 1.076
SD 696004 03 1.071
MT 799065 03 1.000

As expected, the state with the largest population among states with only 3 electors (Montana) gets the shaft and serves as the low point. Wyoming, the least populated state, actually fares quite well, standing as the 23rd most powerful state to vote in. But what is most interesting about this chart are the large gaps between one state and the next, suggesting major disparity in voting equality. For example, at the bottom, Montana voters are 7% less powerful than South Dakota voters, who are in turn 7% less powerful than West Virginia, who are 7% less powerful than Oregon, who are 7% less powerful than Colorado - and so on and so forth. But there is no one between Montana and South Dakota, while there are 14 states between West Virginia and Oregon - suggesting that Montana is even more disenfranchised than you would think, and is thus somewhat inflating the numbers above it. And, of course, it's good to be the king: California, despite its claims of underrepresentation, is a full 330% more powerful than Montana, over 250% more powerful than Alaska, and nearly twice as powerful as Michigan, a fairly powerful state itself. In fact, California is 140% more powerful than second place New York - California is a major outlier here.

So what does this data suggest? Well, it suggests that despite the claims that the Electoral College overrepresents smaller states, in general the larger states have much more sway. This holds true with the idea that the "swing states" are more valuable than the smaller states on the totem pole, because they are more likely to be the determining factor. In short, the BPI puts the small states back in their place - nullifying the overrepresentational factor by adding the perspective of their actual contribution to the 270 votes needed. In this way, at least, the old saying is true: bigger is better.

#### Commentary on BPIs and Electoral Colleges

At the heart of the debate of the electoral college are the ideas of fairness, equality, and representation. There are also a lot of ancillary factors, such as procedural quirks (the faithless electorate), the winner take all state elections, and the problems that might evolve out of a direct election - ignoring rural areas for the heavily populated urban areas. But at the very center of it all is the idea of one person, one vote.

It's pretty foolish to assume that any US state election will be decided by one vote - Florida's 2000 result was the closest one recorded ever, and was still over 300 votes away from being a perfect scale. In essence, our ratios don't reflect a real-world ability of the voter to influence the election. Rather, they reflect the fact that **additional electors** give more power in the electoral college than **additional representation** within your set of electors. This is because in power spectrums as wide as the EC - ranging from 3 to 54 - being a critical player becomes virtually impossible at the lower end of the spectrum.

The reason this correlates with the square root of the population is the fallacy of combinatorics. Perhaps you've heard this old chestnut before: A man has two children. He tells you that at least one of them is a boy. What're the odds that the other one is a boy as well? The answer isn't 50%, but merely 25%. This is because there are four possible results for his children's gender: {G,G}, {B,G}, {G,B}, and {B,B}. BPI does not account for the difference between {G,B} and {B,G} because it lumps them all together in determining critical votes. Take the example of a 5 voter election, with Bob, Jim, Dan, Susie, and Lisa - I'll just show you the coalitions of 3, where every vote is critical:

{B,J,D}, {B,J,S}, {B,J,L}, {B,D,S}, {B,D,L}
{B,S,L}, {J,D,S}, {J,D,L}, {J,S,L}, {D,S,L}

In this case each voter is critical 6 times out of 32 possible outcomes. Their BPIs are all .2, but in fact their BPIs should be .1875. This "dilution" of power is due to the binary nature of elections - once a candidate has "won" an election, he isn't awarded any extra bonus points for extra votes. So there is a small (but real) chance that even though you voted for the winning candidate, your vote didn't matter, because he had already secured the votes he needed. This is the combinatoric law clashing with the rule of discrete voting procedures - once a winner has been declared, why count the rest of the votes?

But this gets even better, because it goes back to the heart of our debate: one man, one vote. Let's say we have a 5 state nation - back to Aksala, Amabala, Ohadi, Saxet, and Nogero. Here's a population listing:

EV1
Aksala 701 12
Amabala 361 8
Nogero 241 6
Saxet 181 4
Ohadi 21 3
Quota: 17/33

Now let's say that a proposal comes up, and here is the popular voting on the proposal:

FOR AGIN EV FOR EV AGIN
Aksala 300 401 12
Amabala 261 100 8
Nogero 201 140 6
Saxet 1 180 4
Ohadi 21 0 3
Totals 784 821 17 16

So an unpopular proposal still passes. But if you readjust electors to the square roots of the populations in our election:

POP EV1 EV2 EV FOR EV AGIN
Aksala 701 12 265 265
Amabala 361 8 190 190
Nogero 241 6 155 155
Saxet 181 4 134 134
Ohadi 21 3 46 46
Totals 790 391 399

Suddenly the people have power again. Giving out electors based on the square roots of population **minimizes the effect of the minority will**, i.e. having an unpopular decision pushed across. There is an excellent paper on this written by a University of London philosophy professor named Moshe Machover, who does a lot of good stuff on game theory. It's called "Minimizing the mean deficit: the second square-root rule", but you can't find it online. It's in Mathematics and Social Science magazine - I used to flip through these all the time for my classes. You can get details on the web.

In any case, square rooting the population creates a system that more fairly awards the state's electors. It spreads the power of the "critical voter" out evenly to every voter, which is good for democracy. The essence of one man, one vote, so to speak.