# US presidential electoral system

## Abstract

We recall the definition of the coefficient of power of an elector in a voting system, and state some elementary facts. Using this formalism, we study the system used for US presidential elections (the electoral college). We show that this system is biased toward populous states, giving a citizen of California roughly four times more power than one of Montana in the choice of the US president.

## Introduction

It is frequent, in various kinds of electoral systems, to give electors a variable number of votes in deciding the outcome. The US presidential electoral system is archetypal in this: the president is de facto elected by the states of the Union, where each state gets a number of votes (in the electoral college) according to its population.

It does not take much reflection to understand that the relation between the number of votes given to an entity, and the power that the entity draws from these votes, is somewhat complex. At any rate, it is not simply linear (though in approximation, for some special cases, it can be). For example, if one entity gets more than half of the votes, then that entity receives all the power. To take a more interesting example, suppose Alice, Bob, Carol and David have, respectively, 4, 4, 2 and 1 votes in an election: a quick check shows that the outcome of a yes/no vote between them (with majority being at 6 votes; and where we assume that parties cannot abstain) will be determined by the majority of Alice, Bob and Carol; so in fact, Alice, Bob and Carol hold equal power, whereas David has none. Real-life examples, of course, need not be so clear-cut, but there's more to an electoral system than meets the eye.

To measure the actual power that an entity holds in such a decision system (rather than simply its number of votes), a simple indicator called the coefficient of power has been devised. It is simply the proportion of all configurations of votes of the other entities (in a yes/no vote without abstention) where the entity's choice will be decisive. It is this indicator that we intend to study in the case of the US presidential election.

## Mathematical background

We have tried to be as extensive as possible in our treatment of decision systems. Consequently, some might find the following catalog of properties somewhat boring. It shouldn't be necessary to understand it all in detail to get the gist of our purpose.

### Prolegomena and generalities on decision systems

By a yes/no decision system we mean a function F:{0,1}I→{0,1}, where I is a set called the set of electors. In other words, for each choice between yes (1) and no (0) of the elements of I (the electors), the decision system yields an answer which is also yes or no. This is also called a boolean function. We make the further assumption that the decision system is monotone (nondecreasing), which means that changing some inputs to F from 0 to 1 (and leaving the others unchanged) can only change the output from 0 to 1 if at all (in more sophisticated terms, F is nondecreasing from the set {0,1}I with the product order, to the set {0,1}).

A decision system is said to be unbiased when changing every input's value (from 0 to 1 or vice versa) also changes the output value. In plain English, this means that the system does not prejudge between the two possible outcomes: it does not favor one over the other. This is a rather strong property: the values 0 and 1 are actually indistinguishable (in sophisticated terms, the permutation of 0 and 1 induces an automorphism of the decision system). Evidently, if F is unbiased, then F is neutral, meaning that there are as many configurations x (elements of {0,1}I) such that F(x)=0 as there are such that F(x)=1. The converse is not true, even if F is monotone: there exist decision systems which are neutral and monotone, but which are still biased. (Note incidentally that being neutral makes sense only when I is finite, whereas being unbiased makes sense for any set of electors. Of course, in real life, I is always finite. If we really insist on the greatest possible generality, we can actually make sense of neutrality for infinite I, on the condition that F is measurable for the product measure on {0,1}I.)

The properties of being “unbiased” or just “neutral” pertain to the treatment of the values (yes or no) by the decision system. We now introduce two concepts: “symmetric” and “fair” which pertain to the system's treatment of electors.

We say that F is symmetric when it satisfies the property that F(x)=F() for any configuration x and any permutation σ of I (in sophisticated terms, any permutation on the electors induces an automorphism of F). Here, refers to the configuration obtained by composing x by σ, in other words permuting the electors' choices as given by σ (or its inverse, but no matter). This means that the electors are indistinguishable in the strong sense: permuting their votes in any way whatsoever will not alter the final decision. More generally, we can consider the set S of all permutations σ of I satisfying the condition that F(x)=F() for all x: this is a subgroup of the group of all permutations of I, which is equal to the full symmetric group precisely when F is symmetric. When S acts transitively on I, in other words when for each i and j of I there exists σ in S verifying σ(i)=j, then we say that F is fair. So fairness is a weaker criterion than symmetry: it means that every elector can be replaced by any other one, but not necessarily all at once (thus, there can be a bias toward certain pairs of electors, for example).

There exist decision systems which are monotone, fair and unbiased, but which still aren't symmetric. For a simple example, suppose we have nine electors, divided in three “castes” of three electors each. Let the decision system be as follows: the outcome is “yes” precisely when there are at least two castes in each one of which at least two electors have voted “yes”. Evidently this is monotone (because the wording carefully used the words at least). Also, this is unbiased, because if the outcome is “no”, then there are less then two castes with two or three “yes” votes, so there are at least two castes with fewer votes, meaning that there are at least two castes with at least two “no” votes in each. Finally, we can see that the decision system is fair. In fact, we can determine exactly what the group S is: it contains those permutations that permute the castes, and permute the electors within each caste, but do not mix the castes; thus, only 1296 of the 362880 possible permutations of nine electors are in S. But these 1296 permutations operate transitively on the nine elements of I.

On the other hands, if F is a monotone and symmetric decision system, then there isn't much room for choice left. Indeed, since F is symmetric, its value on a certain configuration depends only on the cardinality of the set of yes-voters and the cardinality of the set of no-voters (for a finite I, the cardinality of one can be computed from the other, obviously); and since F is monotone, it does so monotonically. Now there aren't many ways it could do that: if I is finite, it just means that F chooses “yes” provided there are more than a certain number of yes-voters. If F is monotone, symmetric and unbiased (or simply neutral, when that makes sense), then F is uniquely determined: in this case, I must be finite, must have an odd number of elements, and F simply makes the decision that the majority of voters (electors) make. We say that F is simply the majority vote decision system.

### Votational decision systems

Assume that the set I of electors is finite. Assume further that for each element i of I (each elector) we are given a nonnegative real number pi, the weight of i. Assume finally that we are given a nonnegative real number t, the voting threshold. From these data we construct a decision system as follows: if x∈{0,1}I is a voting configuration, define F(x)=1 if the sum of those pi for which x(i)=1 is greater or equal to t, and F(x)=0 otherwise. Such a decision system is called votational. It just says that the decision is made by taking votes, summing the votes, each elector having a certain weight of importance in the decision, and comparing the total with a certain threshold.

Note that just knowing that a decision system is votational does not permit one to recover the weights and the threshold. In fact, there are always several possible values for the weights and the threshold that will lead to identical decision systems.

A votational decision system is always monotone (because we declared that the weights must be nonnegative). A votational system that is is neutral is, in fact, unbiased; and one that is fair is, in fact, symmetric. We leave the proofs as exercices (the main difficulty lies in formulating things correctly). These propositions can be used to show that certain decision systems, albeit monotone, are not votational. I do not know whether some simple, intrinsic criterion can be found for a decision system to be votational; nor of a way to constructively produce a set of weights for a decision system that is know to be votational.

Of course, the majority vote decision system (the only monotone symmetric unbiased voting system) is votational: it is obtained by giving every elector the same weight, and choosing a threshold of half the total weight (the number of electors must be odd, remember).

### Coefficients of power

Assume that I is finite (or that the decision system F is a measurable function for the product measure). We assume that F is monotone. Fix an elector iI. For any configuration x, we can define two configurations x0 and x1 which are derived from x by forcing i's vote to 0 and 1 respectively (of course, one of x0 and x1 is already equal to x; besides, they depend only on the votes of the elements of I other than i). We say that i's vote is decisive in the configuration x when F(x0) is not equal to F(x1) (so, because F is monotone, the first must be 0 and the second must be 1). We define the coefficient of power of i to be the probability that i's vote is decisive; here, “probability” is taken in the sense of the product measure: we assume that every elector except i votes yes or no with probability 1/2 for each, and each elector being independent of the others. So in other words, the coefficient of power of i is the fraction of all configurations in which i's vote is decisive.

While the coefficient of power can be defined in all generality (at least for I finite), even when F is not monotone, yet it is only interesting when F is monotone and neutral. However, the system does not have to be votational. Naturally, if F is fair, then all coefficients of power are equal: we then say that F is balanced. Being balanced is a weaker condition than being fair (which is in turn, as we have seen, a weaker condition than being symmetric); but these conditions become all equivalent for a votational system. Later on, we will give an example of a system that is balanced without being fair.

### Calculating the coefficient of power in majority voting

We have already mentioned that the monotone symmetric unbiased voting system (majority voting) is uniquely determined up to the number of electors, which must be a finite odd (positive) integer. Let it be N=2n+1, for n some natural number. The decision system is simply a votational system with all electors having weight 1 and the threshold being n+1/2. We now want to calculate the coefficient of power of an elector in this system (of course, all electors are equivalent, by symmetry). The number of configurations of the other electors' votes is 22n. Of these, the configurations where the chosen elector's vote is decisive are precisely those where n of the other electors vote “yes” and n vote “no”. Now there are C2nn (2n choose n), that is (2n)!/(n!2) such configurations. So the sought coefficient of power is (2n)!/(22nn!2).

We are interested in the behavior of this coefficient when n gets large (or equivalently when the population N=2n+1 gets large). But if we estimate factorials using the Stirling formula, we get such an equivalent. In fact, we find that the coefficient is equivalent to sqrt(1/(πn)), in other words sqrt(2/(πN)). (If your browser supports MathML, that's $\sqrt{\frac{2}{\pi N}}$.) The essential result, here, is that the coefficient of power decreases like the inverse square root of the population of electors. So in a population of a hundred (and one) electors the probability that your ballot makes a difference is 8 percent; and in a population of ten thousand (and one), this probability is just about ten times lower, or 0.8 percent.

Because this fact is so important, let us restate it once more: the coefficient of power in a population of (electoral) peers drops down (for large populations) with the square root of the population. It is this square root which makes the US electoral system unbalanced, as we shall see.

If we want more precision, we can use the following asymptotic expansion (for large N) of the coefficient of power (written in MathML): $\sqrt{\frac{2}{\pi N}}\left(1+\frac{1}{4N}+\frac{1}{32{N}^{2}}-\frac{5}{128{N}^{3}}+O\left(\frac{1}{{N}^{4}}\right)\right)$. However, such extra precision is rarely needed.

It is, of course, the approximation we have given here (the leading term is quite sufficient) that we will use to calculate the coefficient of power of a state's citizen within that state.

### Coefficients of power for votational decision systems

We now consider a votational decision system that is neutral (hence unbiased). As has been pointed out in the introduction, the coefficients of power are not in general proportional to the weights of the system. In general, there does not exist (at least, to the best of the author's knowledge) a method of calculating the coefficients of power from the weights, other than examining every voting configuration (which is inacceptably slow); in practice, however, one can get approximate answers by randomly selecting sufficiently many voting configurations to obtain a meaningful statistical sample.

In the case where every weight is 1, we have obtained an asymptotic approximation of the coefficients of power, when the population is large. This approximation can be carried over, under appropriate hypotheses, for variable weights. The important condition for the approximation to be valid is that the sum of the squares of the weights be large compared to the square of the largest weight. We have the following approximate facts (note that we are prudent in making no actual statement as to the error terms!):

• Each elector's coefficient of power is (approximately) proportional to the corresponding weight. (And not to the square of that weight!)
• The sum of the squares of the coefficients of powers is (approximately) two over pi.
• Consquently: to find the actual value, define a “virtual population count”, N, equal to the sum of squares of the weights divided by the square of the weight of the elector under consideration. Then the coefficient of power of the elector is (approximately) equal to the (approximate) value obtained in a population of peers of that value. (Namely, sqrt(2/(πN)).)

Remember that we are assuming the system to be unbiased (i.e. the threshold is, essentially, one half the sum of the weights).

Actually, the approximation is surprisingly good. In the case of the US electoral college (with “electors” being states, and weights being the number of electoral votes), it works quite well except for the state of California (having weight 54) which has more power than given by proportionality with its weight; the virtual population count is then between 10000 and 11000 (divided by the square of the number of votes of the state in question, naturally).

### Two-stage decision systems

A two-stage decision system is defined as follows: partition the set I of electors in a certain set of castes. Choose a decision system for each caste. Then choose an overall decision system that uses the castes as electors (with each caste's vote determined by its own decision system) to produce the final output. We have given an example of this earlier, with three castes of three electors each, using the monotone symmetric unbiased voting system in each caste and at the global level. This is also, typically, the system used in the US presidential election: each state represents a caste, each caste's decision system is the monotone symmetric unbiased voting system, and the overall decision system is a weighted votational system.

If each caste's decision system is neutral, then it is easy to compute the coefficients of power in such a system. Indeed, the caste's decision system being neutral means that the caste's vote will be randomly yes or no, with equal probability for each, when each of the caste's members vote follows this law. So to compute an individual's coefficient of power, we simply multiply the individual's coefficient of power in the caste by the caste's coefficient of power in the whole.

Of course, in a two-stage decision system, each caste can, in turn, adopt a two-stage decision system on its own. We can use that remark to construct a decision system that is (monotone, unbiased and) balanced, but yet isn't fair (there's nothing remarkable about it; in fact, it's pretty much trivial, but it's an interesting curiosity). Proceed as follows: use 45 electors; divide them in three “castes”. The first caste contains five “tribes” of three electors each. The other two castes contain three “tribes” each, which in turn have five electors each. Each tribe's vote is determined by the majority vote of its electors; each caste's vote is determined by the majority vote of its tribes; and the overall vote is determined by the majority vote of the three castes. The system is evidently monotone; and it is unbiased because changing every elector's vote will change every tribe's vote, hence every caste's vote, hence the overall vote. Since majority vote (with an odd number of electors) is neutral, we see that the system is neutral (in fact, unbiased) at every level. So we can calculate its coefficients of power: an individual's coefficient of power in the tribe is 1/2 if the tribe has three electors, and 3/8 if it has five; a tribe's coefficient of power in the caste is 1/2 if the caste has three tribes, and 3/8 if it has five; but this shows that every individual's coefficient in the caste is 3/16. And of course, every individual's coefficient of power in the whole is then 3/32. Proving that the system is not fair is an easy exercice in group theory (essentially, there are 933120 permutations of the first caste which preserve the decision system, and as many as 10368000 for the two other castes; so the castes cannot be permuted).

## The US presidential election system

### Relevant text

The Executive power shall be vested in a President of the United States of America. He shall hold office during the term of four years, and together with the Vice President, chosen for the same term, be elected as follows:

The Electors shall meet in their respective States and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same State with themselves; they shall name in their ballots the person voted for as President, and in distinct ballots the person voted for as Vice-President, and of the number of votes for each, which lists they shall sign and certify, and transmit sealed to the seat of the Government of the United States, directed to the President of the Senate; the President of the Senate shall, in the presence of the Senate and House of Representatives, open all the certificates and the votes shall then be counted; — The person having the greatest number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed; and if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President. But in choosing the President, the votes shall be taken by States, the representation from each State having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the States, and a majority of all the States shall be necessary to a choice. And if the House of Representatives shall not choose a President whenever the right of choice shall devolve upon them, before the fourth day of March next following, then the Vice-President shall act as President, as in case of the death or other constitutional disability of the President. The person having the greatest number of votes as Vice-President, shall be the Vice-President, if such numbers be a majority of the whole number of electors appointed, and if no person have a majority, then from the two highest numbers on the list, the Senate shall choose the Vice-President; a quorum for the purpose shall consist of two-thirds of the whole number of Senators, and a majority of the whole number shall be necessary to a choice. But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.

The Congress may determine the time of choosing the electors, and the day on which they shall give their votes; which day shall be the same throughout the United States.

No person except a natural born citizen, or a citizen of the United States, at the time of the adoption of this Constitution, shall be eligible to the office of President; neither shall any person be eligible to that office who shall not have attained to the age of thirty-five years, and been fourteen years a resident within the United States.

In case of the removal of the President from office, or of his death, resignation, or inability to discharge the powers and duties of the said office, the same shall devolve on the Vice President, and the Congress may by law provide for the case of removal, death, resignation, or inability, both of the President and Vice President, declaring what officer shall then act as President, and such officer shall act accordingly, until the disability be removed, or a President shall be elected.

The President shall, at stated times, receive for his services, a compensation, which shall neither be increased nor diminished during the period for which he shall have been elected, and he shall not receive within that period any other emolument from the United States, or any of them.

Before he enter on the execution of his office, he shall take the following oath or affirmation: “I do solemnly swear (or affirm) that I will faithfully execute the office of the President of the United States, and will to the best of my ability, preserve, protect and defend the Constitution of the United States.”

Constitution of the United States, article II, section 1, as modified by the twelfth amendment (1804).

### Description

The US presidential electoral system is in effect a two-stage decision system. In the first stage, the voters from every state elect a certain number of members of electoral college, who are pledged, de facto if not de jure, to vote for a certain candidate. In the second stage, the electoral college elects the president.

So in fact, everything happens as though the States of the Union elected the president, where each state receives a certain weight (fixed a priori), and votes according to the result of the popular vote in the state. (Historically, the state legislature could choose to select their electors rather than letting the people vote for them. No longer is this the case anywhere.)

In fact, this is not strictly true. Whereas nearly all states give all their electors to the candidate who got the greatest number of popular votes in the state, yet two states, Maine and Nebraska, proceed differently, in theory at least: two of their electors are chosen by popular vote in the whole state, and the other (two or three) are elected by electoral districts (as representatives are). We have chosen to ignore this phenomenon: this means that the coefficients of power we compute for Maine and Nebraska are wrong (to be precise, they are too high); but it should alter only very slightly the coefficients of the other states, and the ratios of the coefficients of power of the other states should be changed only infinitesimally. In any case, we have chosen to proceed as though the electoral process were strictly two-stage (correcting for this would be difficult).

According to the general formalism we have laid out, we consider a yes/no election. This is not what the presidential election is, but it hardly matters: it could just as well be used for that purpose, it is a handy way of measuring power, and, in any case, third party candidates are, in the United States, unimportant enough to be negligible.

The number of electors each state receives (i.e. its electoral weight) is computed according to rules prescribed by the US constitution. It varies between 3 (Wyoming, Vermont, etc) and 54 (California). It is equal to the number of senators (always 2) plus the number of representatives that the states sends in Congress, the number of representatives being proportional to the state's population. The District of Columbia (seat of the US federal government), although not a state, also gets three electors. Since we are not interested in American politics, or the reasons for the difference in status, the District of Columbia will be deemed a state for our purposes.

### Qualitative description

We must compute two different coefficients of power for each state. The first is the coefficient of power of the state in the Union, i.e. in the electoral college, interpreting the electoral college as a votational system. So it is equal to the number of configurations of yes/no votes among the states, where the given state's vote will be decisive, divided by the total number of configurations (namely 251 because there are 51 states). The computation of the coefficients of power has been done numerically. As we have mentioned, it is very much a linear function of the number of seats, except in the case of California, which has distinctly more power than in proportion to its number of electors.

This first coefficient varies between 46.6% in the case of California, and 2.3% for the states having three electors.

The second coefficient is that of an individual within a state. We are quite within the domain of validity of the asymptotic approximation we have described earlier, according to which this coefficient of power is proportional to the inverse square root of the population.

This second coefficient varies between 0.167% in the least populous state (Wyoming) and 0.0227% in the most populous (California).

And as explained in the general discussion on two-stage decision systems, the overall coefficient of power of an individual of the given state in the Union, is the product of the two aforementioned coefficients of power.

We can already see that there is a problem: the electoral weight of each state is an affine function of its population (two electors for any state plus one for every so many citizens), and the corresponding power is roughly proportional; whereas the coefficient of power of an individual within the state drops down only like the square root of the population. This means, and numerical results confirm it, that citizens of the most populous states of the Union have more power than those of less populous states.

In fact, we find that the overall (product) coefficient of power is highest in California, where it is 0.0106%, and lowest in Montana, where it is 0.00265% — or four times less.

### Numerical protocol

To compute the coefficient of power of each state in the Union, we have used a statistical sampling method: three hundred million voting configurations have been randomly selected, and, in each one, the list of decisive state votes has been determined.

We can compute the error bar for this method. We recall that for a Bernoulli distribution with probability p, the mean deviation is sqrt(p(1-p)); so the relative statistical error on N tries is sqrt((1-p)/(pN)). Here, p is 0.022 at least, and N is 300000000, so the relative statistical error is less than 0.04% on each figure. For a confidence interval of five times the deviation, we can treat the first three digits as significant.

To compute the coefficient of power of an individual in a state, we have used the first two terms of the asymptotic development: but in fact, this is bogus, and the leading term would have been quite sufficient for the level of precision we want. The main difficulty, however, lies in determining what is meant by “population”. In fact, we have chosen the number of registered voters in each state, as given by the US Census Bureau's table of Reported Voting and Registration by Sex, Race and Hispanic Origin, for States: November 1998. We have tabulated the states by total population (as the most significant criterion to list them by), but this figure has not been used in the computations.

### Program used

The program we have used in computing the numerical results was written in C. It is the following:

```#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include <math.h>

#define NBSTATES 51 /* Including DC */

const struct {
const char *name; /* State name */
const char *abbr; /* Abbreviation */
int electors; /* Number of electors in college */
double regist; /* Registered voters */
double popul; /* Total population */
} states[NBSTATES] = {
{ "Alabama", "AL", 9, 2398e3, 4369862 },
{ "Alaska", "AK", 3, 299e3, 619500 },
{ "Arizona", "AZ", 8, 1719e3, 4778332 },
{ "Arkansas", "AR", 6, 1172e3, 2551373 },
{ "California", "CA", 54, 12356e3, 33145121 },
{ "Colorado", "CO", 8, 2024e3, 4056133 },
{ "Connecticut", "CT", 8, 1625e3, 3282031 },
{ "Delaware", "DE", 3, 339e3, 753538 },
{ "District of Columbia", "DC", 3, 256e3, 519000 },
{ "Florida", "FL", 25, 6653e3, 15111244 },
{ "Georgia", "GA", 13, 3462e3, 7788240 },
{ "Hawaii", "HI", 4, 463e3, 1185497 },
{ "Idaho", "ID", 4, 528e3, 1251700 },
{ "Illinois", "IL", 22, 5530e3, 12128370 },
{ "Indiana", "IN", 12, 2702e3, 5942901 },
{ "Iowa", "IA", 7, 1562e3, 2869413 },
{ "Kansas", "KS", 6, 1191e3, 2654052 },
{ "Kentucky", "KY", 8, 1921e3, 3960825 },
{ "Louisiana", "LA", 9, 2261e3, 4372035 },
{ "Maine", "ME", 4, 704e3, 1253040 },
{ "Maryland", "MD", 10, 2331e3, 5171634 },
{ "Massachusetts", "MA", 12, 2947e3, 6175169 },
{ "Michigan", "MI", 18, 5217e3, 9863775 },
{ "Minnesota", "MN", 10, 2787e3, 4775508 },
{ "Mississippi", "MS", 7, 1446e3, 2768619 },
{ "Missouri", "MO", 11, 2848e3, 5468338 },
{ "Montana", "MT", 3, 472e3, 882779 },
{ "Nebraska", "NE", 5, 790e3, 1666028 },
{ "Nevada", "NV", 4, 606e3, 1809253 },
{ "New Hampshire", "NH", 4, 563e3, 1201134 },
{ "New Jersey", "NJ", 15, 3638e3, 8143412 },
{ "New Mexico", "NM", 5, 756e3, 1739844 },
{ "New York", "NY", 33, 7656e3, 18196601 },
{ "North Carolina", "NC", 14, 3468e3, 7650789 },
{ "North Dakota", "ND", 3, 416e3, 633666 },
{ "Ohio", "OH", 21, 5266e3, 11256654 },
{ "Oklahoma", "OK", 8, 1556e3, 3358044 },
{ "Oregon", "OR", 7, 1635e3, 3316154 },
{ "Pennsylvania", "PA", 23, 5452e3, 11994016 },
{ "Rhode Island", "RI", 4, 478e3, 990819 },
{ "South Carolina", "SC", 8, 1907e3, 3885736 },
{ "South Dakota", "SD", 3, 364e3, 733133 },
{ "Tennessee", "TN", 11, 2607e3, 5483535 },
{ "Texas", "TX", 32, 8301e3, 20044141 },
{ "Utah", "UT", 5, 788e3, 2129836 },
{ "Vermont", "VT", 3, 316e3, 593740 },
{ "Virginia", "VA", 13, 2996e3, 6872912 },
{ "Washington", "WA", 11, 2622e3, 5756361 },
{ "West Virginia", "WV", 5, 880e3, 1806928 },
{ "Wisconsin", "WI", 11, 2602e3, 5250446 },
{ "Wyoming", "WY", 3, 227e3, 479602 },
};

double states_power[NBSTATES];  /* Power of state in college */
double ind_state_power[NBSTATES];  /* Power of individual in state */
double ind_union_power[NBSTATES];  /* Power of individual in union */

#define NBTRIES 300000000

void
calc_states_power (void)
{
long n;
int i;
long states_casting[NBSTATES];
char states_voting[NBSTATES];

majority = 0;
for ( i=0 ; i<NBSTATES ; i++ )
{
states_casting[i] = 0;
majority += states[i].electors;
}
fprintf (stderr, "Electoral total is %d.\n", majority);
majority /= 2;  majority++;
fprintf (stderr, "Electoral college majority is %d.\n", majority);
for ( n=0 ; n<NBTRIES ; n++ )
{
for ( i=0 ; i<NBSTATES ; i++ )
if ( (states_voting[i] = ((rand()>>4)&1)) )
for ( i=0 ; i<NBSTATES ; i++ )
{
if ( states_voting[i] )
states_casting[i] += ( votes >= majority
else
states_casting[i] += ( votes < majority
}
}
for ( i=0 ; i<NBSTATES ; i++ )
{
states_power[i] = (double)states_casting[i]/NBTRIES;
}
}

void
calc_ind_state_power (void)
{
int i;

for ( i=0 ; i<NBSTATES ; i++ )
ind_state_power[i] = ((1.+1./(4.*states[i].regist))
/sqrt(M_PI*states[i].regist/2.));
}

void
calc_ind_union_power (void)
{
int i;

for ( i=0 ; i<NBSTATES ; i++ )
ind_union_power[i] = states_power[i]*ind_state_power[i];
}

int
main (void)
{
int i, j, k;
double p, q;

srand (time (NULL));
calc_states_power ();
calc_ind_state_power ();
calc_ind_union_power ();
p = 1e9;
for ( i=0 ; i<NBSTATES ; i++ )
{
q = 0.;
j = i;  /* (useless) */
for ( k=0 ; k<NBSTATES ; k++ )
if ( states[k].popul>q && states[k].popul<p )
{
j = k;
q = states[j].popul;
}
printf ("<tr><th>%s</th><td>%s</td><td><pre>%7.1f</pre></td>"
"<td><pre>%5.0f</pre></td><td><pre>%2d</pre></td>"
"<td><pre>%7.5f</pre></td><td><pre>%9.7f</pre></td>"
"<td><pre>%8.10f</pre></td></tr>\n",
states[j].abbr, states[j].name, states[j].popul/1e3,
states[j].regist/1e3, states[j].electors,
states_power[j],
ind_state_power[j], ind_union_power[j]);
p = q;
}
return 0;
}
```

### Numerical results

The first column (Ab) gives the state's two-letter abbreviation as is commonly used in that country. The second column (Name) is the state's name. The third column (Pop) is the state total population, in thousands: this is the sort key, but it has not been used for computations. The fourth column (Vot) is the number of registered voters, in thousands: this has been used to compute individuals' coefficient of power in the state. The fifth column (El) is the number of electors in college for that state. The sixth column (P st/un) gives the coefficient of power of the state in the Union (in electoral college). The seventh column (P in/st) computes an individual's coefficient of power in the state. The eighth column (P in/un) is the coefficient of power of an individual from that state in the Union.

AbNamePopVotElP st/unP in/stP in/un
CACalifornia
`33145.1`
`12356`
`54`
`0.46566`
`0.0002270`
`0.0001056996`
TXTexas
`20044.1`
` 8301`
`32`
`0.25025`
`0.0002769`
`0.0000693021`
NYNew York
`18196.6`
` 7656`
`33`
`0.25936`
`0.0002884`
`0.0000747902`
FLFlorida
`15111.2`
` 6653`
`25`
`0.19319`
`0.0003093`
`0.0000597596`
ILIllinois
`12128.4`
` 5530`
`22`
`0.16935`
`0.0003393`
`0.0000574592`
PAPennsylvania
`11994.0`
` 5452`
`23`
`0.17770`
`0.0003417`
`0.0000607239`
OHOhio
`11256.7`
` 5266`
`21`
`0.16139`
`0.0003477`
`0.0000561131`
MIMichigan
` 9863.8`
` 5217`
`18`
`0.13786`
`0.0003493`
`0.0000481571`
NJNew Jersey
` 8143.4`
` 3638`
`15`
`0.11460`
`0.0004183`
`0.0000479396`
GAGeorgia
` 7788.2`
` 3462`
`13`
`0.09916`
`0.0004288`
`0.0000425221`
NCNorth Carolina
` 7650.8`
` 3468`
`14`
`0.10683`
`0.0004285`
`0.0000457723`
VAVirginia
` 6872.9`
` 2996`
`13`
`0.09915`
`0.0004610`
`0.0000457053`
MAMassachusetts
` 6175.2`
` 2947`
`12`
`0.09145`
`0.0004648`
`0.0000425061`
INIndiana
` 5942.9`
` 2702`
`12`
`0.09146`
`0.0004854`
`0.0000443965`
WAWashington
` 5756.4`
` 2622`
`11`
`0.08380`
`0.0004927`
`0.0000412944`
TNTennessee
` 5483.5`
` 2607`
`11`
`0.08379`
`0.0004942`
`0.0000414067`
MOMissouri
` 5468.3`
` 2848`
`11`
`0.08375`
`0.0004728`
`0.0000395967`
WIWisconsin
` 5250.4`
` 2602`
`11`
`0.08380`
`0.0004946`
`0.0000414507`
MDMaryland
` 5171.6`
` 2331`
`10`
`0.07612`
`0.0005226`
`0.0000397824`
AZArizona
` 4778.3`
` 1719`
` 8`
`0.06085`
`0.0006086`
`0.0000370294`
MNMinnesota
` 4775.5`
` 2787`
`10`
`0.07613`
`0.0004779`
`0.0000363838`
LALouisiana
` 4372.0`
` 2261`
` 9`
`0.06849`
`0.0005306`
`0.0000363425`
ALAlabama
` 4369.9`
` 2398`
` 9`
`0.06846`
`0.0005152`
`0.0000352749`
` 4056.1`
` 2024`
` 8`
`0.06084`
`0.0005608`
`0.0000341217`
KYKentucky
` 3960.8`
` 1921`
` 8`
`0.06083`
`0.0005757`
`0.0000350187`
SCSouth Carolina
` 3885.7`
` 1907`
` 8`
`0.06083`
`0.0005778`
`0.0000351471`
OKOklahoma
` 3358.0`
` 1556`
` 8`
`0.06085`
`0.0006396`
`0.0000389243`
OROregon
` 3316.2`
` 1635`
` 7`
`0.05321`
`0.0006240`
`0.0000332010`
CTConnecticut
` 3282.0`
` 1625`
` 8`
`0.06084`
`0.0006259`
`0.0000380774`
IAIowa
` 2869.4`
` 1562`
` 7`
`0.05322`
`0.0006384`
`0.0000339755`
MSMississippi
` 2768.6`
` 1446`
` 7`
`0.05322`
`0.0006635`
`0.0000353112`
KSKansas
` 2654.1`
` 1191`
` 6`
`0.04559`
`0.0007311`
`0.0000333323`
ARArkansas
` 2551.4`
` 1172`
` 6`
`0.04561`
`0.0007370`
`0.0000336164`
UTUtah
` 2129.8`
`  788`
` 5`
`0.03798`
`0.0008988`
`0.0000341406`
` 1809.3`
`  606`
` 4`
`0.03037`
`0.0010250`
`0.0000311277`
WVWest Virginia
` 1806.9`
`  880`
` 5`
`0.03798`
`0.0008505`
`0.0000322999`
NMNew Mexico
` 1739.8`
`  756`
` 5`
`0.03799`
`0.0009177`
`0.0000348628`
` 1666.0`
`  790`
` 5`
`0.03798`
`0.0008977`
`0.0000340964`
MEMaine
` 1253.0`
`  704`
` 4`
`0.03037`
`0.0009509`
`0.0000288819`
IDIdaho
` 1251.7`
`  528`
` 4`
`0.03038`
`0.0010981`
`0.0000333620`
NHNew Hampshire
` 1201.1`
`  563`
` 4`
`0.03039`
`0.0010634`
`0.0000323160`
HIHawaii
` 1185.5`
`  463`
` 4`
`0.03037`
`0.0011726`
`0.0000356137`
RIRhode Island
`  990.8`
`  478`
` 4`
`0.03037`
`0.0011541`
`0.0000350484`
MTMontana
`  882.8`
`  472`
` 3`
`0.02278`
`0.0011614`
`0.0000264504`
DEDelaware
`  753.5`
`  339`
` 3`
`0.02284`
`0.0013704`
`0.0000312937`
SDSouth Dakota
`  733.1`
`  364`
` 3`
`0.02277`
`0.0013225`
`0.0000301131`
NDNorth Dakota
`  633.7`
`  416`
` 3`
`0.02279`
`0.0012371`
`0.0000281884`
`  619.5`
`  299`
` 3`
`0.02279`
`0.0014592`
`0.0000332515`
VTVermont
`  593.7`
`  316`
` 3`
`0.02277`
`0.0014194`
`0.0000323232`
DCDistrict of Columbia
`  519.0`
`  256`
` 3`
`0.02278`
`0.0015770`
`0.0000359220`
WYWyoming
`  479.6`
`  227`
` 3`
`0.02277`
`0.0016747`
`0.0000381351`

TODO!

### Conclusion

There are occasional claims that the US electoral college system over-represents small states. This belief probably comes from the observation that the number of electors is not in proportion to the population, since every state has two electors corresponding to its two senators, on top of the electors corresponding to representatives. However, we have just shown that this belief is in error: the bias is in favor of populous states. It is in no way a negligible phenomenon: similar coefficients of power would be obtained by giving as much as 4 votes to every Californian, 2.6 for each Texan, and so on down to 1 for each citizen of Montana (actually the figures for Maine and Nebraska might be even lower, for we have not computed them accurately).

Note that this unfairness (actually, “imbalance” would be a more proper term in view of the formalism) has nothing to do with the existence of the electoral college in se. It would be quite possible to have a balanced electoral system with an electoral college, by giving each state an electoral weight proportional to the square root of its population (not quite true either, but it would be better, at least). We have already explained that some two-level (or higher-level) decision systems can be (monotone,) unbiased and fair, without being (directly) votational.

It is beyond our purpose to discuss whether it is actually desirable to change the US electoral system. However, the present study should at least shed new light on the question.