1. VOTING POWER IN THE U.S. ELECTORAL COLLEGE
Note: this discussion is based on the 2000 apportionment of electoral votes
N. R. Miller

2. Voting Power and the Electoral College
The Voting Power Problem. Does the Electoral College system (as it has operated [“winner-take-all”] since the 1830s) give voters in different states unequal voting power?
If so, voters in which states are favored and which disfavored and by how much?
One obvious answer is that voters in battleground states have more “voting power” than voters in non-battleground states.
They certainly attract more campaign attention.
But the classification of states by battleground status is contingent and varies by voting alignments and historical eras.
Suppose our concern is whether there are intrinsic features of the Electoral College that, from behind a “veil of ignorance” about how voters may “chose up sides,” make some voters more powerful than others.
No features of a national popular vote system that would have this effect.
This is referred to as a priori voting power.

3. Voting Power and the Electoral College (cont.)
The only differences among between voters that are intrinsic to the nature of the Electoral College is whether they live is small or large states.
With respect to the question of whether voters in small or big states are favored by the EC, directly contradictory claims are commonly expressed.
This results from the failure many by many commentators to make two related distinctions:
the theoretical distinction between
voting weight and
voting power, and
the practical distinction between
how electoral votes are apportioned among the states (which determines their voting weights), and
how electoral votes are cast by states (which influences their voting power).

4. There is a significant small-state advantage with respect to the apportionment of electoral votes (voting weight)

5. Weighted Voting Games
Given the winner-take-all system of casting electoral votes, the top-tier of the Electoral College is an example of a weighted voting game.
Instead of casting a single vote, each voter (state) casts a bloc of votes, with some voters (states) casting larger blocs and others casting smaller.
Other examples:
voting by disciplined party groups in multi-party parliaments;
balloting in old-style U.S. party nominating conventions under the “unit rule”;
voting in the EU Council of Ministers, IMF council, etc.;
voting by stockholders (holding varying amounts of stock).

6. Weighted Voting Games (cont.)
A weighted voting game is a “simple game” in which
each player is assigned some weight (e.g., a [typically whole] number of votes); and
a coalition is “winning” if and only if its total weight meets or exceeds some quota
and its complement is “losing.”
Such a game can be written as (q : w1,w2,…,wn).
The (top-tier) of the Electoral College is a weighted voting game in which:
the states are the voters (so n = 51);
electoral votes are the weights;
total weight is 538, and
the quota is 270.
[Based on 2000 apportionment] EC = (270: 55,34,31,27,…,3).
The Electoral College game is almost “strong,” but not quite (because there may be a tie, in which neither opposing coalition is winning).

7. Weighted Voting Games (cont.)
With respect to weighted voting games, the fundamental analytical finding is that voting power is not the same as, and is not proportional to, voting weight; in particular
voters with similar (but not identical) voting weights may have very different voting power; and
voters with quite different voting weights may have identical voting power.
However, it is true that
two voters with equal weight have equal power, and
a voter with less weight has no more voting power than one with greater weight.
Generally, it is impossible to apportion voting power (as opposed to voting weights) in a “refined” fashion,
especially with a small number of voters;
as n increases, the possibility of refinement increases.
As we shall see, n = 51 allows a high degree of refinement.

8. Weighted Voting Example: Parliamentary Coalition Formation
Suppose that four parties receive these vote shares: Party A, 27%; Party B, 25%; Party C, 24%; Party 24%.
Seats are apportioned in a 100-seat parliament according some proportional representation formula. In this case, the apportionment of seats is straightforward:
Party A: 27 seats Party C: 24 seats
Party B: seats Party D: 24 seats
While seats (voting weights) have been apportioned in a way that is precisely proportional to vote support, voting power has not been similarly apportioned (and cannot be).
Since no party controls a majority of 51 seats, a governing coalition of two or more parties must be formed.
A party’s voting power is reflects its opportunity to create (or destroy) winning (governing) coalitions.
But, with a small number of parties, coalition possibilities -- and therefore different patterns in the distribution of voting power -- are highly restricted.

9. Weighted Voting Example (cont.)
A: 27 seats; B: 25 seats; C: 24 seats; D: 24 seats
Once the parties start negotiating, they will find that Party A has voting power that greatly exceeds its slight advantage in seats. This is because:
Party A can form a winning coalition with any one of the other parties; so
the only way to exclude Party A from a winning coalition is for Parties B, C, and D to form a three-party coalition.
The seat allocation above (totaling 100 seats) is strategically equivalent to this smaller and simpler allocation (totaling 5 seats):
Party A: 2 seats;
Parties B, C, and D: 1 seat each;
Total of 5 seats, so a winning coalition requires 3 seats, i.e., (3:2,1,1,1)
So the original seat allocation is strategically equivalent to one in which Party A has twice the weight of each of the other parties (which is not proportional to their vote shares).
Note: while we have determined that Party A has effectively twice the weight of the others, we still haven’t evaluated the voting power of the parties.

10. Weighted Voting Example (cont.)
Suppose at the next election the vote and seat shares change a bit:
Before Now
Party A: 27 Party A: 30
Party B: 25 Party B: 29
Party C: 24 Party C: 22
Party D: 24 Party D: 19
While seats shares have changed only slightly, the strategic situation has changed fundamentally.
Party A can no longer form a winning coalition with Party D.
Parties B and C can now form a winning coalition by themselves.
The seat allocation is equivalent to this much simpler allocation:
Parties A, B, and D: 1 seat each;
Party D: 0 seats
Total of 3 seats, so a winning coalition requires 2 seats, i.e., (2:1,1,1,0)
Party A has lost voting power, despite gaining seats.
Party C has gained voting power, despite losing seats.
Party D has become powerless (a so-called dummy), despite retaining a substantial number of seats.

11. Weighted Voting Example (cont.)
In fact, these are the only possible strong simple games with 4 players:
(3:2,1,1,1);
(2:1,1,1,0); and
(1:1,0,0,0), i.e., the “inessential” game in which one party holds a majority of seats (making all other parties dummies), so that no winning (governing) coalition [in the ordinary sense of two or more parties] needs to be formed.
Expanding the number of players to five produces these additional possibilities:
(5:3,2,2,1,1); and
(4:3,1,1,1,1); and
(4:2,2,1,1,1); and
(3:1,1,1,1,1).
(1:1,0,0,0,0).
With six or more players, coalition possibilities become considerably more numerous and complex.

12. Weighted Voting Example (cont.)
Returning to the four-party example, voting power changes further if the parliamentary decision rule is changed from simple majority to (say) 2/3 majority (i.e., if the quota is increased).
Under 2/3 majority rule, both before and after the election, all three-party coalitions, and no smaller coalitions, are winning, so all four parties are equally powerful, i.e., (3:1,1,1,1)
In particular, under 2/3 majority rule, Party D is no longer a dummy after the election.
All two-party coalitions are “blocking” (neither winning nor losing).
Thus, changing the decision rule (or quota) reallocates voting power, even as voting weights (seats) remain the same.
Making the decision rule more demanding tends to equalize voting power.
In the limit, weighted voting is impossible under unanimity rule.
However, in the Electoral College the decision rule is fixed at (essentially) simple majority rule (quota = 270).

13. Voting Power Indices
Several power indices have been developed that quantify the (share of) power held by voters in weighted (and other) voting games.
These particularly include:
the Shapley-Shubik voting power index; and
the Banzhaf voting power measure.
These power indices provide precise formulas for measuring the a priori voting power of players in weighted (and other) voting games.
A measure of a priori voting power is one that takes account of the structure of the voting rules but of nothing else.

15. The Absolute Banzhaf Measure
Here I use the Absolute Banzhaf measure of voting power, which can be interpreted as follows:
Imagine a random (or Bernoulli) election, in which everyone votes by independently flipping a fair coin.
New Yorker, 1937 =>

16. The Absolute Banzhaf Measure (cont.)
A voter’s absolute Banzhaf power is the probability that his or her vote is decisive, i.e., will decide the outcome of a random election.
In an unweighted voting system, a vote is decisive when it either
breaks what would otherwise be a tie, or
creates a tie that (we may suppose) will be broken by the flip of a fair coin.
In the EC weighted voting system, California’s absolute Banzhaf power of means that, if the states were repeatedly to cast their electoral votes by independently flipping coins, almost half [.475] of the time the other 49 states plus DC would split their 483 votes sufficiently equally that California’s 55 votes will be decisive and determine the winner.

17. State Voting Power in the Existing EC (cont.)
It is evident from the following charts that
only California’s share of voting power substantially deviates from (and exceeds) its share of electoral votes;
the modest large-state advantage in voting power (relative to voting weight) is not sufficient to balance out the small-state advantage in apportionment; indeed,
even California’s distinctive advantage in terms of voting power (relative to voting weight) is not sufficient to give it voting power proportional to its population.

18. Share of Voting Power by Share of Electoral Votes

19. Share of Voting Power by Share of Population

20. Individual A Priori Voting Power
The full Electoral College system is a two-tier voting system, in which
in the bottom tier, there are 51 (unweighted) one-person, one-vote elections (in each state), and
the top tier is the 51-state weighted voting game.
The overall [absolute Banzhaf] voting power of an individual voter in the two-tier voting system is his probability of his “double decisiveness,” i.e.,
the probability that the voter cast a decisive vote in the state election
times
the probability that the state casts a decisive bloc of votes in the Electoral College
in a random election.

21. Individual A Priori Voting Power (cont.)
Clearly a small-state voter has an advantage over a large-state voter in that his votes is more likely to be decisive at the state level,
i.e., the popular vote is more likely to be [essentially] tied in a small state than a large state.
On the other hand, a large-state voter has an advantage over a small-state voter in that, if his vote is decisive, he will be “swinging” a larger bloc of electoral votes in the EC.
The question is how these two factors balance out.

22. Individual A Priori Voting Power (cont.)
On the one hand, we have seen that the voting power of states is approximately proportional to their voting weights (i.e., electoral votes),
and therefore is (somewhat more) approximately proportional to their populations (apart from a relatively small bias in favor of small states).
Probability theory tell us that the probability of an (essentially) even split between Heads and Tails
is not inversely proportional to the number of flips (i.e., voters in a state), but rather
is inversely proportional (to very good approximation) to the square root of the number of voters.
Thus we can conclude that individual voting power under the Electoral College is approximately proportional to the square root of the population of a voters state,
except that voters in small states are somewhat advantaged relative to this general rule.

23. The Small-State Apportionment Advantage is More Than Counterbalanced by the Large-State Advantage Resulting from “Winner-Take-All”

24. Absent the Small-State Apportionment Advantage, the Overall Large-State Advantage Would be Far More Extreme.

25. Individual Voting Power by State Population: Electoral Votes Precisely Proportional to Population

26. Individual Voting Power by State Population: Electoral Votes Proportional Population, plus Two

27. Individual Voting Power under Alternative Rules for Casting Electoral Votes
Calculations for the Pure District Plan are entirely straightforward.
Calculations for the Pure Proportional Plan and the Whole-Number Proportional Plan are relatively straightforward.
But under the Modified District Plan and the National Bonus Plan, each voter casts a single vote that counts two ways:
within the voter’s district (or state) and
“at-large” (i.e., within the voter’s state or the nation as a whole).
Calculating individual voting power in such systems is far from straightforward.
I have found it is necessary make approximations based on large samples of Bernoulli elections.

28. Pure District System

29. Modified District System (Approximate)

30. (Pure) Pure Proportional System

31. The Whole-Number Proportional Plan

32. National Bonus Plan (Bonus = 101)

33. National Bonus Plan (Varying Bonuses)

34. Summary: Individual Voting Power Under EC Variants