# Who’s Got The Power? Measuring How Much Trump Went Banzhaf On Tuesday

The Democratic and Republican Parties each use a weighted voting system to choose their presidential nominees.  This only matters when no candidate has a majority of the delegates, and the details are complicated because the weight a particular candidate has is actually a number of (possibly independent) delegates.  Leaving those details to the side, let’s consider how much Donald Trump’s wins on Tuesday April 26th “mattered.”  The simplest measure of success, for each candidate, is how many additional delegates they each won.  As a result of Tuesday’s primaries, Trump is estimated to have picked up 110 delegates, Senator Cruz is estimated to have picked up 3, and Governor Kasich similarly is estimated to have picked up 5.

A key concept in weighted voting games is that of power.  There are literally countless ways to measure power, but one of the most popular ways is called the Banzhaf index.

If there are $N$ total votes, and a candidate “controls” $K$ of those votes, the Banzhaf index measures the probability, given the distribution of the other $N-K$ votes across the other candidates, that the candidate in question will cast the decisive vote: that is, that he or she will have enough votes to pick the winner, given every way the other candidates could cast their ballots. (I’m skipping some details here.  For the interested, the most important detail is that the index presumes that the other candidates will randomly choose how to vote.)

A higher power index implies that the candidate is more likely to determine the outcome. What is key is that the power index for a candidate with $K$ votes out of $N$ is generally not equal to $\frac{K}{N}$.  For example, if a candidate has over half of the votes,[1] then that candidate’s Banzhaf index is equal to 1 (and those of all other candidates are equal to zero, and we’ll see that come up again below), because that candidate will always cast the decisive vote.

So, back to Tuesday.  Here is the breakdown of how the GOP candidates’ delegates translated into “Banzhaf power” before Tuesday’s primaries.

 Candidate Donald Trump Ted Cruz John Kasich Marco Rubio Ben Carson Jeb Bush Carly Fiorina Rand Paul Mike Huckabee Total Delegates 846 (48.85%) 548 (31.64%) 149 (8.6%) 173 (9.99%) 9 (0.52%) 4 (0.23%) 1 (0.06%) 1 (0.06%) 1 (0.06%) 1,732 Banzhaf Power 0.5 0.1667 0.1667 0.1667 0.1667 0 0 0 0

Going into Tuesday’s primaries, Trump held just under majority of the delegates and held exactly half of the power.  More interesting in this comparison is that Marco Rubio’s power was still significant: in fact, equal to the individual powers of Kasich and Cruz.

Even though Rubio and Kasich each had less than a third of Cruz’s delegates, their voting power as of Monday was equal to Cruz’s. This is due to the fact that Rubio, Kasich, and Cruz could defeat Trump if and only their delegates voted together, regardless of how the other delegate-controlling candidates had their candidates vote.  In other words, Carson, Bush, Fiorina, Paul, and Huckabee truly had—as of Monday (and today)—no bargaining power at a contested convention.

However, after Tuesday’s results, the following happened:

 Candidates Donald Trump Ted Cruz John Kasich Marco Rubio Ben Carson Jeb Bush Carly Fiorina Rand Paul Mike Huckabee Total Delegates 956 (51.68%) 551 (29.78%) 154 (8.32%) 173 (9.35%) 9 (0.49%) 4 (0.22%) 1 (0.05%) 1 (0.05%) 1 (0.05%) 1,850 Banzhaf Power 1 0 0 0 0 0 0 0 0

By securing a majority of the delegates allocated so far, Trump’s power jumped from 0.5 to 1 and all of his opponents’ powers dropped to zero.  If the convention occurred today, they would be powerless to stop Trump.

Now, suppose that the candidates had votes equal to the actual votes (rather than delegates) they receive.  If the convention were held today under such rules, this would result in the following:

 Candidates Donald Trump Ted Cruz John Kasich Marco Rubio Ben Carson Jeb Bush Jim Gilmore Chris Christie Carly Fiorina Rand Paul Mike Huckabee Rick Santorum Total Popular Votes 10,121,996 (39.65%) 6,919,935 (27.10%) 3,677,459 (14.40%) 3,490,748 (13.67%) 722,400 (2.83%) 270,430 (1.06%) 2,901 (0.01%) 55,255 (0.22%) 36,895 (0.14%) 60,587 (0.24%) 49,545 (0.19%) 16,929 (0.07%) 25,530,125 Banzhaf Power 0.5 0.1667 0.1667 0.1667 0 0 0 0 0 0 0 0

If the popular votes were the basis of the GOP nomination and the convention were held today, then the candidates would still have the same “powers” as they did prior to Tuesday’s primaries.  Thus, on Tuesday, we arguably truly witnessed the effect of the “delegate system.”

As a final note, this power calculation clearly indicates something that I think is underappreciated about multicandidate races in majority rule settings.  To break Trump’s lock on the race, it is unimportant which candidate (other than Trump) an “unpledged” delegate decides to support.  Right now, if and only if at least 62 unpledged delegates (and I have no idea how many of them there are left right now) decide to support “other than Trump,” then the Trump’s power drops below.  In addition to (and in line with) the fact that it doesn’t matter how those delegates allocate their support across the other candidates, if 62 such delegates appeared in the hypothetical conference tomorrow in Cleveland, the powers of the candidates would be as follows:

 Candidates Donald Trump Ted Cruz John Kasich Marco Rubio Ben Carson Jeb Bush Carly Fiorina Rand Paul Mike Huckabee Total Delegates 956 (50%) 613 (32.06%) 154 (8.05%) 173 (9.05%) 9 (0.47%) 4 (0.21%) 1 (0.05%) 1 (0.05%) 1 (0.05%) 1,912 Banzhaf Power 0.97 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004

Conclusion. There are two “math of politics” points in here. The first is that votes/delegates are definitely not a one-to-one match: indirect democracy is distinct from direct democracy—it’s always important to remember that.  The second, and more “math-y” is that, when people have different numbers of votes, it is not the case that the number of votes a person has is equal to his or her voting power.[2]

With that, I leave you with this.

PS: If you would like (Mathematica) code to calculate the Banzhaf index for this and other situations, email me.

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[1] I am assuming for simplicity throughout, in line with the rules of the GOP and Democratic Party, that the collective decision is made by simple majority rule.  One can calculate the Banzhaf index for any supermajority requirement as well.  As the supermajority requirement goes up, the power indices of all candidates with a positive number of votes converge to equality (guaranteed to occur when the decision rule is unanimity).

[2] For a great review of how this is important in the real world, see Grofman and Scarrow (1981), who discuss a real-world use of weighted voting in New York State back in the 1970s.