So I have an amateur interest in Game Theory (I’ve taken a course and read some books), and I like messing around with different examples. One that occurred to me is how much power particular coalitions (including coalitions of one) have in the Senate, based on how many people are in each coalition.

This is obviously an oversimplification, but I’m going to assume that the votes we care about are won by the side with the most votes - with the subtlety that the one of the coalitions has a Vice President to break ties in its favor.

There are at least two ways to calculate coalition power indices - The Shapley-Shubik criterion and the Banzhaf criterion - and both of these are simple enough for me to do by hand while watching reruns (for the simple cases I’m dealing with), so I used both.

For the pre-midterm case, we’ll say that the D coalition has 48 votes, the R coalition has 50 votes and the W (wildcard) coalition has two votes. For simplicity’s sake, I’m assuming that all these votes are solid (this is obviously not true, but it’s a useful assumption for figuring out the benefits of working as a coalition). With an R VP, both criterion would have the R’s having all the power in the Senate and the D and W having none (again, I’m oversimplifying), but there was a D VP - so things were a little more complicated.

Both criteria show D, R and W coalitions having 1/3 (each) of the election power in the Senate - because to win any election, you need two of the coalitions to agree with each other. So a coalition of two senators has just as much power as coalitions of 48 or 50 senators.

Assuming the Georgia runoff goes to the Ds, then the situation because D=49, R=49, and W=2 - and the voting power evaluations remain identical.

But I decided to look a bit deeper. What’s up if the W coalition don’t work together? Then before the midterms we had D=48, R=50, M=1, and S=1 for the coalitions. The S-S index gives the result that the D have 5/24 of the election power, the R have 11/24 of the election power, and M and S each have 1/6 of the election power. The R’s have more senators, but the D’s have the tie-breaker - and M and S each still can determine the result of particular votes in the Senate. The B criterion comes up with slightly different results (R=3/7, D=2/7, S=M=1/7), but still have R>D>M and M=S.

Both criteria show that M and S lose a lot of power by not committing to do exactly the same thing all the time.

Again, assuming the Georgia runoff goes to the Ds, we have D=49, R=49, M=1 and S=1.

Now the S-S criterion comes up with D=1/2 and M=S=R=1/6. The B criterion shows D=1/2 (as with the S-S criterion), R=1/3 and M=S=1/12. The power of the tie-breaker makes the D’s significantly more powerful than the R’s even though they have the same number of senators and again both criteria show that M and S lose a lot of power by not committing to do exactly the same thing all the time.