Messing around with Senate Coalition Game Theory

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.

Are you taking into account a filibuster needing only the support of 41 Senators?

Nope. Leaving out the filibuster entirely - just dealing with the kind of voting that requires only a simple majority (confirming judges, setting the rules of the Senate, reconciliation, etc.). Filibuster situations can be very important, but there’s enough importance to the other stuff to make analysis worth a little time (I thought).

P.S. The filibuster case is too simple to analyze anyway. By inspection the the R power = 1/2, the D power is 1/2 and W, L or S are 0. Either the two big groups agree or nothing happens (or more realistically, one or both big groups fracture so simple analysis becomes impossible).

Game room? Ok, I guess.

The Game Room is a reasonable forum (though not the only possible reasonable forum) for game theory discussions, and the structure of the OP is conducive to a Game Room discussion. If the OP would prefer this be moved to some other forum, we can, but meanwhile, a reminder to keep this abstract, without reference to (for instance) specific policies, or assessments of how sane the different factions are, and so on. If you prefer, imagine this to be a question about any generalized voting body, of any polity.

[Not moderating]
For our purposes, the easiest way to consider the Vice President to be simply another Senator, and to count the body as having 101 members. “Only voting in the event of a tie” is exactly as much voting power as “always voting”, because if the vote wasn’t already tied, the VP’s vote wouldn’t matter anyway.

Also, there are some serious issues with the definition of “power” as “how often you cast the deciding vote”. A better definition of “power” would be “how often you get your way”. By either of the power criteria in the OP, the two “wildcards” should always commit to always voting together, because that maximizes their power. But if they disagree on some issue, and one of them has already voted one way on that issue, the other has no reason to vote that same way (that they disagree with).

Got it.

Yeah, I could have conceptualized it that way for equivalent results

That might be true. On the other hand, if the current vote is on issue A, and the two wildcards disagree, one or the other might decide to take the hit on today’s issue, to preserve the W-group reputation as a cohesive group, so that for issues B through Z and beyond in the future, the Rs and Ds treat them as a very powerful team, rather than as two individuals of limited power. Politics is an iterated game.

Then you have to assume every Senator votes because if 99 Senators vote 50 Aye / 49 Nay then a VP qua Senator could vote nay to defeat the motion.

Pedantry: same could be said about any odd number of voters.

Of course, if it’s possible for one senator to abstain, then it’s possible for two to do so as well, or more, and so that’s relevant for even numbers of voters as well. And then you have to model how likely senators are to abstain, and that’s probably quite difficult, because abstentions are probably not independent: A senator might be in favor of something but not want to be on the record as being in favor, and so might abstain if it’ll pass anyway without them, but still vote if their vote would be deciding.