Arrow's Theorem mentioned on BBC Question Time


I thought this was quite interesting. Last week (25th April 2019) the actor John Rhys-Davies brought up Arrow’s Theorem in response to the Green Party’s Caroline Lucas saying that Donald Trump had lost the popular vote. The programme is available here: but you probably have to be in the UK to see it. The relevant bit starts at about 55:21. But I’ll quote the it about Arrow’s Theorem:

John: He represents the American people.

Caroline: No, he doesn’t! He lost the popular vote! He lost the popular vote by nearly three million votes!

John: Oh woman! Do you not - have you never read Kenneth Arrow and Arrow’s Theorem? Any system of election has its problems!

Caroline: Yes, and the American one and ours more than most.

John: Read Arrow’s Theorem please somebody.


Just found it on YouTube -


It may be more dangerous when people use nonpartisan election reform to bolster partisan objectives. Also, Arrow’s theorem doesn’t apply to the systems that really need to pick up steam, so maybe less emphasis should be placed on it regardless.


Arrow’s theorem has nothing to do with the electoral college, though.


The overall point is just that you can’t have a perfect system of elections - there will always be tradeoffs. The Electoral College is one such compromise with what some would consider tradeoffs.


Electoral college has nothing do with voting methods. It deals with who’s represented, not how they vote.


His argument is incoherent.


His point that all systems have flaws is a fair one. Arrow’s theorem is not great to prove it but it does move the discussion in the right direction. Gibbard–Satterthwaite theorem might have been a better theorem to quote even though it is not relevant to the electoral college. The point is that mathematical proofs about election science were discussed by an actor on a major network. I am going to count this as a win.


I believe that his point was that Trump has won by the rules and that, for Trump’s legitimacy, it is irrelevant whether there are election methods where he would not have won.


His point seemed to me to be that because all systems have flaws, any system is as good as any other, and this doesn’t follow. However, he lost his composure somewhat and I don’t think a clear point really came through.

I did, however, find it very interesting that Arrow’s Theorem came up at all in such a programme, and also in such an emotional way rather than by someone dryly explaining the finer points of election flaws.

In any case, I do think that Arrow’s Theorem is very overstated. Aside from the fact that it doesn’t encompass every system anyway (e.g. approval, score), I don’t think it really added that much to our understanding in the first place.

If you look at the criteria that can’t all coexist in a single method according to Arrow’s Theorem, it’s not that some voting methods fail this criterion and others fail that criterion etc. Most of the criteria would never be failed by any sensible voting method, and it really just comes down to one of them.

Informally stated, Arrow’s Theorem says that with a few reasonable background assumptions, any ranked-ballot voting method fails Independence of Irrelevant Alternatives. But this was already known anyway from the Condorcet Paradox.


Rebuttal: You can choose one of the following options:
A: You lose $30.
B: You have a 50% chance to lose $10, and a 50% chance to lose $100.
C: You lose $90.

Even though all three make you lose money, C is clearly worse than A. You will probably also say A is better than B (although there may be cases where B is better). Just because every option has flaws does not mean that some flaws are worse than others.


Arrow’s Theorem has nothing to do with the electoral college. Arrow’s Theorem is about voting methods, and Electoral College is about who gets to vote and how much voting power they have. These issues are orthogonal, and EC only adds to the problems of FPTP.

Arrow: “But certainly the Electoral College only adds. It doesn’t subtract. It only adds to the problem.”