In an older post, I stated that the Brexit impasse was due to a Condorcet paradox in Parliament. Today I'd like to give a short run breakdown of the mechanics of it.
There are broadly three groups of MPs. Hard Leavers who always wanted to leave and are prepared to take no deal as an option if a deal can't be negotiated. Hard Remainers who want to reverse the decision to leave and think any Brexit is a terrible idea. Reluctant Leavers who campaigned for Remain in the referendum but think you have to honour the referendum decision. However they are not prepared to accept no deal exit as an option. The three groups are of roughly equal size (Hard Leave is the smallest, originally Reluctant Leave was the Largest but it may be that now it has lost people to both Hard Leave and Hard Remain).
The preferences align as follows.
Hard Leave WTO > Soft Brexit > Stay In
Hard Remain Stay In > WTO > Soft Brexit
Reluctant Leave Soft Brexit > Stay In > WTO.
If Stay In vs No Deal/WTO then Stay In wins.
If Stay In vs Soft Brexit Soft Brexit Wins.
If Soft Brexit vs No Deal/WTO WTO wins.
It isn't so much that one option wins in any of those pairwise choices as that what you might call the proposition is always voted down no matter what it is. The key factor is that the Hard Remain would rather No Deal than Soft Brexit because (a) they think there's such a big majority against WTO that faced with that as an option everyone will go for Stay In and (b) they genuinely think that in many ways a soft Brexit is the worst of both worlds.
So there's no majority for anything. I thought that as we got to the deadline enough people would peel away from the Hard Remain and Hard Leave camps to produce a cross party majority for Soft Brexit in some form (not May's deal but something like Norway+ or customs union). However the opposite has happened - the two Hard groups have increased. The critical factor though is the refusal of the Hard Remain side to compromise (most of the Hard Leave group were prepared to go as far as to back May's deal) and voting against the customs union or abstaining. That's a very high risk strategy both in the short term (it makes a no deal exit more likely) and the longer term (because of the likely political effects). The fact that this is what has happened is itself very interesting and it tells us something about what is going on both among the political class and wider British society.