Demonstrators protest against Brexit. (AP)

No one was surprised that British Prime Minister Theresa May’s proposed Brexit deal was rejected Tuesday by the House of Commons. What was surprising was that the vote was 432 to 202. Normally, such a humiliating defeat would lead to the resignation of the prime minister. That is highly unlikely to happen: May will continue as prime minister, while Labour leader Jeremy Corbyn’s motion of no confidence is unlikely to succeed. However, it’s hard to see an alternative deal that would pass muster with both the House of Commons and the European Union.

There’s no real room for compromise and negotiation

Scholars of international relations tend to assume that negotiations only happen where there is a plausible agreement that all parties can live with. In technical language, they start from the assumption that there is a “win set” of possible deals that might be reached, some of which are more favorable to one negotiating party, others of which are more favorable to the other. Which of the possible deals in the win set usually depends on the bargaining power of the parties.

The problem with the U.K.-E.U. negotiations over Brexit is that it isn’t at all clear that there is a win set of agreements that both parties can live up to. The European Union thinks that it has gone as far as it reasonably can, but the United Kingdom clearly believes that this isn’t nearly far enough. Both sides have come to an impasse over an issue that played a relatively minor role in the debates leading up to Brexit: the border between Northern Ireland and the Republic of Ireland.

There’s no clear solution to the border issue

The border problem is straightforward, even if the solutions are not. When both the United Kingdom and the Republic of Ireland were members of the European Union, the border between Northern Ireland (which is still part of the U.K.) and the Republic of Ireland, became increasingly less important in politics and economics. Free movement of goods back and forward across the border helped to create deep economic ties, while free movement of people helped soften political divisions between unionists (who want Northern Ireland to stay part of the U.K.) and republicans and nationalists (who want Northern Ireland to be part of the Republic of Ireland).

When the United Kingdom voted to leave the European Union, all that changed. The Republic of Ireland, which feared a resurgence of paramilitary violence, insisted that the U.K. would have to protect the special nature of the border between the North and the Republic as part of the deal for leaving the E.U. Thanks to assiduous Irish diplomacy, other E.U. member states agreed to support this demand, and the U.K. obliged, albeit in somewhat vague terms, accepting a “backstop” under which the open border would be preserved as long as the E.U. and U.K. failed to reach a more satisfactory deal over their new relationship. However, as the U.K. began to realize what the demand meant in practice, it started to try to wriggle out of its commitment.

The reason U.K. negotiators have been backpedaling is that it’s hard to see how one could keep the current border arrangements without either Northern Ireland or the United Kingdom as a whole remaining part of the European Union’s current market institutions. Unionists fear that if Northern Ireland alone were part of these arrangements, its union with the U.K. would be weakened, while the reason that many conservatives wanted Brexit was to escape E.U. market rules. Both Unionists and conservative Brexiters have enough votes to veto any deal. The U.K. has repeatedly sought to water down the backstop and make it legally nonbinding. The E.U., while it has made political statements that it doesn’t see the backstop as a desirable long-run solution, has refused to allow the U.K. the option to withdraw unilaterally from it. May had delayed Tuesday’s vote in the hope of getting substantial concessions from the E.U., but she failed.

There’s still no deal unless someone backs down. No one is prepared to.

It isn’t clear what happens next. The politics are unpromising on both the U.K. and E.U. sides. None of the U.K. major factions are prepared to make any concessions. None really want a May-type deal, and each hopes that it might prevail if chaos erupts. The E.U. is obdurate, too, not least because its negotiators fear that the U.K. is not in a position to reciprocate concessions, but will instead pocket them and demand more. There are some within the E.U. who are unhappy that the Irish border question has blocked the negotiations — but it’s very hard for them to back down. It seems nearly certain that negotiations are going to be extended, which may create its own political problems (it will be very awkward if the U.K. is still a member during the next European Parliament elections, while on its way out the door).

The result is that both the U.K. and E.U. seem to be moving from one kind of bargaining (trying to reach a mutually acceptable agreement) to another (the kinds of nuclear brinkmanship seen during the Cold War). Very few people on either side of the negotiations want a no-deal Brexit, in which the U.K. would lurch out of the European Union without any agreement, creating economic chaos. However, politicians in both the E.U. and U.K. are trying to suggest that their own side would be less badly hurt in case of disaster, in an effort to terrify the other into backing down and accepting a deal that would otherwise be unacceptable. Extending negotiations may ease some of the tensions, but given the continued absence of a mutually acceptable solution, there is still a substantial risk of both sides ending up with a no-deal outcome that neither wants.