The tragedy of Brexit is that neither the United Kingdom nor, especially, the European Union saw fit to convincingly spell out to the public what the future would look like after the referendum. A credible precommitment, not excluding credible threats, ought to have promoted a cooperative outcome in the game. Instead, the uncertainty drove each of the British peoples to what they believed to be the safest option.

Brexit is sometimes argued to represent a situation where mutual cooperation is in everyone’s best interests but still the individual players “defect” and go solo. To model such situations, a game called “prisoner’s dilemma” has been conceived. In such a game, two partners-in-crime are interrogated separately. If both cooperate – with each other that is, not with the authorities – they can only be convicted on a lesser charge; if both defect, they face heavy charges; but in case one partner remains silent and the other betrays his partner, the partner ends up in jail for an even longer time whereas the other is released. Cooperation in the prisoner’s dilemma simply is not a strong equilibrium, i.e. a player can do better by unilaterally changing strategy. Defection is the dominant strategy. That is not the game you want to play.

It is implied that the prisoners have no recourse to rewarding or punishing their partner-in-crime after the game has played out. By advertising beforehand an expected penalty for players that defect against cooperators, the prisoner’s dilemma morphs into another game that has been known since Jean-Jacques Rousseau and David Hume as the stag hunt. Two hunters must decide whether to combine their skills to try and hunt down a stag or go it alone and settle for an easier hare. (For those that dislike hunting: Hume gave the example of two rowing a boat; if one does not cooperate, the other rows in vain.)

Crucially, the stag hunt game has two (pure-strategy) equilibria: the more rewarding stag hunt in the event both cooperate, and the alternative of the hare that is less risky as it does not depend on coordination. The crux is not whether to cooperate is an equilibrium but rather which equilibrium to choose: potential solidarity or perceived safety. Precommitting to a penalty mechanism – ranging from the European Union precluding any cherry-picking to full-out retaliatory measures – allows at least a cooperative equilibrium to arise by transforming the prisoner’s dilemma into a stag hunt. But uncertainty surrounding the willingness to cooperate and the payoff from doing so pushes the hunters to individually settle for the hare. Resolving such uncertainty may in addition nudge them towards the higher-potential stag. We leave it to the reader to identify which is which in brexit country.

Or maybe brexit is best captured by a game called “battle of the sexes”? Suppose, for argument’s sake, that Britain preferred to leave the Union whereas the continentals would have liked the islanders to remain, but that in any case all of Europe’s politicians wanted to be ‘in the same place’: either the United Kingdom leaves and the European Union reluctantly accomodates the departure, or the United Kingdom remains and puts on a happy face. In either coordination equilibrium only one player is pleased, but both equilibria are preferred to the situation in which a mismatch occurs. How then to avoid the parties choosing different strategies?

One possible way out is to “refine” the equilibrium concept by means of a randomising device that correlates the parties’ strategies. If both parties decide to flip a coin (say the United Kingdom organises a referendum) and agree beforehand that “heads” means “remain” and “tails” implies “leave,” neither would rationally want to alter his strategy after the referendum for fear of getting stuck in a coordination failure. Which brings us back to the tragedy we started from.

]]>Firstly, to take supranational organisations. The case for them is not automatically a yes. The UN, WTO, World Bank, yes; FIFA, maybe; the group of People’s Republics of Eastern Europe after World War II, hmmm. You may say that example is a bit far-fetched, but my point is that you should look at the nature of a supranational organisation before embracing it.

You talk of Sovereignty, and I’m going to talk of Democracy, and why people might want decision-making to be taken at the level of the nation state. Not necessarily, is what I say; but for a system to work democratically there has to be a perceived link between the citizen and the decision-maker. Having previously lobbied my UK MP successfully, I assumed that the situation was similar with MEPs. I learned the hard way, being fobbed off by a secretary. And now that MEP’s names don’t appear on our ballot papers – only party loyalties – I could not vote this MEP out without crippling the London’s labour vote, which I am neither willing nor able to do. I am not criticizing the MEP here – I’m sure he is a decent man who works hard – I’m criticizing a system that means he doesn’t need to care what I think.

I also discovered that, as you probably know, the European Parliament is like ours but with no Government present. Imagine how democratic that is! Legislation happens elsewhere, in the European Commission, so our main representative is – what is his name? He’s resigned now anyway, so it doesn’t matter. Again, I’m sure he did the best he could. But democracy it ain’t.

One characteristic of the EU that I really like, and you mention, is the principle of Subsidiarity. Unfortunately, there is another principle which contradicts it – ‘Ever Closer Union’. When these two clash, guess which wins. Subsidiarity produces a flexible diversity, but that’s not what I see the EU doing.

Your discussion about the National Interest, and when you make decisions for your own good and when for the greater good is a bit theoretical for me. But I’d like to point out that if you make decisions that are for the greater good, and in the process alienate sections of your own population, you are heading for disaster. Don’t call a referendum whatever you do! And don’t assume, if you are losing support in many countries across the continent, that it’s because people are getting nastier. It may be something you are doing.

]]>I made a similar post about sports in general: https://blogofmaths.wordpress.com/2016/07/07/a-question-of-sport/

]]>I’ll make some comments addressed solely to you (since I

suspect not many other people will read this post again).

I agree that it would have been helpful to set the problems in a more historical context, but I didn’t do that for the sake of time (the goal I set to myself was to state the theorem precisely, which required most of the talk to consist of definitions and examples, and didn’t allow for a historical overview). I’m still not sure how important the result is compared to other breakthroughs in Thurston’s “program”, and I should be clear, it pales in comparison to Perelman’s resolution of the geometrization conjecture. I think it got some attention since it was “last” of a list of problems of Thurston, which included the geometrization conjecture and the ending lamination conjecture (resolved by Brock-Canary-Minsky). Actually, there are still a few unresolved open-ended questions from this list – there is a nice survey of Thurston’s paper by Otal. http://link.springer.com/article/10.1365%2Fs13291-014-0079-5

I chose to describe Haken manifolds as containing π_1-injective surfaces rather than incompressible surfaces since this is easier to describe (π_1 is taught in any introductory algebraic topology course). It is a non-trivial result that the two notions (for co-orientable surfaces) are equivalent, called the “loop theorem”. Again, I didn’t want to get into these things, because they take some time to describe, even though the loop theorem was one of the most important early results on 3-manifolds of the 20th century (by Pappakyriokopoulos).

I’ll say that the virtual Haken conjecture had some historical importance (in the view of the 3-manifold community) because it would have implied the geometrization conjecture. However, as we know, the implication went the other direction (geometrization was used to prove virtual Haken). By the time geometrization was proved, however, the virtual Haken question was a well-studied open question – Freedman, Lackenby, and Thurston among others had worked on it extensively, and hence it gained some notoriety.

I think the solution I gave (based on the visionary ideas of Wise and his collaborators) will be more important in the future for the applications to cubulated hyperbolic groups, rather than the implications for 3-manifolds (although it does have many applications in 3-manifold topology that were not foreseen when it was conjectured).

The example of the Seifert-Weber dodecahedral space is as you say: taking a hyperbolic dodecahedron with dihedral angles 2π/5 along each edge, the gluing by twists will yield a single vertex with solid angle 4π (one may apply the Poincaré polyhedron theorem to determine the hyperbolic structure). In fact, the cell-decomposition of the Seifert-Weber space is self-dual: put a point in the dodecahedron, an edge through each face, and a pentagon dual to each edge, and the complementary cell structure is isometric.

As for the complements of knots and links (usually in the 3-sphere, so R^3 compactified by a point), these are manifolds for trivial reasons: they are open subsets of a manifold. In fact, the hyperbolic metric is on the open subset obtained by removing the link, and the diameter of the metric is infinite, even though the volume is finite. I like to think of the link as being “pulled” off to infinity like taffy, which gives one a rough idea of what the hyperbolic metric looks like (technically, the end is a warped product of a torus and a ray, with exponential warping function).

]]>For the costs Tim only mentions $10 per submission for the software managing the refereeing process, to be provided by Scholastica. But what about the human side of managing the refereeing process? Some people have to spend time on this. Either this time is free time or it is time paid by their institution, or it means additional costs for the journal to be paid by others.

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