BrettSpiel is a blog about board game design, written by game designer Brett J. Gilbert.

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Citius, Altius, Fortius and All That: The London Olympics and an unexpected hanging

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I am no sports fan, but I have been enjoying the Olympics — and not just out of a sense of patriotic duty. As a games designer it’s fascinating and salutary to reflect on how different sports choose to codify and enforce the rules that surround their athletes’ endeavours. The ‘Olympic spirit’ speaks to the romantic notions of sportsmanship and fair play, but in the arena of international competition that spirit isn’t enough. It’s practical application has to be translated into a written rulebook, and that’s when things start to get complicated.

There have been lots of examples in the past two weeks of how the the romantic spirit of true sportsmanship interacts with the classical application of sporting law, but the most striking was the controversy surrounding the disqualification of eight badminton players for “not using one's best efforts to win”. I shan’t rehearse all of the arguments here; fortunately others have done that for me: London-based game design studio Hide & Seek handily put the two key positions in a single tweet:

The question is this: Were the athletes right to play to lose individual matches so as to better their overall chances, or were the judges right to disqualify them on the grounds that they were breaking the rules by doing so? David Sirlin’s classicist view is that the athletes didn’t break any rules: they were ‘playing to win’ all along, and knew that their best chances of winning the tournament was to lose those matches. Pat Kane’s romantic opinion is that not only did the players break the rules, but they broke something more fundamental and more meaningful too: the social contract of the Olympic ideal.

And it is at this point that the unexpected hanging enters the race. The story goes something like this:

A condemned man is told by a capricious judge that “You will be hanged one day next week, but the hanging will be a surprise. You will not know the day of your execution until you are summoned to the gallows on the day of the executioner’s choosing.”

What is the condemned man to make of this? He reasons that he can’t be hanged on Friday, since in that case you would know this on Thursday and Friday’s hanging would then hardly be a surprise. Since Friday is impossible, he further reasons that he cannot be hanged on Thursday either, since by Wednesday he would know that Thursday must be the day. By the same logic he eliminates in turn every day of the following week and retires to his cell in a remarkably upbeat mood.

The next week the executioner arrives on Wednesday and, much to the condemned man’s surprise, he is hanged after all.

This paradox has been discussed at length since it first appeared in print in the 1940s. Martin Gardner dissects its intricacies in The Unexpected Hanging and Other Mathematical Diversions. In that book, and after after much discussion, he reveals the paradox’s beating heart:

The judge … knows that his prediction is sound. But the prediction cannot be used to support a chain of arguments that results eventually in discrediting the prediction itself. It is this roundabout self-reference that … tosses the monkey wrench into all attempts to prove the prediction unsound.

The problem for the condemned man is that the judge’s statement is both self-referential and self-contradictory, and so the very first step of his reasoning is flawed. Not only could he have been hanged on Friday, it would have been quite the surprise! On that fateful Friday morning, should it have ever come for the prisoner, he would have woken content that no executioner was readying the rope: after all, it would be no surprise if he were. Unfortunately, since the prisoner would then no longer have been expecting him, the executioner’s eventual arrival could only have been all the more shocking!

When presented with a self-contradictory ruleset, how can players rationally play within them, and how can judges rationally and fairly enforce them? They cannot, and the rule itself and any attempt at its interpretation becomes absurd. The Badminton World Federation, in attempting to codify the spirit of sportsmanship, have unintentionally achieved something else: An elegant illustration of just how elusive and uncaged that spirit truly is.

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