# New York Times puzzle is pure game theory

The Upshot is a column in the New York Times based around analytics, data and graphics. (It was conceived around the time when Nate Silver left to work for ESPN). Earlier this week, managing editor David Leonhardt and data journalist Kevin Quealy posted an interesting puzzle, entitled ‘Are You Smarter Than 49,485 other New York Times Readers?’

The puzzle consists of a simple question – you need to pick a number between 0 and 100, and all 49,485 of the responses will be collated (assuming that every single one of the Times’ readership actually enters a number) and averaged. If your guess turns out to be the closest whole number to two-thirds of the average guess, you are clever and you win.

It’s a very well-known classic piece of game theory, sometimes called ‘Guess 2/3 of the Average‘ (mathematicians: they call it like they see it) and has been well analysed by game theorists. It originally appeared in Alain Ledoux’s magazine ‘Jeux Et Stratégie‘ as a tie-breaker question, and is a great example of a problem in which you have to make an educated guess, based on what you think other people will guess. And therein lies the problem (other people. ugh.)

This class of problem is sometimes called the Keynesian Beauty Contest, in reference to John Maynard Keynes’ example in his 1935 book The General Theory of Employment, Interest and Money. In the example, entrants must pick the six most beautiful faces from a collection of photos, and will win a prize if they pick the most popular faces – which means picking the ones you personally think look pretty is actually a bad strategy, if that’s not in line with popular opinion.

The NYT’s ‘2/3 of the average’ problem has a similar structure, and as in all game theory problems, the incentives and pressures on people playing the game will influence their behaviour. This means it’s possible to work out an optimal strategy, or at least have a stab at one. If you don’t want to hear about that, because you want to think about it and go and enter the competition first, go and do that, and don’t read the rest of this until you have. Here’s the competition entry page.

Assuming all the players in the contest behave rationally, and they’re all trying to guess a figure that’s 2/3 of the average, it makes no sense for anyone to pick a number bigger than 67, since there’s very little chance the average is going to be 100, and any number bigger than 67 will never be the correct answer. So, if everyone’s playing rationally, the numbers picked will all be between 0 and 67 – the option of picking numbers bigger than 67 is called a weakly dominated strategy, which means any other choice is as good or better.

This means the 2/3 of the average will never be more than 44. So nobody rational will pick anything bigger than 44. So the range of sensible guesses reduces again, and this thought process can be continued indefinitely until the only logical conclusion is to pick 0 (which works out pretty well, because if everyone is logical they’ll also do this, and the average will be 0, and 2/3 of the average will also be 0, so everyone wins). This is called a Nash Equilibrium, after game theory legend John Nash.

The above all applies to situations where any real number value is allowed – but in the NYT example, the game is restricted to integers. In this case, an interesting (integeresting) thing happens – instead of the only truly rational answer being 0, the options of 0 and 1 are both possible. If you suspect that a lot of people (more than a quarter of the players) will pick 0, then you should pick 0, but if you think more people will pick 1, it’s a more sensible choice. This is then a different game theory problem, where being in the majority means you win.

As the Wikipedia article on ‘Guess 2/3 of the Average’ so nicely puts it, “ordinary people” rarely turn out to be perfect game theorists, and so the NYT competition may find that many entries deviate from the optimal strategy, and instead rely on people’s instinct (and in fact, given this knowledge, the optimal strategy itself changes). It’s almost like they knew this would be a brilliant logical challenge, and presented it as a puzzle oh wait.

It’s also worth noting that all the deductions above rely on the additional assumption that everyone is trying to win – and if they’re not, then their strategy may not be the one most likely to result in them winning, but could be entirely random. You also don’t know what proportion of players have this attitude. Alfred was right when he said that some men just want to watch the world burn.

Past versions of this type of competition include one run by Danish magazine Politiken in 2005, using similar rules, in which 19,196 entrants turned out a winning value of 21.6 (the prize was shared by five people). As to whether NYT readers are more or less smug than Danish people, only time will tell.