If you enjoy that witty sense of humour commonly described as "English" then you'll love the idea of a non-game. This article casts a mathematical eye over a famous non-game enjoyed by thousands of people up and down the UK every week.

Many readers of this article may at some point have listened to the BBC Radio 4 comedy programme I'm Sorry I Haven't a Clue (ISIHAC), and to its extraordinary game Mornington Crescent. In this game four players alternate in naming London Underground stations, and the first player to say "Mornington Crescent" is the winner. No one is really sure if the players are adhering to any rules or not, although they claim to, and the game has a bewildering number of differerent variations and conventions which poke fun at many other games.

However this is not an article on how to play Mornington Crescent but an article on how to win at Mornington Crescent, in which we don't worry about any rules but simply assume that you can choose the stations freely. We hope that by the time you have read this article you will not only be a better Mornington Crescent player, but may even have learned a bit about some new aspects of the mathematics of games.

The essence of the game, and what it means to win

A typical game on ISIHAC progresses as follows:

• There are several players (at least two, and usually four).
• Each player in turn says the name of a London Underground station apparantly at random but with no repeats.
• The first player to say Mornington Crescent is the winner.

So, why doesn't the first player simply say "Mornington Crescent" on their first move, and thus win? There are two objections to this strategy. Firstly, if you always did this then no one would ever play with you again. Secondly, and much more importantly, your pay off measured in terms of laughs and/or comedic impact, increases the longer that the game lasts. This now gives us the clue to finding a strategy for winning at Mornington Crescent without having to know the rules for playing it. Here is the revised version of the game in which we have a pay off function which (for example) might measure the amount of laughs that the player gets.

• Each player in turn says the name of a London Underground station, chosen freely from the list of all possible stations (without repeating earlier choices, but as there are 270 such stations this is not usually an issue).
• The first player to say "Mornington Crescent" is the winner. If this happens on the $n$th move then the pay off to the player is $F(n)$.

The key to winning is now to devise a strategy which maximises your possible pay off $F(n)$. If the pay off does not depend upon $n$, then your best strategy is to call "Mornington Crescent" on your first move. But this is far from the actual case. Instead $F(n)$ increases as $n$ increases, and this alters your strategy. As an example, consider the case where $F(n)$ doubles with each turn. (We will continue to use this example for the rest of this article.) Now let's suppose that we have a two-player game and it is your turn. You could say "Mornington Crescent" and you would win with a certain pay off. If, however, you DON'T say "Mornington Crescent", and (crucially) your opponent also doesn't say it on their turn, then your pay off has increased four-fold if you say it on your next turn. Thus you have signifcantly increased your winnings, with of course the risk that your opponent will say "Mornington Crescent" on their turn.

Here we start to see the real subtlety of this great game. The whole essence of the play is that to maximise their possible pay off, each player wants to wait for as long as possible before they say "Mornington Crescent", but they still have to say it before the other player. Mornington Crescent in this form has similarities to a number of other games involving brinkmanship. One example is a Dutch Auction in which the price of an item goes down steadily and the first person to bid for it wins. So it pays to wait as long as possible, but to still make a bid before anyone else. Mornington Crescent also resembles the issues facing a company that knows it has to spend a lot of money developing a new product, and has to do it before a rival, but wants to wait for as long as possible. Thus playing Mornington Crescent is a good training for real life. This led to NF Stovold's often misquoted remark "She who is tired of Mornington Crescent is tired of life".

There is strong evidence that mathematicans invented Mornington Crescent. An article about non-games, published in the Warwick Mathematics Journal Manifold in 1969, by Ian Stewart and John Jaworski, described an essentially similar game called Finchley Central along with several other interesting non-games. It is well worth a look.

Strategies for winning

We now consider strategies for a simple two-player game, playing against an opponent who also wants to maximise their winnings.

Our first example is a needle match between two perfect logicians. Each reasons that as there are a finite number of underground stations the game must end after 270 moves, with player one having the first move, and all odd moves. Player one reasons as follows. If they say "Mornington Cresecent" on the 269th move then they will win big. So if the game gets to the 268th move then player two will say "Mornington Crescent" to stop this, and they too will win big. Of course if the game gets to the 267th move then player one will need to stop this and will call "Mornington Crescent" then. Accordingly player two will call "Mornington Crescent" on the 266th move. The logic then continues backwards until we get to move one by player one, when they call "Mornington Crescent". This is impeccibly logical. However as $F(1)$ is very low compared to later values, the perfectly logical player one has basically won nothing for their logical efforts. More fool them.

Our second example is a match between two game theorists. (These may or may not be prisoners in a dilemma). Game theorists are interested in Nash equilibria. These are situations in which each player has chosen a particular strategy (which might involve some randomisation to be unpredictable) and no player can benefit from changing their strategy while the other person sticks to theirs. If both players choose not to always call "Mornington Crescent" on their respective first moves, then the player more likely to say "Mornington Crescent" later will benefit from changing their strategy to saying it just before the other player. The Nash equilibrium therefore requires both players to say "Mornington Crescent" as soon as possible, forcing player one to always say it on the first go. Not good.

What's the problem?

Something seems to be wrong here. The problem is that in both these scenarios we have ruled out the possibility either of taking a risk or (heavens above) of the two players cooperating. Let's see what happens if they cooperate. If the expected winnings of the two players are $A$ and $B$ at the start of the match, then after two turns without any mention of Mornington Crescent, we are (thanks to the large number of stations that we can possibly call), essentially back to the same state that we were before the start of the match, except that each persons expected winnings are now $4A$ and $4B$. Cooperate for another two turns and the expected winnings are $16A$ and $16B$. Thus each player's best strategy is to cooperate for as long as possible before ratting on their opponent.

But what does "as long as possible" mean? You can make a judgement on your opponent to see how keen they also are on cooperating. If they have cooperated up to turn $n$ this might incline you to think that they might cooperate further. So your future play depends on the past history of the game. (For those in the know, this means that the expectation of winning is not a Martingale. Mornington Crescent is not a fair game!). Note that in this method of play you don't want to call Mornington Crescent too early. Indeed the optimal time is when your opponent is thinking of calling it themselves. Your judgement of when to call "Mornington Crescent" might then depend on your past experiences of playing against the same player. (This takes us into the realm of Bayesian inference and machine learning — find out more in this article).

An alternative strategy with essentially the same effect is one in which you are willing to risk your opponent calling "Mornington Crescent" early on in the game when $F(n)$ is low, and to then wait for a few turns as the possible winnings mount up with the intention of calling Mornington Crescent as soon as the pay off becomes high enough for you to be interested.

Thus, if you are either prepared to cooperate, or a risk player, one possible strategy for playing Mornington Crescent is to have some predetermined point at which you are prepared to play, and wait till then. As soon as you get to this point (which you may well increase if you think you have a cooperating opponent), say Mornington Crescent and achieve immortality. And, if your opponent plays first, well, as a famous Cambridge Professor is known to remark, that's life in the Big City!

Conclusions

Our hope is that by reading to the end of this article you will have come to appreciate how subtle the game of Mornington Crescent is, and the very high level of skill that is employed by the teams in ISIHAC to play it. We would greatly welcome the comments of the readers of Plus on strategies for how to win, provided that they are backed up with clear mathematical arguments. We eagerly anticipate a postcard from a Mrs. Trellis of North Wales.

---

About the authors

Chris Budd was born in London and grew up on the Northern Line. He then moved West and is now Professor of Applied Mathematics at the University of Bath, where he does research into the applications of mathematics to the real world, including food, trains and climate. A passionate populariser of mathematics, he is a good friend of Sam Parc who edited 50 Visions of Mathematics. His favourite tube station is Fairlop.

Jeremy Budd is Chris Budd's son and is currently studying Mathematics at Trinity College, Cambridge. Mornington Crescent and ISIHAC have been part of his upbringing from a very early age.