24 April 2012
Professor John Barrow FRS
What I want to talk about today are some of the curious scoring systems that exist in sport, and the more you explore sports, both Olympic sports and non-Olympic sports, you find that there is a whole plethora of strange ways in which performances, distances, times, numbers of laps completed and so forth, are converted into points, and those points are then aggregated to determine who wins. Some sports have scoring systems that are so extraordinary, I cannot even begin to try to explain them, sports like professional boxing, and then there are other sports which are continually changing their scoring systems, tinkering with the way they work. Formula One motor racing is an interesting example of that sort.
Let me begin just by looking at one rather odd, but to statisticians rather familiar, paradox, which shows you how careful you have to be when you start to aggregate or join together different performances and scores to get a final result.
Let us focus on cricket, for example. Suppose that we have two bowlers, which I have called Anderson and Warne, for the sake of argument, and they each bowl in the two innings of a test match. In the first innings, Anderson takes three wickets for seventeen runs, and so his average, the number of runs he gives away for each wicket that he takes, is seventeen divided by three, 5.67. But Warne takes seven wickets for 40, and so his average is 5.71. Better average, in this sport, obviously, is the smaller number, the less expensive bowler, so Anderson has got the better bowling average in the first innings.
A few days later, they are bowling again. In the second innings, Anderson takes seven for 110, so his average in the second innings is fairly expensive, 15.71. Warne cannot really take advantage of this – he takes three for 48, so in the second innings, his average is sixteen. So, in the second innings, Anderson has the superior average again.
In both innings, Anderson is better than Warne, but if you add the figures together for the entire match, Anderson takes ten for 127, his average is 12.7, Warne takes ten for 88, so Warne’s average is much better for the match as a whole. Even though Anderson is better in both innings, when you add them together, Warne has the better average for the whole match. So averaging averages is a very dangerous business.
You can set up other examples like this that are perhaps more worrying. Suppose you have a school that is trying to impress Government Ministers and parents about its performance in league tables, and in every subject, the first school, School A, shows that it’ has got superior exam performance to School B – so, if you compare Physics, it is superior, English, it is superior, Maths, it is superior – but when you add all the performances together to get an overall league table, it is possible for School B to be superior to School A, even though it is inferior in every single subject on the average.
This is sometimes known as Simpson’s Paradox and it just leads you to beware of performance league tables that are aggregated in particular ways.
Here is another odd example. We are going to create a strange football league and show how sensitive the results can be to just a small alteration in the points-scoring system. It is a rather realistic alteration, one that did happen in football, worldwide, a long time ago, and that was the difference between giving two points for a win and giving three points for a win.
So, in this league, there are thirteen teams, and they each play the other twelve teams just once. There are two points for a win, and one point for a draw. A team called the All Stars, we are told they win five of their games and they lose seven – every other game is drawn. So, you can work out what happens: the All Stars, they score 5 x 2, they score ten points. The teams they beat lose one game and draw eleven, so all the teams they beat score eleven points. The teams they lose to score two points for beating them and one point for every other game because they draw them all, so the teams they lose to all score thirteen. So you can see, the All Stars have to come bottom of the league. After the end of the last game, they are, as it were, sick as parrots, but when they get back to the dressing room, someone says that the FA has just had a meeting with FIFA and they have decided at this eleventh hour to change the points-scoring system in the league and apply it retrospectively, and there are going to be three points for a win, not two. So, what then happens, if you give three points for a win? Well, the All Stars win their five games, so they have fifteen points. The teams they beat still end up just drawing eleven games, so they have eleven points, and they teams they lose to have three points for their win now, and eleven for their draws, so they score fourteen. So the All Stars have won the league now, with fifteen points. So, this small change in the rules turns the entire league table upside down.
Let us move on and have a look at some specific sports, and we are going to have a look at squash. Squash, strangely, although it is played particularly in the Far East by huge numbers of people, and also in this country, is not an Olympic sport at present. The reason is probably to do with television and television money – it is very difficult to televise it. There have been attempts to televise it using transparent Perspex walls to the court, but it is not an easy thing to show on television and so it does not tend to produce the sort of money that people want to see flowing into the sport.
But it had an old rule, a rather interesting rule mathematically. I think this has changed in many competitions now, but the situation was that, if the scores reached eight-all, then the receiving player had a choice to make: they could choose whether to play to nine or to ten. Squash has a scoring system which you see in a number of other racquet games or in volleyball, where you only score a point if you win a point when you are serving. If you are receiving, to score a point, you have got to win a point to get the serve, and only if you win the next point will you score a point. So, the question is what should you do – should you play to nine or should you play to ten? And the one fact we are going to introduce is that we are going to say, regardless of whether you are serving or receiving, your probability of winning a point is going to be P. If that probability is very high, close to one, you are the better player, if it is close to a half, you are very evenly matched, if it is close to 0, you are the worse player, and it is rather surprising that you are at eight-all.
So, what is the probability? We are going to distinguish two quantities. R is going to be the probability of scoring the next point if you are the receiver, and S is going to be the probability of scoring the next point if you are a server. Well, there is a simple relation between the two because, if you are a receiver, you have first got to win a point to become the server and then you have got to win the point as a server, so R is just equal to P times S. What is S? Well, the probability of winning, of scoring a point, if you are the server is, you could just win the next point or you could lose the next point and then you are going to be the receiver, so your chance of winning a point from then is just multiplied by the probability of winning if you are a receiver.
If we tidy these little formulae up, we can express the probability of scoring the next point if you are a server or if you are a receiver just in terms of this probability, P. And you can see that one is indeed just P times the other, you have got a bit more work to do, another point to win, if you are the receiver.
So what should you now do if it is 8-all? If you say I am going to play to nine, then, because you are the receiver, your chance of winning the match is just R – it is the probability of winning that next point if you are a receiver. But, if you elect to play to ten, there are different routes by which you could win the whole match. You could win by going 9-8 and then to 10-8 and the probability of doing that would be the probability of winning first as a receiver and then as a server. Or you could go 9-8, 9-all, 10-9, and that would be R times 1 minus S, losing as a server and then you are receiving again, so you are back trying to win from being a receiver. Or you could lose the first one 8-9 and then win the next two, so 1 minus R, then you are a receiver again, and then you are a server. So the probability of winning when you play to ten is the sum of the probabilities of going through these three routes to success. All we have to ask is: which is bigger? Is the chance of playing to nine and winning R bigger or less than this lot of three possibilities here?
Well, you are better off playing to ten if this probability here is bigger than R. If you put that in, we can cancel some Rs out, move it around a bit, and there is a simple formula. R is a probability so it never gets bigger than one, so this is positive, so this tells you if something positive times something else is negative, the something else must be negative, and so S must be bigger than a half.
Here is S, just in terms of the Ps, so we can plug that in, and we have this simple condition here. If we multiply up by two, rearrange this, this is an interesting little arithmetical condition. So this condition requires the probability to be bigger than a half times three minus the square root of five, so that is about 0.38. What this is saying is that you are off playing to ten if your chance of winning a point is bigger than about 38%.
Intuitively, what this is saying is that, if you are a good player and your probability of winning a point is quite high, bigger than 38%, then you should play the two points. If your probability of winning a point is very low, then you might fluke one point but you will not fluke two, so if you are the weaker player, play just one more point. If you are the stronger player, then elect to play two more. The difference between the definition of being strong and being weak is this 38%.
Another interesting scoring system, which has changed in recent years and it is one of the reasons it becomes rather interesting, is table tennis. When I used to play table tennis, and maybe when you do in your back room or your garage, you probably still play to 21, and you might play best of five sets, and you serve five shots at a time. But the rules were changed a few years ago, again, to make the game shorter and more predictable, and to reduce the advantage that the server has for the whole period when they are serving. Games are now played to eleven or a score higher than that where you have a two point advantage. You have three serves and you play the best of seven games rather than of five. What is the rationale for these sorts of choices?
Again, let us assume that a player’s chance of winning a point is P, irrespective of whether they are serving or they are receiving, and if you have a match where the opponents are evenly matched, then P is going to be a half plus a little bit, so this number is much, much smaller than a half, and the closer it is to 0, the more evenly matched the opponents are. One of the questions you can ask about scoring systems – we will look at tennis towards the end of the lecture – is: to what extent does skill win out over luck? If that S is just a little bit positive, how does it carry through, through a sequence of games and sets?
If you regard the table tennis game as a random process, where you have got to win N points before you lose N points, then the chance of that happening works out to depend upon S and the square root of the number of points that you need to win before you lose them. So this is what statisticians call the Bernoulli process: the probability of winning all these points before you lose them depends on the square root of the number of points divided by pi.
If you then play M games, you want to know what is the marginal probability of winning a game to N points and then winning M games. This is like a composition of this, so you have this probability, and then the new S for the sequence of M games you just put in here, twice the square root of M over pi. So this compound probability of winning the points and then winning the games, so that you win the whole match, is proportional to S again but it is multiplied by the square root of M times N.
This is not surprising. If you think about it, M times N is the total number of points that are really being played for in the game. Random process, you have dependence on that square root. But you can see how the little imbalance between the two players, the bias away from a half, feeds through here, into this formula.
But what happens if you change the rules? Well, under the old rules, you were playing to 21 points, if we forget about the deuces and so forth, and the number of games that you needed to win the match, if you were playing best of three, say, would be two, and you would have M times N, which would be 42.
Under the new rules, if you were playing best of seven, you would have to win four games, so M would be four, and you would be playing to eleven points, so you have got 44.
Comparing those two, you see there is a very close equivalence between the old rules and the new rules with regard to the reward for skill over luck, that the new set-up is really very, very similar to the old one. Presumably, somebody knew what they were doing under this rue change.
Let us move now to have a look at some sports where you do not win the points, as it were, directly and count them, but you allocate them in some way by some mathematical formula. Here is just a quick example.
A sport of modern pentathlon, which was introduced into the Olympics by the founder of the Olympics in the early 20th Century, Baron Coubertin, the modern pentathlon involves competitors shooting, fencing, swimming, riding and running. Originally, I think this event, like all the equestrian events at the very early Olympics, was only open to members of the Armed Forces or serving members of the Armed Forces of the countries of the world. It changed, with time. But de Coubertin’s rationale was rather strange. So these, he thought, were the five skills that would be required of an Army officer who found himself stuck behind enemy lines and he needed to make his escape: so he might need to shoot his way out, he might need to fence his way, in combat, he might have to swim across the river, he might have to jump on a strange horse and ride off, or, finally, he might have to run for it. So this event has got a strange history. It is changing its rules rather a lot at the moment because I think it is under threat – it is only guaranteed a place in this year’s Olympics. It is not guaranteed that it will be there next time.
The situation now – long ago, there used to be one event a day. Now, they are all done on the same day.
The swim is about 200 metres. It is in the pool. Pentathletes tend to be good swimmers.
There is then a fencing competition, where everybody fights everybody else and there is a complicated points-allocation system then rewarding the tournament placing.
Show-jumping is particularly challenging. You are introduced to a randomly chosen horse twenty minutes before the competition begins. The modern pentathlon horse competition at the Beijing Olympics was a fiasco because there was very heavy rain. The show-jumping arena was sand-based and it was a like a quagmire, and the horses clearly did not want to take part in this event. One of the British women competitors said that she got on her horse, the horse threw her off, jumped out of the arena and ran off down the road and was re-captured a quarter of a mile away! So the performances were very strange, hardly any competitors completing within the time allowed.
There is then a shooting event, with air-pistols. In London, they will not be real guns at all. They will just be laser weapons.
And there is a run of 3,000 metres, and this last part of the event is staged in a very interesting way. So, the competitors start at intervals, where the intervals are determined by the performance in the previous event, and that means that the first competitor to cross the line at the end really is the winner of the whole event. Whereas, in something like the decathlon or the heptathlon in the stadium, the winner is very unusually the first person to cross the finishing line in the 1,500 metres or the 800 metres. But, in London, there is a change to the rules: the shooting and the run are going to be combined, so that you will start shooting at the beginning of the run, you will stop after each 1,000 metres and do some more shooting, you will not carry your gun with you, so it is rather like a biathlon in the Winter Olympics, so it is changed the event rather significantly.
Here is an example from the last Olympics. There is nothing unusual about these percentages. These are the percentages of the points scored of all the competitors in the Olympic event, across disciplines, and you can see what tends to happen. There is some very significant positive weighting towards the swim. As I said, the pentathletes are usually very strong swimmers, swimming significantly under two minutes for 200 metres, the best competitors. Fencing is the weak point of everybody, something that, in most countries of the world, has to be learnt at a rather later age – difficult to find lots of good opponents. So you can see here how, if you altered the points allocation across these sports, you would end up with a rather different final table of medallists.
Let us look at a similar situation but in more detail because it is more familiar – the decathlon.
In the decathlon, you have ten events, spread over two days. They involve throwing and running and jumping, so some of the results are times and some of the results are distances, and they have to be converted into points, which are then added up to determine who is the best overall competitor.
The women’s event, which was once a pentathlon, when people like Mary Rand came second in 1964 at the Olympics, and then the heptathlon now has this extra event added to it, the 800 metres. Mary Peters won it in 1972 at the Munich Olympics, when it was a pentathlon. She would not have won it if there had been an 800 metres.
So, the question is: how do you convert the performances to points, and what would happen if you changed the way that you did it?
Well, there is a history to how this was done. It is done by studying, historically, performances in those events by all athletes and then performances by athletes who are decathletes and performances in decathlons to try to get a statistical picture of what the performance averages and levels are.
Here is a summary of roughly how this is done. There are two formulae that you need to know. In the field events, you get more credit for getting a longer distance, a greater height or distance from a jump or a throw, and the points allocated if you achieve a distance D are given by a power law formula. So there is a constant A, multiplying out the front, and there is a very important constant up here, which is a power C, and there is some other constant, B, here, which tells you if you go below that distance, you will not score any points, so if your distance is B, this is 0, and you will not score anything.
Here is the similar formula for times, in running events. Of course, here, you want to get the shortest time that you possibly can, so this is the other way round. If you take longer than the time B, you will score no points.
If you look at these formulae, you can see the sort of choices that have to be made. A and B are rather boring and simple choices to make, but C is a very important one. C is telling you how your reward changes as your performances get better. If C is bigger than one, then you can see, as your distances get longer and longer and you become a better and better jumper and thrower, you will get proportionally more reward. But if C was less than one, then the opposite would be true: you will be getting proportionally less and less reward for better and better performances. The situation with C bigger than one is usually called a progressive scoring system, and C less than one, regressive, and if C was equal to one, it is neutral and this would just be a straight line.
Here is what these quantities look like for all the different events. Do not worry about A and B – they are just really telling you how poor you would have to be to score nothing. The times are all in seconds and the distances are all in meters. The interesting thing to note, which derives from all the statistics of human performances in these events, is what the values of C look like or what the people who make the tables think they ought to look like from performances.
For the runs, C, you can see, for 100 metres, 400 metres, 1500 metres, it is all about 1.8-ish. Whereas, if you look at the jumps, the long jump, the high jump, and the pole vault, which is slightly different, they are all about 1.4. Whereas, for the throws, 1.05, 1.1, 1.08, they are all rather close to one. So, the runs are highly progressive, the jumps are pretty progressive, but the throws are really rather neutral.
If you look at the heptathlon women’s events, Jessica Ennis, Kelly Sotherton, previous Olympic medallist, the situation is very similar. It is different running events – hurdles, 800 metres, 200 metres – all around 1.8. Here are the jumps, around 1.4, and here are the throws, all around 1.05. The nature of these three events is really rather different, and the reward that is given for them is correspondingly scaled in quite a different way.
Here are some odd facts about this situation. If I was to change the points scoring system and have a really very progressive system and pick C equals two for all events – and there is a reason of physics to do that because the energy that you generate goes like V squared, and if you are a jumper, the distance you achieve goes like V squared over G, so there is an argument for making the reward scale like the square of the distances or the times. Then, what happens is that the present world record-holder, Sebrle, his world record, about ten years old now, is 9026 points, so he would no longer be the world record. The situation would change quite dramatically, and the current number two, Dvorak, who is also from the Czech Republic, would become the new world record-holder, with a hugely increased score, 9468. What is really reflecting is the fact that those throwing events, with their Cs around one, if a competitor is a good thrower and you increase the reward to C is two, then there is a massive increase in their score.
This turning of the tables, every so often, there is a little change in the tables, and a rather strange thing happened in 1984. Daley Thompson won the Olympic decathlon in Los Angeles, and he missed the world record by one point. A few months later, the scoring tables were changed, and his final score in his Olympic winning performance was therefore also very slightly changed and he now found that he was the world record-holder. So, retrospectively, he became the world record-holder.
Here is the current performance level. The way these tables were originally created, there was an idea that about 1000 points would be a world-class performance in each event. Suppose you took the world record in every event today – it would be 12500. If you took the best performances ever in decathlon competitions, you are a lot less, 10485. Individual performances, Bolt’s 100 metre record, would only get him 1202. The best performance across any event against the decathlon tables is the world discus record, 1383.
What do the competitors tend to be like in practice? This is what you would have to do if you want to go out and prepare and train for a decathlon, and you want to win in London, 9000 points will surely win, in London, believe me. So, if you just want to get 900 points in every event, at first it does not look terribly demanding. There are lots of schools’ athletes who can achieve 10.83 for 100 metres – fifteen year olds do that with no trouble. There have been fourteen and fifteen year olds, that can run 48.19, 400 metres, and they can certainly run four minutes seven for 1500 metres. So the individual performances often do not look really dramatic. But being able to put all these different things together is very challenging.
This interesting graph here takes the 100 best performances of all time in the decathlon and works out how the points tended to be won by the individual competitors. You then get a picture of what is the profile of a very good decathlete, on the average, and you can see that they tend to get most of their points from long jumping and sprint events, like the 100 metre hurdles and the 100 metre flat. Pole vault is next. That is slightly unusual. There is no point being sort of slightly good at the pole vault. If you register no height, you have essentially lost the competition, so you have got to take the pole vault very seriously, and the heights go up in fairly discrete jumps. But what you see is that, if you want to score a lot of points in the decathlon, perhaps you have only got a certain amount of training time or you want to introduce a new athlete from an individual event into the decathlon, what you should be looking at is concentrating on people who can sprint because that feeds into long jumping and it feeds into hurdling. Nobody bothers with the 1500 metres – they just tend to run that on general fitness. These people tend to be very big and heavy, and it is very difficult to make significant improvement in the 1500 metres. But what you can see from this is how the event is biased in certain respects, in the way the scoring works, towards certain types of athlete and certain events.
I think in one of my previous lectures I proposed a completely different scoring system, where you did not introduce points at all, but since the events either involve you trying to get large distances or short times, why do you not just multiply together all the distances in the jumps and the throws and then divide by all the times in the running events, because what you want to do is to make the top as big as possible and this as small as possible. This has got units, of course, so you would want all the distances in metres, all the times in seconds, and the answer would be in metres to the power six divided by seconds to the power four.
Interestingly, if I do this with the top two decathlon performances ever, so I will call this the B for Barrow Total, then the world record holder scores 2.29. The number two, Dvorak – you remember this was the chap who suddenly becomes the best if you weight the throws better – he scores 2.4, so he becomes the number one. Pretty much all that happens here is that these top two change performances, but nonetheless, this has got its own biases, just like any other scheme. You can see that, in the events where there is a very large distance that is required, so the javelin for example, where you throw something a long way, you have got much more scope to make that number bigger in the denominator, and by improving in the 1500 metres, you can make a bit improvement, twenty seconds, by training appropriately. With the 100 metres, you will be lucky to make an improvement of 0.1 or 0.15. So this formula has got its own biases. It tends to favour the situations where the times are long and the distances are long, where big improvements can be made. It is just an illustration of how you could have had a completely different system that did not go through a transformation of points to get the final order.
Today’s a big or a small football day, depending on your outlook. Let us say a few things about football scoring.
If you are as old as me, you can remember a time before 1976 when football league tables were handled in a different way to the way they are today. Today, we are familiar with points won but also goal difference as a way of splitting teams that have got the same number of points. Well, back before 1976, they did not use goal difference, they used goal average, so instead of subtracting the goals conceded from the goals scored, they divided them. So you can see this produces something that you might need a slide rule, in those days, or a calculator to work out, so it is slightly more complicated or difficult to see what is going on, and it has a rather different bias to it. It tended to encourage rather defensive play because having a small A really did produce a bigger boost to this average than it does here.
Here is a very interesting situation I remember from 1988. In 1988, at the end of the football league season, Arsenal and Liverpool were tied, with identical performances – so they had both won 22 games, they had both drawn twenty games, and they had both lost six games, so they had rather different goal scoring, for and against, but the differences were identical. They had identical points and they had identical goal difference.
If you had had the old system, which was not in play anymore, if you had worked out goal average, then Arsenal would have had 2.15 goals scored for every one conceded, whereas Liverpool would have had 2.5. So you can see how much they gain by having that better defensive record.
The rule at the time for splitting this tie, which is no longer a rule in the Premier League, was that you took the team with the bigger F, who scored the most goals, and so it was Arsenal who won the league. This was a very rare and unusual historic occurrence that you are not likely to see again in your lifetime.
This is another thought about Premier League, so here is another table with a peculiar outcome, Chelsea at the top in 2009/10, and the point here is that you could argue that you should not bother with points at all in the football league tables. Why bother with points? We saw at the beginning, if you have two points or three points for a win, you can completely change what the performance tables will eventually be, and there are arguments to tinker with points in other ways, that you should give no points for an away draw, and maybe you give no points to anybody if it is a nil-nil draw. So there are lots of little things that you can do, but if you look at the situation here, so this was a league table here, Portsmouth at the bottom, as they always are you see – one of Harry Redknapp’s former teams – and these were the total points, and this was the goal difference. What would have happened here if you gave no points for a home draw? So there is a big incentive for home teams to try and win.
You can see there was not much that happened. The teams at the very top do not tend to draw at home, they win. But as you go down, you get more and more differences in points totals for the teams, but really it is not very much. So this is not a sensible rule change really.
What might be a sensible rule change is just not to have points at all, but to decide the whole league on goal difference, because you can see here that there is a very strong correlation between goal difference and position in the league – you get pretty much the same outcome. After all, the idea of the game is to score more goals than you concede, and if you just reflected that, it might produce a simpler and more interesting way of evaluating what is going on over the season.
I discovered recently that there is one football league in the world that mathematically is more interesting than our own, and it is the Argentinean league – they call it the Primera League, interesting – because, in Argentina, they play two seasons a year, so you have to go through it all twice each year. But the way they work out the relegation of teams is that they relegate the two teams that have got the lowest points scored per match over the past three championship, so there is an average over the last, not three years, but it is 1.5 years. So this is rather interesting. If a team has not been in the league for the past three championships, they take the average from the previous one or two years. So, this is – and then there are various nuances here, that the next two teams are going to play off against the third and fourth teams from the league below. There is a memory in the system, and it is rather more complicated to work out on the spur of the moment who is going to go up and who’s going to go down.
Well, let us look at an idea for a completely different type of scoring system, and this is a type of scoring system that you see in some sort of indoor sports, like chess, but you also find it sometimes in America, in hockey or football small leagues and so forth, and it is a system that you could apply and I have applied here, say, to – this was a world cricket cup tournament in the Caribbean some years back. It was that one, you remember, where someone – they thought that one of the coaches had been murdered and so forth and it was the whole tournament really ground to a halt, with the umpires forgetting the rules in the final and so forth.
So, this was the super 8 stage of the tournament, so the second stage, where you had these eight teams that all played each other at some stage, and they give two points for a win and one for a draw, but there were never any draws, and then, if there had have been teams tied on points, they used the run rate.
Now, because this is a small tournament – England of course did rather badly in this tournament – that what you might replace this simple system with is, if a team beats another team, then you somehow feel that if they beat one of the top teams, so I think Ireland beat South Africa, whereas England tended just to beat rather weak teams at the bottom, like Bangladesh, that you ought to get more credit for beating the better teams than for beating the lesser teams.
Suppose that, if you beat a particular team, your reward is to receive their points score, then Australia, who beat pretty much everybody, their points score would be the sum of the point scores of Sri Lanka, New Zealand and so on, all the way down to Ireland, and Sri Lanka would get a point multiplied by the scores of these teams, and so on all the way down. This is really what, in mathematics, just looks like a matrix equation. If we put brackets around this whole column here, then it is equal to this big matrix of terms here, so SL is the first entry, NZ is the next, SA is the next, and so forth, and all these ones are zeros, multiplied by this same list. So we have a simple big matrix equation for the list of teams, where K is a multiplier time the matrix, what you might call an eigen value for a mathematician, and A is this big matrix which tells you all the win-loss performances. It is a big matrix, it is eight by eight, but all the entries are either zeros, where they do not play each other or where you lose, and ones for wins.
What you want to do is to solve this equation. It is like solving eight simultaneous equations to find what the new points scoring is under this system of giving credit for who you beat as well as how many people that you beat.
Well, you can solve those equations. The mathematics of this, finding this solution, has a particular sort of jargon name – it is finding the first-ranked positive eigenvector of the matrix. This is what that is. This is what the solution, where all the entries are positive, so they are realistic, looks like, and these are the scores. So, Australia has got 0.729, all the way down to South Africa at the end with 0.332. So there is a new ranking here. The first-ranked team, Australia, the next is Sri Lanka, New Zealand, South Africa, Bangladesh, England, West Indies, Ireland. The original league position, in this case, it did not turn out as different as I had hoped, but it starts to be different down the bottom, where, when the weak teams win their game, who they beat makes a difference.
The other interesting thing to know about this system, and why I am spending so much time telling you about it, this is how Google works as a search engine. In effect, when you put something in, it derives a score from it, from the other searches and other links that have been made to that name or that word that you have put in, and why Google has huge factories full of more computers than anywhere else on Earth except the National Security Agency in the US, is because what they are doing with those machines is inverting and solving gigantic matrix problems like this, not with eight by eight matrix but thousands by thousands matrixes, and storing other information and correlation about these matrices. So the Google search engine is really searching for these first-ranked eigenvectors that give the strongest links, the strongest rewards, between two quantities – one that you are searching for and one that it is recorded. It was Brin and Co. who set up Google, this was really their idea about how you could create a new and super-efficient way of searching information on the web using this matrix algebra.
Let us go back to tennis before we finish. You remember, we looked at an example, early on, in table tennis, with its scoring system, about the reward relatively for luck or for skill. Well, in tennis, you have got a very complicated scoring system really. You wonder why do they not just play to 21, or why do they not just play to 100 or something, you know, and the first one to win 50 points, with a two point margin, wins the match. But, instead, they go through this long rigmarole of games and sets – there may be three of them, there may be five of them – and tie-breaks to try to reduce the time. So what is going on there? Why are they doing that?
One of the reasons is to continually maintain interest, both of the players and also of the spectators. So they are worried that, if you played best to 50, you would get games where one player would become fifteen or twenty points ahead, and we would all have to watch a whole sequence of points then to be played which did not really have much of a bearing on the final result, so the game would stagnate. By continually having sets, you re-capture the attention of the spectators and you rejuvenate the energies of the players – there is something new to play for, and you can win a tennis match even though you do not win most of the games. So, you could lose the first set 6-love, and then win 6-4, 6-4, but you would not have won most of the games.
Let us just look rather simply, if we have this same model we have used before, let us suppose we forget about the advantage you gain from being the server – take that into account later if you want. Let us assume your probability of winning a point is P and we are going to be interested in the situation where the players are very evenly matched, so P is very close to a half – it is a half plus U, where U is very, very small, maybe 1%, so it is much, much smaller than a half.
What are the ways in which you can win a set? Well, you could win 40-love – there is one way to do it like that. There are four ways in which you could win 40-15, so that fifteen could come at all sorts of different points in the game. To win 40-30, there are ten different ways in which you could win 40-30 – you can check these at your leisure on the journey home. And there are twenty ways in which you could get to deuce.
Now, once you get to deuce, there is an interesting situation. I have seen books that then set up a sort of an infinite recursion relation to figure out what is the probability of winning from deuce, but if you think carefully, there is a trick to work out what is the probability of winning from deuce. Let us call it D, D for deuce. Well, how could you win from deuce? The first way is you could just win the next two points – advantage, and then you win – and the probability of doing that is P squared. Or you could win the next point, probability P, lose the next one, probability 1 minus P, and then you are back to deuce, so the probability just multiplies by D again. Or you could lose the first one, 1 minus P, win the next one, you are back at deuce, multiply by D. So the probability of winning from deuce is just P squared plus twice P, 1 minus P times D. So that is getting to advantage, losing the next point, being back at deuce, so you are winning from deuce. So if we solve this, this is the probability of winning from deuce.
If we put all these together, the probability of winning the set, that is the 40-love route, that is the 40-15 route, so there are four times P to the fourth, times 1 minus P, you have lost one, plus 10 times P to the fourth, times 1 minus P squared, plus the probability of winning from deuce – and there should be a 20 in front of that which I left out.
If you add all these up, you get a messy, rather complicated formula that only depends on P. So if we then substitute in P is a half plus U, and U is very, very small, so you can forget about U to the power four compared with U squared or U, what the probability of winning the set reduces to is a half plus five U over two. So you can see this set structure means that your chance of winning the set has been amplified a little bit. It is not just half plus U, which was the chance of winning a single point, so there is more reward for skill over luck.
If you played three sets and you work all this out again, the chance of winning best of three is a half plus eleven U – if I forget about tie-breaks to deal with, a little messy.
If you go to five sets, best of five, it is up to thirteen U. So you can see, the more sets you play, then having a little bit of positive U, a little imbalance between the players, tends to win out, with greater and greater probability. There is a greater reward for skill over pure luck.
Originally, long ago, when rules were set up that men were going to play best of five at Wimbledon and women best of three, the women’s game was not that competitive. There were not many evenly matched top-class women’s players, and so it was believed that it was sufficient to play best of three. Somehow, that is still remained the case at Wimbledon, and even though men and women are now paid the same prize money, even though men play far more games to win it than women, there is no reason why top women players should not be playing best of five.
There is another amusing factor that we might add into this, that we all know, in any sporting event where you are playing for points like this, one player against another, psychological factors play a very big role, and if you win a point, you might feel that you get a boost, that you are more likely to win the next point, and if you lose a point, you are maybe more likely to lose the next one after that.
Suppose we try to factor this in in some way, and we think about one set after the other, so the odds on winning a set – what you mean by the odds, it is the probability that you win it divided by the probability that you lose it. So, if the probability of winning a set is S, the probability of losing it is 1 minus S, so the odds are S over 1 minus S. Now, you could assume that, if you win a set, you get a big confidence boost, and your odds of winning the next set become B times the odds of winning that first set, so you are more likely to win it – B is bigger than 1 – but if you lose, then your odds of winning the next set are reduced to O divided by B. You can see that what happens, if your odds of winning the first set are O, if you win it, your chance of winning the next one is BO, and if then you win that, the chance of winning the third one is B squared O, even bigger. Whereas, if you lose that one at the beginning, it is O over B, if you lose that, it is O over B squared, but you have a chance for a bit of a recovery, if you win here and you lose, or you lose here and you win, the odds are back to O. So you can see you could, if you wished, fold this into the other analysis, and also take into account the fact you are more likely to win a point if you are serving, perhaps, than if you are receiving.
Finally, the last picture, today is a momentous Olympic football day, I saw as I came in, on the screen. So England have their draw for the Olympic football tournament, playing Senegal, Uruguay, and somebody else who I cannot remember.
I thought I would show you what happened the first time, in 1908, when Great Britain played in the Olympic football tournament. The tournament was played essentially at what was the old White City Stadium, down near Shepherd’s Bush, and, in those days, it was very hard to get teams to take part in the Olympic football tournament, for financial reasons. In the previous Olympics, in St Louis, in the US, hardly any teams went. It was just so expensive to take a team across the Atlantic. Here, they were expecting to have eight teams competing, but unfortunately two of them withdrew at the last moment – Hungary and Bohemia, essentially I think for financial reasons again. So this was the draw for the first round, the Netherlands, France entered two teams - I do not know if this was in some way to sort of make up for the situation -Great Britain, Denmark and Sweden.
In the first round, the Netherlands and the French A teams had “byes”. Britain did rather well, beat Sweden 12-1 in the first round, and Denmark beat France B nine-nil. The Danish team is very interesting. It included, which is why I am telling you this, included a famous mathematician, Harold Bohr, who was the brother of Nils Bohr, the even more famous physicist. Nils Bohr was also a very good footballer, at almost international level at some period, goalkeeper. Well, Harold Bohr scored two goals in this game, and then went on to play in the other games in the tournament.
You can see, in the semi-finals, Britain beat the Netherlands. Denmark had a fairly easy route here – they beat France 17-1! So they beat the B team nine-nil, and they beat the A team 17-1 – they must have been managed by Harry Redknapp!
Then, in the final, Britain beat Denmark two-nil, and exactly the same thing happened in the next Olympics, the same final. So, it can all work out, but perhaps the teams involved are rather different nowadays. Britain has not played in the Olympic football tournament since 1960, for essentially political reasons, and so, for the first time since then, we will have a Great British football team in the Olympic tournament, at least for a while.
Well, that is all I have got to say, to thank you all for faithfully coming along to this lecture series this year which, in some ways, is a bit specialised, more specialised than in the previous years of the maths series, and I hope you have learnt something about mathematics and sport that will invigorate your time sitting on the couch during July and August, because I am sure you do not have any tickets, like me – I do not have any tickets – and that it will give you some insight into some of the things that are going on in some of these events that might not have occurred to you. And, if you are a school student working on mathematics, there are lots of interesting applications of mathematics, both in statistics, in mechanics, that we have looked at in past lectures, and other areas of mathematics, applied to sport, so there are interesting hard problems and, for teachers, stimulating and motivating examples of many areas of mathematics which are both inspiring and of great interest to young people. I hope this connection between sport and mathematics will be something that you will remember and, if you want to remember it even further, you can still purchase copies of my book from the bookshop upstairs. It is the last opportunity for me to sign them for you.
© Professor John Barrow 2012