WILL THE UNDERDOG WIN?
The maths behind memorable sporting moments
Everybody has a favourite sporting moment. Maybe it’s England’s 1966 World Cup victory against Germany, John McEnroe’s thrilling tie-break with Bjorn Borg, or Steve Redgrave’s fifth Olympic Gold for rowing.
How the TV executives long for such thrilling moments, since this is what causes sudden surges in the number of viewers. The sports authorities love the big moments, too, and over the years they have not been averse to the odd rule change here and there to improve the chance of more thrills in a shorter time interval.
What, then, are the key factors that make a thrilling sporting moment? If you trawl through all of the great moments, certain themes seem to crop up time and again.
Victory for the underdog
Crowds – especially English crowds – love to see the little guy win. The nation was gripped when second division Sunderland beat the mighty Leeds in the 1973 FA Cup final. Every year, too, some unknown Brit upsets a stronger player at Wimbledon, before disappearing into oblivion. And in 1997, a littlefavoured European Ryder Cup golf team managed to overcome the mighty Americans.
Although there don’t seem to be any hard statistics on this subject, it does seem that some sports throw up more victories by ‘underdogs’ than others. There are many examples of the proclaimed underdogs winning in football, and examples are also quite common in cricket, tennis and golf. On the other hand, in rugby, athletics, rowing and many other sports it is unusual for the outsider to win.
Before we see why this might be the case, we need some definition of what is meant by an underdog. Since underdogs aren’t given much chance of winning, you might define them as competitors to whom the bookmakers give only a slim chance of winning – less than 10 per cent, perhaps. However, this simple definition doesn’t work if there are more than two competitors. Even the favourite in the Grand National only has about a 10 per cent chance of winning.
In any case, it seems wrong to use probability as a way of defining an underdog. Suppose we defined an underdog as a competitor with only a 1 per cent chance of winning. We would expect that the number of underdog victories would be the same in all sports (i.e., they would win 1 per cent of the time). The fact is, however, that underdog victories happen far more often in some sports than in others.
So instead, we won’t define underdogs by their chance of winning, but by their relative ‘weakness’ compared with their opponents. Some sports are merciless on the weak, who don’t stand a chance. Other sports, however, thanks to the scoring format or the opportunity for flukes, are much more favourable to the weak. These are the sports where underdogs are more likely to come out on top.
Tennis is a good example of a sport where the most talented player doesn’t get all the credit he or she deserves. If you really wanted the technically better player to win a tennis match, then you would have a scoring system such as ‘first to reach 100 points wins’. Bjorn Borg and Martina Navratilova would have been unbeatable. But this would make for very dull sport (see the later section on ‘lead-swapping’).
Instead of this, the hundred or more points in a major tournament match are broken down into ‘big points’, more commonly known as sets. Regardless of whether a player wins a set by six games to love, or by seven games to six, they still only get one set. Because of this, there are many examples in tennis where the player who won fewer points won the match. In fact, it is hypothetically possible for a player to score almost twice as many points as the opponent, and still lose. Suppose Tim Henman loses a match 6-0, 6-0, 6-7, 6-7, 4-6. If you are familiar with tennis scoring, you can figure out the extreme numbers of points he could have won and lost in each game and the match overall. It’s possible for the loser Henman to have ‘won’ this match by a whopping 158 points to 86. Is this the biggest margin of ‘winning’ defeat possible in any sport?3
Underdogs are also favoured in sporting contests where there are very few scoring opportunities. One of the lowest scoring of all sports is football. A typical match has only two or three goals. There are, however, far more scoring chances than goals. Although it is simplifying matters a little, there is some truth in the argument that, while the number of scoring chances is a direct reflection of the team’s skill, the number of chances that are converted into goals is more a matter of luck. Everton might therefore ‘beat’ Peterborough by fifteen chances to four, but, if the probability of a chance going in the net is only about 1 in 5, this means an expected result more like 3-1. The winning margin is just two goals, and, when margins are this small, random variation says that there is a decent chance of this becoming a 2-2 draw or even a 2-3 defeat.
There is one other major factor that can benefit an underdog, and that is the freakish accident or fluke. In 1967, a horse called Foinavon won the Grand National, despite being the rank outsider at 100-1 against. Had the horses all completed the course, Foinavon wouldn’t have had a chance. On this occasion, however, about twenty horses fell or refused at one fence. Foinavon was so far behind the rest that he was able to avoid the melee and had a free run for victory. In this case being hopeless turned out to be a huge advantage to the underdog.
Motor racing is also prone to accidents or car problems that randomly hit the strong as much as the weak. Almost always the reason why a non-favourite wins a grand prix is because something other than driver skill disabled the frontrunners.
Ill fortune can have a disproportionate effect in golf, too. Certain golf courses, such as Carnoustie in 1999, have so many hazards on them that even a highly skilled golfer has a very high chance of hitting a treacherous bunker or losing a ball in the rough. In these circumstances, a game of skill (akin to chess) is turned into a game of chance (more like snakes and ladders), and the more there is of the latter, the greater the opportunity there is for a relatively weak contestant to come through the field and win.
Why underdogs aren’t always underdogs
Of course, it might just be that the so-called underdogs are no such thing. There is a lovely illustration of how we can be fooled into thinking that a team is an underdog when this is not the case.
Here is the argument. Since 10 is greater than 7 (symbol ‘>’) and 7 > 3, it follows of course that 10 > 3. And in fact this can be generalised to the statement that if A > B and B > C, then A > C. This is known as transitivity.
We sometimes wrongly assume that transitivity applies to other situations. For example, if team A usually beats B, and B usually beats C, then surely A will usually beat C? Not so. To discover a situation that defies the norm, make four dice to represent four different football teams. On the six faces of the dice, put the numbers shown in the table below. For example, Chelsea’s die must have four faces with number 4 on them, and two with 0s. Everton should have a 3 on every face.
According to this rather artificial simulation, when Chelsea play Everton, there are only two possible scores: 4-3 to Chelsea, and 3-0 to Everton. If you play several dice games between these two teams, Chelsea will win about two-thirds of the matches.
When Everton plays Blackburn, Everton wins 3-2 or Blackburn wins 6-3, and in this case Everton wins about two-thirds of the matches.
Blackburn-Fulham has more possible results: 6-5, 6-1 or 2-1 to Blackburn or 5-2 to Fulham. Again, Blackburn wins two -thirds of the games.
Chelsea usually beats Everton, Everton beats Blackburn and Blackburn beats Fulham. So Chelsea will surely thrash Fulham? Not in this game, they won’t. Rolling Chelsea’s dice against Fulham, Fulham amazingly wins two -thirds of the games.
This is an illustration of a non-transitive system. And, if it is possible here, perhaps something similar is possible when the real Chelsea and Fulham play a football match.
Random walks, deuce and snooker leads
Imagine a long straight road with a line down the middle. There is a hedge to the left of the road and a stream to the right. A drunk starts in the middle of the road, and attempts to walk along it. With each step he randomly lurches to the left or the right, with an equal chance of either direction. One question immediately comes to mind. How long before he falls into the hedge or the stream? It depends, of course, how many left or right steps it takes to reach the edge of the road. But, if the edge is N steps from the centre, then it turns out that on average he will take N2 steps before he finishes his walk, though it could be fewer or more than this.
The drunk’s progress is known as a random walk, and is the metaphor for a whole range of interesting problems in probability, including sport. The example above of falling into a hedge is directly analogous to a tennis game at deuce. In this case it takes two steps either way for the game to end (the equivalent of the drunk falling over), so on average – if each point has a 50-50 chance of going to either player – it takes 2 2(i.e. 4) points for the game to end.
A second question of the drunk could be ‘How often does he cross over the white line in the road?’ This is the same as asking ‘What is the chance in a snooker match that the lead swaps from one player to the other?’ If the number of frames played in the match is F and the players are of equal strength, the answer turns out to be, on average, roughly (√F)/3. So, after nine frames have been played, the lead is expected to have changed over only once. Over 100 frames, the average number of lead swaps is about three – fewer than intuition would suggest.
One-sided contests are rarely memorable. There isn’t much excitement in a rugby game in which England beats Japan by 50 points to nil. It’s the close ones that we remember, especially if the lead changed hands towards the end of the match.
If one team is stronger than the other, then once it establishes a lead the chances are that it will hold on to it. But, if the competitors are roughly the same strength, then the frequency of the change of lead depends on the effect of the lead on the competitors’ relative strength.
Lead changes are much more likely in some sports than in others. In some contests, being ahead influences the relative strengths of the competitors; in others it has no effect.
There are two well-known contests where establishing a lead actually has a reinforcing effect on the leader. One is the Oxford-versus-Cambridge boat race. In this case, once one boat is sufficiently far ahead of the other, the conditions actually favour the boat that is ahead, since it can occupy the middle water. Lead changes are almost unheard of. The Monaco Grand Prix is similar, because overtaking is so difficult. Even football can show these tendencies: if the away team scores a goal, it then plays defensively and the chance of any more goals in the match is reduced.
In other situations, however, establishing a lead probably has very little influence on what happens next. These situations can be analysed by what is known to statisticians as a random walk (see the box).
Some of the mathematics of random walks is complicated, but the conclusions make interesting reading. Once one side has established a lead, random-walk analysis says that the lead will change relatively rarely.
Another conclusion from random-walk theory is that, in an even contest, whoever is leading halfway through the game has a 50 per cent chance of continuing to lead throughout the rest of the game. Is this enough to make the second half exciting? Probably just enough. In the Oxford-Cambridge boat race, however, which is not a random walk, the leader at halfway has probably more than a 90 per cent chance of staying ahead thereafter, which is why it is normally such an anticlimax.
One way to increase the chance of a change of lead at the end of the game would be to make more points available in the later stages. If a goal counted as two points instead of one in the last ten minutes of a match, this would generate more occasions when there is a late switch of leader. Interestingly this is how some TV game-shows work. In Channel 4’s Countdown, for example, most of the word rounds produce only 5 or 6 points for a player, but the final round – the Conundrum (an anagram) -is worth 10 points. This gives the trailing player one final opportunity to overhaul their opponent.
Children’s fiction has thrown up a more extreme example of how the points scored in the final stage of a game can have a disproportionately high value. In Harry Potter’s wizard sport, Quidditch, goals count for 10 points but catching the ‘snitch’ at the end of the game scores 150 points. For goals to influence the result, one team therefore has to be at least 15 goals ahead, and none of the games in the early Potter books come anywhere close to this. Catching the snitch therefore counts for almost everything, though goal difference might come into play.
The disadvantage of increasing the points for later stages of the contest is that it rather negates the point of the early stages. The contest turns into a last-lap sprint with the rest of the contest becoming almost meaningless – the frenetic end to cycle track races after many laps of sedate cruising comes to mind. So far, sports administrators have held back from awarding more points for the final contests, but maybe the day will come.
The favourites meet for the decider
People may enjoy seeing the underdogs win, but the greatest sporting occasions are usually the battles between the giants, especially if the winner of these contests receives the ultimate trophy or gold medal.
The ideal type of tournament for building up to a big-match decider is the knockout. The FA Cup is one of the longeststanding examples of this. In the FA Cup, before each round the teams are drawn randomly from a bag. This means that there is a chance that the two ‘favourites’ might be drawn to play each other long before the final, thus knocking one of them out. Likewise, weak sides might be fortunate and end up being drawn against other weak sides, thus making far more progress in the tournament than they had any right to expect.
What the knockout cup doesn’t do, therefore, is guarantee that the best two teams meet in the final. In fact, there is never more than a 2-in-3 chance that the best teams will meet in a Cup final – even if both of them get through to the semifinals.
Suppose that the favourites Rangers and Celtic are both through to the last four, along with two ‘minnows’, Falkirk and Alloa. The names are now drawn out of the hat. Here are the possible semifinals:
All of these three sets of fixtures are equally likely, and in one of them Rangers and Celtic meet. So the chance that Rangers and Celtic will meet in the final is 2/3, assuming that both are near certain to beat the other teams.
At the start of the competition, it is even less likely that the ‘big guns’ are in different halves of the draw. If there are eight teams left in the contest, the chance of two chosen teams being in different halves of the draw is 4/7, or 57 per cent. If there are sixteen teams, the chance drops to 8/15. In fact, with N teams left, the probability of teams staying apart until the final turns out to be N/(2N-1), which tends towards 50 per cent as N becomes large. So, in about half of all tournaments, Rangers and Celtic will meet in an earlier round than the final, assuming that one of them isn’t knocked out first.
What this means is that knockout tournaments are a bit of a lottery, they don’t necessarily reward the strongest teams. Some sports do something to correct this. The Wimbledon tennis tournament, for example, seeds the top players (32 of them at last count), and designs the draw so that these players cannot meet each other until the tournament has reached the last 32.
Furthermore, the top sixteen seeds are in separate parts of the draw until the last sixteen, the top eight seeds cannot meet until the last eight, right on to the top two seeds, who are in separate halves of the draw and cannot meet until the final. As a result, it is extremely rare for a non-seed to make it to a tennis final or even to the last four, in contrast with the FA Cup where non-Premiership sides have made the semifinals on a number of occasions.
The fairest way of identifying the best team is almost always a league, where each team has to play every other team at least once. The perfect climax to a league would be if the final match of the season was between the best two sides, but since the league fixtures are arranged before the season begins this rarely happens.
It’s a knockout
Including qualifiers, 596 teams participate in the FA Cup; 282 men compete for the singles at Wimbledon. In both cases, the strongest participants don’t enter until the later stages. With this limited information, how quickly can you work out how many matches there are in each tournament (ignoring any replays)?
The answer is surprisingly simple. The number of matches in a knockout tournament is always one smaller than the number of participants, making 595 matches in the FA Cup and 281 at Wimbledon. The reason is that each match knocks out exactly one participant, and at the end of the tournament only the winner has not been knocked out.
However, some sports have successfully combined the fairness of a league with the excitement of a knockout by having play-offs at the end of the season. Typically, the top few sides in the league go on to the knockout phase, which is set up so that the top two teams cannot meet until the end. This league/knockout combination is a fundamental part of not only the FIFA World Cup, but also many of the team sports in the USA. The authorities may not be able to guarantee a thrilling climax, but they know how to give themselves a great chance of one.
3. Some wise-cracker once pointed out that in tennis the winner actually has to win only one point – the last one.