Steve Davis, friend of the USA 10 Kit (also: my father), happens to be something of a whiz when it comes to statistics. After a rash of 0-0 draws in the first days of the 2010 FIFA World Cup, I asked him if he thought it made sense for teams to play for a tie. He did, of course. Below, he uses a concept called the Poisson distribution to explain why squads and their managers play conservatively when they are not overwhelming favorites.
This set of observations relies on the assumption that soccer teams and coaches can shift their strategies to be slightly more offense-oriented or more defense-oriented.
Suppose that is true, and suppose that a team’s typical formation or strategy would give them an average of one goal per game. With an average of one goal per game, sometimes the team would score one goal, sometimes they would score none, sometimes two, and sometimes more than two. If your average goal per game is one, you might expect your team to score one goal in the most number of games (not exactly true: tied with no goals), but it is a bit harder to predict how many times that you would score no goals or two goals or more than two goals. And how many times really would you expect to score one goal in the game?
As you might have guessed, there is a mathematical formula called the Poisson distribution that predicts the percentage of times that you will score each number of goals in a game given a particular average. The Poisson distribution is designed to work with events that are unlikely to happen, such as a soccer goal on any given play, but for which there are many chances for it to happen, such as plays in a game. It was originally used for Prussian cavalry soldiers being mortally kicked by their horses—unlikely, but lots of chances.
If you apply the math, the Poisson formula predicts that if you average one goal per game, then you can expect to score one goal in 36.8 % of the games. (Remember that 36.8% number.) Surprisingly, also from the prediction, you can expect to score no goals in 36.8% of the games. You can expect to score two goals in 18.4% of the games, and more than two in 8.0%. So, even though your chance of scoring one goal is the same as your chance of scoring no goals, your chance of scoring (of not scoring no goals) is 63.2% (100 – 36.8%).
Suppose that you have a game against a high-scoring team, and your coach wants to emphasize defense. This emphasis on defense causes your offensive capabilities to be reduced so that your average goals per game with this line-up is only 0.9. Surprisingly, the Poisson distribution predicts that you have almost exactly the same chance of scoring one goal, 36.6% (as compared to the 36.8% above with an average of 1 goal per game). The tradeoff is, however, that you have a higher chance of scoring no goals (40.7% vs. 36.8%) and a lower chance of scoring more than 1 goal (22.7% vs. 26.4%).
If your coach is even more conservative and shifts your emphasis so that you are stronger defensively but your average scoring ability is lower at 0.75 goals per game, your chance of scoring one goal is 35.4%, only 1.4% less than your basic team arrangement. The other numbers become 47.2% for no goals and only 17.4% for more than one goal. But surprisingly, your chance of one goal is still almost the same (35.4% vs. 36.8%).
Similarly, if your coach decides to let loose and design your attack to increase your offensive strength to 1.25 goals per game, you will have a 35.8% chance of scoring one goal. Fortunately, your chance of scoring two goals is 22.4% and your chance of scoring more than two goals is 13.1%. (Also, the chance of no goals is still considerable at 28.7%). These increased chances of scoring will be needed since your defense may be suffering.
If your coach decides to be not so cavalier and adjusts the team so that the average goal expectation is 1.1 goals per game, then, perhaps not surprisingly, your chance of one goal is 36.6%, still in the same range, your chance of no goals is 33.3%, your chance of two goals is 20.1%, and your chance of more than two goals is 10.0%.
So, there you are: a coach who wants to maximize his or her chance of winning but definitely does not want to lose. You consider that your chance of scoring one goal is almost exactly the same (~36%) if you play your standard game or if you structure your team in a defensive posture. The latter presumably keeps your opponent’s score down. You do it, and play for a 1-0 win.
On the other hand, if you structure your team for a slightly more powerful offense (at 1.25 goals per game), you have about the same chance of scoring just one goal (35.8%), and you still have a considerable chance of scoring no goals (28.7%). So, even with the more potent offensive line-up, you have at 64.5% chance of scoring one goal or none, and, consequently, only a 35.5% chance of scoring more than one goal. Since your defense is presumably weaker, you run the risk of your opponents getting more than the one goal (or less) that you are likely to score and winning the game.
Soccer coaches, who are so averse to losing, especially in the qualifying rounds of the World Cup, behave as though they have these averages in their heads.