This week I posed a simple question to the students in the “Sports Analytics in Practice” course that I teach with Philip Maymin at New York University’s Stern School of Business: “Would you rather have two 50% chances to score, or five 20% chances?” It’s a question that teams in the English Premier League and other competitions might want to ask themselves, too, in light of Leicester City’s performance this season.
If the question was simple, the answer was not – and there was no consensus in the class. Some students thought a team should be indifferent. Others argued that in a low-scoring sport, it was most important to be fairly sure of scoring once. And still others thought that having a chance to score several goals was superior, since a win was worth three points versus one for a draw.
In truth, this is a difficult question whose answer requires a bit of game theory as well as probability theory. Let’s say that teams can play one of two strategies: “Quality” will yield two 50% chances, and “Quantity” will yield five 20% chances. We can simulate the outcomes of the matches for each possible combination: Quality v Quantity, Quality v Quality, and Quantity v Quantity.
Over hundreds of thousands of simulations, the average points won by each team were as follows:
These are the payoffs of a Prisoner’s Dilemma. Not knowing what the other team’s strategy will be, each team has a dominant choice: Quality. But if both teams choose Quality, the expected payoff for each will be less than if they had both chosen Quantity. This last observation makes sense intuitively; more shots mean more variance in the scoreline, more variance means fewer draws, and fewer draws mean more points per match.
Of course, the setting in soccer/football is different from the one in the classic example from game theory. The teams may not be able to choose between the two strategies; each squad may be set up for one strategy or the other. And it would be unsavory (if not illegal) for the teams to collude in order to reach the cooperative outcome, by both promising to play Quantity.
But the example is striking nonetheless. It means that among two teams with the same total chances of scoring – the same total expected goals – the one that accumulates this total with fewer chances will come out on top in the long term. Imagine that one team plays Quality in a 20-team home-and-away league where every other team plays Quantity. Then the Quality team would take 1.38 points per match, while all the others would take 1.34 in 36 matches and 1.28 in the remaining two. At the end of the season, the Quality team would be expected to take 52.4 points, versus 50.8 for the rest.
This is where Leicester comes in. Currently, the Foxes have about the same expected goals total per match as Tottenham in a simple shots-based model. But they also take about a quarter fewer shots per match than Tottenham. And, as you’d expect, their expected goals per shot are about a third higher (and also the highest in the league).
Are these differences enough to generate more points for Leicester? The gaps in quantity and quality aren’t as stark as in my example, and a previous look at this question offered results that were only suggestive. Moreover, as I’ve said before, the Foxes’ lead in the Premier League also owes something to luck. But perhaps being the main exponents of Quality in a league obsessed with Quantity has given them one more little edge.