# Tournament Design: Designing the length of a playoff series

How long should a playoff series be? This is a question that involves different dimensions ranging from practical/logistics (e.g., traveling, length of the series etc.) to ones related with the league objective (e.g., notion of “fairness”, value of upsets etc.). In this blog post I will make an attempt to show some of the tradeoffs among various series length.  There will be some assumptions made of course with the major ones being: (i) teams’ win probability is not affected from a longer series (e.g., through fatigue etc.), and (ii) there is no home edge (this last assumption affects only the absolute values of the results and not the shape/trend).

The main tradeoff here is how likely it is for an upset to appear in the series. Some fans like the underdog beating the favorite and hence, leagues can embrace this.  However, on the other hand, fairness is important and thus, leagues want also to assure that the better team wins more often than not. So let’s see what we get with different length series.

Let’s first start by assuming that the favorite has a real chance of winning equal to p > 0.5. We want to estimate the probability that the underdog will win the series (as a function of the series length l).  So let’s see how the favorite is going to lose a series of best-of-five (as an illustration):

• Being swept by the underdog, i.e., have 0 wins in the 3 first games
• Lose 3-1, i.e., have exactly 1 win in the 4 first games
• Lose 3-2, i.e., have exactly 2 wins in the five games

Note that the above events are all mutually exclusive.  For example, in order to lose by 3-2 the favorite should have won exactly 2 games in the first 4.  This allows us to calculate the probability of the underdog winning a best-of-l series as:

where U is the random variable representing whether the underdog won the series or not, and f(*) is the binomial probability function. We have used different values of p and l in order to obtain the corresponding curves and explore the tradeoff between upsets and fairness. For the values of l we are using 1-9, even though 9 is not used in any league to the best of our knowledge.  The figure below presents the results.

As we can see for “weak” favorites (i.e., p closer to 0.5) a series (rather than a single game as in NFL) leads to higher probability of an upset.  Of course, as the number of the games in the series increase, then the probability of an upset is smaller and it gets close to the probability of a single game.  However, for stronger favorites (i.e., larger p), a longer series reduces quickly the probability of an upset.  For example, at a best of 7 series (which is the case at NBA and NHL for example), there is really small room for frequent surprises as compared to the NFL (single elimination games in playoffs) for the same type of favorite-underdog matchup.

Of course as aforementioned, there are several assumptions that have been made to obtain these numbers, but it is evident that the design of a tournament has an impact itself on the outcome of a tournament.  The best example is a tournament like the soccer world cup where the short duration and single game eliminations make for some interesting upsets compared say to a year-long championship!

## 4 thoughts on “Tournament Design: Designing the length of a playoff series”

1. […] for the quarter finals.  The probability of seeing this matchup was about 2.2%, but in a short tournament underdogs will stun the favorites more often than in a full-fledged league. Following the games in top-16 we have the following updated ratings.  Notice the emergence of […]

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2. This does not seem correct. I must be misunderstanding something but it seems to me that the Condorcet jury theorem establishes that the probability of an upset decreases monotonically with series length.

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1. That’s a good comment. Even though I am not sure the assumptions and the setting of Condorcet jury theorem directly apply here, I believe that I have a mistake in the binomial equation (the exponents are reversed). I will look into it more carefully and update the post. Thanks for checking it

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2. And you were right! Thanks for pointing it out!

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