This week’s article in The Star looked at “across-season parity”. In a nutshell, across-season parity refers to all teams spending equal time being playoff teams, and winning an equal number of Stanley Cups. A lack of across-season parity occurs when the same teams make the playoffs each year, and when the same team wins the Stanley Cup over and over again (a dynasty).

How can this be measured?

As it turns out, there is a well-established way of measuring across-season parity that sports economists use. It is based on the measure economists use for the competitiveness of an industry, called the Hefindahl-Hirschman Index (HHI). The HHI works as follows. It takes each firm’s market share, squares it, and then adds them all up. If all firms have equal market share, then the HHI yields a measure of 1/N, where N is the number of firms. As the industry becomes concentrated in a single firm, the measure moves closer to 1. Thus, the HHI represents more concentration the higher the measure.

Consider the following examples. Suppose there is an industry with 6 firms. Equal distribution of market share would mean each firm serves 1/6 of the market. Squaring 1/6 yields 1/36, and adding up over the six firms gives 6/36 = 1/6. Now suppose that a single firm makes all the sales (even though the other 5 were still in business somehow). In that case, the share of the dominant firm is 1, while the share of the other 5 is 0. Squaring each yields 1 for the dominant firm and 0 for the rest, and the sum is 1. The following table depicts these scenarios along with two others.

Firm 1 | Firm 2 | Firm 3 | Firm 4 | Firm 5 | Firm 6 | HHI | |

Market Share – case 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 |

Market Share – case 2 | 2/3 | 1/12 | 1/12 | 1/12 | 1/24 | 1/24 | 0.47 |

Market Share – case 3 | 1/4 | 1/4 | 1/4 | 1/12 | 1/24 | 1/24 | 0.20 |

Market Share – case 4 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |

Now let’s consider how this can be applied to across-season parity. First, let’s consider Stanley Cups. Let’s suppose that there were 6 teams in the league, and that the league operated at this number of teams for 24 years. If there were perfect parity, then each team would have won 4 Stanley Cups, or 1/6 of the total possible. This corresponds to the 4^{th} case in the table above, and the HHI of Stanley Cups would be 1/6. If one team won all 24 (nice try, Habs fans, the Canadiens were never that dominant), then we would be in case 1 above. The second case would entail one teams that won 16 Cups, three teams that won 2, and one team that won 1 Cup. The third case describes a scenario in which three teams won 6 Cups each, another team that won 2, and two teams that won 1 Cup.

Things are a little more complicated when it comes to playoff berths. Let’s suppose that 4 teams made the playoffs each year during this time period. In that case, 96 playoff berths would have been allocated over the 24 year period. With perfect parity, each team would have made the playoffs 16 times, which is the same as saying they received 1/6 of the total number of playoff berths. A complete lack of parity, however, would have had the same 4 teams making the playoffs each year, meaning that four teams would have received ¼ of the total number of playoff berths, and two teams would have had zero.

There is one additional complication to deal with. You will note that the measure for perfect parity depends on the number of teams. If there had been 12 teams in my example above, then perfect parity would have had each team winning 2 Cups, or having a 1/12 share, over the 24 years, yielding an HHI of 1/12.

In fact, it’s a little more complicated than that. Suppose there had been 48 teams over that 24 year period. Given that it’s impossible to win ½ of a Stanley Cup, perfect parity would have had 24 teams winning the Stanley Cup once, which yields an HHI of 1/24. This means that when comparing eras with differing numbers of teams and comprising differing numbers of years, adjustments have to be made to account for the fact that perfect parity would correspond to different numbers.

The adjustment I made, in this case, was to take the observed measure of parity (as given by the HHI) and to divide it by the number corresponding to perfect parity for that particular case. So, for the examples above, case 4 corresponds to perfect parity, which is a measure of 1/6. If I were to observe that case, I would give it a score of 1 (1/6 divided by 1/6). The case of the dominant team, case 1, would get a score of 6 (1 divided by 1/6), and the two intermediate cases would get scores of 2.81 and 1.19, respectively.

Finally, the time period needs to be set such that the number of teams in the league is constant, and the time period should be reasonably long. Perfect parity does not mean an equal allocation of Stanley Cups (or playoff berths) if some of the teams were not in existence for some of the time period.

In the NHL, there are only three eras where the number of teams has stayed constant for more than a decade: the Original Six era (25 seasons from 1942-67), the first Post-Expansion Era with 21 teams (12 seasons from 1979-91), and the Modern Era, with 30 teams (13 seasons from 2000 to present). These were the eras used for the analysis. The breakdown for Stanley Cups and playoff berths over these time periods are given in the tables below. First, let us consider the Original Size era.

Original 6 - 25 years | Playoffs | Stanley Cups |

Montreal | 24 | 10 |

Toronto | 21 | 9 |

Detroit | 22 | 5 |

Boston | 14 | 0 |

Chicago | 12 | 1 |

NY Rangers | 7 | 0 |

100 | 25 |

Note that this era comprises 6 teams and 25 years, so it is actually quite close to the example used above, just with one extra year. This means that perfect parity would entail 5 of the teams winning 4 Cups, and one team winning 5. This would yield an HHI of 0.168, so very close to 1/6. The HHI of the actual distribution, however, is 0.3312. Dividing this number by the HHI for the case of perfect parity yields the number 1.971 that was reported in The Star article.

In terms of playoff berths, perfect parity would have 4 teams going to the playoffs 17 times, and 2 teams going 16, for an HHI of 0.1668. The HHI of the observed distribution is 0.189, which when divided by 0.1668, gives us the number of 1.133 that was in The Star.

Next, let us consider the 12 years where there were 21 teams in the NHL.

Post-Expansion (21 teams, 12 years) | Playoffs | Stanley Cups |

Philadelphia | 10 | 0 |

Edmonton | 12 | 5 |

Washington | 9 | 0 |

Winnipeg | 8 | 0 |

Calgary | 12 | 1 |

Montreal | 12 | 1 |

Quebec | 7 | 0 |

Buffalo | 10 | 0 |

NY Islanders | 10 | 4 |

St Louis | 11 | 0 |

Chicago | 12 | 0 |

Boston | 12 | 0 |

Los Angeles | 10 | 0 |

Hartford | 7 | 0 |

Detroit | 6 | 0 |

NY Rangers | 11 | 0 |

Minnesota | 10 | 0 |

Vancouver | 8 | 0 |

New Jersey | 3 | 0 |

Pittsburgh | 5 | 1 |

Toronto | 7 | 0 |

192 | 12 |

In this case, we have more teams than years of observations, so perfect parity in terms of Stanley Cups would mean that 12 different teams won the Cup once. This would yield an HHI of 1/12. The distribution above, however, yields an HHI of 0.306, which gives us an adjusted score of 3.667.

For playoff berths, perfect parity would have 18 teams making the playoffs 9 times, and 3 teams making it 10 times. This yields an HHI of 0.048. The actual distribution of playoff berths, however, gives us an HHI of 0.051, which leads to an adjusted score of 1.074.

Finally, let us look at the 13 seasons in which there have been the current 30 teams.

Modern Era (30 teams, 13 years) | Playoffs | Stanley Cups |

Philadelphia | 11 | 0 |

Edmonton | 3 | 0 |

Washington | 8 | 0 |

Winnipeg | 1 | 0 |

Calgary | 5 | 0 |

Montreal | 9 | 0 |

Colorado | 8 | 1 |

Buffalo | 5 | 0 |

NY Islanders | 5 | 0 |

St Louis | 8 | 0 |

Chicago | 7 | 2 |

Boston | 10 | 1 |

Los Angeles | 7 | 2 |

Carolina | 4 | 1 |

Detroit | 13 | 2 |

NY Rangers | 8 | 0 |

Minnesota | 5 | 0 |

Vancouver | 10 | 0 |

New Jersey | 10 | 1 |

Pittsburgh | 9 | 1 |

Toronto | 5 | 0 |

San Jose | 12 | 0 |

Ottawa | 10 | 0 |

Tampa Bay | 6 | 1 |

Anaheim | 8 | 1 |

Dallas | 7 | 0 |

Arizona | 4 | 0 |

Nashville | 7 | 0 |

Columbus | 2 | 0 |

Florida | 1 | 0 |

208 | 13 |

Again, there are more teams than years, so perfect parity would have 13 teams winning the Stanley Cup exactly once. This would yield an HHI of 1/13. The actual distribution of Stanley Cups gives us an HHI of 0.112. Dividing by 1/13 gives us the score of 1.462 reported in The Star.

Finally, looking at playoff berths, if they were distributed equally, then 28 teams would have made the playoffs 7 times, while 2 teams would have made it 6 times. This corresponds to an HHI of 0.033. The actual distribution of playoff berths, however, has an HHI of 0.040, giving us the adjusted score of 1.191.

Given these measures of across-season parity, it is certainly worthwhile to ask what is going on, especially with playoff berths. As mentioned in The Star article, one possible explanation is that the salary cap era has meant that teams with incompetent front offices cannot buy their way out of their mistakes, so it takes bad teams longer to recover from mistakes. The fact that some teams have been slow to adopt analytics could also make for a widening gap between good and bad teams.

While I (or any economist, really) would generally not be opposed to punishing incompetence, the fact of the matter is that the GMs whose bad decisions condemn teams to mediocrity are rarely the ones in charge of the rebuild. The people who really suffer when rebuilds become harder are the fans. As a Leafs fan, I speak from experience on this.