## Dominetrics

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In the February 2011 edition of the *Journal of Sports Economics*, Bob Elder of Beloit College and Scott Beaulier of Arizona State University published their article “Employing ‘Dominetrics’ to Impose Greater Discipline on Performance Rankings.” In this paper, Elder and Beaulier generate an alternative type of college basketball ranking by adapting a methodology pioneered by Laurens Cherchye of Leuven University (Belgium) and Frederic Vermeulen of Tilberg University (the Netherlands). Elder and Beaulier argue that the *ordinal* approach used by Cherchye and Vermeulen to rank cyclists from the Tour de France provides an opportunity to improve upon the *cardinal* approach that the NCAA uses to rank college basketball teams through its Ratings Percentage Index (RPI). To make their case, Elder and Beaulier first draw attention to the components of a team’s RPI:

W%_{1} = A Team’s Opponents’ Winning Percentage, sometimes called “strength of schedule’

W%_{adjusted} = ^{A Team’s Own Weighted Wins}/ _{A Team’s Own Weighed Wins + A Team’s Own Weighted Losses}

W%_{3} = A Team’s Opponents’ Opponents’ Winning Percentage, sometimes called “opponents’ strength of schedule”

A team’s Weighted Wins = 0.6 Home Wins + 1.0 Neutral Wins + 1.4 Road Wins, while a team’s Weighted Losses = 1.4 Home Losses + 1.0 Neutral Losses + 0.6 Road Losses. Once these weighted wins and weighted losses are used to compute a team’s own W%_{adjusted}, the NCAA computes the team’s RPI as 0.25W%_{adjusted} + 0.5W%_{1} + 0.25W%_{3}.

Elder and Beaulier emphasize that the array of *cardinal weights* (0.6, 1.0, 1.4, 0.25, and 0.5) impose subjectivity on the RPI and the rankings it generates. For example, the assertion that a road win is ^{1.4}/_{0.6} = ^{8}/_{3} times as important as a home win is a subjective judgment, the claim that a team’s strength of schedule is ^{0.5}/_{0.25} = 2 times as important as a team’s own adjusted winning percentage is a subjective judgment, etc.

Thus, Elder and Beaulier seek to minimize subjectivity’s impact on the rankings by replacing the NCAA’s *cardinal methodology* with an *ordinal methodology*. To employ an *ordinal methodology*, all that Elder and Beaulier require is an importance-ordering, and to obtain this importance-ordering they simply go with what is implied by the RPI’s weighting scheme. From most important to least important, this importance ordering emerges as follows:

W%_{1} = A Team’s Opponents’ Winning Percentage, sometimes called “strength of schedule’

W%_{2} = A Team’s Own Winning Percentage in non-home games

W%_{3} = A Team’s Opponents’ Opponents’ Winning Percentage, sometimes called “opponents’ strength of schedule”

W%_{4} = A Team’s Own Winning Percentage in home games

Equipped with this importance-ordering, the Elder-Beaulier rankings can be understood through an appreciation of four concepts: *Pareto-superiority*, *domination*, a *compensation principle*, and *dominetrics*.

**PARETO-SUPERIORITY**

Team A is *Pareto-superior* to Team B if at least one of Team A’s four winning percentages listed above exceeds the corresponding Team B winning percentage (for example, W%^{A}_{1} > W%^{B}_{1}), and if none of Team A’s four winning percentages is less than the corresponding Team B winning percentage.

Before we can relate the concept of *Pareto-superiority* to the concept of *domination*, we must define the following four sums as functions of the four winning percentages listed above.

Σ _{1} = W%_{1}

Σ _{2} = W%_{1} + W%_{2}

Σ _{3} = W%_{1} + W%_{2} + W%_{3}

Σ _{4} = W%_{1} + W%_{2} + W%_{3} + W%_{4}

**DOMINATION**

Team A *dominates* Team B if Team A’s four sums listed above are each greater than or equal to the corresponding Team B sum:

Σ^{A}_{1} ≥ Σ^{B}_{1} and Σ^{A}_{2} ≥ Σ^{B}_{2} and Σ^{A}_{3} ≥ Σ^{B}_{3} and Σ^{A}_{4} ≥ Σ^{B}_{4}

**COMPENSATION PRINCIPLE**

Due to the Cherchye-Vermeulen *compensation principle*, *domination* is easier to achieve than *Pareto-superiority*. For example, if W%^{A}_{1} > W%^{B}_{1} but W%^{A}_{2} < W%^{B}_{2} , then teams A and B are *Pareto-unrankable*. In contrast, Team A could still *dominate* team B even though W%^{A}_{1} > W%^{B}_{1} but W%^{A}_{2} < W%^{B}_{2} if the amount by which Team A exceeds Team B in the more important category *compensates* for the amount by which Team B exceeds Team A in the less important category, i.e., if W%^{A}_{1} - W%^{B}_{1} ≥ W%^{B}_{2} - W%^{A}_{2}. Indeed, notice that pertinent *domination* criteria embody this *compensation principle*:

Σ^{A}_{2} ≥ Σ^{B}_{2} → W%^{A}_{1} + W%^{A}_{2} ≥ W%^{B}_{1} + W%^{B}_{2} → W%^{A}_{1} - W%^{B}_{1} ≥ W%^{B}_{2} - W%^{A}_{2}

** **

**DOMINETRICS**

For any given Tour de France cyclist, Cherchye and Vermeulen define his Net Dominance Metric as the number of cyclists he dominates minus the number of cyclists who dominate him. Elder and Beaulier have contracted the term Net Dominance Metric into the single word *Dominetric*, and they let any given team’s *Dominetric* equal the number of teams they dominate minus the number of teams that dominate them*. * Finally, it is on the basis of these *Dominetrics* that Elder and Beaulier rank the 344 men's Division 1 NCAA basketball teams.

**DECOMPOSITION OF DOMINETRICS**

Recall that Team A *dominates* Team B if the following four relationships hold true:

Σ^{A}_{1} ≥ Σ^{B}_{1} and Σ^{A}_{2} ≥ Σ^{B}_{2} and Σ^{A}_{3} ≥ Σ^{B}_{3} and Σ^{A}_{4} ≥ Σ^{B}_{4}

Again, the ensuing *Dominetric* for any given team equals the number of teams it dominates by these four criteria minus the number of teams that dominate it by these four criteria. To look for the sources of any team's overall Dominetric, it is interesting to decompose their Dominetric as follows.

For example, suppose that the only basis for determining domination is the Opponents' Winning %. Then Team A would dominate Team B if the following single relationship holds true.

Σ^{A}_{1} ≥ Σ^{B}_{1}

In the second table of rankings that we publish on this site, D_{1} is the Dominetric that emerges for each team if Opponents' Winning % is the only dimension of performance used to determine domination.

Suppose that there are just two dimensions: in order of importance, (1) Opponents' Winning % and (2) Own non-home Winning %. Then Team A would dominate Team B if the following two relationships hold true:

Σ^{A}_{1} ≥ Σ^{B}_{1} and Σ^{A}_{2} ≥ Σ^{B}_{2}

In the second table of rankings, D_{2} is the Dominetric that emerges for each team if these first two ordered criteria are used to determine domination.

Suppose that there are three dimensions: in order of importance, (1) Opponents' Winning %, (2) Own non-home Winning %, and (3) Opponents' Opponents Winning %. Then Team A would dominate Team B if the following three relationships hold true:

Σ^{A}_{1} ≥ Σ^{B}_{1} and Σ^{A}_{2} ≥ Σ^{B}_{2} and Σ^{A}_{3} ≥ Σ^{B}_{3}

In the second table of rankings, D_{3} is the Dominetric that emerges for each team if these first three ordered criteria are used to determine domination.

Finally, recall once more that the overall Dominetric uses four dimensions of performance: (1) Opponents' Winning %, (2) Own non-home Winning %, (3) Opponents' Opponents Winning %, and (4) Own home Winning %. Team A dominates Team B if the following four relationships hold true:

Σ^{A}_{1} ≥ Σ^{B}_{1} and Σ^{A}_{2} ≥ Σ^{B}_{2} and Σ^{A}_{3} ≥ Σ^{B}_{3} and Σ^{A}_{4} ≥ Σ^{B}_{4}

In the second table of rankings, D_{4} is the overall Dominetric that emerges for each team if all four ordered criteria are used to determine domination. This is the same Dominetric that appears in column 3 of our first table of rankings, and it is what determines those rankings.

In sum, the second table shows how this final and overall Dominetric, which is based on all four performance criteria, evolves from the Dominetrics that are based on the three (importance order preserving) subsets of these performance criteria.

**PARETO DOMINETRICS**

Recall once more the four winning percentages used by the NCAA to construct a team's RPI: (1) opponents' winning %, (2) own non-home winning %, (3) opponents' opponents' winning %, and (4) own home winning %. Recall also the earlier observation that Team A is Pareto-superior to Team B (i.e., Team A Pareto-dominates Team B) if Team A's winning percentage exceeds Team B's winning percentage in at least one of these four categories, and Team A's winning percentages in each of the other three categories are not worse than Team B's winning percentages in each of the other three categories.

Using these criteria, a Pareto dominetric can be computed for any given team, like Team A, by taking the quantity of teams that Team A Pareto-dominates minus the quantity of teams that Pareto-dominate Team A. The third table of rankings on this site is determined by such Pareto dominetrics.

Observe that while the Dominetrics rankings in the first two tables on this site remove the subjectivity of the cardinal weights from the RPI, they retain an element of subjectivity by preserving the importance-ordering implied by those cardinal weights. Pareto-dominetrics therefore become interesting to consider because they remove that subjective importance-ordering as well.