Given the equivalence of the Markov rating of (12) and the diffusion process of (22), we define a family of rankings that depends on a single parameter and has a natural interpretation as a diffusion process. Suppose there are *n* teams to rank, and define win and loss matrices

$${W}_{ij}=\{\begin{array}{cc}-{w}_{ij}\hfill & \text{if}i\ne j\hfill \\ {L}_{i}\hfill & \text{if}i=j\hfill \end{array},$$(23)

and

$${L}_{ij}=\{\begin{array}{cc}-{l}_{ij}\hfill & \text{if}i\ne j\hfill \\ {W}_{i}\hfill & \text{if}i=j\hfill \end{array},$$(24)

where *w*_{ij} and *l*_{ij} are, respectively, the number of wins and losses for team *i* against team *j*, and *W*_{i} and *L*_{i} are the total number of wins and losses for team *i*. For *p* a parameter with 0 ≤ *p* ≤ 1, define

$${\mathcal{L}}_{p}=W+pL,$$(25)

and take

$$\mathbf{s}=\left[\begin{array}{c}\hfill {W}_{1}-{L}_{1}\hfill \\ \hfill {W}_{2}-{L}_{2}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {W}_{n}-{L}_{n}\hfill \end{array}\right].$$(26)

We begin by trying to define a rating vector **v**_{p} as a solution to

$${\mathcal{L}}_{p}\mathbf{x}=p\mathbf{s}.$$(27)

When *p* = 0, note that we get precisely the Markov equation in (12), or equivalently, the net-flow-zero equation of (22). Requiring that **x** = **v**_{0} be a probability vector, therefore, recovers the Markov rating when *p* = 0.

Also note that for *p* > 0, any rating vector satisfying (27) will be a scalar multiple of a rating vector satisfying *ℒ*_{p}**x** = **s**. From the perspective of the underlying ranking problem, therefore, we can define

$$\mathbf{s}{\prime}_{p}=\{\begin{array}{cc}0\hfill & \text{if}p=0\hfill \\ \mathbf{s}\hfill & \text{if}p>0\hfill \end{array}.$$(28)

and rewrite (27) as

$${\mathcal{L}}_{p}\mathbf{x}=\mathbf{s}{\prime}_{p}.$$(29)

Finally, note that when *p* > 0 the columns of *ℒ*_{p} are still linearly dependent, so a constraint is still required to identify a rating vector. While the particular constraint choosen is irrelevant from the perspective of the resulting ranking, we follow the common convention [and the one used in (5)] that the rankings sum to 0:

$$\left[\begin{array}{c}\frac{\hfill {\mathcal{L}}_{p}\hfill}{}\hfill 1\mathrm{\cdots}1\hfill \end{array}\right]\mathbf{x}=\left[\begin{array}{c}\frac{\hfill \mathbf{s}{\prime}_{p}\hfill}{}\hfill 0\hfill \end{array}\right].$$(30)

Thus, taking **x** = **v**_{p} to be the rating vector solving (30), and letting ${\mathcal{L}}_{p}^{+}$ and **s**_{p} be, respectively, the augmented matrix and right-hand-side in (30), the rating equation can be more succinctly written as

$${\mathcal{L}}_{p}^{+}{\mathbf{v}}_{p}={\mathbf{s}}_{p}.$$(31)

For *p* > 0, the system (31) is still interpreted using the diffusion paradigm of Section 3.2. As before, the (*i*, *j*)-entry of the matrix *ℒ*_{p} represents the flow from *j* to *i* [compare with the left hand side of (22)]. Now, however, the flow is not determined solely by wins for *i* versus *j*. Indeed, in this more generous process even losses to team *j* contribute some flow to team *i* as regulated by the parameter *p*. Thus, increasing *p* from 0 moves the diffusion process away from a pure meritocracy, toward a ranking where teams get some credit for simply playing against other teams, and especially for playing against other good teams (with high rankings). As *p* grows we weaken the importance of the result of the interaction between teams from the perspective of rank flow. To compensate, we introduce a measure of overall team success based on aggregate outcomes and represented by the vector **s**_{p} on the right hand side of (31). Continuing the diffusion analogy, the vector **s**_{p} represents an external infusion of rank (possibly negative) at each vertex *i*.

The particular choice for the infusion of rank here is a win percentage proxy given by wins minus losses, though there is no reason why other choices could not be made. (Massey’s original method, for example, uses cumulative point differential and would be a natural choice.) The rank infusion vector can be considered an external success metric in that it is not network dependent: two teams with the same record will get the same infusion regardless of which teams they played. The network dependent rank-flow update to **s**_{p} is given by the left-hand-side of (31). With this **s**, note that the rankings obtained by solving *ℒ*_{0}**x** = **0** in the Markov method are the same as the rankings from solving *ℒ*_{0}**x** = **s** (though the ratings are different). This is specific, however, to this particular choice of **s** and wouldn’t hold for other infusion vectors like Massey’s point differential. The choice of infusion used here will clarify the connection with Colley’s and Massey’s methods as seen below, while still giving a continuous family of ratings. Finally, we also note that the ranking problem as described here bears a strong resemblance to problems of current flow on electrical networks as described in Doyle and Snell (2000).

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.