To estimate a player’s weekly performance, we assume that each offensive player *i* has an innate talent for achieving each relevant fantasy statistic independent of his opponent, and we measure this as $${u}_{i}^{s}$$ for each statistic *s*, such as passing/rushing touchdowns, passing/rushing yards, fumbles, field goals, etc. summarizes the offensive and defensive statistics used in the prediction.

Table 1 Offensive and defensive statistics.

Similarly, we assume that every defensive team *j* has an innate ability to defend against these statistics, denoted as $${w}_{j}^{Ds}$$ where *Ds* stands for defense against statistic *s*. The projection for the level of each statistic achieved by a player in a given week is the product of his innate ability and his opponent’s ability to defend. We use the product of the two competing statistics as a simple model to capture the first order interaction between the two matched up agents. For instance, if Peyton Manning, the QB for the Indianapolis Colts, plays against the New York Jets in week 4, we would project him to pass for $${u}_{\text{PeytonManning}}^{\text{passYards}}\times {w}_{\text{Jets}}^{\text{DpassYards}}$$ yards in week 4. Then, *f*(PeytonManning, 4) is calculated by adding the projected points for each statistic.

Following the above idea, we use a two-step procedure to derive *f*̅(*i*, *t*). In the first step, we estimate the offensive and defensive strength of each team for each fantasy statistic, i.e. $${u}_{k}^{s}$$ and $${w}_{k}^{Ds}$$ for each team *k*. To do this, we use a weighted average of historical performance with recent performance weighted more heavily as our data, and a least squares estimation as the methodology. In the second step, we compute the innate abilities of each offensive player for each fantasy statistic, i.e. $${u}_{k}^{s}$$ for each player *k*.

For each statistic, we know the aggregate achievement in each historical season. For example, the Minnesota Vikings gave up 986 rushing yards in 2006. We also know which opponents each team faced in each season. To estimate the innate offensive and defensive abilities of each team *k*, we would therefore like to solve the following systems in the least squares sense:

$$\forall \text{Team\hspace{0.17em}}k\text{\hspace{0.17em}and\hspace{0.17em}statistic\hspace{0.17em}}s\mathrm{:}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\displaystyle \sum _{\tau \in \mathscr{H}\mathcal{S}}}{\displaystyle \sum _{j\in {\text{Opp}}_{\tau}\mathrm{(}k\mathrm{)}}}{2}^{\tau}\cdot {u}_{k}^{s}\cdot {w}_{j}^{Ds}=H{T}_{k}^{s}\mathrm{,}\text{\hspace{1em}(2)}$$(2)

$$\forall \text{Team\hspace{0.17em}}k\text{\hspace{0.17em}and\hspace{0.17em}statistic\hspace{0.17em}}s\mathrm{:}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\displaystyle \sum _{\tau \in \mathscr{H}\mathcal{S}}}{\displaystyle \sum _{j\in {\text{Opp}}_{\tau}\mathrm{(}k\mathrm{)}}}{2}^{\tau}\cdot {w}_{k}^{Ds}\cdot {u}_{j}^{s}=H{T}_{k}^{Ds}\mathrm{,}\text{\hspace{1em}(3)}$$(3)

where Opp_{τ
} (*k*) is the set of opponents that team *k* faced in a historical season *τ*, and ℋS is the index set of the historical seasons, where, for example, *τ*=0 indexes the season 1 year ago from the current season, *τ*=−1 for the season 2 years ago, etc. The quantity $$H{T}_{k}^{s}$$ is the weighted historical total points that team *k* achieved for statistic *s*, calculated as $$H{T}_{k}^{s}={\displaystyle {\sum}_{\tau \in \mathscr{H}\mathcal{S}}}{2}^{\tau}H{T}_{k\mathrm{,}\tau}^{s},$$ where $$H{T}_{k\mathrm{,}\tau}^{s}$$ is the total points that team *k* achieved for statistic *s* in the specific historical season *τ*. In a similarly way, we calculate $$H{T}_{k}^{Ds}$$ for the weighted historical total points that team *k* achieved for defensive statistic *Ds*.

Formally, we want to solve the following optimization problem:

$$\begin{array}{l}\underset{u,w\ge 0}{\mathrm{min}}{\displaystyle \sum _{k\mathrm{,}s}}[{\mathrm{(}{\displaystyle \sum _{\begin{array}{c}\tau \in \mathscr{H}\mathcal{S}\\ j\in {\text{Opp}}_{\tau}\mathrm{(}k\mathrm{)}\end{array}}}{2}^{\tau}\cdot {u}_{k}^{s}\cdot {w}_{j}^{Ds}-H{T}_{k}^{s}\mathrm{)}}^{2}+\\ \text{}{\mathrm{(}{\displaystyle \sum _{\begin{array}{c}\tau \in \mathscr{H}\mathcal{S}\\ j\in {\text{Opp}}_{\tau}\mathrm{(}k\mathrm{)}\end{array}}}{2}^{\tau}\cdot {w}_{k}^{Ds}\cdot {u}_{j}^{s}-H{T}_{k}^{Ds}\mathrm{)}}^{2}]\end{array}\text{\hspace{1em}(4)}$$(4)

This is a nonlinear least squares problem. However, we notice that the system of (2) and (3) is composed of bilinear terms in *u* and *w*. Therefore, for a fixed defensive multipliers $${w}_{j}^{Ds},$$ (2) becomes a linear system in the offensive multipliers $${u}_{k}^{s}$$ and vice versa. Using this observation, we implement an alternating minimization algorithm for solving the nonlinear least squares problem (4), as outlined in .

Algorithm 2 Alternating minimization algorithm.

The principle of alternating optimization has a long history, dating back to the early work of alternating projection methods for solving linear systems by Agmon (1954) and Motzkin and Schoenberg (1954). Our proposed method is also closely related to the hill-climbing method for solving bilinear optimization problems, which is used as the first-step in a cutting problem algorithm proposed by Konno (1976). Note that with one set of variables fixed (e.g. $${w}_{j}^{Ds}$$), the nonlinear least squares problem (4) becomes convex quadratic over the other set of variables (e.g. $${u}_{k}^{s}$$).

The following argument shows that the proposed algorithm converges. Denote the objective function in (4) as *h*(*u*, *w*). Then, the iterations indexed by *t* of the algorithm generate a sequence of decreasing objective values as *h*(*u*^{t}, *w*^{t})≥*h*(*u*^{t}, *w*^{t+1})≥*h*(*u*^{t+1}, *w*^{t+1}), due to the alternating minimization procedure. Since *h*(*u*, *w*)≥0, this decreasing sequence *h*(*u*^{t}, *w*^{t}) is bounded from below. Therefore, by the monotone convergence theorem (see e.g. Rudin (1976)), {*h*(*u*^{t}, *w*^{t})} converges.

Of course, due to the nonconvex nature of the above optimization problem, we cannot guarantee global optimality of the convergent solution. However, for our data set, convergence is typically achieved within two or three iterations of the above algorithm, and *R*^{2} values for the goodness-of-fit are very high around 0.999 across all estimation entries. Therefore, we believe the above simple algorithm is sufficient.

Once we have obtained offensive and defensive multipliers for each team, we must estimate the innate ability of offensive players whom we could potentially draft. Again, for each historical season, we have aggregate data for every fantasy statistic as well as knowledge of what opposing teams each player faced. The multipliers for player *i*’s statistic *s* are then given by:

$${u}_{i}^{s}=\frac{H{T}_{i}^{s}}{{\displaystyle {\sum}_{\tau \in \mathscr{H}\mathcal{S}}{\displaystyle {\sum}_{j\in {\text{Opp}}_{\tau}\mathrm{(}i\mathrm{)}}{w}_{j}^{Ds}}}}\mathrm{,}$$

where $$H{T}_{i}^{s}$$ is the weighted historical total points of statistic *s* obtained by player *i*, Opp_{τ} (*i*) is the set of defensive teams that player *i* played against in the historical season *τ*, and $${w}_{j}^{Ds}$$ is the defensive team *j*’s multiplier for defending against statistic *s*, computed from the first step.

Finally, the estimate of how much total fantasy points player *i* will achieve in week *t* is then

$$\overline{f}\mathrm{(}i\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}={\displaystyle \sum _{s}}{u}_{i}^{s}\cdot {w}_{\text{Opp}\mathrm{(}i\mathrm{,}t\mathrm{)}}^{Ds}\mathrm{,}$$

where Opp(*i*, *t*) is the opponent defensive team that player *i* meets in week *t* of the current season.

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