An Analytical Approach for Fantasy Football Draft and Lineup Management

J. Quant. Anal. Sports 2016; 12(1): 17–30

Adrian Becker and Xu Andy Sun* An analytical approach for fantasy football draft and lineup management

DOI 10.1515/jqas-2013-0009 playing fantasy sports online.1 This number has risen significantly from a 2006 survey of 19.4 million online Abstract: In this paper, we consider fantasy football, an players in North America.2 Its annual economic impact increasingly-popular online game based on the actual, across the sports industry in 2006 is estimated to be $3–4 on-the-field performances of players in the National billion.3 About 85 percent of all fantasy sports participants Football League. It is estimated by the Fantasy Sports play fantasy football, most of whom have their games set Trade Association that in 2011 there were 35 million peo- up in major media websites such as Yahoo!, ESPN, MSN, ple in the US and Canada playing fantasy sports online. and NFL.4 Numerous websites have specialized in report- About 85 percent of all fantasy sports participants play ing NFL games, providing preseason rankings, fantasy fantasy football, most of whom have their games set up point projections, team and player statistics, and expert in major media websites such as Yahoo!, ESPN, MSN, draft opinions. However, despite the vast popularity of and NFL. Numerous websites specialize in reporting NFL the game, the intensive analysis by experts, and various games, providing preseason rankings, fantasy points online tools that offer prediction for the values of players, projections, team and player statistics, and expert draft to the best of our knowledge, there is no method that opinions. However, despite the vast popularity of the provides a comprehensive strategy for the entire fantasy game, the intensive analysis by experts, and various football season. Thus, winning a league is, by and large, online tools that offer prediction for the values of play- still more of an art than a science. ers, to the best of our knowledge, there is no method that We set out to develop such an approach that predicts provides a comprehensive optimization strategy for the team and player performance based on the rich histori- entire Fantasy Football season. We set out to develop cal data, and builds a mixed integer programming (MIP) such a methodology that predicts team and player per- model using such predictions for the draft selection as formance based on the rich historical data, and builds well as weekly lineup management, incorporating the a mixed-integer optimization model using such predic- entire objective of winning a fantasy football season. Due tions for the draft selection as well as weekly line-up to the special structure of our model, the MIP formula- management that incorporates the entire objective of tion can be solved very efficiently, which is crucial for an winning a fantasy football season. Numerical tests of our on-line environment as the fantasy football draft process. model show promising performance. We train our model using the data of 2004–2006 seasons Keywords: fantasy sports; mixed integer optimization; and simulate the 2007 season and the 2008 season. The performance prediction; sports draft. result is encouraging and shows an edge of our method over the conventional strategy.

1 Introduction 1 See Fantasy Sports Trade Association’s official website http://www. In this paper, we consider fantasy football, which, as one fsta.org/. of the fantasy sports, has become increasingly popular. It 2 See report “Fantasy Sports Conference Demographic Survey Shows is estimated by the Fantasy Sports Trade Association that Continued Growth” at http://www.prweb.com/releases/2007/08/ in 2011 there were 35 million people in the US and Canada prweb543818.htm. 3 See report “The fantasy football phenomenon” at http://www. theacorn.com/news/2006-08-03/Sports/076.html. 4 See websites at http://football.fantasysports.yahoo.com/, http:// *Corresponding author: Xu Andy Sun, Georgia Institute of games.espn.go.com/frontpage/football, http://msn.foxsports.com/ Technology – Industrial and Systems Engineering, 755 Ferst Drive, fantasy/football/commissioner/, http://www.nfl.com/fantasyfoot- Atlanta, GA 30332, USA, e-mail: [email protected] ball, and report “CNN Money: Fantasy football...real money” at Adrian Becker: Dynamic Ideas LLC, 465 Waverley Oaks Road, http://money.cnn.com/2006/08/11/news/companies/fantasyfoot- Suite 315, Waltham, MA 02452, USA ball/. 18 A. Becker and X.A. Sun: An analytical approach for fantasy football draft and lineup management

Two core methodologies are presented in this paper: presented in this paper. In the following, we review pre- 1. A holistic optimization model which manages a team vious work in both real-world sports drafting and fantasy through draft construction and weekly management. sports drafting and provide detailed comparison with 2. The analysis of a player’s historical statistical perfor- our proposal. mance on a weekly basis in the context of the player’s The paper by Summers, Swartz, and Lockhart (2007) opponents; and the ability to make predictions on this considers the problem of optimal drafting in hockey analysis. pools, which is similar to the drafting process in fantasy football. The authors take a statistical approach and esti- While we apply these methodologies to fantasy football, mate, at each stage of the drafting, the probability that a they have potential use to general managers in the real lineup drafted by a player beats one of other lineups. The world outside the realm of fantasy sports. Often general optimal drafting is to choose an available hockey player managers with a positional need will need to evaluate that maximizes this probability. another team’s reserve players in the off season for a Fry, Lundberg, and Ohlmann (2007) propose a sto- potential trade. Since these reserve players’ experience chastic dynamic programming (DP) model for the player at the professional level is typically limited, it is crucial selection draft of a single real-world NFL franchise, that it be evaluated in the context of their opponents. where the best choice of drafting at each round is deter- Furthermore, like fantasy managers, real world general mined by the DP recursion that maximizes the sum of the managers must build and manage a team to win weekly value added by the drafted player and the total expected matchups against known opponents and advance value added to the team in the future rounds. To produce through the playoffs to be successful. We also acknowl- a computationally tractable model, some simplifying edge that the real world general managers have to con- assumptions are introduced to remove stochasticity from sider several factors that are not present in the fantasy the model (mainly the uncertainty in opponent teams’ games, such as salary cap, multi-year contracts, player behavior) and reduce the size of the state space. The characters, team chemistry, etc; however, we believe resulting deterministic DP can be efficiently solved as understanding the trade-offs in the “stylized” fantasy linear programs. sports settings could potentially factor into the real- Gibson, Ohlmann, and Fry (2010) extend the above world decision-making process. work of Fry et al. (2007) to a more general situation where The paper is organized as follows. Section 2 reviews the decision maker (DM) executes a sequence of resource and analyzes some related work in sports drafting and allocation decisions under the uncertainty of resource prediction. Section 3 introduces background knowledge availability due to actions of competitors. The paper intro- of fantasy football and the dynamics of a fantasy football duces a new type of stochastic knapsack problem with season. Section 4 discusses in detail our integer opti- sequential competition and proposes a stochastic ruler mization model for draft selection. Section 5 proposes approach and agent-based modeling framework. The a new estimation methodology for NFL player perfor- numerical test compares favorably with the deterministic mance prediction, which plays an essential role in the DP approach proposed in Fry et al. (2007). draft selection model. Section 6 discusses available data Among these three papers, the work of Summers et al. and simulation procedure, presents the calibration and (2007) concerns hockey drafting in a fantasy environ- model evaluation results, and provides detailed analy- ment, which is also potentially applicable to other fantasy sis on several aspects of the proposed model. Section 7 sports drafting. However, the model and methodology concludes the paper with a discussion on possible exten- proposed in Summers et al. (2007) mainly use statisti- sions of our model. cal analysis tools, which is fundamentally different from our approach. Gibson et al. (2010) extend the work in Fry et al. (2007) to general situations of sequential resource allocation problems. The stochastic optimization frame- 2 Literature review work and agent-based simulation approach are also qualitatively different from our proposal. Comparing to Fry Fantasy sports drafting and lineup management is a et al. (2007), our proposed model uses a forward-looking relatively new area of sports analytics. The related area approach, which avoids the computational difficulty of of real-world sports drafting is also relatively unex- the DP model; the proposed model has a mixed-integer plored. We are not aware of any existing work that pro- optimization formulation that incorporates a comprehen- poses similar methods for fantasy sports drafting as sive objective of winning the entire season; and our model A. Becker and X.A. Sun: An analytical approach for fantasy football draft and lineup management 19 also considers the uncertainty in opponent owners’ draft- 3 The dynamics of fantasy football ing behavior and model it through robust optimization. In addition to the optimization model for drafting, we Fantasy football is an online game where 10–20 individu- propose a prediction methodology that estimates player als (called “owners”) create and manage teams composed and team performance using historical data augmented of real-life NFL players. Each week, owners are paired by expert opinions. There is a rather extensive literature against each other and score points based on the actual on sports game outcome prediction. For example, linear performance of players on their teams. The owner with regression Markov chain models proposed by Kvam and the highest total point for a week records a win. A fantasy Sokol (2006) and Brown and Sokol (2010) have proved to football season is typically divided into three phases: be successful for NCAA basketball prediction. The task for draft, weekly play, and playoffs. There are many varia- our model is more fine-grained in that it needs to predict tions on the specifics of a fantasy football league, such fantasy points of each player and team in each game as as the number of owners participating in the league, the well as the points needed for a team to win a game against maximum size of the team roster, the number of slots for a specific opponent. The distinctive feature of our model is starting positions, how many points are rewarded for an that it introduces the notion of innate abilities of a player NFL player’s specific real-world accomplishments, the or a team in various statistics, and makes prediction based length and dynamics of the playoff phase, and so on. on them. To facilitate exposition, we develop and evaluate our The artificial neural network model proposed in David methodology using the default rules of the Yahoo public et al. (2011) has a component that uses statistical differ- fantasy league.5 The proposed methodology is generally entials to compare teams, such as the offensive passing applicable to other variations of these rules. Throughout yards gained by a team versus the defensive passing yard the paper, we assume the perspective of a team owner, allowed by an opposing team. This is similar in spirit to and call it the DM, while other owners are called the our notion of opponent specific comparison. However, opponents. both the prediction methodology and the way the model is constructed are qualitatively different. The work by Berri and Simmons (2011) investigates the relationship 3.1 Draft phase between the draft position of a quaterback (QB) and his subsequent performance in the NFL. Their study focuses The draft phase occurs before the NFL season officially on predicting the overall value of a QB, independent of starts. Owners are randomly assigned a pick order and the opponent faced in each game, whereas our method then, according to this order, select real-world NFL players offers prediction of players performance at all positions, to be placed on their respective fantasy teams. Each NFL in the context of competing against the opponent teams. player can only be drafted on at most one owner’s team, Massey and Thaler (2013) finds similar conclusions as so if a player is drafted, it precludes all other owners from in Berri and Simmons (2011) that top draft picks in the including that player on their teams. The draft continues real-world NFL drafting are significantly overvalued in according to the pick order until each owner has filled his a manner that is inconsistent with the notion of rational or her roster. As a side note, the draft typically follows expectation and efficient markets. Again, their work a snaking order. For example, in a league of 20 owners, focuses on estimating the market values or surplus values if the DM is assigned pick number 19, the DM makes the of players, rather than game-by-game performance. 19th pick, and the next owner makes the 20th and the 21st The doctoral thesis of Young (2010) develops a hybrid- picks, then it wraps back and the DM makes the 22nd pick, intelligent decision support system for determining the then the 59th and the 62nd picks of the draft and so on. expected contribution value of a potential NFL candi- date. The model is based on machine learning and statistical methodologies. Works by Harville (1980) and Reid 3.2 Weekly play phase (2003) on predicting team scores and game margins have a similar principle to our work in that they account for the The regular NFL season consists of 17 weeks. Each NFL intricacies of a particular matchup including the offen- team plays 16 games with one per week and also has sive and defensive characteristics of opposing teams. Our work can be seen as an extension of this principle to the prediction of the individual player’s statistics rather than 5 For details, see http://help.yahoo.com/l/us/yahoo/sports/fanta- the overall team score. sysports/football/rules/. 20 A. Becker and X.A. Sun: An analytical approach for fantasy football draft and lineup management one bye week, i.e. the week they do not play. Typically value of players for a fantasy football season. However, the first 15 weeks are devoted to the weekly play phase. we do not know any methodologies that plan a compre- For each week in this phase, owners are matched head hensive strategy for the entire fantasy football season. to head. Each owner must select players from his or her Our model for draft selection as well as weekly and playoff roster to start that week. For each position, there is a limit management offers such a methodology. to the number of players that can start. For the analy- In particular, we believe that draft selections should sis, it is assumed that each owner can start with up to 1 fully incorporate the dynamics of the weekly play and quarterback, 2 running backs, 3 wide receivers, 1 tight playoff phases. Moreover, during the draft phase, owners end, 1 kicker, and 1 team defense. Owners then score typically have a very limited time frame of one to three points for that week according to the actual performance minutes in which they must make their draft picks. So of their starting lineup. For example, according to the any methodology developed should incorporate a com- Yahoo league’s rule, an owner would score 6 points for putationally tractable model which can be solved quickly. each touchdown and 1 point for each 25 yards that the The proposed integer optimization model is quickly solv- owner’s starting quarterback achieves in his game that able to accomplish this task. As a preface, we introduce week. For each head to head pairing, the owner with the the parameters and decision variables used in the integer highest total points records a win. It is important to note program. that players on NFL teams which have a bye week score no points and that any particular NFL team may be facing a strong or weak opponent in any given week. Therefore, an 4.1 Parameters and decision variables owner will likely want to adjust his or her starting lineup from week to week. 1. Parameters: N: The set of NFL players and defensive teams. M: The set of positions M = {QB, RB, WR, TE, K, Def}. 3.3 Playoff phase T: The set of weeks in the NFL regular and playoff seasons. Pos(i): The position of player i, e.g. Pos(PeytonManning) The playoff phase typically consists of the last 2 weeks = QB. of the regular season. The four owners with the highest PosLimit(j): The upper bound on the number of starting winning records during the weekly play phase enter the players for position j ∈ M during the weekly play and playoff phase. The playoff phase proceeds in a similar playoff phases, e.g. PosLimit(QB) = 1. manner as the weekly phase: in the first week, the 1st and nk: The overall pick number of the DM’s k-th draft pick. 4th as well as the 2nd and 3rd ranked owners are matched DMPlayer(k): The set of players that the DM has drafted head to head. The two winners are matched head to head by her k-th pick. in the second week to determine the ultimate winner. In OppPlayer(k): The set of players that the opponents have the event of a tie, the owner who has scored the most drafted by the DM’s k-th pick. total points throughout the season is declared the winner. Rk(i): Anticipated ranking of unselected player i at the Again, bye weeks and NFL team matchups influence the DM’s k-th draft pick. starting lineup for owners in the playoff phase. Owners f(i, t): An estimate of the number of fantasy points player i ranked 5th–8th also compete in a consolation playoff. will score in week t of the NFL regular season. Public fantasy football league challenges often charge β(t): The predicted amount of fantasy points the DM an entry fee and offer cash prizes to winners, and private needs so to be reasonably confident to win in week t ∈ T leagues are often operated by groups of individuals who against the DM’s matchup opponent. place entry fees in a “pot” which is paid out to the top two γj: The number of players the DM must draft at position j. owners in the end of the season. 2. Decision variables: The optimization model has two sets of binary decision variables for picking players in the draft phase and starting 4 An integer optimization model drafted players in each week, respectively. In particular, for draft selection –– yi ∈ {0, 1}: yi = 1 if the DM picks player i in the draft phase, and yi = 0 otherwise. tt As we have mentioned, there are many tools available on –– xxii∈={0, 1} : 1 if the DM starts player i in week t, t the internet that offer a prediction methodology for the and xi = 0 otherwise. A. Becker and X.A. Sun: An analytical approach for fantasy football draft and lineup management 21

–– zt ∈ {0, 1}: zt = 1 if the total estimated fantasy points be viewed as the uncertainty set in the robust optimization of the DM’s line-up in week t is greater than β(t) (i.e. framework. In particular, suppose that it is currently time fi(, tx)(t > β t )), and z = 0 otherwise. for the DM to make her k-th pick in the draft. Several players ∑i∈N i t will have already been drafted by owners, imposing the

There are roughly 590 offensive players and 32 defensive constraints yj = 0 for players already picked by opponents teams in the NFL, which determines the size of N. In a and yj = 1 for players already picked by the DM. Using the league of 20 owners, if the DM is assigned pick number 19, draft order, we can create a new ranking for the remain- the overall pick number of her first draft pick n1 = 19, her ing players, denoted as Rk. After the DM picks her k-th second draft pick n2 = 22, her third draft pick n3 = 59 and so player, nk + 1 − nk − 1 players will be selected by opponents on, following a snaking draft order as mentioned earlier. before the DM’s next pick. If we assume that these picks

DMPlayer(k) and OppPlayer(k) are updated after each draft will come from the top α · (nk + 1 – nk − 1) ranked players in pick. Using various online expert articles and preseason Rk for some parameter α ≥ 1, then for the DM’s next pick, predictions, we create an initial player ranking R0(i) for she can pick at most one player from the top α · (nk + 1 − nk) each player i. In particular, R0(i) is calculated by ordering ranked players in the current ranking Rk, and for the DM’s the average ranking of available expert data sources. If a next two picks, she can pick at most two players from the particular source does not rank a particular player, that top α · (nk + 2 − nk) ranked players in the current ranking player’s rank is taken to be one greater than the number of Rk, and so on for all her remaining picks. Note that this players ranked by the source. After each draft pick k, the uncertainty set imposes constraints on the DM’s drafting, selected players are eliminated from the set Rk, and the not on the opponent owners’ drafting. As an example, the ranking of the remaining players are adjusted and form uncertainty set for the DM’s (k + 1)-th pick does not impose

Rk + 1. More will be discussed in the following subsection a rigid rule that the opponent owners must draft players on modeling the opponent owners’ behavior. Quantities ranked at the top α · (nk + 1 − nk − 1), but rather it prohibits f(i, t) and β(t) are derived from an estimation methodology the DM from drafting these top α · (nk + 1 − nk − 1) players at proposed in Section 5. her (k + 1)-th round, regardless if these players have been drafted by the opponents. The opponent owners still draft according to the probability distribution mentioned in the 4.2 Modeling the drafting strategy of the previous paragraph. opponent owners

Requisite in developing an optimization model for the 4.3 The draft selection model and algorithm draft selection is anticipating how opponent owners will draft players. In this paper, we propose a simple model for At each round of the DM’s draft pick, the DM makes the opponents’ drafting behavior using the media prediction draft decision by solving the following optimization and opinions. The media publishes numerous expert arti- problem that incorporates the objective of maximizing cles and websites during the NFL preseason that indicate the total number of winning games in the entire season a measure of players’ value and a suggested draft order.6 as well as the total points scored by her team, subject to Using these suggested draft orders we can assign each the constraints describing the fantasy football dynam- player i an initial anticipated draft ranking R0(i). This ics, opponents’ drafting behavior, and logic relationship ranking indicates the expected overall pick number for between drafting decisions. In particular, we propose the player i. In other words, when the opponent owners pick following integer optimization model, denoted as (Draftk), their draft, they pick players according to a probability solved at the DM’s k-th draft pick: distribution whose expectation is equal to R0(i) for each 17 15 17 t player i. In terms of model development, we feel that it is (Draftki) max λλ01⋅⋅∑∑((fi, tx))+⋅∑∑zztt+⋅λ2 reasonable to assume that opponent owners will approxi- ti=∈11N tt==16 mately adhere to this probability distribution for drafting. (1a) To account for the uncertainty in the opponents’ s.t. yk≤−kk, ∀≥k + 1, drafting behavior, we propose a simple model, which can ∑ i (1b) ∈≤N α⋅− iR:(kkin)( nk ) 6 See for example http://sports.yahoo.com/fantasy, http://www. rotosource.com, http://games.espn.go.com/frontpage/football, and ≥∀∈M ∑ yjijγ , , (1c) http://fantasyfootball.com. ii∈=N :Pos() j 22 A. Becker and X.A. Sun: An analytical approach for fantasy football draft and lineup management

t ≤∀∈∈M T ∑ xji PosLimit(), jt, (1d) –– Constraints (1g)–(1h) specify the existing drafting ii∈=N :Pos() j results of which players have already been taken by

t the opponents and by the DM’s team. xyii≤∀, it∈∈NT, , (1e)

fi(, tx) ⋅ t It is important to note that due to the special structure of ∑i i ztt ≤∀, ∈T , (1f) this IP, we can relax the binary constraints on the varia- β()t t t bles xi and assume without loss of generality that xi are continuous between 0 and 1. This leads to a mixed-integer yi =∀1, ik∈DMPlayer(), (1g) program (MIP). Since there are in total approximately 620 offensive players and defensive teams in the NFL and 17 yi =∀0, ik∈OppPlayer( ), (1h) weeks in a season, this MIP formulation has about 640 t t integer variables (without relaxation on xi , there would yii, xz, t ∈∀{0, 1}, it∈∈NT, . (1i) be more than 11,000 binary variables), 10,550 continuous In the following, we explain in details each component of variables, and 10,700 linear constraints. Also notice that, this integer optimization model. as the draft progresses, the number of fixed yj increases, –– The objective function aims to win as many head-to- therefore the number of unknown variables decreases. As head matchups as possible during the weekly play shown in our computational experiments, this MIP can be

and playoff phases [the terms associated with λ1 and solved very fast within a minute on a normal laptop, and

λ2 in (1a)], as well as to maximize the total number of accelerates as the draft progresses. fantasy points scored throughout the season (the term Using the above MIP model, the draft selection

associated with λ0), where zt’s are indicator variables algorithm works as follows. At the DM’s k-th draft pick,

modeled in constraint (1f). The weights λ0, λ1, λ2 will the DM solves the above MIP problem (Draftk). Denote k be obtained from the calibration phase discussed in the optimal solution yi as yi for each i ∈ N. Define k Section 6. Briefly speaking, the available evaluation Pki=={:iy 1, ik∉DMPlayer()} as the set of players that data is divided into a training period and a valida- (Draftk) recommends that the DM should select as its k-th

tion period. The training period is used to search the draft pick, (k + 1)-th draft pick, …, nD-th draft pick, where

parameter space and select the parameter values. The nD denotes the total number of draft picks allowed for each final parameter choice is then used in the validation owner. Of course, the DM only executes the selection for period. the current (k-th) draft pick (the other picks are modeled –– Constraint (1b) restricts that, up to the k-th̅ pick, the to help evaluate the impact of the choice at the current

DM can only pick no more than k − ̅k players from draft pick). Specifically, the DM selects the player i ∈ Pk

the top α · (nk − ̅ nk) players ranked in the list Rk. This with the highest ranking Rk(i) as her k-th draft pick. The essentially defines an uncertainty set for the oppo- draft then moves on to the DM’s (k + 1)-th pick and repeats nents’ possible draft selections between the DM’s k-th the same procedure. The draft selection algorithm is sum- and the k-th̅ picks, and makes the DM’s future drafts marized in Algorithm 1. robust to any variation of the opponents’ picks within Note that the f(i, t) is an internal metric that the esti-

this uncertainty set. The parameter α ≥ 1 can be used mation algorithm generates, while Rk is a public signal; to control the conservativeness of the DM’s drafting there is incidental correlation between the two but it is decision, with larger α implying a more risk-averse not direct. The above optimization model uses f(i, t) in attitude on the DM’s side. the objective. Once the optimal yk is found, the players are

–– Constraint (1c) requires that the DM drafts a certain selected based on Rk. The reason for such a procedure is

number of players γj in position j. For example, the DM may want to draft at least five wide receivers to prepare for bye weeks and possible injuries, although only Algorithm 1: Draft selection algorithm. three wide receivers can be started at any given week. 1: for k = 1, 2, ..., nD do –– Constraint (1d) enforces the upper bound on the num- 2: Solve the draft model (Draftk). Denote the optimal solution by ber of players of a given position j that the DM’s team k yi for all i ∈ N can start in any week t. 3: Draft the newly selected player ik with the highest rank, that –– Constraint (1e) imposes the logic relation that the DM is, ik = arg maxi ∈ Pk Rk(i). can only start a player in week t who has been drafted 4: Update DMPlayer(k + 1) = DMPlayer(k) ∪ {ik}. 5: end for on her roster. A. Becker and X.A. Sun: An analytical approach for fantasy football draft and lineup management 23 the following. The presumption of the model is that oppo- is to win weekly matchups, it is important to provide a nent owners are making their decision based on the public methodology that estimates fantasy points of players and

Rk, and this is the behavior that the optimization model is teams scored on a weekly basis. robust to, by constraints (1b). Note that the DM only makes Since individual players and defensive teams all have one pick each round. Picking based on Rk ensures that, relative strengths and weaknesses, and any particular if the assumption on the uncertainty sets holds, the ulti- player or defensive team will face a different opposing mate team the DM ends up with will be no worse than the team from week to week, we propose an index f(̅i, t) to anticipated team calculated in the k-th round optimiza- capture the notion of innate ability of a player or defensive tion. However, picking based on f(i, t) would not ensure team, and use this index to allocate experts’ projections of this. As an example, assume that the player with the fantasy points for the entire season to each week. Eventu- highest f(i, t) has a poor ranking in Rk and the player best ally, the weekly fantasy points f(i, t) is given as ranked in R has the second highest f(i, t). The k-th round k fi(, t ) draft optimization (Draft ) anticipates that the DM can get f (,it )(=⋅gi), k fi(, t) ∑t∈T both players by picking the player with high ranking in Rk now and the other player with high f(i, t) in a later round. where g(i) is the expert prediction of player i’s total However, if the DM picks based on f(i, t), the player with fantasy points over an entire season. The index f̅(i, t) can high ranking in R will certainly be drafted by opponent k also be viewed as the prediction of “when” the player i owners by the time the DM picks again. will score. In the following, we discuss in details the algorithm for estimating the index f̅(i, t) and the procedure to obtain β(t). 4.4 Weekly play and playoff phases: greedy algorithm 5.1 An algorithm for estimating f̅ (i, t) During the weekly play and playoff phases, we use a simple greedy algorithm to select the starting lineups. To estimate a player’s weekly performance, we assume In particular, since we know which opponents the DM’s that each offensive player i has an innate talent for players and defensive teams face during that week, we achieving each relevant fantasy statistic independent of select the starting lineup from the DM’s roster with the his opponent, and we measure this as us for each sta- highest f(i, t) by position. Throughout the season, we keep i tistic s, such as passing/rushing touchdowns, passing/ f(i, t) updated using the new data available up to week t. rushing yards, fumbles, field goals, etc. Table 1 summa- rizes the offensive and defensive statistics used in the prediction. 5 Estimating f(i, t) and β(t) Similarly, we assume that every defensive team j has an innate ability to defend against these statistics, Ds In this section, we propose a prediction method to esti- denoted as wj where Ds stands for defense against mate the weekly fantasy points f(i, t) of a player or a defen- statistic s. The projection for the level of each statistic sive team i in week t, and fantasy points β(t) needed by the achieved by a player in a given week is the product of DM’s team to reasonably ensure a victory in week t. his innate ability and his opponent’s ability to defend. Various websites publish expert opinions on players’ We use the product of the two competing statistics as and teams’ performance. Such expert predictions incorpo- a simple model to capture the first order interaction rate valuable information about rookie players and roster between the two matched up agents. For instance, if changes, which is impossible to anticipate by historical data. However, a drawback of the expert prediction is that Table 1: Offensive and defensive statistics. they are usually given in the form of total season points of a player or a team.7 Since the objective of fantasy football QB Passing/rushing yards, passing/rushing touchdowns interceptions, fumbles, 2-point conversions RB/WR/TE Receiving/rushing yards, receiving/rushing touchdowns fumbles, 2-point conversions 7 Some websites provide weekly performance predictions at the start Kicker XPM, FGM by distance of a season, such as Yahoo! and www.accuscore.com. These data can Def team Points allowed, sacks, turnovers, safeties, blocked be used in place of f (̅i, t). However, they may not be easy to access kicks and process. Furthermore, their accuracy may not be vetted. 24 A. Becker and X.A. Sun: An analytical approach for fantasy football draft and lineup management

Peyton Manning, the QB for the Indianapolis Colts, plays Algorithm 2: Alternating minimization algorithm. against the New York Jets in week 4, we would project passYards DpassYards 1: Initialization : t = 0 and set ()w Ds =1 for each defensive team j him to pass for uwPeytonManning × Jets yards in week 4. j 0 and statistic Ds Then, f(PeytonManning, 4) is calculated by adding the 2: repeat projected points for each statistic. s 3: Solve the linear least-squares problem (4) over uk with fixed Ds Following the above idea, we use a two-step pro- ()wjt, using the Singular Value Decomposition (SVD). s cedure to derive f(̅i, t). In the first step, we estimate the Denote the solution as ()ukt+1. Ds offensive and defensive strength of each team for each 4: Solve the linear least-squares problem (4) over wj with s Ds fixed s using SVD. Denote the solution as Ds fantasy statistic, i.e. uk and wk for each team k. To do ()ukt+1, ()wjt+1. this, we use a weighted average of historical performance 5: t ← t + 1 6: until convergence criterion is met. 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 s offensive player for each fantasy statistic, i.e. uk for each This is a nonlinear least squares problem. However, player k. we notice that the system of (2) and (3) is composed of For each statistic, we know the aggregate achievement bilinear terms in u and w. Therefore, for a fixed defensive Ds in each historical season. For example, the Minnesota multipliers wj , (2) becomes a linear system in the offen- s Vikings gave up 986 rushing yards in 2006. We also know sive multipliers uk and vice versa. Using this observation, which opponents each team faced in each season. To esti- we implement an alternating minimization algorithm for mate the innate offensive and defensive abilities of each solving the nonlinear least squares problem (4), as out- team k, we would therefore like to solve the following lined in Algorithm 2. systems in the least squares sense: The principle of alternating optimization has a long history, dating back to the early work of alternating pro- τ sDss ∀⋅Team ks and statistic : ∑∑2,uwkj⋅=HTk jection methods for solving linear systems by Agmon τ∈∈HS jkOpp(τ ) (1954) and Motzkin and Schoenberg (1954). Our proposed (2) method is also closely related to the hill-climbing method for solving bilinear optimization problems, which is used τ Ds sDs ∀⋅Team ks and statistic : ∑∑2,wukj⋅=HTk as the first-step in a cutting problem algorithm proposed τ∈∈HS jkOpp(τ ) by Konno (1976). Note that with one set of variables Ds (3) fixed (e.g. wj ), the nonlinear least squares problem (4) becomes convex quadratic over the other set of variables s where Oppτ(k) is the set of opponents that team k faced (e.g. uk ). in a historical season τ, and HS is the index set of the The following argument shows that the proposed historical seasons, where, for example, τ = 0 indexes the algorithm converges. Denote the objective function in (4) season 1 year ago from the current season, τ = −1 for the as h(u, w). Then, the iterations indexed by t of the algo- s season 2 years ago, etc. The quantity HTk is the weighted rithm generate a sequence of decreasing objective values historical total points that team k achieved for statistic s, as h(ut, wt) ≥ h(ut, wt + 1) ≥ h(ut + 1, wt + 1), due to the alter- ss= τ s ≥ calculated as HTHkk∑ τ∈HS 2,T ,τ where HTk,τ is the nating minimization procedure. Since h(u, w) 0, this total points that team k achieved for statistic s in the spe- decreasing sequence h(ut, wt) is bounded from below. cific historical season τ. In a similarly way, we calculate Therefore, by the monotone convergence theorem (see e.g. Ds t t HTk for the weighted historical total points that team k Rudin (1976)), {h(u , w )} converges. achieved for defensive statistic Ds. Of course, due to the nonconvex nature of the above Formally, we want to solve the following optimization optimization problem, we cannot guarantee global opti- problem: mality of the convergent solution. However, for our data set, convergence is typically achieved within two or three 2 2 τ sDss iterations of the above algorithm, and R values for the min2∑∑ ⋅⋅uwkj−+HTk uw,0≥ goodness-of-fit are very high around 0.999 across all esti- ks, τ∈HS jk∈Opp(τ ) mation entries. Therefore, we believe the above simple 2 τ Ds sDs  algorithm is sufficient. ∑ 2 ⋅⋅wukj− HTk  (4) τ∈HS  Once we have obtained offensive and defensive mul- jk∈Opp(τ )  tipliers for each team, we must estimate the innate ability A. Becker and X.A. Sun: An analytical approach for fantasy football draft and lineup management 25 of offensive players whom we could potentially draft. 6 Model calibration and evaluation Again, for each historical season, we have aggregate data for every fantasy statistic as well as knowledge of what To evaluate the MIP draft model and the estimation meth- opposing teams each player faced. The multipliers for odology, we conduct extensive numerical simulation player i’s statistic s are then given by: using historical data. In the following, we discuss the data sources, the simulation engine that we created to simulate HT s us = i , i Ds fantasy football seasons, and calibration techniques to wj ∑∑τ∈∈HS jiOpp(τ ) deal with limited data for rookie players. Then, we present simulation results for 2007 and 2008 seasons. The perfor- s where HTi is the weighted historical total points of sta- mance of our model is very promising for both seasons. tistic s obtained by player i, Oppτ(i) is the set of defensive teams that player i played against in the historical season Ds τ, and wj is the defensive team j’s multiplier for defend- 6.1 Data ing against statistic s, computed from the first step. Finally, the estimate of how much total fantasy points There are numerous internet websites that provide exten- player i will achieve in week t is then sive data on career statistics of nearly every player in professional football history, game-by-game statistics f (,it ),=⋅uwsDs ∑ i Opp( it,) of each player going back to early or middle 1990’s, pre- s season ranking of each position, and many rankings for where Opp(i, t) is the opponent defensive team that player fantasy football. For the purpose of predicting players i meets in week t of the current season. and defense team performance, we obtained data corresponding to the overall performance of individual players and NFL teams for the 2004–2007 NFL seasons, as well 5.2 Estimating β(t) as performance for the 2008 season from Yahoo!8 and NFL.9 We also obtained preseason fantasy draft rankings The quantity β(t) represents the number of points that are for each position from expert articles. Furthermore, we needed by the DM’s team to reasonably assure a victory obtained summary reports of actual owner draft behav- against its opponent owner in week t. At the DM’s pick ior for several public fantasy contests for the 2007 and in each round k, β(t) is re-estimated in the following way. 2008 seasons. For simulation purposes, we obtained more We distinguish between the weekly play phase (Week 1 detailed game-by-game box scores for each player of the to Week 15) and the playoff season (Weeks 16, 17). For 2007 and 2008 season. each week t in the weekly play phase, the DM knows which opponent owner she will face before she solves the (Draftk) problem. She also has the current ranking of 6.2 The simulation engine players Rk. With this knowledge, we simulate the remaining draft from the current round by assigning players to To calibrate the model parameters and assess perfor- each team according to the rank order Rk. In this process, mance, we develop a simulation engine for the fantasy we skip over a pick for a particular opponent if one of football problem. This engine simulates the draft, weekly the following holds: 1) that pick of position j would play, and playoff phase on data available for the current put the opponent’s roster over PosLimit(j); 2) that pick or a historical season. We incorporate three aspects of would cause the opponent to reach PosLimit(j) and that variability: player’s bye week is shared with assigned picks of that (a) The DM’s draft order, position, leaving the opponent without enough starters. (b) Weekly head-to-head match-up schedule during the Then, we calculate the sum of the f(i, t) values over the weekly play phase, starting lineup of the week t opponent’s team with the (c) Opposing owners’ draft picks. highest ranking by Rk. For the playoff phase, because we do not know in advance which opponent owner the DM’s The first two aspects are known at the start of the draft team will play, to be robust, we calculate the projected phase, while the third is realized as the draft progresses. fantasy points for each opposing owner in Weeks 16 and 17, then take the maximum point projection for any pos- 8 Data extracted from http://sports.yahoo.com/nfl/stats. sible opponent. 9 Data extracted from http://www.nfl.com/stats/player. 26 A. Becker and X.A. Sun: An analytical approach for fantasy football draft and lineup management

For each simulation trial, we assign the DM a uniform parameters: α = 1, γj = PosLimit(j) + 1, λ0 = 1, λ1 = 100, and random order number between 1 and 10 to simulate a λ2 = 150. In Section 6.5.1, we will study the performance of league of 10 owners. We then select one match-up sched- the model with an alternative choice of γj. In general, γj’s ule uniformly from all possible regular season phase are a parameter setting that can be calibrated according to schedules with the following characteristics: the DM’s preference for buffering injury risk and accumu- 1. Each owner plays all other owners at least once, lating tradeable assets. 2. No owner plays another more than twice.

The draft then proceeds as follows: 6.4 Model evaluation (1) When it is the DM’s turn to pick, use the Draft Selec- tion Algorithm outlined in the Section 4.3. For this 6.4.1 2007 Simulation results phase, we calculate f(i, t) based on historical data available at the beginning of the simulation season. Simulating 300 trials with the final model construction, (2) When it is an opponent’s turn to draft, examine each we obtained the following results shown in Figure 1. Recall

remaining player i in order of descending rank in Rk: that the top two owners are typically financially com- –– If opponent already has enough players with the pensated with the first place owner receiving the largest same position as i to fill a starting lineup as well reward. Thus, for 2007 our approach won its fantasy as a backup player for that position to cover bye football league 16.7% of the time, coming in at least the week, opponent rejects player i. second best at nearly 27.3% of the time. The average rank –– Otherwise, accept or reject player i based on the achieved was 4.60 whereas a uniform approach, using the empirical probability that player i was selected same drafting methodology as our baseline opponents with a similar pick number in mock draft data. would be expected to yield an average rank of 5.5, achieving each rank 10% of the time. Therefore, this study finds

We also note that the γj values used in the DM’s model do that the probability of winning money using the IP draft- not apply to the opposing teams in step (2). ing approach is statistically significantly higher than the For each week in the weekly play phase, the DM starts baseline approach at the 99.75% confidence level. the player on her roster with the highest f(i, t). Opponents As an example, Table 2 shows the draft result of the start the players playing this week whom the opponents DM’s team in a trial run and its performance in the 2007 drafted first for each position. Scores for the starting season. In each pair of scores, the first one is the DM’s lineups are then given according to the actual historical team’s fantasy points, and the second one is the opposing performance of that week during the simulation season. owner’s points. Again, this shows a specific example of

At the end of the weekly play phase, the playoff phase using γj = PosLimit(j) + 1 in the model. Other γj values can schedule is constructed and simulated using the same be chosen according to the DM’s preferences. dynamics. To assess the value added by considering each Each simulation trial was performed on a 1.5GHZ player’s ability to score points in a given week (projecting Pentium M 2GB RAM machine running AMPL-CPLEX 11.2, f(i, t)) rather than simply their overall point scoring ability the first and second round of the draft consumed the (projecting f(i)), we also simulated 300 trials in which most time, taking on average 42 seconds with the longest time observed being 75 seconds. The 13 subsequent draft 2007 Season performance rounds took under 30 seconds a piece on average. Data 18% processing, weekly play, and playoff phases took in aggre- 16% 14% gate 10 seconds on average. 12% 10% 8% ical probability 6.3 Model calibration 6% 4% Empir 2% For model calibration, we derived f(i, t) from the 2004 0% to 2006 season data, the anticipated draft rankings 12345678910 Ranking Rk(i) using expert websites with 2007 draft ranking and then simulated performance on the 2007 season. This Figure 1: The empirical distribution of the final ranks achieved by calibration allowed us to select the following values for our model in 2007 season. A. Becker and X.A. Sun: An analytical approach for fantasy football draft and lineup management 27

Table 2: An example of the DM’s draft and its performance in the 2007 season (score format: DM – opponent) for weeks 1–17.

1 Peyton Manning 2 Chad Johnson 3 Marshawn Lynch 4 Donald Driver 5 Laveranues Coles 6 Warrick Dunn 7 Denver Broncos 8 Alge Crumpler 9 Brandon Jackson 10 Nate Kaeding 11 Jeff Wilkins 13 Derrick Mason 13 New York Giants 14 Heath Miller 15 Brady Quinn

Week 1 2 3 4 5 6 7 8 9 Score 104–84 110–100 107–91 112–45 79–28 32–75 131–64 47–116 87–75 Week 10 11 12 13 14 15 16 17 Score 65–73 60–90 99–84 64–72 66–62 76–40 109–95 102–90

2008 Season performance f(i, t) was simply set to g(i)/16 during non-bye weeks, 16% where g(i) is the third-party expert prediction. Using this 14% uniform method of calculating f(i, t), we placed first or 12% second only 15% of the time, see Figure 2. This study finds 10% that using a weekly allocation increases our probability 8% of winning money by a statistically significant amount at ical probability 6% 4%

the 97.8% confidence level. Empir 2% 0% 12345678910 Ranking 6.4.2 2008 Simulation results Figure 3: The empirical distribution of the final ranks achieved by For the 2008 season, we derived f(i, t) from the 2004–2007 our model in 2008 season. season data, the anticipated draft rankings Rk(i) using expert websites with 2008 draft ranking and then simu- bye-week, is plotted in Figure 4, where f *(i, t) is the actual lated performance on the 2008 season. The results are fantasy points obtained by player i at week t. shown in Figure 3. Here we place first or second 20.7% of On average, a player’s fantasy point score for each the time. The average rank achieved was 5.14, beating the week is overestimated by 4.5 with f(i, t). Recall that each baseline approach by a statistically significant amount at f(i, t) has been reweighed by the expert total season pro- the 99.25% confidence level. jections, so this suggests that experts tend to over predict players’ fantasy scoring abilities. The spread of the empirical distribution indicates that there is room to improve on 6.4.3 Distribution of projection accuracy the prediction method for f(i, t). From Figure 4, we can see that much of the predic- For the 2007 and 2008 seasons, the distribution of pre- tion error is driven by over-optimism in predicting points diction error (f(i, t) − f *(i, t)), excluding each player’s for skill position players. While predicting Kicker and

2007 Season performance using just 3rd party projections 25% Prediction error 7% 20% 6% 5% 15% 4%

10% 3% ical probability ical probability 2% 5% 1% Empir Empir 0%

0% 3 8 –7 –2 13 18 23 28 33 38 –1 7 –1 2 12345678910 –4 2 –3 7 –3 2 –2 7 –2 2 Ranking f (i, t) – f *(i, t)

Figure 2: The empirical distribution of the final ranks achieved by Figure 4: The empirical distribution of prediction error f(i, t) − f*(i, t) our model with just the 3rd party projection. for the 2007 and 2008 seasons. 28 A. Becker and X.A. Sun: An analytical approach for fantasy football draft and lineup management

Table 3: The average actual fantasy points (Avg points), i.e. values of γj’s can be chosen according to the DM’s prefer- * average of f (i, t) for each position, the mean absolute error (MAE) of ences. For example, the following strategy also seems rea- prediction, i.e. average of |f(i, t) − f*(i, t)|, and the sample standard sonable: select γj = PosLimit(j) + 1 for QB, RB, and WR, and deviation (Std Dev) of the absolute error of prediction over the 2007 γ = and 2008 seasons. select j PosLimit(j) for TE, Kicker, and Defense team. Figures 5 and 6 show the performance of the strategy of

QB RB WR TE K DEF choosing γj for the years 2007 and 2008, respectively. For 2007, the new strategy achieves the first or second place Avg points 10.92 7.46 6.56 4.72 7.18 8.00 MAE 8.18 6.77 5.70 4.61 3.64 5.18 with probability 28% and average ranking 4.53; for 2008, Std Dev 5.67 9.32 4.22 3.39 2.66 3.91 the new strategy achieves the first or second place with probability 20.3% and average ranking 5.11, both of which are close to the performance of the original strategy of

Defensive scoring is generally thought of as a more chal- γj = PosLimit(j) + 1 for all position j. lenging endeavor, Table 3 shows that prediction errors for Kicker and Defensive scoring are actually smaller than QB, RB, and WR. This demonstrates that expert bias in 6.5.2 Sensitivity on α wanting to predict the next breakout skill position player may be an overwhelming influence in prediction error. As explained in Section 4.3, the constraint (1b) for each Based on the average actual fantasy points for each posi- draft round ensures the robustness of the DM’s draft tion as shown in the first row of Table 3, we can see that selection against the uncertainty in the opponent owners’ the relative prediction error for QB is in fact lower than RB, drafting. The parameter α controls the degree of con- TE, and WR, and higher than Defensive and Kicker. servativeness of the DM’s assumption on how much the

2007 Season performance 6.5 Further discussions 20% 18% 16% In this subsection, we present further discussions on three 14% relevant questions. The first one is to study the effect of 12% the lower bounds on the number of drafted players for 10% 8% ical probability each position, i.e. γj in constraint (1c). The second one 6% 4% is a sensitivity analysis of the model performance with Empir 2% respect to the parameter α. The third one is to test the 0% hypothetical case where all the owners have access to the 12345678910 Ranking actual performance data of that season, i.e. owners have perfect information. This would give an indication on how Figure 5: The empirical distribution of the final ranks achieved by the performance of the proposed model is affected by the our model in 2007 season using γj = PosLimit(j) + 1 for QB, RB, and errors in the f(i, t) forecast. All the following experiments WR, and γj = PosLimit(j) for TE, K, and Def. have 300 simulation trials.

2008 Season performance 18% 16% 6.5.1 Choice of γj 14% 12% In the proposed model (Draftk), constraints (1c) set the 10% requirement that the DM at least drafts a certain number 8% ical probability of players (or defensive teams) γj for each position j. The 6%

Empir 4% actual values of the parameters in the (Draftk) model, 2% including γj’s, are calibrated using the 2004–2006 season 0% data to achieve an optimized performance. The calibra- 12345678910 Ranking tion selects γj = PosLimit(j) + 1 for all j, which means that, for each position, the model will draft at least one more Figure 6: The empirical distribution of the final ranks achieved by player than the number at that position required to fill the our model in 2008 season using γj = PosLimit(j) + 1 for QB, RB, and starting lineup. We would like to emphasize that other WR, and γj = PosLimit(j) for TE, K, and Def. A. Becker and X.A. Sun: An analytical approach for fantasy football draft and lineup management 29

Table 4: Sensitivity of α. improvement in our model’s draft performance empha- sizing both the ability of our model to improve drafting α Avg Place 1st (%) Place 1st or 2nd (%) Place top half (%) decisions and the challenge that forecast error creates 1.0 4.60 16.7 27.3 64.7 in this problem. 1.1 4.76 15.3 23.0 62.7 1.2 5.06 10.0 19.7 58.3 1.3 5.07 12.0 24.7 55.7 1.4 5.09 12.7 20.3 57.3 7 Conclusion and possible 1.5 5.38 12.0 21.7 52.7 extensions opponents could deviate from the publically available The proposed mixed-integer programming based method ranking Rk at each round k. The calibrated model has for draft selection outperforms the baseline strategy of α = 1, which means the DM’s drafting model will assume using publicly available expert rankings by a statistically the opponents draft according to Rk in expectation. Here significant margin under simulation. Planning for the we provide a sensitive analysis on the performance of the weekly matchups that NFL players will face rather than model for different α’s. their overall fantasy point scoring ability was also found In Table 4, the second to the fifth columns are, respec- to increase the likelihood of winning. We feel that this tively, the average rank achieved by the proposed model method would perform well in fantasy football league for different α’s, the empirical probability of coming in play against real-life opponents by providing a quantita- the 1st place, the empirical probability of placing in the tive edge; however there are still several aspects of the first two places, and the probability of placing in the top model that could be improved through future research. half (top 5 in a league of 10 owners). Again, 300 simula- Future work directions could include the following tion trials are conducted for each value of α. For α ≤ 1.4, extensions: the probability of placing in the top half beats the base- 1. As seen in the Section 6.4.3, it may be possible to line approach (50%) by a statistically significant amount improve upon the prediction methodology used or above the 97.5% confidence level. to improve results buy purchasing expert 3rd party From this figure, we can see that as α increases, the weekly predictions for players’ fantasy points. Group- model behaves more conservatively, which results to a ing players’ forecast points into tiers and assigning all worse average achieved rank, and lower probability of players in the same tier the cluster average may high- winning top places. light the significant differences in forecast points and help improve the performance of the model. 2. Given a methodology for projecting f(i, t), we can 6.5.3 Impact of perfect information observe the distribution of historical prediction error for players in various classes. This empirical distribu- We have also tested the case where the owners have access tion could be used to create an uncertainty set around to perfect information about the players and teams perfor- f(i, t). Stochastic or robust optimization counterparts mance. In this case, the opponents draft according to a to our model could then be evaluated. ranking list using the perfect information of the players’ 3. A significant source of uncertainty in fantasy football performance. In particular, we use the 3rd party ranking is player injuries. If the DM drafts a player that is sub- as the overall ranking of players, and then the relative sequently injured, the DM would receive no points for rankings for each position within the overall ranking list starting that player on a week in which they are sitting are ordered using the actual points. out due to injury. We could incorporate a robust for- Even with perfect information on player perfor- mulation where an adversary is allowed to pick one or mance, a fantasy sports draft still remains a challenging more of our players to injure for the season, ensuring optimization problem due to the positional requirements that we do not rely too heavily on the point contribu- of the roster and the varying week-to-week point pro- tions of a few key players that may be injured. duction and opponent matchups. The simulation results 4. The calculation of β(t) may be improved. In particular, are the following: Average achieved rank is 3.32, 24.7% there is room for a possible model extension by solv- of the time coming in the 1st place, and 15.0% the time ing an auxiliary optimization problem to maximize coming in the 2nd place, therefore winning the league β(t) for the opposing owner’s team in round t over the with 39.7% of the time. This represents a significant same pick uncertainty set described in Sections 4.3, or 30 A. Becker and X.A. Sun: An analytical approach for fantasy football draft and lineup management

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