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Surviving a National Football League Survivor Poolhttp://orcid.org/0000-0002-5566-5224David Bergman, Jason Imbrogno

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OPERATIONS RESEARCHArticles in Advance, pp. 1–12

http://pubsonline.informs.org/journal/opre/ ISSN 0030-364X (print), ISSN 1526-5463 (online)

Surviving a National Football League Survivor PoolDavid Bergman,a Jason Imbrognob

aDepartment of Operations and Information Management, University of Connecticut, Stamford, Connecticut 06901; bDepartment ofEconomics & Finance, University of North Alabama, Florence, Alabama 35632Contact: david.bergman@uconn.edu, http://orcid.org/0000-0002-5566-5224 (DB); jimbrogno@una.edu (JI)

Received: January 5, 2016Revised: November 11, 2016; January 19, 2017Accepted: March 30, 2017Published Online in Articles in Advance:August 2, 2017

Subject Classifications: programming: integer:applications; games/group decisions: gambling;industries: recreation/sportsArea of Review: Optimization

https://doi.org/10.1287/opre.2017.1633

Copyright: © 2017 INFORMS

Abstract. In this paper, an analytical approach to National Football League (NFL) survivalpools is investigated. This paper introduces into the literature NFL survival pools andpresents optimization models for determining strategies. Computational results indicatethat planning only partway through the season yields the highest survival probabilities,which dominate millions of randomly generated strategies.

Keywords: sports analytics • NFL survival pools • programming • integer • applications • betting pools

1. IntroductionBetting on sports games has grown rapidly in theUnited States and abroad. According to the Ameri-can Gaming Association, sports fans were expected towager over $95 billion on football games in theNationalFootball League (NFL) or at the collegiate level in 2015alone (American Gaming Association 2015). Some esti-mate the size of this market to be even greater—forexample, theNevadaGaming Commission in 2011 esti-mated that nearly $400 billion is wagered on footballannually (Green 2012).

Betting pools encompass a portion of this market andcome in a variety of flavors. Two popular pools includethe National Collegiate Athletic Association (NCAA) bas-ketball tournament and single-week NFL pools. The ever-popular NCAA basketball tournament occurs in earlyspring, when 68 college basketball teams play a single-elimination tournament. In single-week NFL pools,participants are required to select the winner of eachgame played that week (13–16 games). The participantwho selects the most teams correctly wins the pool,with ties paying out equal pots to all winners.This paper addresses another popular version of

sports betting pools, NFL survival pools, which differsubstantially in format from those addressed previ-ously. In an NFL survivor pool, each participant paysan entrance fee and must select a winning team ineach week of the NFL season. Each team in the leaguecan only be chosen one time throughout the season.The participant selects the winning team for that weekat the beginning of each week’s slate of games. Theterm “survivor pool” comes from the fact that a par-ticipant is eliminated as soon as one week’s selection

loses or ties. The goal of the survivor pool is to sur-vive as long as possible into the 17-week season, whichrequires a participant to be correct in his selection of awinning team each week. This leads to a complex opti-mization problem where, given probability estimatesfor the remaining games of the season, a participantmust select a team to win while also considering futurematchups. The complexity is expanded when partici-pants pay for (a limited number of) multiple entries.

Bettor pools have received attention from the aca-demic community because of their relation with othercompetitive real-world optimization problems, in addi-tion to the immediate application to sports betting. Asdescribed by Clair and Letscher (2007), a sports bet-ting pool is an example of a more general competitivesetting where agents must select from a set of choices.Examples given in their paper include airline routesand combinatorial auctions (Clair and Letscher 2007).

More related to the present study is the edge-weightedonline bipartite matching problem (EWBMP) (Khulleret al. 1994, Raghvendra 2016, Bansal et al. 2014,Anthony and Chung 2014), a variant of the sequen-tial stochastic assignment problem (SSAP) (Derman et al.1972; Gershkov and Moldovanu 2010; Nikolaev andJacobson 2010; Khatibi et al. 2015, 2014). The EWBMPis defined on a bipartite graph G (A∪ B,E). The ver-tices in B appear sequentially, along with the weightsof the edges incident to the vertex and its neighborsin A. Upon the arrival of a vertex b ∈ B, a vertex a ∈ Amust be irrevocably assigned to b with the goal of max-imizing (or minimizing) the total weight of all edgesselected. Associating the teams with A and the weekswith B, the decision maker in an NFL survival poolis faced with a similar decision—selecting team a for

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week b has a probability of success, and the goal is tosucceed in the selection for every week without know-ing ahead of time the probabilities in future weeks. Theapplications for these problem classes include selec-tion of passengers for airport screening, kidney assign-ment, and accepting bids in real estate markets (Lee2009). The literature on the EWBMP and the SSAP hasfocused on developing algorithmswith theoretical per-formance guarantees assuming a distribution on thestochastic inputs. This paper focuses on data-drivencomputational models assuming only point estimateson the win probabilities in the future, which moveaccording to external, unpredictable events. Addition-ally, we study a generalization of the EWBMP, wheremultiple edges are to be selected by separate agentseach time a vertex in B arrives.Clair and Letscher (2007) address both the NCAA

tournament and single-week NFL pools. They presentprobabilistic models, establish a closed-form solutionfor expected return in the single-week football pool,and show an approximation to the expected return fora fixed set of picks in the context of the NCAA tourna-ment. The results indicate that when the total numberof entries is small, a single participant should selectteams most favored to win, but when the number ofentries grows, the entrant should pick teams less likelyto win in order to gain a competitive advantage overthe masses. The present paper incorporates a similartype of strategy, where the amount of “forward think-ing” required in making picks and saving teams forlater in the season depends on the number of entries.The popularity of the NCAA tournament has resultedin a stream of publications (Kaplan and Garstka 2001,Breiter and Carlin 1997, Kaplan and Magazine 2003,Metrick 1996, Caudill and Godwin 2002, Kvam andSokol 2006, Stekler and Klein 2012, Niemi et al. 2008).However, the literature on football pools has been lim-ited, despite the growing popularity of single-weekpools andNFL survivor pools (RunYourPool.com 2015,OfficePoolStop.com 2015, Politi 2015).

This paper addresses the following question: Givena set of probability estimates for the remaining gamesof the NFL season, what strategy should a participantemploy to maximize the probability of winning a sur-vivor pool? The standard greedy approach amountsto selecting each week the team that has the highestprobability of winning among those teams that havenot been previously selected. This strategy is oftenemployed because the ultimate goal is to never losein any given week. The problem, as will be shownthrough computational experimentation, is that plan-ning only week to week results in obtaining strategieswith a low probability of lasting the entire season, incomparison with long-term planning strategies.

For the single-entry case, an integer programming(IP) model is formulated for obtaining such long-term

strategies. The solutions obtained are compared withstandard greedy approaches and randomly generatedstrategies. It is shown experimentally that planningonly partway through the season leads to strategiesthat greatly increase the probability of surviving theentire season.

Because of the complex dependencies between theselection decisions made in the multiple-entry case, amore complex model is required to solve the optimiza-tion problem. In this case, based on the principle ofinclusion-exclusion, a nonlinear optimization model isformulated. This novel model is able to capture theintricate dependencies in order to accurately model theprobability of at least one entry surviving the season,or some fixed time horizon. The model is able to iden-tify solutions to the multiple-entry case that lead to farsuperior strategies to standard greedy approaches andmillions of other strategies obtained through a struc-tured random-sampling procedure.

The rest of this paper is organized as follows. Sec-tion 2 describes how the NFL regular season scheduleis organized and the standard rules of NFL survivorpools. The model used to estimate NFL probabilitiesis then discussed in Section 3. The underlying opti-mization problem is formulated in Section 4. Optimiza-tion models for the single-entry and multientry vari-ants are described in Sections 5 and 6, respectively, andan experimental evaluation is provided in Section 7.Finally, future work and a conclusion is provided inSection 8.

2. The NFL Season andSurvivor Pool Details

The NFL season consists of 32 teams playing 16 gameseach. The schedule revolves around the concept ofweeks, where a givenweek of games are played betweenThursday and Monday. Each team plays just one gameper week and receives one “bye week” in which itdoes not play a game. The organization of the seasonby weeks has resulted in the development of survivorpools, where betting does not involve point spreads andstretches over multiple weeks of the season.

A survival pool is an iterative betting pool. It beginswith each participant picking a team in week 1. Shouldthe team chosen win its week 1 game, the participantcontinues to week 2 and is otherwise eliminated. Theprocess continues until the end of the season; havingcorrectly selected a winning team in each of the first wweeks, a participant again must choose a team to winin week w + 1 and is eliminated if that team loses orties. The only restriction is that a participant may notchoose a team he chose earlier in the season. The win-ner of a survival pool is the participant who lasts thegreatest number of weeks. If multiple participants lastthe full season, the winnings are split among them. It is

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possible to win a survival pool, in the most extremeexample, by correctly picking a week 1 team and see-ing all other participants incorrectly choose in the firstweek. Generally speaking, the greater the number ofparticipants, the longer a participant will have to lastin order to win the pool. A participant’s choice for ateam in any given week is made after all of the previ-ous week’s games conclude and before any games inthe next week begin.The need to last many weeks into the season in order

to win the survival pool, combined with the require-ment that any given team be chosen nomore than once,creates an obvious trade-off for the participant in mak-ing his decision each week; if he chooses a team to winin the current week, and it does, he advances to thenext week of the pool at the expense of having lost thatteam as a choice in later weeks. Participants then haveincentive to save good teams until later in the seasoninstead of choosing them early; then again, an early sea-son loss that eliminates the participant from the poolwhen he failed to use the best teams in his selections isan unhelpful result.

Assuming the participant in the survivor pool canreliably estimate the probabilities for any game ofthe season in any given week, this paper investigatesstrategies geared toward maximizing the probabilityof lasting the entire season. An additional complica-tion in choosing whether or not to select a given teamin a given week is that estimated win probabilitiesfrequently change, sometimes substantially, over thecourse of a season.

Each participant is required to pay an entrance fee inorder to participate in the pool. The money collectedis then available to be won, with the payouts typicallyoccurring according to some type of scale decided bythe organizer. The payout scale can be dramatically dif-ferent from survivor pool to survivor pool, but in allcases, lasting longer is certainly better and hence morelucrative. Unlike the study of Clair and Letscher (2007),which focuses on winning an NCAA tournament poolby selectively choosing underdogs that other partici-pants will avoid, this study focuses on the probabilityof surviving the longest in an NFL survivor pool.

A complicating factor is that each participant may beallowed to pay an entrance fee for more than one entry.Each entry plays separately but all are commonly con-trolled by the participant paying the entrance fees. Typ-ically, the number of entries per participant is capped,often to five or so. Expanding the analysis to multi-ple entries was mentioned as a possible generalizationin the conclusion of the paper by Clair and Letscher(2007) and is considered in the present paper.

3. Probability ModelThe academic literature on gambling onNFL outcomesfocuses on the results of bets using previous game out-comes and point spreads. Sauer (1998) provides an

extensive background on the characteristics and anal-ysis of these betting markets. The focus of this paperis not on finding a precise mechanism for estimatingwin probabilities; rather, this paper focuses on mod-eling and identifying strategies for winning NFL sur-vivor pools based on a given set of reliable probabilityestimates. A participant can derive his own win prob-ability estimates via any approach and then adapt thecomputational models presented in the paper to iden-tify winning strategies.

Clair and Letscher (2007) adapt team win proba-bilities based on Massey Ratings (Massey 2004). Thispaper assumes a similar technique for estimating thewin probabilities, based on ESPN NFL Power Rank-ings, a weekly updated ranking of the 32 teams pub-lished by popular sports-broadcast network ESPN.Based on the rankings, a logistic regression model isemployed to find, given the team that is playing athome and the relative ranks of the teams, the probabil-ity that the home team will win. The most importantreason for choosing this mechanism for approximat-ing win probabilities is that the computational mod-els formulated in this paper require probability esti-mates for each remaining game in the season, whichmust be updated as the season progresses. Basing thewin probabilities off another measure, such as pointspreads, requires the current spread of each remainingNFL game in a season, updated weekly. To the best ofour knowledge, such data are not available, at least notdating back to the 2002 NFL season. We also choosethese data to estimate win probabilities because theexpert rankings would account for all off-season ros-ter changes (such as retirements or trades) in week 1and then change weekly to reflect team performance,injuries, suspensions, player acquisitions, and coach-ing changes as the season progressed.

There are a variety of reasons why a participant’sestimate of win probabilities may change, most notablybecause of the performance of a team throughout theseason. Consider the game in the last week of the 2013season between the Houston Texans and TennesseeTitans. The Texans finished the 2012 season with arecord of 12 wins and 4 losses (12–4), the third-bestrecord in the entire NFL, while the Titans went 6–10.Expectations in week 1 of the 2013 season were that theTexans would win that game, but as the season pro-gressed, with the Texans winning 2 of 13 games beforeplaying the Titans, probability estimates for the Texanswinning that game decreased substantially.

Historical ESPN NFL Power Rankings are availableonline (ESPN.com 2015) and can be downloaded forreproducibility. It is important to evaluate the accuracyof the probability model adapted in order to assess thevalidity of the survivor-pool strategies. The ESPNNFLPower Rankings became available in 2002, which wasalso the year that the current set of NFL teams and

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Table 1. Evaluation of Probability Model

Probability Number of Games correctlyrange games predicted Correct

0.5 to <0.6 96 51 53.1%0.6 to <0.7 88 54 61.4%0.7 to <0.8 57 43 75.4%0.8 to <0.9 15 14 93.3%0.9 to <1.0 0 0 n/aTotal 256 162 63.3%

divisional breakdown were established. The estimatesused for year X were based on models using results ofthe previous seasons 2002, . . . ,X−1 because the resultsfrom the present year would not have been available.A logistic regression was run using HomeRank andAwayRank as input variables and HomeTeamWin as theoutput. This yields the estimated probability of thehome team winning.To evaluate the accuracy of the probability estimates,

a similar analysis to that done by Clair and Letscher(2007) is presented for the year 2014, which is basedon data from 2002 to 2013 and presented in Table 1.The first column corresponds to ranges of expectedwinprobabilities. Each game is placed into the bin corre-sponding to the probability that the favored team (theteam with the higher probability of winning) was pre-dicted to win. The second column reports the numberof games in this bin. The third and fourth columnsreport the number and percentage of games withineach bin that are correctly predicted, respectively. Thetable indicates that as the estimated win probability ofthe favored team increases, so too does the percentageof games accurately predicted by the model. Amongall games, 162 (63.3%) were correctly predicted. Thiscompares nearly identically with the predictive powerof the model used by Clair and Letscher (2007) with areported 63.7% accuracy.

4. The Underlying Optimization ProblemLet nt , ne , and nw be the number of teams, entries,and weeks, respectively, in a survivor pool. Let T

1, . . . , nt be the set of teams, let E 1, . . . , ne bethe set of entries that a given participant is controlling,and let W 1, . . . , nw be the set of weeks. For teamt ∈ T, let ow(t) ∈ T be the opponent of t in week w,only defined on those weeks where t is not enjoying abye week. For 1 6 w′ 6 w 6 nw , let aw′ ,w , t be the actualreal-world probability in week w′ that team t will winin week w. The values aw′ ,w , t are unknown and so aparticipant must determine estimates of the probabil-ities, which will be denoted as pw′ ,w , t . For entry e ∈ Eand w ∈ W , let tw

e be the team chosen by entry e inweek w, and let Se[t1

e , t2e , . . . , t

we ] be the event that entry

e survives until the target week w ∈W , given the set ofteams chosen each week. Finally, for a , b ∈ , a 6 b, let[a , b] : a , a + 1, . . . , b − 1, b.

A participant’s goal is to maximize the probabilitythat one of his entries will outlast all other entries. De-pending on ne , the number of consecutive weeks thatan entry has to win in order to outlast all other entriesvaries. The underlying optimization problem faced bythe participant is

max Pr(∨

e∈ESe[t1

e , t2e , . . . , t

we ]

)subject to tw1

e , tw2e , ∀w1 ,w2 ∈ [1, w], w1 , w2.

(1)

The optimization problem (1) seeks to maximize theprobability that at least one of the participant’s entrieswill survive until the target week w, subject to a setof constraints enforcing the condition that each entrycannot choose the same team in different weeks, perthe rules of survivor pools.

There are a variety of complicating factors associatedwith identifying the optimal solution of the underlyingoptimization problem (1), themost prominent of whichare the following.

1. Factor F1: The actual real-world probabilities areincomputable and change throughout the season,whereas the participant must choose one team perweek.

2. Factor F2: The events are highly dependent on oneanother.

Factor F1 makes the optimal solution incomputable.Therefore, one must rely on estimates of the proba-bilities that, as discussed earlier, can be calculated ina variety of ways. Additionally, the participant mustdecide on a team at the start of each week. This placesan onus on him to plan in advance, but, because theprobabilities will change throughout the year, it maybe beneficial not to plan the entire season in a givenweek. This will be incorporated in the computationalmodels.

Factor F2 arises because mutual dependence of theevents exists between the teams chosen by differententries in a given week and from outcomes in oneweek to another. For the single-entry case, the depen-dence only appears from week to week, and the prob-ability estimates that are updated from week to weekallow for modeling the effect of one week’s perfor-mance on the next week’s performance, as reflectedin the relative ranking of the teams. For the multiple-entry case, the calculation of the probabilities is moreinvolved and requires highly nonlinear objective func-tion terms. These appear because the model must takeinto account that a participant can select team t in weekw for one entry and team ow(t) in another entry in thesame week (and this may be optimal).

5. The Single-Entry ModelThe single-entry case is discussed first, where an opti-mization model is formulated that seeks to maximize

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the probability of winning until a target week w. Fort ∈ T and w ∈ [1, w], define binary variable xw , t thatindicates whether team t is selected in week w bythe single entry. Optimization problem (1) can then bemodeled as a (nonlinear) binary IP:

maxxw , t∈0,1

w∏w1

(∑t∈T

pw ,w , t xw , t

)subject to

∑t∈T

xw , t 1, ∀w ∈ [1, w],

w∑w1

xw , t 6 1, ∀ t ∈ T.

(2)

The objective function seeks to maximize the probabil-ity of winning in each week until the target week, sub-ject to constraints enforcing that each team is selectedat most once. For a fixed w ∈ W , the first set of con-straints require that exactly one of the variables xw , tequals 1, resulting in ∑

t∈T pw ,w , t xw , t evaluating to theprobability that the team chosen by the participantwins in that week. Multiplying these terms results inthe probability that the entry survives until w. Opti-mization problem (2) assumes full knowledge of pw ,w , t ;however, these probability estimates are only known tothe participant in the week w. For example, in the firstdecision period, the participant only knows p1,w , t andmust rely on these estimates whenmaking the decisionin week 1, which is based on incomplete informationabout how the teams are projected to perform in futuregames.This can be handled computationally by breaking

the participant’s decisions into stages. The naturalstages to consider are the w weeks, where in eachstage an optimization problem will be solved and theselection of one team to pick for that week is made.To account for the shifting probabilities throughoutthe season, a parameter L ∈ [1, w] will represent thenumber of weeks to look ahead for planning. Namely,L 1 corresponds to considering only the current weekwhen making the decision in each stage, while, at theother end of the spectrum, if L w − w, the entireremaining season-long horizon is taken into accountwhen making the decision in stage w.Themodel is formulated as follows. Suppose the par-

ticipant is making a decision in week w′. The decisionsin all stages w < w′ to select teams tw must be used torestrict the decision inweek w′. For a given L, the objec-tive in stage w′ will be to maximize the probability ofthe entry surviving in the pool until week L minw′+L − 1, w given the decisions made in previous weeks.In the optimization problem, and thereby in the opti-mal solution, a set of teams to select in weeks w′, . . . , Lwill be determined, yet only the decision correspond-ing to the current week w′ will actually be made, withall remaining decisions left for future weeks.

The optimization problem (3) in week w′ with anL-week look ahead can then be formulated as follows,where tw′ is passed as input from previous stagesto subsequent stages, where xw′ ,w , t indicates whetherteam t is chosen in week w′ for week w:

maxxw′ ,w , t∈0,1

L∏ww′

(∑t∈T

pw′ ,w , t xw′ ,w , t

)subject to

∑t∈T

xw′ ,w , t 1, ∀w ∈ [w′, L],

L∑ww′

xw′ ,w , t 6 1, ∀ t ∈ T,

xw′ ,w , t 0,∀w ∈ [w′, L], t ∈ tw : w ∈ [1,w′− 1],

tw′ ∑t∈T

t · xw′ ,w , t .

(3)The objective function maximizes the probability ofselecting a winning team in each of the weeks w ∈[w′, L]. The constraints, in order, enforce that (1) exactlyone team is selected in each week in the planning hori-zon, (2) each team is selected at most once over thecourse of the weeks in the planning horizon, (3) noteam previously selected can be selected in the plan-ning horizon, and (4) the team selected in the currentweek w′ (tw′) is equivalent to the team whose corre-sponding binary variable in the week takes value 1.

This model allows for changing decisions whenmore information is made available. In particular, inweek w′, a plan for the next L weeks is made, butonly one team is actually selected—the team for whichxw′ ,w′ , t 1. For example, supposing L > 1 and w′ < nw ,the team t′ for which xw′ ,w′+1, t′ 1 of the stage w′ opti-mization problem will not necessarily be the team cho-sen in week w′+ 1. The team will certainly be availableand can be chosen, but should probabilities changeand a better decision become available in subsequentweeks, the decision is free to be altered. The modelsimply lays out a plan for selections in the currentand future weeks. The current week selection mustbe made, but if the participant wins that week andsurvives to the subsequent week, a different choicethan was laid out in previous plans can still be made.Indeed, if the win probability estimates change sub-stantially, the planwill bemodified by future optimiza-tion models.

To implement a practical computational solution tosolve optimization problem (3), the problem is lin-earized in order to arrive at a binary linear program-ming model by applying the log function to the objec-tive function. Because the objective function is theproduct of terms, each of which will be greater than0 because some team is chosen for each week in theplanning horizon, and the objective is to maximize thisfunction, the concave log function can be applied to the

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objective function, resulting in the function (OF). Theoptimal value will change, but the optimal solutionswill remain optimal after this transformation becauselog is an increasing, concave function:

L∑ww′

log(∑

t∈Tpw′ ,w , t xw′ ,w , t

)

L∑ww′

(∑t∈T

log(pw′ ,w , t)xw′ ,w , t

).

(OF)Furthermore, because each of the internal sums will

amount to exactly one of the probabilities being cho-sen, the logarithm can be distributed within the sum,resulting in the equivalent objective function assumingthat only those variables corresponding to teams thatdo not have byes are defined (or some very small prob-ability is chosen for teams in weeks in which they donot play). The model proposed for the single entry isthereby given by the parametrized w-stage binary lin-ear integer programs (3) with the modified objectivefunction (OF). Note that the problem reduces to themaximumweight bipartite matching problemwith thetwo shores of the graph corresponding to the teamsand weeks, respectively, and the weight of an edge tothe logarithm of the win probability.

6. The Multiple-Entry ModelMany NFL survivor pools allow participants to controlmultiple entries, typically on the order of 5 or so. Thisrequires a more complex computational model, which,unfortunately, does not admit a simple linearization.The model must take into account that different entriescould choose opponents in a given game. For example,suppose w 2 and that a participant has four entries.An optimal strategy is to choose four different teams inweek 1, one team per entry, with pairs of teams playingone another. Exactly two entries will then proceed toweek 2, and again, choosing opponentswill ensure thatone entry survives.A further complication is that the model requires the

calculation of the union of a set of mutually depen-dent events and so relies on the principal of inclusion-exclusion (PIE), which can be stated as follows. LetA1 , . . . ,An be arbitrary events in a probability space(Ω,F,P). Then,

Pr( n⋃

i1Ai

)

n∑k1

((−1)k−1

∑I∈(1,...,nk )

Pr(AI)),

where AI ⋂

i∈I Ai and(1,...,n

k

)is the family of k ele-

ment subsets of 1, . . . , n. This allows a union of theprobability of events to be recast as a sum of the inter-section of all subsets of those events.The objective function with multiple entries will

still assume an L-stage look ahead in which decisionsin the current week will be transferred as input tothe model in subsequent weeks. The set of events

under consideration in a given week w′ will be Ai :Si[t i

w′ , tiw′+1 , . . . , t

iw′+L−1].

The PIE decomposes the probability calculation intoa sum of joint probabilities for each possible subsetof E, the set of entries. Let E′ be any subset of E ofsize b, where, for the moment and without loss ofgenerality, assume that E′ [1, b]. Then Pr(AE′) cor-responds to the probability that each team in the set⋃b

i1(⋃

w∈[w′ ,w′+L−1]t iw) wins. Formulating this in the

context of an optimization problem requires ensuringthat the probability evaluates to 0 if two entries selectopponents in anyweekwithin the planning horizon, aswell as ensuring that each team is only calculated in theprobability once—if two entries select the same team,then the probability calculation should only considerthat team once.

The expanded variable set for the case of multipleentries is

• xew′ ,w , t ∈ 0, 1: indicates whether entry e plans to

select team t in week w when planning in week w′;• ze

w ∈ T: indicates the team that entry e selects inweek w;

• oe1 , e2w ∈ 0, 1: indicates whether the teams chosen

by entry e1 and e2 in week w are opponents;• ce1 , e2

w ∈ 0, 1: indicates whether the teams chosenby entry e1 and e2 in week w coincide.

Then, with the above integrality constraints, the opti-mization problem can be cast as

maxne∑

k1

(−1)k−1

∑I∈(1,...,nk )

( L∏ww′

∏e1<e2∈I

(1− oe1 , e2w )

)·(∏

e∈I

[ L∏ww′

(1−

∏e′<e , e′∈I

(1− ce′ , ew )

)·(1−

∑t∈T

pw′ ,w , t xew′ ,w , t

)]+

∑t∈T

pw′ ,w , t xew′ ,w , t

)s.t.

∑t∈T

xew′ ,w , t 1, ∀ e ∈E, w ∈ [w′, L],

L∑ww′

xew′ ,w , t 6 1, ∀ e ∈E, t ∈T,

(zew t)↔(xe

w′ ,w , t 1), ∀ e ∈E, t ∈T, w ∈ [w′, L],xe

w′ ,w , t 0,∀ e ∈E, w ∈ [w′, L], t ∈ t e

w : w ∈ [1,w′−1],t e

w′ ∑t∈T

t · xew′ ,w′ , t , ∀ e ∈E, t ∈T,

xe1w′ ,w , t + xe2

w′ ,w , t −16 oe1 , e2w ,

∀ e1 , e2 ∈E, t ∈T, w ∈ [w′, L],oe1 , e2

w 6∑t∈T(xe1

w′ ,w , t + xe2w′ ,w , ow (t)

2),

∀ e1 , e2 ∈E, w ∈ [w′, L],(ze1

w ze2w )↔ ce1 , e2

w ,

∀ e1 , e2 ∈E, e1 < e2 , t ∈T, w ∈ [w′, L].(4)

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Theorem 1. Constraint programming (CP) model (4) ac-curately models a participant’s survivor pool optimizationproblem, seeking to find the selection of teams for ne entriesthat maximizes the probability of at least one entry survivingthrough week L, starting in week w′.

Proof. Consider the set of constraints sequentially. Thefirst two constraints ensure that each entry selects someteam in each week and that an entry cannot pick agiven team inmore than oneweek. The next constraintsenforce that ze

w 1 if and only if xew′ ,w , t 1, and the

following constraints enforce that the teams alreadyselected by an entry before week w′ are not selectedagain within the current planning horizon. The follow-ing constraints select the teams chosen to be passed asinput to subsequent models. The next two constraintslink the o variables with the x variables. The first setof constraints enforces that if two entries e1 and e2pick the same team in a given week w, then the cor-responding oe1 , e2

w variable is equal to 1. The followingconstraints enforce that if oe1 , e2

w 1, then, for at least onepair of opponents, theymust be chosen by the entries e1and e2 in week w. The right-hand side of this constraintcounts the number of times that the equality within theparentheses is satisfied. Finally, the last constraints linkthe z variables with the c variables.Regarding the objective function, the outer sum∑ne

k1and the first inner sum ∑

I∈(1,...,nk ) are an instantiationof the PIE, so it suffices to show that for a fixed I ⊆ E,the term within the summation properly calculates thejoint probability of all of the entries successfully sur-viving up to week L.The term∏L

ww′∏

e1 , e2∈I(1−oe1 , e2w ) evaluates to 0 if and

only if one of the terms oe1 , e2w equals 1. This variable

indicates that entries e1 and e2 have selected opponentsin week w, thereby making the probability vacuousthat both entries will survive the entire span of weeks,as desired. If all entries select different teams in allof the weeks considered, this entire term becomes 1,making the term irrelevant in the final evaluation ofthe objective function.The remaining portion of the objective function com-

putes the product of the probabilities of the teamsselected by any entry. This is accomplished via the twoterms separated by the plus sign. For a fixed entry eand week w, (1 −∏

e′<e , e′∈I(1 − ce′ , ew )) evaluates to 1 if

some entry e′ ∈ I, indexed lower than e (i.e., e′ < e),selects the same team as e in week w and evaluates to0 otherwise. In the former case, the second two lines inthe objective function evaluate to

1 ·(1−

∑t∈T

pw′ ,w , t xew′ ,w , t

)+

∑t∈T

pw′ ,w , t xew′ ,w , t 1.

This is desired because the selected team has alreadybeen selected by some other entry so the probability is

already incorporated in the product. In the latter case,this evaluates to

0 ·(1−

∑t∈T

pw′ ,w , t xew′ ,w , t

)+

∑t∈T

pw′ ,w , t xew′ ,w , t

∑t∈T

pw′ ,w , t xew′ ,w , t ,

which coincides with the probability that the teamselected by entry e for week w will win. The productof these terms is therefore the probability of each teamselected over the planning horizon by some entry in Iwinning, as desired. Q.E.D.

7. Experimental ResultsAll code is written in C++ (gcc version 4.8.2) and runin Ubuntu 14.04.1 LTS on a machine with an Intel®Core™ i7-4770 CPU @ 3.40 GHz processor and 32 GB ofmemory. The solvers used are IBM ILOG CPLEX 12.6for IP models and IBM ILO CPO 12.6 for CP models.

7.1. Single-Entry ResultsTo provide an evaluation of the strategies identified,random strategies are generated for comparison. Eachweek, a team, among the set of teams still available, T′,is selected based on a probability distribution scaledby the win probabilities. In particular, for week w, letpmin be the minimumwin probability of any team in T′,let qt pw ,w , t − pmin, and let Q

∑t∈T′ qt . The probabili-

ties qt were calculated as qt qt/Q. These probabilitiesresulted in strategies that were far worse than thosestrategies generated by redefining T′ T′∩ t | pw ,w , t >0.5 (i.e., using only those available teams that arefavored), and so the latter was used in the experiments.

Additionally, a one-week look-ahead strategy isidentical to the participant picking the team, amongthose still available, with the highest chances of win-ning in each week. Since this is likely a standard strat-egy used by some participants in survivor pools, it isalso included for comparison.

Finally, for the single-entry case, it is possible to iden-tify the Ideal strategy. Consider model (3) with w′ 1, L ∈ 1, . . . , 17 and, instead of p1,w , t , use probabilitiespw ,w , t , which are the win probabilities for the teams atthe start of week w. These probabilities cannot be usedin identifying a strategy because pw ,w , t is not knownto the participant until week w; however, if these prob-abilities were known, the participant could make theoptimal selection. No strategy can have a survival prob-ability surpass the survival probability of Ideal, but thecloser a strategy gets, the better.

First, a comparison of strategies for only the mostrecently completed NFL season (2014) is presented.Figure 1(a) compares the survival probability of theeight-week look-ahead strategy with the one-week

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Figure 1. Comparison with Simulated Strategies for 2014 NFL Season

0.0001

0.001

0.01

0.1

1

1 5 9 13 17

0.01760.01450.01070.0086

0.0020

0.0001

Pro

babi

lity

Week

(a) One million simulated strategies

0.2

0.4

0.6

0.8

1.0

1 5 9 13 17

Sca

led

prob

abili

ty (

over

Idea

l)

Week

(b) One million simulated strategies

Ideal

1Week

8Week

106Sim(Max)106Sim(Avg)106Sim(Min)

Ideal-Scaled1Week-Scaled8Week-Scaled

106Sim(Max)-Scaled

104Sim(Max)-Scaled

102Sim(Max)-Scaled

look-ahead strategy and one million randomly gener-ated strategies. Each line corresponds to a strategy. Thex axis corresponds to the number of weeks into the sea-son, and the y axis shows the probability, in log scale, ofsurviving until that week given the employment of thegiven strategy. The plot shows the survival probabili-ties of the randomly generated strategy with the worst,max, and average survival probability. The single strat-egy that was determined by looking eight weeks inadvance had a higher probability of survival for thefull season than both the 1000000Sim(Max) and greedy(or one-week look-ahead) strategies; it even approachesthe success of Ideal. The one-week look-ahead strategyis competitive, though inferior to 1000000Sim(Max).

Figure 1(b) shows another comparison with ran-domly generated strategies. The plot depicts the scaledsurvival probability, calculated each week as the ratioof the survival probability of the given strategy untilthat week divided by the probability of the Ideal strat-egy in that week. A line is included for the Ideal strat-egy, one-week look-ahead strategy, eight-week look-

Figure 2. Comparison of Increasing Look-Ahead Strategies for 2014 NFL Season

1 5 9 13 17

Week

1 5 9 13 17

Week

0.01

0.10

Pro

babi

lity

Ideal1WeekLA2WeekLA5WeekLA8WeekLA11WeekLA14and17WeekLA

(a) Survival probability

0.4

0.6

0.8

1.0

Pro

babi

lity

(b) (scaled) Survival probability

ahead strategy, and maximum survival probabilitystrategy for an increasing number of randomly gen-erated strategies—from one to one million, in powersof 10. The plot depicts how the survival probabilitythrough any given week increases as more entries jointhe survival pool. It also elucidates how the eight-weeklook-ahead strategy dominates the other strategies onlywhen progressing into the later part of the season,which is necessary as the number of entries in the poolgrows. If a 2014 pool had a relatively small numberof entries, the greedy one-week look-ahead strategymay have performed better because, until week 10, thesurvival probability was higher for the one-week look-ahead strategy. However, in larger survival pools it isoften necessary to survive the entire season, and so alonger-term planning horizon would have been moredesirable.

The reason for choosing L 8 is addressed compu-tationally next. Figure 2(a) depicts the survival prob-abilities for L ∈ 1, 2, 5, 8, 11, 14, 17, and Figure 2(b)depicts the same data but scaled based on Ideal. These

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Figure 3. Comparison of Increasing Look-Ahead Strategies for Multiple Seasons

0.8

1.0

Pro

babi

lity

(a) Early seasons (2003–2008)

0.6

0.8

1.0

Pro

babi

lity

(b) Recent seasons (2009–2014)

1 5 9 13 17

Week

1 5 9 13 17

Week

Ideal 1WeekLA 2WeekLA 5WeekLA 8WeekLA 11WeekLA 14WeekLA 17WeekLA

plots further elucidate the strengths and weakness ofa greedy strategy. When seeking a strategy to surviveseven or fewer weeks, the greedy strategy works well.As the season progresses, this strategy eventually isdominated by other strategies, ending with a final sur-vival probability approximately half that of the eight-week look-ahead strategy. The eight-week look-aheadstrategy became the best strategy to adopt if the par-ticipant anticipates needing to last more than halfwaythrough the season.These results conform in the aggregate, when evalu-

ated over the most recent seasons. Figures 3(a) and 3(b)depict the scaled surviving probability per week, aver-aged over 2003–2008 and 2009–2014, respectively. Inthe more recent NFL seasons, it is apparent that along-term strategy is superior to the greedy one-weeklook-ahead strategy. This can be attributed to the factthat the probability model has more data for train-ing. For any given year, the probability model is cali-brated using only the previous set of years, so as theyears progress, more input data are accumulated. Also,ESPN started publishing their rankings in 2002, andthe ESPN NFL Power Rankings have become moreaccurate in predicting future game outcomes. Consid-ering the 100 highest probability predictions from 2003to 2008, 83 are correctly predicted; from 2008 to 2014,93 are correctly predicted. Because the optimizationmodels for survivor pools seek to select the teamsmostlikely to win, it is in this region that the probabilityestimates should be most accurate.

Figure 3 shows that the greedy one-week look-aheadstrategy performs well, as it does for the 2014 sea-son alone, when the participant is concerned withsurviving only a few weeks. As the season progresses,however, the probability of surviving is only increasedby looking further into the future. Looking the full 17weeks into the future does not fare well though. Look-ing partway through the season yields the superiorsurvival probability. In the more recent seasons, when

the probability model is more accurate, an eight-weeklook ahead yields the best survival probability startingonce the survivor pool winner reaches at least week 13.

7.2. Multiple-Entry ResultsUnlike the single-entry case where each model is sol-ved in a fraction of a second, the time required to solveCP model (4) is significant. For example, solving allstages for the multientry model with ne 2 and L 2takes over 20minutes, and the run times increase expo-nentially as the parameters increase. Therefore, a timelimit of 100 seconds per stage is imposed (1,700 sec-onds total for all stages), and the best solution foundby that time is used for the analysis.

Additionally, since the search for solutions willbe limited by time, it is of interest to investigatewhether adding constraints to (4) can help in find-ing high-quality solutions quickly. Three constraintsinvestigated by the authors were (1) preventing theentries from picking the same team in a given week,(2) restricting the set of entries from picking opponentsin the sameweek, and (3) allowing entries only to selectteams with win probabilities greater than 0.5. Theserestrictions can be added for only the current week orfor the entire planning horizon. Through preliminaryexperimentation, it was determined that limiting theentries to only choose teams favored to win through-out the entire planning horizon and adding constraintsthat restrict any pair of entries from picking opponentsin the current week led to the best results. That secondrestriction only impacts the set of available solutionswhen the probability of winning is exactly 50%, but,even when all win probabilities are not 50%, addingredundant constraints can affect search and solver deci-sions and, ultimately, the best solutions found. Thebest configuration added the following additional con-straints on (4): (A) xw ,w′ , t 0, ∀w ∈ [w′, L] for whichpw ,w′ , t < 0.5, and (B) oe1 , e2

w 0, ∀ e1 , e2 ∈ E, t ∈ T, w ∈[w′, L]. As an example, for the 2014 NFL season with

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Table 2. 2014 NFL Season Survival Probabilities for Five Entries

L 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 0.999 0.985 0.950 0.874 0.780 0.656 0.546 0.429 0.346 0.277 0.196 0.148 0.106 0.076 0.058 0.040 0.0302 0.999 0.985 0.947 0.869 0.775 0.652 0.540 0.427 0.343 0.275 0.194 0.147 0.104 0.076 0.057 0.040 0.0305 0.992 0.974 0.943 0.877 0.802 0.692 0.591 0.493 0.417 0.336 0.241 0.191 0.138 0.105 0.080 0.054 0.0428 0.992 0.967 0.934 0.857 0.788 0.643 0.565 0.482 0.411 0.325 0.247 0.200 0.150 0.116 0.087 0.059 0.04611 0.998 0.973 0.917 0.853 0.784 0.685 0.579 0.483 0.404 0.329 0.238 0.190 0.145 0.110 0.082 0.054 0.04214 0.992 0.942 0.889 0.793 0.710 0.587 0.508 0.424 0.357 0.286 0.217 0.177 0.128 0.098 0.074 0.051 0.04017 0.992 0.939 0.869 0.782 0.707 0.604 0.500 0.411 0.345 0.273 0.206 0.164 0.122 0.095 0.072 0.049 0.038

ne 5 and L 5, adding no additional restrictionsyields a solution with survival probability 0.01902, butwith these imposed restrictions, the solution obtainedhas a survival probability of 0.04304. This configura-tion was therefore fixed for the remainder of the exper-imental results.First, an analysis of how many weeks to look ahead

in the season is provided for the 2014 NFL season andby aggregating all seasons. Table 2 reports the survivalprobabilities in the 2014 NFL season, given L weeklook-ahead strategies. If surviving only a few weeks isthe goal, then a single-week look-ahead strategy workswell, but, as in the case of one entry, if more weeks arenecessary to win the survivor pool, a more long-termstrategy should be adopted.Figure 4 depicts a plot of the probabilities in Table 2,

scaled by the greedy one-week look-ahead strategy (forthe 2014 NFL season) and then the same data averagedover all seasons in the test set, scaling each year bythe greedy one-week look-ahead strategy in that year.These plots more readily display the advantage that aparticipant can get by planning more than one weekin the future, and they show that adopting approxi-mately a half-season look ahead yields strategies withthe highest survival probability. We therefore fix L 8for the remaining experiments.Figure 5 depicts the amount by which, for the 2014

NFL season, the survival probability grows as a par-ticipant adds entries. Figure 5(a) shows the survivalprobability per week for ne 1, 2, . . . , 5, and Figure 5(b)

Figure 4. Comparison of Increasing Look-Ahead Strategies for Five Entries

–10

0

10

20

30

40

50

Sca

led

prob

abili

ty (

over

1Wee

kLA

)

1WeekLA2WeekLA5WeekLA8WeekLA11WeekLA14WeekLA17WeekLA

(a) 2014 NFL season

–10

0

10

20

30

40

50

60

70

Sca

led

prob

abili

ty (

over

1Wee

kLA

)

(b) Average over all seasons

1 5 9 13 17

Week

1 5 9 13 17

Week

shows the same data but as a percent increase over thegreedy one-week look-ahead strategy, indicating thatthe survival probability more than triples when thenumber of entries is increased from 1 to 5, increasingthe season-long survival probability from 1.4% to 4.6%.

A comparison to one million random strategies isprovided in Figure 6. Five million single-entry strate-gies were generated and broken into groups of five,resulting in one million strategies, to simulate one mil-lion competitors. Figure 6 depicts the minimum, aver-age, and maximum survival probabilities over all com-petitors. The survival probabilities per week for thesingle-entry and five-entry participant strategies withL 8 are included, along with the one-entry Ideal strat-egy. The plot indicates that the eight-week look-aheadfive-entry strategy is far better than the best five-entryrandom strategy, and it also shows that a participantshould pay for extra entries if possible, since Ideal(with one entrant) has an even lower survival proba-bility.

A brief discussion of the expected monetary gain isin order. Assuming that from year to year the outcomesof games are independent, with a yearly success prob-ability of 0.04588 with five entries, a bettor reaches aprobability of 0.506 of lasting the entire season at leastonce after 15 seasons. After 30 seasons, this probabil-ity increases to over 0.75. Since the money won in asurvival pool far exceeds the entry fees, winning oncewill result in substantial monetary gain. For compar-ison, even using the best among the million random

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Figure 5. Increase in Survival Probability by Adding Entries

0.01

0.10

1.00

0.04590.04110.03460.0246

0.0145

Pro

babi

lity

(a) Survival probability

0

50

100

150

200

250

Per

cent

incr

ease

ove

r gr

eedy

(b) % increase over greedy strategy

1 5 9 13 17

Week

1 5 9 13 17

Week

2Entrant 4Entrant

1Entrant 3Entrant 5Entrant 1Entrant-%Inc

2Entrant-%Inc

3Entrant-%Inc

4Entrant-%Inc

5Entrant-%Inc

Figure 6. Comparison with One Million Random Strategieswith Five Entries

0.01

0.10

1.00

0.0459

0.02510.01760.01450.0098

0.0034

Pro

babi

lity

8WeekLA-5Entrants8WeekLA-1EntrantIdeal-1EntrantMinAvgMax

1 5 9 13 17

Week

strategies for five entries would take 32 years to reacha 0.5 probability of surviving the entire season at leastonce.

8. ConclusionThis paper investigates computational approaches foridentifying strategies for NFL survivor pools. Theauthors formulate various optimization models andprovide computational results comparing the strate-gies obtained. The experimental analysis determinedthat planning picks eight weeks into the future yieldsthe highest probabilities for long-term survival—andhence victory—in the pool. Planning only partwaythrough a season balances future uncertainty at thetime that the decision maker must act. An estimate onthe probability of a team winning a game 17 weeksinto the future can change dramatically as the seasonprogresses. Nonetheless, taking a myopic approach ofonly looking at the games in the current week doesnot allow for any future planning. In the context of

NFL survivor pools, an eight-week look-ahead pro-vides the right balance. This happens to be halfwaythrough the NFL season; it is left for future work toinvestigate whether in other sports leagues—and morebroadly, for other applications where probability esti-mates change over time—planning halfway throughthe horizon is the appropriate time window.

This paper lays the groundwork for identifyingstrategies for NFL survival pools. Several interestingfollow-up research questions arise, including the ques-tion of computational complexity. Consider the follow-ing decision problem: For a fixed ne and L, does thereexist a selection of teams for which the probability ofsome entry surviving is greater than or equal to K?For ne 1, the problem reduces to a maximum weightbipartite matching problem, but the same reductioncannot be directly applied for general ne . Whether thisproblem is NP-hard or not is left for future work.

Additionally, since the solution times are beyondreasonable limits, investigating dedicated solution ap-proaches to the problem may lead to substantial im-provements. Also, using different objective functions(e.g., maximizing the expected number of weeks) orconsidering variants of the problem for other leaguesover longer time horizons to see whether lookinghalfway through the season is still best is of interest,though survival pool competitions as such seem toonly exist for the NFL. Finally, using the optimizationmodels developed in this paper to other forms of sportsbetting pools, and to other optimization problemswith objectives on probability spaces with dependentevents, may lead to novel optimization frameworks ina variety of contexts.

AcknowledgmentsThe authors thank the reviewers and editors for their invalu-able comments that improved the paper.

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David Bergman is an assistant professor of operations andinformation management at the University of Connecticut.His research interests revolve around the development ofnovel methodology for discrete optimization.

Jason Imbrogno is an assistant professor of economics atthe University of North Alabama. His research interests lie ineducation policy evaluation and sports analytics.

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