DTIC ADA593083: Improving Offensive Performance through Opponent Modeling

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by  Defense Technical Information Center

Improving Offensive Performance through Opponent Modeling
KennardLaviers GitaSukthankar MatthewMolineaux DavidW.Aha
SchoolofEECS SchoolofEECS KnexusResearch NCARAI
U.ofCentralFlorida U.ofCentralFlorida Springfield,VA NavalResearchLab
Orlando,FL Orlando,FL matthew.molineaux@ Washington,DC
[email protected] [email protected] knexusresearch.com [email protected]
Abstract
Althoughintheoryopponentmodelingcanbeusefulinany
adversarialdomain,inpracticeitisbothdifficulttodoaccu-
rately and to use effectively to improve game play. In this
paper,wepresentanapproachforonlineopponentmodeling
andillustratehowitcanbeusedtoimproveoffensiveperfor-
manceintheRush2008footballgame. Infootball,teambe-
haviorshaveanobservablespatio-temporalstructure,defined
bytherelativephysicalpositionsofteammembersovertime;
wedemonstratethatthisstructurecanbeexploitedtorecog-
nizefootballplaysataveryearlystageoftheplayusingasu-
pervisedlearningmethod. Basedontheteams’playhistory,
oursystemevaluatesthecompetitiveadvantageofexecuting
aplayswitchbasedonthepotentialofotherplaystoincrease
theyardagegainedandthesimilarityofthecandidateplays
to the current play. In this paper, we investigate two types
ofplayswitches: 1)wholeteamand2)subgroupswitching.
Bothtypesofplayswitchesimproveoffensiveperformance,
Figure 1: Screenshot of the Rush 2008 football simulator.
butbyonlymodifyingthebehaviorofakeysubgroupofof-
Theoffenseteam(showninred)isusingtheplaysplit8and
fensiveplayers,weimproveontheyardagegained.
being countered by the defense (shown in blue) using a 31
formation(variant1).
Introduction
By accessing the play history of your opponent, it is pos-
sible to glean critical insights about future plays. This was (militaryorathletic), teambehaviorsoftenhaveanobserv-
recently demonstrated at a soccer match by an innovative, ablespatio-temporalstructure,definedbytherelativephys-
well-prepared goalkeeper who used his iPod to review a ical positions of team members. This structure can be ex-
videoplayhistoryoftheplayertakingapenaltykick;iden- ploited to perform behavior recognition on traces of agent
tifying the player’s tendency to kick to the left allowed the activityovertime. Thispaperdescribesamethodforrecog-
goalkeeper to successfully block the shot (Bennett 2009). nizingdefensiveplaysfromspatio-temporaltracesofplayer
Althoughplayhistorycanbeausefulsourceofinformation, movement in the Rush 2008 football game (see Figure 1)
itisdifficulttoutilizeeffectivelyinasituationwithalarge andusingthisinformationtoimproveoffensiveplay.
numberofmulti-agentinteractions.Opponentmodelingcan
TosucceedatAmericanfootball, ateammustbeableto
bedividedintothreecategories: 1)onlinetracking2)online
successfullyexecuteclosely-coordinatedphysicalbehavior.
strategyrecognitionand3)off-linereview. Inonlinetrack-
Toachievethistightphysicalcoordination,teamsrelyupon
ing, immediatefutureactionsofindividualplayers(passes,
apre-existingplaybookofoffensivemaneuverstomovethe
feints)arepredicted,whereasinonlinestrategyrecognition,
balldownthefieldanddefensivestrategiestocountertheop-
the observer attempts to recognize the high-level strategy
posing team’s attempts to make yardage gains. Rush 2008
usedbytheentireteam. Inofflinereview,generalstrengths,
simulatesamodifiedversionofAmericanfootball;playsin
weaknesses, and tendencies are identified in an offline set-
Rusharecomposedofastartingformationandinstructions
tingandusedaspartofthetraining/learningregimen.
foreachplayerintheformation. Theseinstructionsaresim-
This paper addresses the problem of online strategy
ilartoaconditionalplanandincludechoicepointswherethe
recognitioninadversarialteamgames. Inphysicaldomains
playerscanmakeindividualdecisionsaswellaspre-defined
Copyright(cid:13)c 2009,AssociationfortheAdvancementofArtificial behaviorsthattheplayerexecutestothebestofitsphysical
Intelligence(www.aaai.org).Allrightsreserved. capability. Rush2008wasdevelopedfromtheopensource
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Improving Offensive Performance through Opponent Modeling
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14. ABSTRACT
Although in theory opponent modeling can be useful in any adversarial domain, in practice it is both
difficult to do accurately and to use effectively to improve game play. In this paper, we present an
approach for online opponent modeling and illustrate how it can be used to improve offensive performance
in the Rush 2008 football game. In football, team behaviors have an observable spatio-temporal structure,
defined by the relative physical positions of team members over time we demonstrate that this structure
can be exploited to recognize football plays at a very early stage of the play using a supervised learning
method. Based on the teams? play history our system evaluates the competitive advantage of executing a
play switch based on the potential of other plays to increase the yardage gained and the similarity of the
candidate plays to the current play. In this paper, we investigate two types of play switches: 1) whole team
and 2) subgroup switching. Both types of play switches improve offensive performance but by only
modifying the behavior of a key subgroup of offensive players, we improve on the yardage gained.
15. SUBJECT TERMS
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ABSTRACT OF PAGES RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE Same as 7
unclassified unclassified unclassified Report (SAR)
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
Rush 2005 game, which is similar in spirit to Tecmo Bowl field is 100 yards by 63 yards. The game’s objective is to
andNFLBlitz(Rush20052005). out-score the opponent, where the offense (i.e., the team
Although there have been other studies examining the with possession of the ball), attempts to advance the ball
problemofrecognizingcompletedfootballplays,wepresent fromthelineofscrimmageintotheiropponent’sendzone.
resultsonrecognizingfootballplaysonlineatanearlystage In a full game, the offensive team has four attempts to get
of play, and demonstrate a mechanism for exploiting this afirstdownbymovingtheball10yardsdownthefield. If
knowledge to improve a team’s offense. Our system eval- the ball is intercepted or fumbled, ball possession transfers
uates the competitive advantage of executing a play switch tothedefensiveteam.
basedonthepotentialofotherplaystoimprovetheyardage
gained and the similarity of the candidate plays to the cur-
rentplay.Ourplayswitchselectionmechanismoutperforms
boththebuilt-inoffenseandagreedyyardage-basedswitch-
ingstrategy.Calculatingtherelativesimilarityofthecurrent
play compared to the proposed play is shown to be a nec-
essary step to reduce confusion on the field and effectively
boostperformance. Additionallyweinvestigatedtheutility
Provs23 Powervs31 Splitvs2222
oflimitingtheplayswitchtoasmallsubgroupofplayers;by
onlymodifyingtheactionsofsmallsubgroupofkeyplayers,
Figure2: Threeoffensiveanddefensiveconfigurations. Of-
wecanimproveonthetotalteamswitch.
fensiveplayersareshowninwhiteandthedefenseinblue.
RelatedWork
ARushplayiscomposedof(1)astartingformationand
Previous work on team behavior recognition has been pri- (2)instructionsforeachplayerinthatformation. Aforma-
marilyevaluatedwithinathleticdomains,includingAmeri- tion is a set of (x,y) offsets from the center of the line of
canfootball(IntilleandBobick1999),basketball(Bhandari scrimmage. Bydefault,directionsforeachplayerconsistof
et al. 1997; Jug et al. 2003), and Robocup soccer simula- (a)anoffset/destinationpointonthefieldtorunto,and(b)
tions(RileyandVeloso2000;2002;Kuhlmannetal. 2006). abehaviortoexecutewhentheygetthere. Playinstructions
To recognize athletic behaviors, researchers have exploited are similar to a conditional plan and include choice points
simple region-based (Intille and Bobick 1999) or distance- wheretheplayerscanmakeindividualdecisionsaswellas
based (Riley and Veloso 2002) heuristics to build accurate, pre-definedbehaviorsthattheplayerexecutestothebestof
but domain-specific classifiers. For instance, based on the theirphysicalcapability. Rushincludesthreeoffensivefor-
premisethatallbehaviorsalwaysoccuronthesameplaying mations(power,pro,andsplit)andfourdefensiveones(23,
field with a known number of entities, it is often possible 31,2222,2231)2. Eachformationhaseightdifferentplays
to divide the playing field into grids or typed regions (e.g., (numbered 1-8) that can be executed from that formation.
goal,scrimmageline)thatcanbeusedtoclassifyplayerac- Offensive plays typically include a handoff to the running
tions.Incontrast,wetrainourclassifiersonrawobservation back/fullback or a pass executed by the quarterback to one
tracesanddonotrelyonafield-basedmarkersystem. ofthereceivers,alongwithinstructionsforarunningpattern
In Robocup, there has been some research on team in- tobefollowedbyallthereceivers.
tentrecognitiongearedtowardstheRobocupcoachcompe-
tition. Techniques have been developed to extract specific PlayRecognitionusingSVM
information, such as home areas (Riley et al. 2002), op-
Inthispaperwefocusonintentrecognitionfromtheview-
ponent positions during set-plays (Riley and Veloso 2002),
pointoftheoffense: givenaseriesofobservations,ourgoal
andadversarialmodels(RileyandVeloso2000), fromlogs
is to recognize the defensive play as quickly as possible
ofRobocupsimulationleaguegames. Thisinformationcan
in order to maximize our team’s ability to intelligently re-
be utilized by the coach agent to improve the team’s scor-
spondwiththebestoffense. Thus,theobservationsequence
ingperformance. Forinstance,informationaboutopponent
grows with time unlike in standard offline activity recogni-
agent home areas can be used triggers for coaching advice
tion where the entire set of observations is available. We
andfordoing“formation-basedmarking”,inwhichdifferent
approachtheproblembytrainingaseriesofmulti-classdis-
teammembersareassignedtotrackmembersoftheoppos-
criminative classifiers, each of which is designed to handle
ing team. However, the focus of the coaching agents is to
observationsequencesofaparticularlength. Ingeneral,we
improveperformanceofteamsinfuturegames; oursystem
expectthattheearlyclassifiersshouldbelessaccuratesince
immediatelytakesactionontherecognizedplaytoevaluate
theyareoperatingwithashorterobservationvectorandbe-
possibleplayswitches.
cause the positions of the players have deviated little from
theinitialformation.
RushFootball
We perform this classification using support vector ma-
Football is a contest of two teams played on a rectangular chines (Vapnik 1998). Support vector machines (SVM)
field that is bordered on lengthwise sides by an end zone. are a supervised binary classification algorithm that have
Unlike American football, Rush teams only have 8 players been demonstrated to perform well on a variety of pattern
on the field at a time out of a roster of 18 players. and the classificationtasks,particularlywhenthedimensionalityof
the data is high (as in our case). Intuitively the support currentstartingconfigurationandtimestep. Anobservation
vector machine projects data points into a higher dimen- vector of the correct length is generated (this can be done
sional space, specified by a kernel function, and computes incrementally during game play) and fed to the multi-class
a maximum-margin hyperplane decision surface that sepa- SVM. The output of the intent recognizer is the system’s
ratesthetwoclasses. Supportvectorsarethosedatapoints best guess (at the current time step) about the opponent’s
that lie closest to this decision surface; if these data points choice ofdefensive playand can helpus toselect the most
were removed from the training data, the decision surface appropriateoffense,asdiscussedbelow.
would change. More formally, given a labeled training set Table 1 summarizes the experimental results for differ-
{(x ,y ),(x ,y ),...,(x ,y )}, where x ∈ (cid:60)N is a fea- ent lengths of the observation vector (time from start of
1 1 2 2 l l i
ture vector and y ∈ {−1,+1} is its binary class label, an play), averaging classification accuracy across all starting
i
SVMrequiressolvingthefollowingoptimizationproblem: formation choices and defense choices. We see that at the
earliest timestep, our classification accuracy is at the base-
l
1 (cid:88) line but jumps sharply near perfect levels at t = 3. This
min wTw+C ξ
w,b,ξ 2 i strongly confirms the feasibility of accurate intent recogni-
i=1 tion in Rush, even during very early stages of a play. At
constrainedby: t = 2, there is insufficient information to discriminate be-
tweenoffenseplays(perceptualaliasing),howeverbyt=3,
yi(wTφ(xi)+b) ≥ 1−ξi, thepositionsoftheoffensiveteamaredistinctiveenoughto
ξ ≥ 0. bereliablyrecognized.
i
The function φ(.) that maps data points into the higher di- OffensivePlaySwitches
mensionalspaceisnotexplicitlyrepresented; rather, aker-
Toimproveoffensiveperformance,oursystemevaluatesthe
nelfunction,K(x ,x )≡φ(x )φ(x ),isusedtoimplicitly
i j i j competitive advantage of executing a play switch based on
specifythismapping. Inourapplication,weusethepopular
1)thepotentialofotherplaystoimprovetheyardagegained
radialbasisfunction(RBF)kernel:
and 2) the similarity of the candidate plays to the current
K(x ,x )=exp(−γ||x −x ||2),γ >0. play. First, we train a set of SVM models to recognize de-
i j i j
fensiveplaysataparticulartimehorizonasdescribedinthe
Several extensions have been proposed to enable SVMs previous section; this training data is then used to identity
to operate on multi-class problems (with k rather than 2 promisingplayswitches. Aplayswitchisexecuted:
classes),suchasone-vs-all,one-vs-one,anderror-correcting
1. after the defensive play has been identified by the SVM
output codes. We employ a standard one-vs-one voting
classifier;
schemewhereallpairwisebinaryclassifiers,k(k−1)/2 =
28foreverymulti-classprobleminourcase,aretrainedand 2. ifthereisastrongeralternateplaybasedontheyardage
the most popular class is selected. When multiple classes historyofthatplayvs.thedefense;
receivethehighestvote,weselectthewinningonewiththe
3. ifthecandidateplayissufficientlysimilartothecurrent
lowest index; the benefit of this approach is that classifica-
playtobefeasibleforimmediateexecution.
tion is deterministic but it can bias our classification in fa-
vor of lower-numbered plays. For a real game system, we To determine whether to execute the play switch for a par-
wouldemployarandomizedtie-breakingstrategy. Manyef- ticularcombinationofplays,theagentconsidersN,theset
ficientimplementationsofSVMsarepubliclyavailable;we of all offensive plays shown to gain more than a threshold
useLIBSVM(ChangandLin2001). (cid:15) value. The agent then selects Min(n ∈ N), the play in
We train our classifiers using a collection of simulated thelistmostlikethecurrentplayforeachplayconfiguration
games in Rush collected under controlled conditions: 40 andcachesthepreferredplayinalookuptable.
instances of every possible combination of offense (8) and When a play is executed, the agent will use all observa-
defense plays (8), from each of the 12 starting formation tions up to and including observation 3 to determine what
configurations. Since the starting configuration is known, playthedefenseisexecutingbeforeperformingalookupto
each series of SVMs is only trained with data that could determine the play switch to make. The process is ended
beobservedstartingfromitsgivenconfiguration. Foreach with execution of a change order to all members of the of-
configuration, we create a series of training sequences that fensive team. Calculating the feasibility of the play switch
accumulates spatio-temporal traces from t = 0 up to t ∈ based on play similarity is a crucial part of improving the
{2,...,10}timesteps. AmulticlassSVM(i.e.,acollection team’s performance; in the results section, we evaluate our
of 28 binary SVMs) is trained for each of these cases. Al- similarity-based play switch mechanism vs. a greedy play
though the aggregate number of binary classifiers is large, switchingalgorithmthatfocusessolelyonthepotentialfor
each classifier only employs a small fraction of the dataset yardagegained.
andisthereforeefficient(andhighlyparalellizable). Cross-
PlaySimilarityMetric
validationwasusedtotunetheSVMparameters(C andσ)
foralloftheSVMs. To calculate play similarities, we create a feature matrix
Classificationattestingtimeisveryfastandproceedsas for all offensive formation/play combinations based on the
follows. WeselectthemulticlassSVMthatisrelevanttothe trainingdata.
Table1: Playrecognitionresults(accuracyoverallplaycombinations)
t=2 3 4 5 6 7 8 9 10
12.50 96.88 96.87 96.85 96.84 96.87 96.89 96.83 96.81
ThefeaturescollectedforeachathleteAare history (based on earlier observation) of scoring the most
yardage. This process is accomplished for every offensive
Max(X): TherightmostpositiontraveledtobyA
play formation against every defensive play formation
Max(Y): ThehighestpositiontraveledtobyA and play combination. When the agent is constructing
Min(X): TheleftmostpositiontraveledtobyA the lookup table and needs to determine the most similar
play from a list, given current play i, it calls the method,
Min(Y): ThelowestpositiontraveledtobyA
min(O D M )whichreturnsthemostsimilarplay.
β αp i
Mean(X): = PNi=−01Xi
N
Mean(Y): = PNi=−01Yi ImprovingtheOffense
N
Our algorithm for improving Rush offensive play has two
Median(X): = Sort(X)
i/2 mainphases, apreprocessstagewhichyieldsaplayswitch
Median(Y): = Sort(Y) lookuptableandanexecutionstagewherethedefensiveplay
i/2
is recognized and the offense responds with an appropriate
FirstToLastAngle: Angle from starting point (x1,y1), to
(cid:16) (cid:17) playswitchforthatdefensiveplay. AsdescribedinSection
endingpoint(x2,y2),isdefinedasatan (cid:52)y wetrainasetofSVMclassifiersusing40instancesofevery
(cid:52)x
possible combination of offense (8) and defense plays (8),
Start Angle: Angle from the starting point (x ,y ) to
0 0 fromeachofthe12startingformationconfigurations. This
(x ,y ),definedasatan(cid:0)y(cid:1)
1 1 x stageyieldsasetofmodelsusedforplayrecognitionduring
End Angle: Anglefromthestartingpoint(x ,y )to thegame. Next,wecalculateandcacheplayswitchesusing
n−1 n−1
(cid:16) (cid:17) thefollowingprocedure:
(x ,y ),definedasatan (cid:52)y
n n (cid:52)x
Step1: Collect data by running the RUSH 2008 football
(cid:16) (cid:17)
Total Angle: =(cid:80)N−1atan yi+1−yi simulator50timesforeveryplaycombination.
i=0 xi+1−xi
Step2: Createyardagelookuptablesforeachplaycombi-
Total Path Distance: =(cid:80)N−1(cid:16)(cid:112)2 x 2+y 2(cid:17) nation.Thisinformationaloneisinsufficienttodetermine
i=0 i i
how good a potential play is to perform the play switch
Feature set F for a given play c contains all the features actionon. Thetransitionplaymustresembleourcurrent
foreachoffensiveplayerintheplayandisdescribedas offensiveplayortheoffensiveteamwillspendtoomuch
−→ timeretracingstepsandperformverypoorly.
F ={A ∪A ∪A ∪···∪A }
c c1 c1 c2 c8 Step3: Create feature matrix for all offensive forma-
These features are similar to the ones used in (Rubine tion/play combinations using the probabilistic trace rep-
1991) and more recently, by (Wobbrock et al. 2007) to resentation.
matchpentrajectoriesinsketch-basedrecognitiontasks,but
Step4: Create the final play switch lookup table based on
generalizedtohandlemulti-playertrajectories. Tocompare
boththeyardageinformationandtheplaysimilarity.
playsweusethesumoftheabsolutevalueofthedifferences
(L norm) between each feature F . This information is To create the play switch lookup table, the agent first
1 ci
used to build a similarity matrix M for each possible of- extracts a list of offensive plays L given the requirement
ij
fensiveplaycombinationasdefinedbelow. yards(Li) > (cid:15) where (cid:15) is the smallest yardage gained
in which the agent does not consider changing the cur-
‚‚‚−F→c‚‚‚−1 rent offensive play to another. We used (cid:15) = 1.95 based
(cid:88) −→ on a quadratic polynomial fit of total yardage gained in
M = ∆F
ij c 6 tests with (cid:15) = {MIN,1.1,1.6,2.1,2.6,MAX} where
c=0 MIN is small enough no plays are selected to change and
i,j =1...8 MAX whereallplaysareselectedforchangetothehighest
yardage play with no similarity comparison. Second, from
There is one matrix M for each offensive formation
thelistLfindtheplaymostsimilar(smallestvalueinthema-
O , where β = {pro, power, split} are the offensive
β trix)toourcurrentplayiusingMin(O D M )andaddit
formations. Defensive formation/play combinations are β αp i
tothelookupfile.
indicated by D , where α = {23, 31, 2222, 2231} and
αp
During execution, the offense uses the following proce-
p represents plays 1..8. M for a specific play configuration
dure:
is expressed as O D M , given i (1...8) is our current
β αp i
offensive play. The purpose of this algorithm is to find a 1. Ateachobservationlessthan4,collectmovementtraces
value j (play) most similar to i (our current play), with a foreachplay.
2. Atobservation3,useLIBSVMwiththecollectedmove-
menttracesandpreviouslytrainedSVMmodelstoiden-
tifythedefensiveplay.
3. Accessthelookupfiletofindbest(i)forourcurrentplay
i.
4. Send a change order command to the offensive team to
changetoplaybest(i).
However it is not necessary (or always desirable) to
change all of the players to the new play. We also inves-
tigated the performance of subgroup switching, modifying
the play of small group of key players while leaving the
remaining players alone. By segmenting the team in this
fashionweareabletoinessencecombinetwoplayswhich
hadpreviouslybeenidentifiedasaliketoeachotherwithre-
Figure4:Comparisonofplayswitchselectionmethods.Our
gardtospatio-temporaldata,butdifferentinregardstoyards
play switch method (shown in red) outperforms both base-
gained.Thefootballoffensiveteamlendsitselftothreemain
line Rush offense (blue) and a greedy play switch metric
groups based on domain knowledge of football. Group 1
(green).
contains the QB, RB, and FB; group 2 has LG, C and RG;
andgroup3consistsoftheremainingplayersLWR,RWR,
RTE,andLTE.
Figure 3 is a good example of a successful merging of
twoplaystoproduceasuperiorplaygiventhisdefense. The
green line represents the average yardage gained. The left
image is the most likely path of the baseline case (a run-
ningplaywhichyieldslittleyardageonaverage). Themid-
dleimageisthemostlikelyexecutiontraceproducedbythe
total play switch method. The play produced by the total
playswitchwasnotmuchmoresuccessfulthanthebaseline
case;howeverwhenonlyGroup1(QB,RB,FB)ismodified
thesuccessoftheplayincreasesgreatlyandthenewplayis
showntobeverycoordinatedandeffective.
EmpiricalEvaluation
The algorithm was tested using the RUSH 2008 simulator Figure5: Theplay-yardagegainoverbaselineRushoffense
for ten plays on each possible play configuration in three yieldedbyourplayswitchstrategy.
separate trials. We compared our play switch model (us-
ing the yardage threshold (cid:15) = 1.95 as determined by the
quadratic fit) to the baseline Rush offense and to a greedy about .2 to .4 yards more than the gains in the baseline
play switch strategy ((cid:15) = MAX) based solely on the sample. In 6 we see all the split configurations do quite
yardage. well; this is unsurprising given our calculations of the best
Overall, the average performance of the offense went response. However,whenthethresholdisnotinuseandthe
from2.82 yardsper playto3.65 yardsper play((cid:15) = 1.95) plays are allowed to change regardless of current yardage,
with an overall increase of 29%, ±1.5% based on sam- the results are drastically reduced. The reason seems to be
pling of three sets of ten trials. An analysis of each of the associated player miscoordinations accidentally induced
formation combinations (Figure 6) shows the yardage gain by the play switch; by maximizing the play similarity
varies from as much as 100% to as little as 0.1%. Over- simultaneously, the possibility of miscoordinations is
all, performance is consistently better for every configura- reduced. Figure 5 shows yardage gained by the best play
tiontested. Inallcases,thenewaverageyardageisover2.3 switch strategy over the Rush baseline offense. Power
yards per play with no weak plays as seen in the baseline. vs.23experiencesthegreatestenhancementandSplitvs.31
Forexample,Powervs.23(1.4averageyardsperplay)and the least. It is interesting to note Split formations in the
Power vs. 2222 (1.3 average yards per play). Results with baseline performed best and improved the least while the
(cid:15) = MAX clearly shows simply changing to the greatest Power formations performed the worst in the baseline and
yardage generally results in poor performance from the of- improvedthemost. Thisindicatesaninverselyproportional
fense. expectedgainbythealgorithm.
Powervs.23isdramaticallyboostedfromabout1.5yards
to about 3 yards per play, doubling yards gained. Other To evaluate the subgroup switching, we ran the simula-
combinations, such as Split vs. 23 and Pro vs. 32 already tion over all three groups and compared them to the base-
scored good yardage and improved less dramatically at lineyardagegainedandtheresultsoftotalplayswitch. Test
Figure3: Subgroupswitching
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