Mechanism Design for Team Formation

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by  Mason Wright

Mechanism Design for Team Formation
MasonWright YevgeniyVorobeychik
ComputerScience&Engineering ElectricalEngineeringandComputerScience
UniversityofMichigan VanderbiltUniversity
AnnArbor,MI Nashville,TN
[email protected] [email protected]
5 Abstract Pastresearchonhedonicgameshasfocusedontheprob-
1 lemofformingstablecoalitions,fromwhichnosetofagents
0 Team formation is a core problem in AI. Remarkably, would prefer to defect. Since a core partition may not ex-
2 little prior work has addressed the problem of mech- ist in a hedonic game, even when preferences of players
anism design for team formation, accounting for the are additively separable (Banerjee, Konishi, and So¨nmez
n need to elicit agents’ preferences over potential team-
a mates.Coalitionformationintherelatedhedonicgames 2001), much research is focused on alternative notions of
J stability, or on highly restricted agent preferences (Bo-
hasreceivedmuchattention,butonlyfromtheperspec-
4 tive of coalition stability, with little emphasis on the gomolnaia and Jackson 2002; Alcalde and Revilla 2004;
mechanism design objectives of true preference elici- CechlarovaandRomero-Medina2001),oronthetimecom-

] tation, social welfare, and equity. We present the first plexityoftestingcoreemptiness(Ballester2004;Sungand
T
formal mechanism design framework for team forma- Dimitrov2010).
G tion,buildingonrecentcombinatorialmatchingmarket Weconsiderteamformationasamechanismdesignprob-
. design literature. We exhibit four mechanisms for this lem,whereindividualshavepreferencesoverteammates,as
s problem, two novel, two simple extensions of known
c inhedonicgames.Asintraditionalmechanismdesign(and
mechanisms from other domains. Two of these (one
[ unlike hedonic games), we assume that these preferences
new,oneknown)havedesirabletheoreticalproperties.
are private and must be elicited in order to partition play-
1 However, we use extensive experiments to show our
v secondnovelmechanism,despitehavingnotheoretical ersreasonablyintoteams.Wedrawaconnectiontoanother
5 guarantees, empirically achieves good incentive com- budding literature, that of combinatorial matching market
1 patibility,welfare,andfairness. design, which has course allocation as a typical applica-
7 tion(BudishandCantillon2012).
0 An important concern in combinatorial matching, which
0 Introduction we inherit, is the ex post fairness of allocations. For ex-
.
1 Teamwork has beenan important and often-studied area of ample, consider a simple randomized mechanism, random
0 artificialintelligenceresearch.Typically,thefocusisonco- serial dictatorship, which has been proposed for course al-
5 ordinating agents to achieve a common goal. The comple- location and is readily adapted to team formation. In ran-
1 mentary problemof teamformation considershow to form domserialdictatorship,agentsarerandomlyorderedbythe
v: high-qualityteams,whoseagentshaveskillsthatarejointly mechanismandthentaketurns,inorder,selectingtheiren-
i well suited for a task (Marcolino, Jiang, and Tambe 2013). tire teams from among the remaining agents. Random se-
X Notable team formation applications include formation of rialdictatorshipisstrategyproof,meaningthatitisadomi-
r researchteams,classprojectgroups,groupsofroommates, nantstrategyforanyagenttoreportitstruepreferencesover
a
ordisasterreliefteams. teams.RandomserialdictatorshipisalsoexpostParetoef-
ficient, in that any allocation it returns cannot be modified
Many prior team formation studies have assumed that
toimproveanagent’swelfarewithoutreducingsomeother
agents are indifferent about which other agents they are
agent’s(assumingnoindifferences).Butthismechanismre-
teamedwith,orhavepreferencesknowntotheteamforma-
sultsinahighlyinequitabledistributionofoutcomesexpost.
tion mechanism. Models dealing with known agent prefer-
encesoverteammates,termedhedonicgames,haveseenan Budish and Cantillon (2012) proposed a more sophisti-
extensive literature since being introduced by Aumann and catedalternative,approximatecompetitiveequilibriumfrom
Dreze (1974). In a hedonic game, the mechanism is given equal incomes (A-CEEI), which is strategyproof-in-the-
a set of agents, each having public preferences over which large (i.e., when the number of players becomes infinite),
others might be on its team; the mechanism must partition and provably approximately fair (Budish and Cantillon
theagentsintoteamsbasedontheirpreferences. 2012; Budish 2011). The work on combinatorial match-
ing in turn follows earlier work on bipartite matching and
Copyright(cid:13)c 2015,AssociationfortheAdvancementofArtificial school choice (Roth and Peranson 1999; Abdulkadiroglu
Intelligence(www.aaai.org).Allrightsreserved. andSo¨nmez2003).
Ourcontributionsareasfollows. non-negative, additive separable preferences, as in the B-
preferences of (Cechlarova and Romero-Medina 2001) and
1. We present the problem of mechanism design for team
inthebiddingpointsauction.
formation,focusedonachieving(near-)incentivecompat-
Most prior work on hedonic games focuses on coalition
iblepreferencereporting,highsocialwelfare,andfairal-
stability. Our goal is distinct: We take as input player pref-
location.Thisproblemiscloselyrelatedtobothcombina-
erences over teams (that is, over others that they could be
torialandbipartitematchingmarketdesign,butisdistinct
teamed with), which we assume to be additive with non-
from both in two senses: first, the matching is not bipar-
negative values, and output a partition of the players into
tite(playersmatchtootherplayers),andthereforetypical
teams. We assume that it is subsequently difficult for play-
matchingalgorithmswhichonlyguaranteestrategyproof-
erstoalterteammembership.Ourprimarychallenge,there-
ness for one side are unsatisfactory; and second, mecha-
fore,istoencourageplayerstoreporttheirpreferenceshon-
nismsusedincombinatorialexchangestoprovidefairness
estly,andformteamsthatarefairandyieldgoodteammate
guaranteesarenotdirectlyapplicable,astheyrelyonhav-
matchings;allthreenotionsshallbemadeprecisepresently.
ingafixedsetofitemswhicharethesubjectofthematch
Notethatinthisconstructionweassumethatnomoneycan
andwhicharenotthemselvesstrategic;
changehands(unliketheworkbyLietal.(2004)).
2. we extend two well-known mechanisms (random serial
Observethatinourmodel,allplayersalwaysprefertobe
dictatorshipandHarvardBusinessSchooldraft)usedfor
putonasingleteam(sincevaluesforallpotentialteammates
combinatorialmatchingtooursetting;
arepositive).Inreality,manyteamformationproblemshave
3. we propose two novel mechanisms for our setting (A- hardconstraintsonteamsizes(or,equivalently,onthenum-
CEEIforteamformation,orA-CEEI-TF,andone-player- ber of teams), particularly when multiple tasks need to be
one-pickdraft,orOPOP); accomplished. For example, project teams usually have an
upper bound on size. We capture this by introducing team
4. we prove that A-CEEI-TF is approximately fair and
size constraints; formally, the size of any team must be in
strategyproof-in-the-large;
the interval [k,k], with k ≥ 1, k ≤ |N|, and k ≤ k. For
5. we offer empirical analysis of all mechanisms, which example, if a classroom with 25 students must be divided
shows that our second mechanism, OPOP, outperforms into6approximatelyequal-sizeteams,wecouldhavek =4
others on most metrics, and has better incentive proper- andk =5.Weassumethroughoutthatthespecificvaluesof
tiesthanA-CEEI-TF.
k and k admit a feasible allocation. (This is not always the
An important and surprising finding of our investigation is case;seesupplementalmaterialfordetails.)
that the simple draft mechanism we propose empirically Incontrastwithatypicalapproachinmechanismdesign,
outperforms the more complex A-CEEI-TF alternative by which seeks to maximize a single objective such as social
alargemargininfairnessandincentivecompatibility,even welfare or designer revenue, subject to a constraint set, we
whileA-CEEI-TFhasmorecompellingtheoreticalguaran- take an approach from the matching market design litera-
tees. ture,andseekacollectionofdesirableproperties(see,e.g.,
Budish (2012)). Specifically, we consider three properties:
MechanismDesignProblem incentive compatibility, social welfare, and fairness. Given
the fact that all three cannot be achieved simultaneously in
Our point of departure is the formalism of hedonic games.
oursetting,wewillanalyzetheextenttowhicheachcanbe
We define a hedonic game as a tuple (N,(cid:31)), where N is
achievedthroughspecificmechanisms.
thesetofplayers,and(cid:31)isavectorcontainingeachplayer’s
preference order over sets of other players that it could be
teamedwith.ThetaskistopartitiontheplayersinN intoa Incentive Compatibility Incentive compatibility holds if
coalitionstructure,whereeachplayerisinexactlyonecoali- thereisnoincentiveforanagenttomisreportitspreferences.
tion. We consider two forms of incentive compatibility: strate-
Weassumethatplayerpreferencesareadditivelysepara- gyproofness,whichmeansthatitisadominantstrategyfor
ble(Aziz,Brandt,andSeedig2011),whichmeansthatthere anyagenttoreportitstruepreferences,andexpostequilib-
exists an assignment of values ui(j) for all players i and rium,whichmeansthatitisaNashequilibriumforallagents
theirpotentialteammatesj,sothati’stotalutilityofasubset to report their true preferences. The former will be consid-
(cid:80)
of others S is j∈Sui(j) (which induces a corresponding eredintheoreticalanalysis,whilethelatterwillbethefocus
preferenceorderingoversubsetsofpossibleteammates).In of empirical incentive assessment. In particular, our theory
addition,weassumethatu (j)≥0foralli,j. willfocusonstrategyproofness-in-the-large(Budish2011),
i
These assumptions are useful for two reasons. First, in definedasfollows.Consideramarketwhereeachagenthas
many data sets that record preferences of individuals over beenreplacedwithameasure-onecontinuumofreplicasof
others, the preferences are entered as non-negative values itself,suchthateachindividualagenthaszeromeasureand
forindividuals,asinrankorderlistsorLikertratingsofindi- allagentsarepricetakers.Amechanismisstrategyproof-in-
viduals.Additiveseparablepreferencesarethemostnatural the-large if, in such a market, it is a dominant strategy for
waytoinducepreferencesovergroupsfromsuchdata.Sec- each agent to reveal its true preferences. An example of a
ond, many prior studies in team formation and the related mechanismthatisnotstrategyproof-in-the-largeistheHar-
domainofcourseallocationhaveassumedthatagentshave vard Business School draft considered below, in which an
agentmaybenefitfrommisreportingitspreferences,regard- either.Someotheragentj mustthenbeonateamwiththe
less of its own measure relative to the market size (Budish two most-preferred agents of the player i. By construction,
and Cantillon 2012). In empirical analysis, in contrast, we player i is on a team of value 3 or less, while the team of
determinealowerboundontheregretoftruthfulreporting, agentjhasvalue12toagenti,andvalue4toagentiwithits
whichisthemostanyagentcangainexpostbymisreporting more valuable player (player A) removed. Therefore, envy
preferenceswhenallothersaretruthful. cannotbeboundedbyasingleteammateforallagents.
TeamFormationMechanisms
Social Welfare As in traditional mechanism design, we
consider social welfare as one of our primary design cri- We describe four mechanisms for team formation: two are
teria. Social welfare is just the sum of player utilities straightforward applications of known mechanisms, while
achieved by a specific partition of players into teams. For- twoarenovel.
mally, if Q is a partition of players, social welfare is de-
fined as SW(Q) = 1 (cid:80) (cid:80) u (j). In addition, RandomSerialDictatorship
|N| S∈Q i,j∈S i
weconsidertheweakernotionofexpostParetooptimality Randomserialdictatorship(RSD)haspreviouslybeenpro-
when discussing alternative mechanisms and their theoret- posedinassociationwithschoolchoiceproblems(Abdulka-
ical properties. A partition of players Q is ex post Pareto diroglu and So¨nmez 2003). In RSD, players are randomly
optimal if noother partition strictly improves some agent’s ordered,andeachplayerchoseninthisorderselectshisteam
utilitywithoutloweringtheutilityofanyotheragent. (with players thereby chosen dropping out from the order).
Theprocessisrepeateduntilallplayersareteamedup.
Fairness The measure of fairness we consider is envy- Proposition2. Randomserialdictatorshipisstrategyproof,
freeness. An allocation is envy-free if each agent weakly andexpostParetoefficientaslongasplayerschoosinglater
prefersitsownallocationtothatofanyotheragent.Anap- cannotchoosealargerteam.2
proximatenotionofenvy-freenessthatweadoptfromBud-
WhileRSDisexpostParetoefficient,thisturnsouttobea
ish(2011)isenvyboundedbyasingleteammate,inwhich
weakguarantee,anddoesnotingeneralimplysocialwelfare
anyallocationanagentpreferstotheirownceasestobepre-
maximization,somethingthatbecomesimmediatelyappar-
ferred through removal of a single teammate from it.1 The
ent in the experiments below. Envy-freeness is, of course,
following negative result makes apparent the considerable
outofthequestionduetoProposition1.
challengeassociatedwiththedesignproblemwepose.
Proposition 1. There may not exist a partition of players HarvardBusinessSchool(HBS)Draft
thatboundsenvybyasingleteammate.
Players are randomly ordered, with the first T assigned as
Proof Consider a team formation problem with 6 agents, captains. We then iterate over captains, first in the random
{A,B,C,D,E,F}, k = 3, k = 3, so that two equal-size order,theninreverse,alternating.Thecurrentteamcaptain
teamsmustbeformed.Theagents’additiveseparablepref- selectsitsmost-preferredremainingplayertojoinitsteam,
erencesareencodedinTable8. basedonitsreportedpreferences.
Proposition 3. HBS draft is not strategyproof or ex post
A B C D E F
Paretoefficient.
A x 0 1 2 4 8
B 8 x 4 2 1 0
One-Player-One-Pick(OPOP)Draft
C 8 0 x 4 2 1
D 8 1 0 x 4 2 Players are randomly ordered. Given the list of team sizes,
E 8 2 1 0 x 4 thefirstT playersareassignedtobecaptainsoftherespec-
F 8 4 2 1 0 x tiveteams.Theniterateoverthecompleteplayerlist.Ifthe
next player is a team captain, it selects its favorite unas-
Table1:Eachrowiencodestheadditiveseparablevaluefor signedagenttojoinitsteam.Ifthenextplayerisunassigned,
agentiofeachotheragent. it will be assigned to join its favorite incomplete team (as
defined below), and if the team still has space, this player
chooses its favorite unassigned agent to join them. We de-
Nopartitionoftheseagentsintotwoteamsofsize3gives
fine a “favorite” incomplete team for an agent as follows.
everyagentenvyboundedbyasingleteammate.Toseethis,
Let S be an incomplete team with v vacancies. Let the
considerthateachagentotherthanAhasablisspointona S
meanvaluetoplayerioftheunassignedplayersbeµ .We
teamwithAandoneotheragent,wherethesecondagentis i
thenassignthefollowingutilityofanincompleteteamS to
C for agent B, D for agent C, and so on until “wrapping (cid:80)
agenti: u (j)+(v −1)µ .
around”withB foragentF.Threeoftheagentswillnotbe j∈S i S i
onateamwithagentA,andatleastoneoftheseagents,say Proposition4. TheOne-Player-One-Pickdraftisnotstrat-
agenti,willnotbeonateamwithitssecond-favoriteagent egyprooforexpostParetoefficient.
1In the supplemental material we discuss another measure of 2The proofs of this and other results are in the supplemental
fairness. material.
CompetitiveEquilibriumfromEqualIncomes Algorithm1A-CEEI-TFAlgorithmOutline.
Require: (N,(cid:31),k,k)
We now propose a more complex mechanism, based on 1: Randomlyassignapproximatelyequalbudgetsbi tothe
the Competitive Equilibrium from Equal Incomes (CEEI), agents,b ∈[1,¯b],¯b<1+1/|N|.
i
which is explicitly designed to achieve allocations that are 2: Randomlyordertheagents.
moreexpostfairthanthealternativemechanisms. 3: SearchforapricevectorpinpricespaceP =[0,¯b]|N(cid:48)|
that approximately clears the (relaxed) market among
We begin by defining CEEI, previously introduced by
Varian(1974).GivenasetofagentsN,asetofgoodsC,and theN(cid:48)remainingagents,givenagentbudgetsb.
agentpreferencesoverbundlesofgoods(cid:31),aCEEImecha- 4: Takethenextunmatchedagentintherandomorder,and
nismfindsabudgetb∈R andpricevectorp∗ ∈R|C|,such assign it to join its favorite bundle of other free agents
+ + that it can afford at the current prices, and that leaves
thatifeachagentisallocateditsfavoritebundleofgoodsthat
a feasible subproblem—i.e., feasible (N(cid:48),k,k). If the
costsnomorethanb,theneachgoodinC isallocatedtoex-
agent cannot afford any remaining bundle of legal size
actly one agent in N, or divided in fractions summing to 1
thatleavesafeasiblesubproblem,theagentisassigned
amongtheN.Incombinatorialallocationproblems,suchas
its favorite remaining bundle of legal size that leaves a
course allocation, goods (seats in a class) are not divisible,
feasiblesubproblem.
and certain bundles of goods (class schedules) are not al-
5: Repeatsteps3and4untileachagentisonateam.
lowedtobeassignedtoanagent.Asaresult,anexactmar-
ket clearing tuple (b,p∗) may not exist. To deal with this
difficulty,CEEIwasrelaxedbyBudish(2011)toanapprox-
imateversion,termedA-CEEI.A-CEEIworksbyassigning Wenowdefineacandidatepriceupdatefunction,f :
TF
nearlyequalbudgetstoallagents,thensearchingforanap-
proximately market clearing price vector and returning the (1+(cid:15)−((cid:15)/¯b)t(p˜) )D −U
f (p˜) =t(p˜) + j j j (1)
allocationinducedbythoseprices.Theresultmaynotclear TF j j |N(cid:48)|
themarketexactly,butthereisanupperboundontheworst-
casemarketclearingerror.Theresultingallocationsatisfies whereD isthenumberofagentsthatdemandj butwhom
j
anapproximateformofenvy-freeness(Budish2011). j does not demand, U = 1 if and only if no other agent
j
BothCEEIandA-CEEItakeadvantageofthedichotomy demandsj,and0otherwise,¯bisthesupremumofallowable
betweenagentsanditemswhichagentsdemand.Thismakes agentbudgets,andt(·)isatruncationfunction,whichtakes
our setting distinct: agents’ demand in team formation is apricevectorp˜andtruncatesittothe[0,¯b]interval.
over subsets of other agents. A technical consequence is
Proposition5. f (·)isadmissible.
that this gives rise to a hard constraint for CEEI that if an TF
agent i is paired with agent j, than j must also be paired While admissibility of f (·) alone does not guarantee
TF
(assigned to) agent i; any relaxation of this constraint fails convergence of the iterative process, it does guarantee that
toyieldapartitionontheagentsandconsequentlydoesnot if convergence happens, we have a solution. The following
result in an admissible mechanism. We therefore design an proposition characterizes some of the properties such solu-
approximation of CEEI, termed A-CEEI-TF, that accounts tionspossess.
forthespecificpeculiaritiesofoursetting.Conceptually,the
Proposition6. A-CEEI-TFisstrategyproof-in-the-large.In
A-CEEI-TF mechanism works by alternating between two
addition,ifA-CEEI-TFyieldsexactmarketclearingandin-
steps. First, it searches in price space for approximate re-
duces the same allocation at each stage of price search, it
laxedmarket-clearingprices.Second,itassignsarandomly
yieldsenvyboundedbyasingleteammate.
selected unmatched agent to form a team with its favorite
bundle of free agents that is affordable, based on current
Proof Wesketchaproofofstrategyproofness-in-the-large.
prices. The result is a mechanism that is strategyproof-in-
the-large,andmorefairthanrandomserialdictatorship. If a team formation problem is modified such that each
agentisreplacedwithameasure-onesetofcopiesofitself,
A key part of A-CEEI-TF is a price update function,
each copy being measure zero, we arrive at what is called
which reflects the constraints of the team formation prob-
a continuum economy. If we run A-CEEI-TF in the con-
lem. We use a taˆtonnement-like price update function f in
tinuumeconomy,anyindividualagent,beingzero-measure,
anauxiliarypricespaceP˜ =[−1,1+¯b]|N(cid:48)|,whereN(cid:48)isthe hasnoinfluenceontheapproximateequilibriumpricevector
setofagentsremaining(unassigned)ataniterationoftheal- arrivedatbyupdatefunctionf (·),atanyiterationofthe
gorithm,and¯bisthesupremumofallowableagentbudgets. A-CEEI-TFmechanism.TherefToFre,theonlyeffecttheagent
We make two requirements of a price update function, one can have on the outcome is that, if the agent is randomly
ensuringthattheiterativeupdatesarewell-defined,another selected to choose its favorite affordable team of available
toensurethatfixedpointsoftheprocessareactualsolutions. agents that leaves a feasible subproblem, the agent’s re-
ported preferences determine which team the agent is as-
Definition1. Apriceupdatefunctionf isadmissibleif(a) signed.Thus,itisadominantstrategyfortheagenttoreport
itsfixedpointscorrespondto(relaxed)marketclearing,and itstruepreferences,sothatinthiscasetheagentwillbeas-
(b)P˜ isclosedunderf. signeditsmost-preferredallowableteam.
Experiments Tomeasureagentpreferencesimilarityinadataset,welet
Cequalthemeancosinesimilarityamongallpairsofdistinct
Although RSD and A-CEEI-TF possess desirable theoreti-
agentsinthedataset.Eachagentassignsitselfavalueof0or
cal properties, these results are loose, and the only approx-
undefined,sowetakethecosinesimilaritybetweenagentsi
imate fairness guarantee, shown for A-CEEI-TF, requires
andj onlyovertheirvaluesforagentsinN \{i,j}:
strongassumptionsontheenvironment.Wenowassessallof
the proposed mechanisms empirically through simulations u =u \{u ,u }
i−ij i ii ij
basedbothonrandomlygeneratedclassesofpreferences,as
wellasreal-worlddata.Ourempiricalresultsturnouttobe (cid:80)|N| (cid:80)|N| ui−ij·uj−ij
C = i=1 j=i+1 (cid:107)ui−ij(cid:107)(cid:107)uj−ij(cid:107)
bothone-sided(ifoneisinterestedinachievingallthreede-
(|N|2−|N|)/2
sired properties) and surprising: OPOP, a mechanism with
noprovabletheoreticalguarantees,tendstooutperformoth-
ersinfairness,andtoperformnearlyaswellasthebestother
InTable2,wepresentthemeancosinesimilarityforeach
mechanismintruthfulnessandsocialwelfare.
datasetwediscussinthispaper.Highercosinesimilarities
DataSets: Weusebothrandomlygenerateddataanddata indicate greater agreement among agents about the relative
from prior studies on preferences of human subjects over values of other agents. We also show the number of agents
eachother: ineachdataset.
• Random-similar (R-sim) (Othman, Sandholm, and
C |N| k k
Budish 2010): Each agent i, i ∈ {1,2,...,|N|}, is as-
Random-similar20 0.914 20 5 5
signedthepublicvaluei.Aprivateerrortermisaddedto
Random-scattered20 0.499 20 5 5
thepublicvalueofitoderivethevalueofitoeachother
Newfrat 0.877 17 4 5
agentj,drawnindependentlyfromanormaldistribution
Freeman 0.551 32 5 6
withzeromeanandstandarddeviation|N|/5.Theprivate
error term is redrawn until the sum of the private error
Table 2: Mean cosine similarity over all pairs of distinct
termandpublicvalueisnon-negative.Thenthevalueofi
agents; number of agents; minimum team size; and maxi-
toj isthesumofiandtheprivateerrorterm.
mumteamsize.Forrandomdatasetclasses,C asshownis
• Random-scattered (R-sca): In this data set class, the themeanover20randomlygeneratedinstancesoftheclass.
value of a player i is generated independently by each
otherplayerj.Todeterminethevalueofotherplayersto
EmpiricalAnalysisofIncentiveCompatibility: Tostudy
playerj,atotalvalueof100isdividedatrandomamong
the incentive compatibility of the mechanisms, we used a
theotherplayersasfollows.Uniformlyrandomnumbers
protocol similar to that used by Vorobeychik and Engel
∈ [0,100]aretaken,todividetheregioninto|N|−1re-
for estimating the regret of a strategy profile (Vorobeychik
gions. Therandom drawsfor agentj aresorted, produc-
andEngel2011).Weraneachmechanismon8versionsof
ing|N|−1valuesfortheotheragents,asthedifferences
each data set, with different random orders over the play-
betweenconsecutivedrawsinsortedorder.
ers,whichweheldincommonacrossdatasets.Wegenerate
• Newfrat: Thisdatasetcomesfromawidelycitedstudy deviations from truthful reporting for agent j one at a time
by Newcomb, in which 17 students at the University of until25uniquedeviationshavebeenproduced.Toproducea
Michiganin1956rankedeachotherintermsoffriendship deviationfromanagent’struthfulvaluesforotheragents,we
ties. We use the data set from the final week, NEWC15 firstrandomlyselectanumberofpairsofvaluestoswapac-
(Newcomb1958).Weletk =4,k =5. cordingtoaPoissondistributionwithλ = 1,with1added.
• Freeman: Thedataarefromastudyofemailmessages For each pair of values to swap, we first select the rank of
sentamong32researchersin1978.Weusethethirdma- oneofthem,withlower(better)ranksmorelikely.
trix of values from the study. The data show how many TheresultsareshowninTable3.3
emailseachresearchersenttoeachotherduringthestudy, RSD is not shown, since it is provably strategyproof,
whichweuseasaproxyforthestrengthofdirectedsocial but, remarkably, A-CEEI-TF empirically produces higher
links(FreemanandFreeman1979).Weletk =5,k =6. (worse) regret of truthful reporting than HBS or OPOP,
even though A-CEEI-TF is strategyproof-in-the-large, and
Fortherandomlygenerateddatasets,weset|N| = 20and
theothersarenot.BothHBSandOPOPappeartoofferplay-
k = k = 5,andourresultsareaveragedover20generated ersonlysmallincentivestolie,withHBSslightlybetter.
preferencerankingsforallplayers.
SocialWelfare: Tofacilitatecomparison,wenormalizethe
The four classes of data set we analyze differ most
totalutilityofallteammatesforeachagentto1,sothatso-
saliently in their number of agents, |N|, and in the degree
cialwelfare(alreadynormalizedforthenumberofplayers)
of similarity among the player preferences. For example,
falls in the [0,1] interval. For comparison, we also include
Random-similaragentslargelyagreeonwhichotheragents
aremostvaluable,whileRandom-scatteredagentshavelit- 3We only report results for the two random data sets and
tleagreement.Differencesindegreeofpreferencesimilarity Newfrat,asitwasnotfeasibletorigorouslyanalyzeregretforthe
leadtomarkeddifferencesintheperformanceoutcomesof farlargerFreemandataset.(However,theFreemandatasetissim-
thevariousmechanisms. ilartoRandom-scattered;seesupplementalmaterialfordetails.)
R-sim. R-sca. Newfrat Newfrat),itdominatesallothersonthedissimilardatasets
HBS 0.02±0.02 0.04±0.02 0.03±0.02 (Random-scatteredandFreeman).
OPOP 0.07±0.02 0.10±0.05 0.06±0.02
A-CEEI-TF 0.19±0.04 0.29±0.07 0.19±0.03 R-sim. R-sca.
Max-welfare 0.20±0.03 0.29±0.07 0.22±0.03 RSD 0.43±0.03 0.59±0.05
A-CEEI-TF 0.66±0.05 0.62±0.05
Table3:Meanmaximumobservedregretoftruthfulreport-
HBS 0.71±0.04 0.61±0.06
ing,with95%confidenceintervals.
OPOP 0.70±0.06 0.79±0.04
Table 6: Mean fraction of agents with envy bounded by a
optimalsocialwelfareforbothRandomdatasets,aswellas
single teammate for the two Random data sets, with 95%
Newfrat.4
confidenceintervals.
The only related theoretical result is that RSD is ex post
Paretooptimal;theotherthreemechanismsdonotevenpos-
sess this guarantee. This makes our results, shown in Ta-
Newfrat Freeman
bles 4 and 5, remarkable: on Random-similar and Newfrat
RSD 0.36±0.02 0.43±0.05
data sets (both with preferences relatively similar across
A-CEEI-TF 0.57±0.03 0.55±0.05
players),thereislittledifferenceinwelfaregeneratedbythe
HBS 0.67±0.05 0.64±0.04
different mechanisms, but on Random-scattered and Free-
OPOP 0.68±0.07 0.78±0.05
man data sets, OPOP statistically significantly outperforms
theothers.
Table 7: Mean fraction of agents with envy bounded by a
singleteammatefortheNewfratandFreemandatasets,with
R-sim. R-sca.
95%confidenceintervals.
RSD 0.22±0.004 0.25±0.01
A-CEEI-TF 0.22±0.004 0.25±0.01
Next,weconsiderinformalperspectivesonthefairnessof
HBS 0.22±0.004 0.25±0.02
thevariousmechanisms.Formechanismsthatusearandom
OPOP 0.22±0.003 0.27±0.01
serialorderoverplayers,wecanstudytypicaloutcomesfor
Max-welfare 0.25±0.001 0.35±0.01
aplayergivenitsserialindex.Ifplayerswithlower(better)
Table4:MeansocialwelfareforthetwoRandomdatasets, serialindexesreceivedrasticallybetteroutcomesthanagents
with95%confidenceintervals. with higher (worse) indexes, such a mechanism is not very
fair. In Figure 1 (left), we plot a smoothed version of the
meanfractionoftotalutilityachievedbyagentsateachran-
Newfrat Freeman dom serial index from 0 to 19, for the mechanisms RSD,
RSD 0.23±0.01 0.20±0.01 HBSdraft,OPOPdraft,andA-CEEI-TF.Resultsarebased
A-CEEI-TF 0.23±0.01 0.19±0.01 on 20 instances of Random-scattered preferences, held in
HBS 0.22±0.05 0.20±0.01 commonacrossthemechanisms,withdifferentserialorders
OPOP 0.22±0.05 0.24±0.02 over the players. From this Figure, it is apparent that ran-
Max-welfare 0.27±0.00 - domserialdictatorshipgivesfarbetteroutcomestothebest-
ranked agents than to any others. Surprisingly, A-CEEI-TF
Table 5: Mean social welfare for the Newfrat and Freeman follows a similar pattern in this case, although we did ob-
datasets,with95%confidenceintervals. servethatforsomeothergametypes(notshown),A-CEEI-
TF’scurvegivesbetteroutcomestolow-rankedagentsthan
RSD.IntheHBSdraft,a“shelf”ofhighutilityforthesev-
Fairness: Fairness of an allocation (in our case, a parti-
eralbest-rankedagentsistypical,asalloftheteamcaptains
tionofplayers)canbeconceptuallydescribedastherelative
receivesimilarlyhighutility,withasteepdrop-offinutility
utility of best- and worst-off agents. Formally, we measure
fornon-captainagents.IntheOPOPdraft,incontrasttoall
fairness in the experiments as the fraction of agents whose
others, the utility curve is far more flat across random se-
envyisboundedbyasingleteammate(asdefinedabove).
rial indexes: even the agents with high (bad) serial indexes
Our fairness results, shown in Tables 6 and 7 are unam-
achievemoderatelygoodoutcomesforthemselves.
biguous: RSD is always worse, typically by a significant
margin,thentheothermechanisms.Thisisintuitive,andis
Somemechanismsforteamformationtendtogivebetter
preciselythereasonwhyalternativestoRSDarecommonly
outcomestoanagentthatis“popular,”havingahighmean
considered.WhatisfarmoresurprisingisthatA-CEEI-TF,
valuetotheotheragents.Forexample,randomserialdicta-
inspiteofsometheoreticalpromiseonthefairnessfront,and
torshipbiasesoutcomesinfavorofpopularplayers,because
inspiteofbeingexplicitlydesignedforfairness,isinallbut
even if a popular player is not a team captain, it is likely
onecasethesecondworst.WhileHBSandOPOParecom-
thatthisplayerwillbeselectedbysometeamcaptainalong
parableonthehigh-similaritydatasets(Random-similarand
withotherdesirableplayers.Anunpopularplayer,however,
4ItwasinfeasibletocomputethisfortheFreemandatasetdue will likely be left until near the final iteration of RSD, to
toitssize. be selected along with other unpopular players. Therefore,
20 References
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ij ji
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ii
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2011. Average-case analysis of VCG with approximate resource beinthisrange.Finally,werequirethatanytworowsinxeither
allocationalgorithms. DecisionSupportSystems51(3):648–656. nothavea1inanyofthesamecolumns,ormustbeidentical;this
meansthatifsomeagentiappearsontwoteams(i.e.,intworows),
Appendix thoseteamsmustcontainexactlythesameagents.
TeamSizeConstraintsthatAdmitaFeasible
|N| |N|
Partition (cid:88)(cid:88)
max: x u
ij ij
x
Sometuples(N,k,k)arenotfeasible,meaningthatitisnotpossi- i=1j=1
bletodivide|N|playersintoteamswithsizesin[k,k].Recallthat subjectto:
werequirebydefinition1 ≤ k ≤ k ≤ |N|.Foraminimalexam- x∈{0,1}|N|×|N|
ple,itisnotpossibletodivide3playersintoteamswithk = 2,
x =1:∀i
k=2. ii
x −x =0:∀i,j
Observation1. Atuple(N,k,k)isfeasibleifandonlyif:k di- ij ji
vides|N|,kdivides|N|,or(|N|\k)>(|N|\k),where\signifies (cid:88)|N|
integerdivision. k≤ xij ≤k:∀i
j=1
ComputingASocialWelfare-MaximizingPartition xij+xi(cid:48)j+xij(cid:48) −xi(cid:48)j(cid:48) ≤2:∀i,i(cid:48) (cid:54)=i,j,j(cid:48) (cid:54)=j
Inordertofindasocialwelfare-maximizingpartition,weformulate
the problem as an MIP and solve it using CPLEX. We are given
a feasible team formation problem as (N,(cid:31),k,k), where either
ProofofProposition1
k=k,ork+1=kandneitherknorkdivides|N|.LetT equal
thenumberofteamsthatresultswhenasmanyteamsofsizekas Consider a team formation problem with 6 agents,
possibleareformed,therestbeingofsizek. {A,B,C,D,E,F}, k = 3, k = 3, so that two equal-size
Weintroduceamatrixx∈{0,1}T×|N|,whereeachrowcorre- teamsmustbeformed.Theagents’additiveseparablepreferences
sponds to one team, and the 1 values in the row indicate which areencodedinTable8.
agents are on that team. We also introduce a dummy variable,
S ∈{0,1}T×|N|×|N|,whereS =1ifagentsiandjarebothon A B C D E F
tij
teamt,otherwise0.S(cid:48) ∈{0,1}|N|×|N|is1ifandonlyifagents A x 0 1 2 4 8
ij
iandjareonthesameteam,andisthesumoverT valuesofS B 8 x 4 2 1 0
tij
foriandj. C 8 0 x 4 2 1
TheMIPformaximizesocialwelfareisthen: D 8 1 0 x 4 2
E 8 2 1 0 x 4
F 8 4 2 1 0 x
|N| |N|
max:(cid:88) (cid:88) S(cid:48) (u +u )
x ij ij ji Table8:Eachrowiencodestheadditiveseparablevaluefor
i=1j=i+1 agentiofeachotheragent.
subjectto:
x∈{0,1}T×|N| Nopartitionoftheseagentsintotwoteamsofsize3givesevery
S ∈{0,1}T×|N|×|N| agent envy bounded by a single teammate. To see this, consider
that each agent other than A has a bliss point on a team with A
S(cid:48) ∈{0,1}|N|×|N| andoneotheragent,wherethesecondagentisC foragentB,D
foragentC,andsoonuntil“wrappingaround”withB foragent
2S ≤x +x :∀t,i,j
tij ti tj
F.ThreeoftheagentswillnotbeonateamwithagentA,andat
S ≥x +x −1:∀t,i,j
tij ti tj leastoneoftheseagents,sayagenti,willnotbeonateamwith
T itssecond-favoriteagenteither.Someotheragentj mustthenbe
Si(cid:48)j =(cid:88)Stij :∀i,j onateamwiththetwomost-preferredagentsoftheplayeri.By
t=1 construction,playeriisonateamofvalue3orless,whiletheteam
|N| ofagentj hasvalue12toagenti,andvalue4toagentiwithits
k≤(cid:88)x ≤k:∀t morevaluableplayer(playerA)removed.Therefore,envycannot
tj
beboundedbyasingleteammateforallagents.
j=1
MaximinShareGuarantee Themaximinshareguaranteeis forthisgameprovideseveryagentwithamaximinshare.Incon-
a concept from multi-unit assignment that can be applied to he- sequence,nopartitionforthisproblemisproportionalorenvy-free
donic games; in multi-unit assignment, “bundles” of “items” are either.
allocated to agents instead of “teams.” A maximin share for an
agent is the agent’s least-preferred bundle in a maximin split for ProofofProposition2
theagent.Amaximinsplitforagentiisapartitionofallitemsinto
Random serial dictatorship for team formation is strategyproof,
bundlessuchthateachagentcanreceiveonebundle,wherethepar-
meaningthatitisadominantstrategyforeachagenttoreportits
titionmaximizestheutilitytoioftheleast-valuablebundle.Note
truevaluesforotheragents,regardlessoftheotheragents’reports.
that envy-freeness in a split for an agent implies proportionality,
ConsiderthateachagentaffectsthepartitionreturnedbyRSDonly
andproportionalityimpliesmaximinshares(ProcacciaandWang
iftheagentisateamcaptain,basedontheserialorderoverplay-
2014).
ersandthechoicesofearlierteamcaptains.Therefore,ifanagent
Weintroduceamodifiedversionofthemaximinshareguaran- is not a team captain, the agent’s report makes no difference. If
tee in our setting, the maximin share guarantee for team forma- anagentisateamcaptain,theagentisassigneditsmost-preferred
tion. A partition in a team formation problem provides maximin teambasedontheremainingplayers,teamsizeconstraints,andthe
sharesforteamformation,ifeachagentweaklyprefersitsteamto agent’s reported preferences. Therefore, it is a dominant strategy
its maximin share. Given (N,(cid:31)i,k,k), consider all partitions of fortheagenttoreportitstruepreferences.
theagentsN withteamsizesin[k,k].Amaximinsplitforagent RandomserialdictatorshipisexpostParetoefficientforMDTFs
iisanysuchpartitionthatmaximizesthevalueforioftheleast- withstrictpreferencesoverteams(i.e.,withoutindifferencesover
preferredteamcontainingi,whichwouldresultfromswappingi teams).Thefirstteamcaptainisassigneditsstrictlymost-preferred
withsomeagentj,wherej mayequali.Amaximinshareisthe team, of the largest feasible team size. Thus, for non-negatively
least-preferredteaminamaximinsplit.Thisdefinitionofthemax- valuedMDTFs,nootheragentcouldswaportakeagentsfromthe
iminshareguaranteepreservesausefulpropertyoftheguarantee firstcaptain’steamwithoutdecreasingitsvaluetothecaptain.The
formulti-unitassignment,whichisthatanagentmaybeassigned sameargumentholds,byinduction,forlaterteamcaptains.Thus,
oneof|N|teamsfromitsmaximinsplit(uptothelossoftheother thepartitionreturnedbyRSDisexpostParetoefficient.
agentintheswapfromateam,inthecaseofteamformation). Randomserialdictatorshipforteamformationisnotenvy-free,
The maximin share guarantee for team formation, in the case becauseforsomeprobleminstances,noenvy-freepartitionexists,
ofmechanismdesignforteamformationproblems(MDTFs)with asshowninProposition7.
non-negativevalues,istriviallysatisfiablefork=|N|:Ifthegrand
ProofofProposition3
coalitionisassigned,everyagent’spayoffismaximized,soenvy-
freeness is achieved. The maximin share guarantee for team for- TheHarvardBusinessSchooldraftforteamformationisnotenvy-
mation, in the case of MDTFs with non-negative values, is also free, because for some problem instances, no envy-free partition
triviallysatisfiablefork = 2,where|N|iseven.Inthemaximin exists,asshowninProposition7.
splitforanagentwithk = 2,allagentsaregroupedinpairs(as- TheHBSdraftforteamformationisnotexpostParetoefficient.
suming|N|iseven),andtheagent’smaximinshareisthepairwith For a counter-example, consider the following team formation
itsleast-favoriteotheragent.Thus,anygroupingoftheagentsinto problem,withk = k = 3andplayerorder(A,B,C,D,E,F),
pairssatisfiesthemaximinshareguarantee. suchthatAandBwillbetheteamcaptains.Theagentpreferences
areencodedinTable10.
Proposition7. ForsomeMDTFswithnon-negativevalues,k≥3,
andk < |N|,themaximinshareguaranteeforteamformationis A B C D E F
not satisfiable. As a result, proportionality and envy-freeness are A x 1 8 0 6 4
notsatisfiableeitherinsuchcases. B 1 x 10 0 5 3
C 0 8 x 5 4 2
Proof. Consider a team formation problem with 6 agents, D 0 8 5 x 4 2
{A,B,C,D,E,F},k = 3,k = 3,sothattwoequal-sizeteams E 8 0 5 4 x 2
mustbeformed.Theagents’additiveseparablepreferencesareen- F 8 0 5 4 2 x
codedinTable9.
Table 10: Each row i encodes the additive separable value
A B C D E F foragentiofeachotheragent.
A x 1 0 2 3 2
B 2 x 1 0 3 2
Inthisexample,thefollowingselectionsaremadeintheHBS
C 2 2 x 1 0 3
draft.AselectsC,B selectsE,B selectsF,andthenAselects
D 1 2 2 x 2 1 D.Theresultingpartitionis{(ACD),(BEF)}.Thispartitionis
E 0 2 3 1 x 2 notexpostParetoefficient,becausealltheagentswouldpreferthe
F 2 1 2 2 1 x partition{(AEF),(BCD)}.
The HBS draft for team formation is not strategyproof. Con-
Table9:Eachrowiencodestheadditiveseparablevaluefor sider a minimal-size example of a problem instance with non-
agentiofeachotheragent. negative values, where the HBS draft is not strategyproof. Let
N = {A,B,C,D,E,F}, k = 3, and k = 3. The players’ ad-
ditiveseparablevaluestoeachotherareencodedinTable11.
Itiseasytoseethatintheproblemdescribedabove,eachagent’s IntheproblemshowninTable11,agentAgetsgreaterutility
maximinshareforteamformationhasvalue3.Furthermore,anyof bymisreportingitspreferencesasinA(cid:48) for24serialordersover
the10possiblepartitionsoftheagentsintotwoequal-sizeteams the players, and never receives worse utility than when reporting
leavessomeagentwithutilityof2orless.Therefore,nopartition A.Ineachofthe24caseswhereAgainsbyreportingA(cid:48),utility
A B C D E F
TheOne-Player-One-PickdraftforMDTFsisnotenvy-free,be-
A x 5.0 4.9 7.0 0.2 0.0 cause for some team formation problem instances, no envy-free
A(cid:48) x 5.0 6.0 7.0 0.2 0.0 partitionexists,asshowninProposition7.
B 0.0 x 1.1 1.6 1.2 1.3 The OPOP draft for MDTFs is not ex post Pareto effi-
C 0.0 1.1 x 1.6 1.2 1.3 cient, as shown in the following minimal example. Let N =
D 0.0 1.1 1.6 x 1.2 1.3 {A,B,C,D,E,F}, k = 3, and k = 3. Let the serial order of
E 0.0 1.1 1.6 1.2 x 1.3 theplayersbealphabetical,soAselectsfirst,andsoon.Theplay-
F 0.0 1.1 1.6 1.2 1.3 x ers’ additive separable values to each other are encoded in Table
13.
Table 11: Each row i encodes the additive separable value
A B C D E F
foragentiofeachotheragent.RowA(cid:48) showsthefalsere-
A x 2 10 9 6 0
portofagentA.
B 0 x 10 9 2 6
C 0 10 x 2 6 9
D 10 0 2 x 6 9
improvesfrom5.2to9.9,asthereceivedteamshiftsfrom(ABE)
E 10 0 2 6 x 9
to (ABC). This occurs when the first player to choose is A and
thesecondisD.Therefore,theHBSdraftisnotstrategyproof,be- F 0 10 2 6 9 x
causeintheexampleshown,playerAgetsbetterexpectedutility
bymisreportingitspreferences. Table 13: Each row i encodes the additive separable value
foragentiofeachotheragent.
ProofofProposition4
Consider a minimal-size example of a team formation problem In the example problem in Table 13, the following selections
with non-negative values, where the OPOP draft is not strate- aremade,inorder.TeamcaptainAselectsC,teamcaptainB se-
gyproof.LetN = {A,B,C,D,E,F},k = 3,andk = 3.The lectsD,C (alreadyteamedwithA)selectsF,andasaresultD
players’additiveseparablevaluestoeachotherareencodedinTa- (already teamed with B) must select E. This produces the parti-
ble12. tion{(ACF),(BDE)}.Everyplayerwouldreceivegreaterutil-
ity from the alternative partition {(ADE),(BCF)}. Therefore,
A B C D E F theOPOPdraftisnotexpostParetoefficient.
A x 5.0 4.9 7.0 0.2 0.0
A(cid:48) x 5.0 6.0 7.0 0.2 0.0 DetailsofA-CEEI-TF
B 0.0 x 1.1 1.6 1.2 1.3 Eachagentreportstothemechanismtheadditiveseparableutility
C 0.0 1.1 x 1.6 1.2 1.3 itgainsfromeachotheragent.Themechanismusesthesereports
D 0.0 1.1 1.6 x 1.2 1.3 to derive which affordable team each agent demands, given each
E 0.0 1.1 1.6 1.2 x 1.3 agent’sbudgetandprice.Thetotalutilityforanagentiofateam
F 0.0 1.1 1.6 1.2 1.3 x ofotheragentsofsizein[k−1,k−1],isthesumoftheutilities
foragentioftheagentsinthatteam.Theclearinghousewillpro-
Table 12: Each row i encodes the additive separable value visionallyassignagentitheteamofhighestutilityforithatitcan
foragentiofeachotheragent.RowA(cid:48) showsthefalsere- afford,ortheemptyteamifitcannotaffordanyteamoflegalsize.
Just as in combinatorial matching, we cannot in general hope
portofagentA.
foranexactmarketclearingsolution(and,consequently,anexact
CEEI)inoursetting:
TheOPOPdraftisarandomizedmechanism,whichworksby Proposition8. Thereexistteamformationsettingswherenoprice
uniformly randomly choosing a permutation of the players, then andbudgetvectors(p,b)existthatinduceexactmarketclearing.
actingdeterministicallybasedonthatorder.Strategyproofnessfor
arandomizedmechanismmeansthataplayer’sexpectedpayoffis Proof. We define exact market clearing for team formation to
maximized by truthful reporting, regardless of other players’ ac- mean that when each agent is allocated its favorite team that is
tions.ToshowthattheOPOPdraftisnotstrategyproof,itsuffices affordable:
toshowthatplayerAgainsinexpectedpayoff,basedonthetrue
• Each agent is assigned to a team with size in [k,k], including
preferencesinrowAofthetable,bymisreportingitspreferences
asinrowA(cid:48)inthetable,ifotherplayersreporttheirpreferencesas itselfintheteamsize.
displayedinthetable. • Foranytwodistinctagentsiandj,idemandsjifandonlyifj
Theexampleproblemhas6players,sothereare6!=720per- demandsi.
mutations,and720equallylikelyoutcomesfromtheOPOPdraft,
Thisresultmarksadistinctionfromthecourseallocationprob-
someofwhichareidenticalintheirinducedpartitions.Itturnsout
that the payoff for player A is better when reporting A(cid:48) instead lem (also known as combinatorial assignment), where with suffi-
cientlyunequalbudgets,somepricevectormustexistthatinduces
of A for 18 of 720 permutations, and worse in 6 others. Specifi-
exactmarketclearing(Budish2011).
cally,playerAperformsbetterbylyingifandonlyiftheorderof
Example. Consider a team formation problem with 4 agents,
the first three players to choose is (ADB), (ADE), or (ADF).
PlayerAdoesworsebyreportingA(cid:48)ifandonlyiftheorderofthe {A,B,C,D},withk = 2andk = 2.Theagents’additivesepa-
firstthreeplayersis(ADC).Afterfactoringinthevalueofthere- rablepreferencesareencodedinTable14.
sultingpartitionstoplayerA,itresultsthattheexpectedgainfrom In the example in Table 14, only two resulting partitions
defectingtoA(cid:48)is0.08utilspergame.Thus,theOPOPdraftisnot must be considered, due to symmetry: {{A,B},{C,D}} and
strategyproof. {{A,C},{B,D}}.