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2nd ReadingJuly 8, 2013 13:53 WSPC/0219-1989 151-IGTR 1340021

International Game Theory ReviewVol. 15, No. 3 (2013) 1340021 (10 pages)c World Scientic Publishing CompanyDOI: 10.1142/S0219198913400215

COOPERATIVE GAME THEORY IN SPORTS

CONRADO MANUEL, ENRIQUE GONZALEZ-ARANGUENAand MONICA DEL POZO

Escuela Universitaria de EstadisticaUniversidad Complutense de Madrid

Av. Puerta de Hierro s/n 28040 Madrid, [email protected]@estad.ucm.es

[email protected]

Received 31 August 2011Revised 30 March 2012Accepted 18 March 2013Published 9 July 2013

This paper contains a survey of cooperative game theory applied to a sports environment.The variety of these applications serves us as a proof of the strength of cooperative gametheory introducing successful strategies in sports and explaining the behavior of dierentactors.

Keywords: Sports; cooperative game theory.

Subject Classication: 22E46, 53C35, 57S20

1. Introduction

In recent years there has been an increasing interest in applying the ideas of gametheory to analyze the environment of sports. The seminal work on the economics ofsport is debt to Rottemberg (1956). Since that date it is frequent to model problemsrelated to sports using an economic approach and specially the game theory toolswhich are now standard in most of the economic literature. The Handbook on theEconomics of Sport (Andre and Szymanski, 2006) contains an excellent collectionof papers devoted to analyze the environment of sports from an economic point ofview.

Ordinary people, and not only scientists, are interested in sports. Moreover,game theory is based on a set of ideas that can be explained to the layman. Soit is not surprising that the use of game theory to analyze or to explain problemsrelated to sports appears frequently in the popular press and on the Internet. Seefor example Ronfeldt (2000), Islam (2000), Waters (2001), Varma (2004) or Gupta(2005).

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C. Manuel, E. Gonzalez-Aranguena & M. del Pozo

Most of previous cited works use noncooperative models of game theory toanalyze sports. This is so because sport has traditionally been described in termsof competition. Games are often called battles and fans frequently say that the rivalteam must be demolished, using aggressive vocabulary. Nevertheless the cooperativegame theory has something to say in describing the sport environment. From nowon, this paper is devoted to describe those scientic papers that use cooperativemodels to analyze some particular aspects in sports, emphasizing the particularmodel used. Section 2 deals with the applications to sports that use cooperativegames and digraphs, these digraphs representing domination in competitions orpasses inside a team in dierent plays in team sports. Sections 3, 4, 5 and 6 arerespectively devoted to applications using status games, team games, multi-choicegames and competitive-cooperative games. The paper ends with a brief section ofconclusions.

2. Cooperative Games and Directed Communication Networks

In this section, we deal with the applications of game theory to the sports eld inwhich the framework can be modeled with a directed graph. In a directed networkeach relationship has an initiator and a respondent. These networks have been usedin the context of sports to describe all type of sport competitions or to describeplays among players in team sports. Cooperative games with restrictions givenby a digraph have contributed to obtain measures of performances of teams insuch sports tournaments or to measure centrality of players in matches of soccer,basketball, handball, and so on.

Let us recall that a game in characteristic function form (a coalitional game ora TU game) is a pair (N, v) where v (the characteristic function) is a real functiondened on 2N , the set of all subsets of N (coalitions), that satises v() = 0.For each S 2N , v(S) represents the utility that players in S can obtain if theydecide to cooperate. Sometimes, the formation of coalitions is a process in whichnot only the members of the coalitions are important but also the order in whichthey appear. As an example we have the order in which players participate, in aparticular play of soccer. Obviously this order is very relevant when trying to obtaina goal. A game in generalized characteristic function form (Nowak and Radzik,1994) is a pair (N,w), w being a function dened on (N) =

SN (S), with

(S) the set of all permutations of players in S, for all S N , and satisfyingw() = 0.

If we denote GN the vector space of all generalized cooperative games withplayers set N , the set GN of all TU games can be identied with the subspace ofGN consisting of all games w GN for which w(S) = w(T ) if players in S are thosein T . A family of point solutions for generalized games is:

i (N,w) =

T(N),iTw(T )

ti(T )

t!t1

l=0 l,

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Cooperative Game Theory in Sports

where i(T ) is the position of i in the ordered coalition T , [0, 1] and w(T ) =w(T ) RT,R=T w(R) =

RT (1)trw(R). For all [0, 1], extends

the classical Shapley value (Shapley, 1953) for games in GN . For 0 is dened inNowak and Radzik (1994) and 1 is dened in Sanchez and Bergantinos (1997).

A directed network (directed graph or digraph) is a pair (N, d), N = {1, 2 . . . , n}being a set of nodes and d a subset of the collection of all ordered pairs (i, j), i = jof elements in N . Each pair (i, j) d is called an arc. It is said that i is a predecessorof j and that j is a successor of i. For our purposes, a successor of team i in sportcompetitions is a team that has been beaten by i, whereas a predecessor is a teamthat defeated i. When nodes represent players in team sports, the arc (i, j) modelsa pass from i to j in a particular play of a match. We will denote DN for the set ofall possible digraphs with nodes set N .

Given d DN and i N , d(i) is the set of i-successors (those beaten by i orthose to which i pass the ball during a sequence of passes in a given play) andd1(i) is the set of i-predecessors. An ordered set of nodes in N , (i1, i2, . . . , ir), isconnected in d if the arcs (ik, ik+1) are all in d for k = 1, 2, . . . , r1. Given d DN ,d DN is a simple subnetwork of d if d(i) d(i) for all i N and |d1(i)| = 1 forevery dominated player i d(N) = {j N | d1(j) = }.

2.1. Cooperative games and domination networks

Domination networks have been used to describe sports competitions. A dominationnetwork is a digraph in which each arc represents the victory of the initiator node(team) over the receiver one.

Cooperative game theory allows us to measure how well dierent teams performin sport competition which is represented via a directed network. We will describetwo measures here. The rst one and the simplest one is the degree of a playerin the network. The second one, the -measure, was introduced by van den Brinkand Gilles (1994) and later analyzed in van den Brink and Gilles (2000). As for themeasuring of the performance, the dominance over a player beaten by many otherplayers weights less than the dominance over a player who has only few dominators.

2.1.1. The degree measure

The degree measure is probably the most traditional representation of scorings insport competitions. It measures the dominance of each player by the number of winsthe player obtains. It is dened as the map : DN Rn such that i(d) = |d(i)|for each (N, d) GN and each i N .

Van den Brink and Gilles (2000) characterize this measure in terms of the follow-ing properties for an arbitrary measure, m, dened for the collection of all directednetworks:

degree normalization: for every d DN it holds that iN mi(d) = |d|

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dummy position property: for all d DN and all i N such that d(i) = , itholds that mi(d) = 0.

symmetry: for all d DN and all i, j N such that d(i) = d(j) and d1(i) =d1(j) it holds that mi(d) = mj(d).

additivity over independent partitionsa: for all d DN and every independentpartition {d1, . . . , dr} DN it holds:

m(d) =r

k=1

m(dk).

Of course, it can be said that apparently there is no relationship between pre-vious results on the degree measure and game theory. This relation is establishedin what follows.

2.1.2. The -measure

The -measure is the function : DN Rn dened: i(d) =

jd(i)|

|d1(j)| foreach i N . The idea is that in any tournament the value of a teams victory isdecreasing with the number of defeats suered by the beaten team. Besides thedegree measure has interesting properties and in fact is very used to score theperformance in competitions, the -measure is more related to the cooperativegame theory tools. In particular, given d DN , consider the following two TUgames:

the optimistic successor game vd GN , dened

vd(S) =

iSd(i)

,

for all S N . Thus, the value of a coalition is the number of teams that hasbeen beaten by teams in the coalition (van den Brink and Gilles, 2000)

the conservative successor game ud GN denedud(S) = |{j N | d1(j) S}|,

for every S N . The value of a coalition of teams is the number of teams thathave been beaten for all the members in the coalition.

Van den Brink and Gilles (2000) proved that the -measure of each d DN isequal to the Shapley value of the optimistic successor game vd and van den Brinkand Borm (2002) showed that it coincides as well with the Shapley value of theconservative successor game ud.

aAn independent partition of a directed network d is a partition {d1, . . . , dr} DN such that forall i N , (a) d(i) = Srk=1 dk(i), (b) dk(i) dl(i) = , k = l and (c) each player is dominated atmost in one subnetwork in this position.

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Moreover, the -measure of each d DN is proportional to the sum of the degreemeasures of all simple subnetworks of d. All previous results can be extended tothe modied -measure. See, for details, van den Brink and Borm (2002) or Gilles(2010). To show the implications of that measure, authors apply it to rank teamsin the group C of the UEFA European Soccer Championship (EURO 2000). Thisgroup consisted of the following national teams: Spain, Yugoslavia, Norway andSlovenia. As it is known, the UEFA ranks the teams by rewarding them 3, 1 and 0points for a win, a draw and a loss, respectively (in the digraph authors representa draw with two opposite arcs). Because of the dierent matches results the UEFAranking was: Spain, 6 points; Yugoslavia, 4 points; Norway, 4 points and Slovenia,2 points. The -measure ranks Norway higher than Yugoslavia and even equal toSpain, as Norway won the match from the stronger team in the group, Spain. Ofcourse, it will be very dicult to convince the UEFA to apply such a ranking. TheUEFA is a very conservative organism, even if several years ago they changed theranking to promote a more oensive play. Moreover, the clubs probably would refusea system of ranking under which dierent victories have dierent reward. We thinkthat it is not clear if this ranking system would aect the behavior of teams duringmatches. There are dierent theories supporting every side: some people think thatteams play very motivated and they do their best, when they play against a verygood team; other people think that in many situations, weak teams save energyand even their best players to play against strong teams so that they can be freshand have their best players when they play against equal strong teams.

2.2. Centrality of players in team sports

Let us consider now the situation in which the digraph represents connections orpasses among players in team sports. Then the arc (i, j) will model the situationin which player i passes the ball to player j in a particular play of a match ofsoccer, basketball, handball or a similar team sport. Dierent plays give us dier-ent digraphs. Del Pozo et al. (2011a, 2011b) develop a game-theoretical central-ity measure that, in particular, can be used to measure centrality of players inteam sports and to describe the type of strategy used in order to win matches.Given a game v GN representing the interests among players, v can be expressedas a linear combination of games in the unanimity basis {uS}=SN as follows:v =

=SN v(S)uS . The coecients {v(S)}SN are known as Harsanyi div-

idends (Harsanyi, 1963). Authors propose to assign for each digraph d DN andeach unanimity game uS, = S N , the restricted game udS GN (the game ofto connect S in d) dened by:

udS =

T(S),T connected in dwT .

The idea behind this denition is that the connection of players in S admits dierentpossible orders that correspond to the dierent sequences of passes among players

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in S. The centrality for player i in a particular play d is dened as i (N, udS). In this

case [0, 1] is a parameter that measures the relative diculty of passing the ballwith respect to controlling it. In general, it is assumed that to pass is easier than toreceive because of the pressing of the defenders. The total centrality of a player canbe obtained as some weighted combination of his centralities in dierent plays dk. Toobtain a measure for all games, authors extend previous denition using linearity.The so dened measures are stable (nondecreasing for both incident players whenadding an arc) for almost positive games and can be characterized in terms ofcomponent eciency and -directed fairness (except for = 0). Embedded in thecentrality of each play, it can be distinguished a part caused by its communicationactivity (to pass the ball and to receive it) and another one explained by its controlon others activity. As a consequence, the centrality of each player is the city-blockmodulus of the three-dimensional vector of emission, betweenness and receptioncentralities. If we apply this to players in a soccer match we obtain a description ofthe type of participation in the game for dierent positions in the team. Typicallythe centrality of defenders will be higher for the emission component as they usuallystart plays. A middle eld player will have grater betweenness centrality (as his jobgenerally consists of connecting defenders with forwards, and these ones have morereception centrality because they tend to nish the dierent plays. The total sumof centralities of players in a team gives us some idea of the strategy followed bythat team. For teams that dominate the game and the ball possessing time the totalcentrality will be higher than the corresponding total one for teams with a moredirect style based on long passes or in fast counterattacks.

In a related context, Amer et al. (2010) establish conditions on cooperativegames so that they can be used to measure accessibility to the nodes in a directedgraph. In their model, they use generalized TU games (games in generalized char-acteristic function form) as they were introduced by Nowak and Radzik (1994)and studied later by Sanchez and Bergantinos (1997) and del Pozo et al. (2011b).Moreover, a detailed study of accessibility in a concrete example, the EuropeanBasketball Championship (EUROBASKET 2009), is oered.

3. Status Games

With the aim of analyzing status using the mathematical tools of game theory,Quint and Shubik (1999) introduced the status games: n-player cooperative gamesin which the outcomes are orderings of the actors within a hierarchy, i.e., permu-tations of N . In many frameworks the amount of money that receives an actor isless important than his position or rank in relation to others. Sports appear in thepaper as examples of the importance of the position relative to others. In general,tennis players place more importance on their position in ATP ranking than onthe amount of money they earn. Sometimes a soccer player leaves a team becausehe wants to be the highest paid player (when Etoo left the FC Barcelona, it wassaid that he wanted to earn more money than Messi). The theory of status-games

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analyzes the status, instead of the wealth, using the mathematical tools of gametheory. In a status game with set of players N , outcomes are represented by permu-tations of N where, if i occupies the position j in the permutation, this means thatplayer i attains the jth best position, and it is assumed that players will alwaysdesire to be placed as far up as possible in the hierarchy, i.e., the closer to the rstplace, the more possible.

Later, Quint and Shubik (2001) distinguish two classes of status games basedon what is assumed about the capabilities of a coalition S N : exo-status gamesand endo-status games. In exo-status games, it is assumed that such a coalition iscapable of enacting orderings over all n players. In endo-status games, each coalitionS is only capable of guaranteeing certain positions for its s members, but it hasno control over the positions that the remaining n s players can occupy in thehierarchy. In these games each coalition has an exogenously given set S of rankingsof positions (matrices) that its members can enforce. Authors dene a conditionof balance on the family {S}SN to guarantee that the core of the game isnonempty.

In this last paper, authors also dene another class of games: the one-to-oneordinal preference (OOP) games. This is to be the largest possible class of gamesin which players may express their preferences ordinary over a set of single objectswith which they are to be matched. This broader class includes the exo- and endo-status games. Formally an OOP game is a quadruple (N, J,R, {(S)}SN), inwhich N = {1, 2, . . . , n} is the players set, J = {1, 2, . . . , n} is a set of individualobjects (physical goods, as cars, or else positions in a hierarchy), R is a matrixof rankings and (S) is the set of all orderings that S can eect. Authors makecertain consistency assumptions about the sets {(S)}SN : (i) disjoint coalitionscannot both have the power to assign objets to the same player, (ii) they cannotboth assign the same object, (iii) super-additivity (if PS1 (S1), PS2 (S2) andS1 S2 = , then PS1 + PS2 (S1 S2).

Finally, authors dene the core of an OOP game and prove that this core isnonempty.

4. Team Games

Section 4 in Hernandez-Lamoneda and Sanchez-Sanchez (2010) deals with teamgames. They dene a team game as a game that vanishes on all coalitions S Nbut those of a certain xed cardinality. For example, if we think of N as the set ofall players in the Spanish Soccer League and v(S) is the worth of a soccer team onthe ground then only makes sense to consider v(S) for coalitions of cardinality 11.Then, they look at the power indices ranking of the players for these team games,and show that there is essentially one possible power index ranking (up to sign) forthem. Authors use results in Hernandez-Lamoneda and Juarez (2007), in which thespace of games was decomposed as a direct sum of three orthogonal subspaces: thesubspace of symmetric games, another subspace that they call U (and that does

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not have a natural denition in terms of well-known game theoretic considerations),and the common kernel of all linear symmetric solutions.

To illustrate their results, authors use the basketball example, with n basketballplayers and s = 5 for the cardinality of the only relevant coalitions. They dene atheoretical team game of ve players for basketball which then implies a rankingfor the players, and they provide, in this way, an answer to The Michael Jordanproblem (Saari and Sieberg, 2001): how it is possible that, even though everyoneagrees Michael Jordan is the best basketball player in the NBA, dierent indiceswhen applied to all the players rank them dierently and not all of them ranksMichael Jordan on the top?

5. Multi-Choice Games

Chou and Hsiao (2010) use multi-choice games to provide some sport applicationson the 21st Summer Deaympics. A multi-choice game (Hsiao and Raghavan, 1993)is a generalization of standard cooperative TU game. For TU games the activitylevel of each player is dichotomous: he is fully or not at all involved in the game.In a multi-choice TU game, each actor has a nite set of dierent activity levels.The duplicate core (Hwang and Liao, 2009) for multi-choice games is an extensionof the core solutions for TU games.

Then, Chou and Hsiao (2010) relate some of the critical factors that Chen andChen (2009) cite as primordial in the success of Taiwan holding international sportgames (political situation, political operation, the sponsor plan and the arrangementexperience) to the axioms of the duplicate core.

Chen and Lin (2010) also use the multi-choice TU games and the duplicate coreto analyze the sport management: as an example, they propose the setting of sportleagues such as Major League Baseball (MLB) or National Basketball Association(NBA). This problem can be modeled as a multi-choice game in which players arethe sport teams and the activity levels, the family of marketing strategies availableto each one.

6. Competitive-Cooperative Games

In a recent paper (Sindik and Vidak, 2008), externally competitive (competitionbetween two teams) but internally cooperative (inside the same team) team sportsare analyzed. These are the so-called sport games (basketball, football, soccer, vol-leyball, etc). Authors apply game theory in the analysis of the consequences ofmaking a decision on tactical performance of an individual player in a team sportfor both the players and opponents teams. The problem of cooperation betweenthe players in the same team during the sport competition (between two teams) issimultaneously considered.

The central part of the article is devoted to analyzing hypothetical situationsin relation to the predictability of a players tactical performance during a game,and the eect onto the tactical performance eciency of the players of his own or

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opponent team. It is assumed that predictability in team sports can be explainedas an asymmetric (individuals do not have the same importance in team perfor-mance), nonzero sum, sequential game of imperfect information (all players havesome knowledge about the moves previously made by all other players, but not alltheir moves). A player cannot make decisions only depending on opponent teamtactical performances (competitive game). He must simultaneously consider thetactical performance of his co-players in the same team (cooperative game). Fourdierent possible situations during team sport competition appear when consider-ing predictability or nonpredictability (for the individuals tactical and technicalactions) in relation with players in the same team or opponents (players in theopponents team). From the (very simplied) hypothesis made by the authors, pre-dictability will generally be better than unpredictability. This is so for the playersin the same team and for the ones in the opponents team.

To sum up, in spite of the authors claims, we think, cooperative game theorydoes not play a relevant role in their model.

7. Conclusion

We hope that readers take some advantage from our eort. Of course, there areprobably papers that have been omitted and deserve to be heard. We apologizeto their authors for this. Our goal is mainly to provide the examples we know inwhich the models of cooperative game theory apply to the analysis of the sportsenvironment. The variety of applications is useful as a proof of the strength ofgame theory introducing successful models to explain the behavior of sportsmen,coaches and teams in many situations. Nevertheless, because of the versatility ofgame theory we hope that, in the future, more contributions based on the use ofcooperative games appear.

Acknowledgments

We would like to thank three anonymous referees for their helpful comments. Thiswork has been supported by the Plan Nacional de I+D+i of the Spanish Gov-ernment, under the project MTM2008-06778-C02-02/MTM.

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