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Ranking players in multiple tournaments
Computers Ops Res. Vol. 23, No. 9, pp. 869-880, 1996
Pergamon
RANKING
0305-0M8(95)00082-8
PLAYERS
IN MULTIPLE
Copyright © 1996 ElsevierScienceLtd Printed in Great Britain. All rights reserved 0305-0548/96 $15.00+ 0.00
TOURNAMENTS
Wade D. Cook, lt:~ John Doyle,2§ Rodney Green2¶ and Moshe Kress 311 1Faculty of Administrative Studies, York University, 4700 Keele Street, North York, Toronto, Ontario, Canada M3J 1P3, 2School of Management, University of Bath, Bath BA2 7AY, U.K. and 3Center for Military Analyses, P.O. Box 2250, Haifa, Israel
(Received December 1994; in revisedform August 1995) Scope and Purpose--In a round robin tournament each of a set of n players competes exactly once with every other player, with the outcome from each match being a win of player i over player j, a loss or a tie. The point of much of the previous research in this area has been to derive a ranking of the players based on their performance in the matches. An example of related work appears in a W. D. Cook, I. Golan and M. Kress, Heuristics for ranking players in a round robin tournament. Computers Ops Res. 15, 135-144 (1988). The tournament structure is not unique to the sports environment, but rather appears in a variety of other application areas. Consumer responses regarding preferences for a set of products, for example, are often expressed via pairwise comparisons--"product i is preferred to product j." Thus, any situation in which elements (players, products, projects. . . . ) are compared in a pairwise sense, may give rise to such a "tournament" structure. In the present paper the single competition setup is extended to a multiple comparison environment. Consider a situation in which a set of products are to be evaluated in terms of K criteria or attributes. For criterion no. 1 (e.g. flavor), suppose that only a subset of the full set of products can be compared (e.g. flavor is not an attribute of some of the products). For criterion No. 2, a different subset of the products are compared, although some members of this second subset may have been in the first comparison as well. So each product is involved in various comparisons, with some products in more sets than others. In addition, some tournaments (subsets) are more difficult than others. The purpose of the research here is to develop models for deriving an overall ranking of the products given the performance in the various tournaments, and taking account of the relative difficulty of the tournaments. Abstract--This paper examines a generalization of the standard round robin tournament. First we consider a set of tournaments wherein each player competes in each member of the set, and the tournaments can be ordinally ranked in order of difficulty. The data envelopment analysis method is used to obtain a ranking for each player, while taking account not only of the players' strength within each tournament, but also the differential difficulty of tournaments. Second, we extend this concept to the case where each player may only compete in a subset of the tournaments. Hence, any given player may have fewer matches than other players. Copyright © 1996 Elsevier Science Ltd
1. I N T R O D U C T I O N
Tournament ranking has been the subject of substantial research efforts over the past several decades. In its most basic form, the round robin tournament consists of a set of pair-wise matches among n players in which all pairs compete exactly once, and each match ends in either a strict fWade D. Cook is a Professor of Management Science in the Faculty of Administrative Studies at York University, Toronto, Canada. He received his Ph.D. in Operations Research from Dalhousie University. His research interests are in the areas of ordinal and multiple criteria decision models, preference structures, and consensus formation. His publications have appeared in a number of academic and professional journals including Management Science, EJOR, CORS, Operations Research, and JORS. :~To whom correspondence should be addressed. §John Doyle is lecturer at the School of Management, University of Bath, U.K., where he teaches courses in psychology, quantitative methods, and information systems. He is currently researching haman aspects of the decision process and techniques (both computer-based and noncomputer-based) to support decision making. ¶Rodney Green is a lecturer in the School of Management, University of Bath. After qualifying as a chemical engineer he worked on the design of computer-based process control systems. The inevitable compromises in such systems led to an interest in Operational Research and he now teaches and researches in that area. IIMoshe Kress is a senior researcher at the Center for Military Analyses (CEMA). He received his Ph.D. in Operations Research from the University of Texas at Austin. His research interests are in the areas of ordinal decision models and preference structures and military operations research. His publications have appeared in a number of Management Science/Operations Research journals, including Management Science, European Journal of Operations Research and
Naval Research Logistics Quarterly.
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Wade D. Cook et al.
decision (a winner and loser) or a draw (tied game). The central problem is to rank order the players in a manner which best reflects the tournament results. Numerous authors including Bent [1] and Moon [2] have concentrated on the structural problems and properties of tournaments, while Goddard [3], Kendall [4] and others have dealt with computational problems of ranking the players based on the outcomes from the matches. Cook et al. [5] present a number of heuristics generally based on row sums or Kendall scores from the tournament matrix, and Ali et al. [6] present a structure for examining distance on tournaments. Attempts to capture player strength were discussed originally by Zermelo [7] and later by Bradley and Terry [8]. Both of these approaches were based on maximum likelihood methods. Wei [9] re-examined the player strength issue by taking kth powers of the tournament matrix. It can be shown that these kth powers converge to the right eigenvector of the tournament matrix. Building on Wei's ideas, Cook and Kress [10], use an mth generation approach to incorporate player strength. Weights on the m generations are derived using DEA (data envelopment analysis) principles adopted from the efficiency analysis concepts of Charnes et al. [11]. For a full discussion of the various approaches to tournament ranking see Cook and Kress [12]. In the present paper we examine a generalization of the basic tournament idea by looking at a set o f tournaments whereby in each member of the set, a subset of the players compete in a round robin sense. We assume that the tournaments are of unequal and unknown difficulty, but can be ranked in order of difficulty. We develop a model for ranking the players that takes account not only of player strength within a tournament, but also of the differential difficulty of competing in the various tournaments. The model also accounts for the fact that a player i may compete in only a subset of the tournaments, hence participating in a different number of matches than may be the case for another player j. The models presented here have application to problem areas outside of the conventional tournament arena: 1.1. Using consumer preferences to evaluate products
Consider the situation faced by a market researcher who is collecting consumer responses pertaining to preferences among formulations of a product (e.g. a pudding mix). A particularly convenient format for soliciting preferences in such a case is pairwise comparisons ("do you prefer product a to product b?"). The outcome from such comparisons can be presented in the form of a binary tournament matrix. Furthermore, when making comparisons among products on the basis of any given attribute, e.g. flavor, it may be that only a subset of the products can be evaluated. 1.2. Evaluating energy R&D projects
The problem of selecting from a set of projects relating to energy research and development is important when limited funds are available to finance such projects. Comparisons are often easiest when pairs of projects are evaluated relative to one another in an ordinal sense. An evaluation committee is commonly charged with the task of prioritizing the projects, basing judgments on information supplied by the proposer. Here, when projects come from different departments, certain pairs may not be comparable on particular criteria, hence an incomplete or partial tournament arises. 1.3. General multiple criteria problems
Clearly, many multiple criteria problems involve making pairwise comparisons of members of a set of alternatives in terms of several criteria. The required outcome from this exercise is generally to obtain a final ranking of those alternatives. As with the R&D project example above, certain pairs of alternatives may not be comparable on certain criteria. Methods developed previously (e.g. the outranking method of Roubens [13]) generally presume that all pairs of alternatives are compared on all criteria. Such methods could likely also be extended to include partial comparisons. 2. COMPLETE TOURNAMENTS 2.1. The single tournament case
Assume that each of n players participates in T tournaments { T/}r=l, with each match between a pair of players in a tournament resulting in either a decision or a draw. Let A t denote the binary
Ranking players in multiple tournaments
871
incidence matrix for tournament Tt, that is a,~-
j" 1
if player i defeats player j in tournament Tt
/ 0,
otherwise
In Cook and Kress [10, 14] player strength within a single tournament is viewed from the perspective o f mth generation wins. Specifically, if we take the mth power (A) m of any tournament (incidence) matrix A, then the entry a U(m) represents the number of Hamiltonian paths of length m from player i to playerj. That is, each of these paths has the property that there are m - 1 players ii, i2,.., im-l such that i >/il >/i2 >/ • .. >/ira-1 ~>J-t It can then be argued that one way o f accounting for player strength is to compute the row sums a i. (m) = E~=I aij(m) of each of the mth generation matrices A(m)(= (A)m), and take as the score or worth for player i, a weighted total M Ri = ~ o~(m)ai.(m). m:
(1)
I
Here, M denotes the numbers of powers of A (generations) computed, and {a(m)}~=l is a set of weights representing the importance of these generations to the player's total score. As proposed by Cook and Kress [14], it is reasonable to assume that the importance c~(m) attached to an ruth generation win should be at least as great as the importance o~(m + 1) associated with an m + 1st generation win. In this regard it is proposed that any reasonable set of multipliers or weights should satisfy a ( m ) - c~(m+ 1)~> e,
m= 1,...,M-1
and a ( M ) >t e, for some infinitesimal e. Clearly, as will be seen below, one could impose additional restrictions on the a(m), and as well e might be replaced by an actual positive lower bound that is more than an infinitesimal. We will, however, not concern ourselves with such additional restrictions here. In the case of the single tournament, Cook and Kress propose rating the players on the basis of the R i as defined by equation (1), but where the multipliers c~(m) are chosen in accordance with an optimization procedure that gives each player i the best possible opportunity to make Ri as large as possible. Specifically, we solve for each player i0 the linear programming problem: M R~ = max y. a(m)aio.(m) {or(m) } m'~----I
(e)
s.t.
M ~, a(m)ai.(m) ~< 1, m=l
ct(m)-a(m+l)>le
(2)
i= 1,...,n
(3)
m=l,...,M-1
(4)
e.
(5)
In this formulation, constraints (3) are imposed to bound the problem, although the outcomes (at least the ordering of the Ri) are independent of the choice of constant C (= 1 in this case). Clearly, this approach, in the spirit of the D E A methodology as proposed by Charnes et al. [11], is destined to produce a different set of multipliers {a(m)} for each player i0. In that sense, notational correctness would demand that we label the c~(m) as am(m ). We have, however, for simplicity of presentation, omitted the subscript to. In the development below we propose a mechanism for determining a common set of weights if it is desired to evaluate all players according to the same setting. To illustrate model (P) and as a preamble to the multiple tournament case, consider the case of a 4 ?The notation i >t k means player i defeated or played to a draw with player k.
Wade D. Cook et al. tournament in which the outcomes from the matches are as follows:
A=
1
2
3
4
1
0
1
-
1
2
0
-
1
1
3
0
0
-
1
4
1
0
0
-
:h powers for m = 2, 3 are 2
1 1
3
4
-
0
1
2
1
-
0
1
3
1
0
-
4
0
1
1
A2=2
1
2
3
4
1
-
0
0
1
2
1
-
1
0
0
3
0
1
-
0
-
4
0
0
1
-
A3=
1at the row sums for these three matrices are given by:
2 RSA = 3
RSA 2
4
2
2 = 3
R S A3 = 3
4
4
)wing (P) for this particular tournament, Player l's optimization problem becomes: Ri = m a x 2 a l + 3a2 + a3 Subject to:
2al + 2 a 2 + 2c~ 3 ~< 1 2al + 3t~2 + c~3 ~< 1 o~1 -~- o~2 -I--o~3 ~ 1
al + 2 a 2 + a 3 ~< 1
a 2 -- a 3 ~ e
a3 >~e )ptimum Ri ---- 1 or 100%. If this procedure is repeated for each of the other three players the s R 1 = 1,
R 2 = 0.99,
R3 = 0.50,
R4 = 0.67.
the players are ranked 1 > 2 > 4 > 3. lem (P) essentially tries to capture the overall picture from the three different postures for the as represented by the row sums R S A , RSA:, RSA3 for A, A 2 and A 3 respectively. For example, 3 and 4 are tied in the actual tournament A (i.e. the row sums for 3 and 4 are both 1 in RSA). g at the next higher generation, however, Player 4 gets a larger score (2) than is true of Player re of 1). In A 3 they are tied again. So on the basis o f these three postures, Player 4 seems to be roger of the 2 players and should be ranked higher than Player 3. Similar arguments can be ar other pairs of players. ~, problem (P) is aimed at finding the best overall rating for each player. dtiple tournaments
~e case where data from multiple tournaments are available, the problem of rating players ;s more complex. N o t only is there the fact that not all players compete in all tournaments, re is also an issue of tournament difficulty. In the present subsection we will consider only the
Ranking players in multiple tournaments
873
case of complete tournaments, i.e. where all players compete in each tournament. Tournament difficulty (or importance), which we do wish to look at here, can be viewed from at least two perspectives. First, tournament difficulty may be player dependent--competing against better players makes a tournament more diffcult than would be true of one with lesser competitors. Second, the importance of a tournament may in some cases be viewed from a prestige standpoint. Wins in an internationally recognized tournament may be seen as being more important than those in lower profile matches. Thus, the issue of difficulty or importance can involve several considerations. While we do not propose to delve into the matter of how relative importance should be decided, we do wish to look at how one should evaluate players when the relative importance has been expressed. Consider then the case of T tournaments which can be arranged in order of difficulty?f Without loss of generality we assume TI > T2 > " " > Tt > " " T r ,
that is tournament T1 is the most difficult followed by T2,..., and so on. This being the case, we assume that a player is awarded more credit or weight for a win over a given player in a more difficult tournament Tt, than in a less difficult tournament Tt2 (i.e. where TII > Tt2.) In that regard, let wt be a variable (whose value is to be determined) that expresses the level of difficulty of tournament T r In the case of multiple tournaments where the level of difficulty cannot be precisely quantified, there is no clear and definitive approach to finding an overall ordering of the players. A number of approaches are possible:
Approach No. 1: First, evaluate each tournament separately using the DEA model (P), obtaining T different sets of player ratings R~ (and as well T different sets of multipliers {at(m)},n,t .) Second, determine, using DEA concepts again, a best set of weights wt for each player, i.e. a set of weights which give the best overall rating R i for the players. If these weights wt are to express the level of difficulty of the tournaments Tt, then we require, according to the above ordering of the tournaments, that wt > wt+l. Along these lines it is then reasonable to solve for each player the optimization problem: T
Rio = max ~ wtRtio
(6)
t=l T
subject to:
~, wtR: <~ 1,
i= 1,...,n
(7)
t=l,...,T-1
(8)
t=l
w t - w t + l l>e, Wr >t e
(9)
Approach No. 2: If we treat the resulting ratings R[ from each tournament evaluation as providing only a rank ordering r~ of the players i (i.e. replace R: by the implied ordinal rank position r~ that player i occupies in tournament Tt), then an ordinal multi-criteria model along the lines of Cook and Kress [16] could be utilized to rank the players. Specifically, define
da( i) =
1 if player i ranks lth in tournament Tt 0 otherwise.
Let Wtl denote the worth or value (to be determined from the model) associated with being ranked lth in tournament T , and consider the n linear optimization problems: T
n
Rio = max ~ E dtl(io)wtl
(10)
t=l 1=1 t i t should be pointed out that the idea of multiple match tournaments (MMT) was examined earlier by Cook and Kress [15], but under the assumption that all matches between a pair of players were equally difficult or o f equal value.
Wade D. Cook et al.
874 T
n
~. ~" dtt(i)wtt ~< 1,
subject to:
i= 1,...,n
(11)
t = l I=1
Wtl - - Wtl+l /> C,
t = 1 , . . . , T,
wtnl>e, I,V t l - - W t + l l ~ £
l = 1,...,n - 1
t=l,...,T t=l,...,T-1,
,
wrt >>,~,
(12) (13)
l=l,...,n
l= 1,...,n
(14) (15)
For such problems, Cook and Kress [16] suggest that since this creates a different set of weights wit for each player i0, and since it may be desirable in certain circumstances to have a single (or common) set against which all players are measured, equations (10) to (15) might be replaced by max e
(16)
subject to: (11) to (15). In this latter case the n left hand sides of the constraints (11) provide a ranking of the n players, where the common set of weights wa provide maximum discrimination in a certain sense. It is observed that it may in some circumstances be desirable to use a different E between rank positions 1 than between tournaments t. Clearly, a different e could be used in equations (12) to (13) than in (14) to (15). In this way, we could choose to maximize the discrimination between rank positions and not tournaments, for example. With no loss of generality we assume here a common e. Two important observations should be made with regard to these two approaches. First, it is the case that both of these approaches are based on a two-stage, sequential analysis. That is, first the best (highest) rating for each player in each tournament is determined; this is followed by a second stage in which an overall rating for the player across all tournaments is desired. The problem with these approaches is that in determining the best set o f weights w t for a given player i, that player does not benefit from using the most favorable (from his/her standpoint) set of alto I to use to evaluate the other players in the optimization process. In the simultaneous approach given below the aggregate rating for a player will tend to be better or higher, hence giving a truer, best possible evaluation for each player. The second point is that the solution to equations (10) to (15) will trivially force all wt as high as possible, hence making them all separated by e. Thus, it isn't really necessary to solve an optimization problem under Approach No. 2.
Approach No. 3: The above two approaches are based on a two-stage sequential analysis, in that an evaluation is first done on each tournament, followed by a second stage procedure which attaches a worth to the numerical or rank order scores arising out of that first stage. An alternative approach and one that we will adopt herein, (and one that will be more appropriate in the case of partial tournaments to be examined in the next section), involves viewing the aforementioned two stages in a simultaneous rather than sequential manner. Specifically, the approach advocates finding a set of ruth generation multipliers { o ~ t ( m ) ) m for each tournament t = 1 , . . . , T together with a set of tournament difficulty multipliers {wt} t through the following multi-tournament generalization of problem (P): T
Rio = T
subject to:
M
max y ~ wtc~t (m)aio. t (m) {~'(,~)t,{w,t t---'lm=l
(17)
M
~ E wtc~t(m)a~.(m) <~ 1,
i= 1,...,n
(18)
t=l m=l M
at(rn)4.(m)<~ 1,
i= 1,...,n;
t=l,...,T
(19)
rn=l
(IT)
o~t(m)--ctt(m+l)>~e,
m=l,...,M-l;
t=l,...,T
(20)
Ranking players in multiple tournaments
a'(M)>le, W t -- Wt+ I ~ £,
875
t= 1,...,T
(21)
l = 1,..., T - 1
(22)
wr >1 e
(23)
Here, we again provide each player with the opportunity to choose not only the most favorable ctt(m) on the m generations, but also to weigh the importance of the tournaments (while respecting the constraints) in a manner that makes his/her rating R~ as high as possible. It is noted that we require the performance in individual tournaments [equation (19)] also be bounded by 1. In this way, R~ becomes a weighted average of the performances in the tournaments t. Similarly, the left hand side of equation (18) becomes the weighted score being assigned to player i (by player/0). Problem (PT), unlike problem (P), is nonlinear in the presence of products of variables, i.e. wtt~t(m). Generally, such nonlinearities would render the problem very difficult to solve, but in the present case an equivalent linear formulation is at hand.
weights
2.3. A linear representation An equivalent linear representation of problem (PT) can be accomplished through a simple change of variables. Specifically, define (24)
firm = wtolt(m),
and replace equations (19), (20) and (21) by the equivalent constraints M
wt ~ at(m)a~.(m)<...wt,
t=l,...,T
i=l,...,m,
rn=l
m=l,...,M-1,
wtott(m)--wtctt(mq-1)>~wt, wto~t(M) >1 £wt,
t=l,...,T
t = 1,..., T
Now, rewrite problem (PT) in the form T
M
(25)
R;0 = max }-" ~ 3,ma~.(m) {3tin}t=l m=l T
subject to:
M
E Z ,maI.(m)
l,
i = 1,...,n
(26)
t=l m=l M
(PTL)
~ 3tma~.(m)-wt <<,O,
i= l,...,n,
t= l,...,T
(27)
rn=l
]~tm --~tm+l --f-Wt ~ 0 ,
m = 1,...,M-
1,
t= 1,...,T
(28)
3tM-ewt>lO,
t=I,...,T
(29)
wt-wt+l />e,
t= 1,...,T-1
(30)
WT >1 e
(31)
Since all wt are strictly positive in view of equation (31), then an optimal solution to the LP problem (PTL) immediately yields an optimum to (PT) as given by the following theorem: Theorem 1: If (/3t*,w~) is an optimal solution to the linear problem (PTL), then (at*(m),w~) where at*(m) = ~tm/W~, is an optimal solution to (PT).
2.4. Common weights across tournaments In the above model a different set of multipliers at(m) arises for each tournament Tt. If it is desired to obtain a single (common) set a(m) that applies to all tournaments, then one would need to solve the n quadratic problems (PT) where we replace at(m) by a(m). Unfortunately, in this case, the linearization procedure presented above doesn't work. In the section following an example is given for both tournament specific and common weight cases.
Wade D. Cook et al.
876
3. P A R T I A L
TOURNAMENTS
In the previous section it was assumed that each of n players participated in the same set of T tournaments. Here we examine a generalization of this idea. Assume each player ic{l,..., n} competes in some subset Ki C_ {T1, T2,..., Tr} of the T tournaments. Further, it is assumed that any tournament where i competes is complete in the sense that each player in that tournament has exactly one match with each of the other players in the tournament. So, each of these tournaments is round robin in the usual sense.~" Finally, we assume for simplicity that the number of generations M that is used to capture player strength is common across all tournaments. As will be pointed out later, this assumption can be removed without changing the general approach. The problem with attempting to model player performance in the partial tournament setting using equations (17) to (23) is that a player /o who competes in only a small subset Kio of tournaments will tend to be dominated by a player il where Kil is a much larger set; i.e. Rio < Rfi, simply because of the numbers of matches played in the two cases. Hence, this formulation fails to account for the differential numbers of matches Ki played. To accomplish this we propose the following generalization of (PT) M
E
•
W
tEK~ m=l
RIO =
max
{¢~'(m)},{w,}
t t ,a(m)aio.(m)
(32)
E Wt t e Kto
M
E subject to:
E wtat(m)a~o.(m)
t E K i m=l
~ w,
~<1,
i = 1,...,n
l,
i= l,...,n,
(33)
tEKi M
Z
(m)
tEKi
(34)
m=l
(P'm)
at(m) - a'(m + 1) >/e,
m=l,...,M-1,
t=l,...,T
(35)
a'(m) I> E, t = 1 , . . . , T W t -- Wt+ I >1 £~
(36)
t = 1,..., T - 1
(37)
(38)
Wr>~e
Problem (PTK), a fractional linear problem, therefore accounts for tournament participation by way of normalization. In this manner, players can be properly compared regardless of the number of tournaments in which each is involved. In Charnes and Cooper [17] it is shown that through a suitable change of variables such a problem can be converted to a linear programming equivalent. The requirement that the denominator be strictly positive clearly holds in the present case, and the appropriate change of variables here is
I TIO--
(39)
E Wt t E Kio
Following the earlier change of variables 13t,, = a t ( m ) w t , and letting vt = riowt and #tin problem (PTK) is equivalent to the linear programming problem
=
3"iot~tm,
M
RIO = max )-" E
#tmOltio'(m)
(39)
tEKio m=l
? It is noted that this definition o f partial involvement is different than that used by Cook and Kress [15] where in a single tournament each player i may compete with only a subset o f other participants.
Ranking players in multiple toumarnents subject to:
~
877
v, = 1
(40)
i= 1,...,n
(41)
t E Ki0 M
~, E lZtma~.(m)-- E vt <~O, tE Ki ra=l
tE K l M
i= l,...n,
E #tmal. (m) - vt ~ O,
(PTKL)
t E Ki
(42)
m=l
~ t m - - ~trn+l ~ ~.Vt,
Vt-
m=l,...,M-l,
]~tM ~ El~t,
t = 1,...,T
Vt+ I ~ ~Tlo,
t = 1. . . . , T -
t= 1,...,T
(43) (44)
1
(45) (46)
V T >1 Cl'io
In solving problem (PTK) we m a y impose a restriction of the form ~ t e r,0 wt <<,0 for some chosen scaler 0. Since e is an infinitesimal we may with no loss of generality choose 0 = 1. The m a x i m u m value o f R~0 for 0 > 1 is only greater than R/0 at 0 = 1 by an amount o f the order of e. T h a t is, & ( 0 > 1) = & ( 0 ) + 0(~). Thus, from a practical point of view, problem (PTK) is equivalent to a problem with a constraint ~t e rio wt ~< 1 added. Thus, we m a y augment problem ( P T K L ) by the additional restriction r~o >/ 1
(47)
In Section 4 we look at imposing different lower bounds qio on %. Again, given a solution (r~,/~*, v*) to (PTKL), then w; = v;/r~ a n d c/*(m) = to (PTK). It is noted that at the o p t i m u m r$ = ~ . In the following section we illustrate these models with a full scale example.
~;,./v7
is a solution
4. EXAMPLE
T o demonstrate model (PTKL), consider the case of 6 players competing in 4 tournaments. The involvement of the players is shown in Table 1. Table 1. Players assigned to tournaments
Players Tournament
1
2
3
4
l
X
2 3 4
x x x
X
X
X
x x
x
x x
5
6
x x x
x x
Assume the tournaments are already arranged in order of importance (strength), i.e.
TI>T2>T3>T4. Because only 4 players are involved in each of the tournaments, except for T o u r n a m e n t 3, we work with only 3 generations o f wins, i.e. M = 3. The four tournaments and their ruth powers are as follows. The row sums for each ruth power matrix are shown as well:
4.1. Tournament 1 1
TI
2
3
4
RS
1
-
1
1
0
2
2 3 4
0 0 1
0 0
1 0
1 1 -
2 1 1
1 .,.2 "/1
2
3
4
RS
1
-
o
1
2
3
2 3 4
1 1 0
0 1
0 1
1 0 -
2 1 2
1 .,.3 "/1
2
3
4
RS
1
-
0
0
1
1
2 3 4
1 0 0
1 0
1 1
0 0 -
2 1 1
Wade D. Cook et al.
878 4.2. T o u r n a m e n t 2
"2 =
1 2 4 5
1
2
4
5
RS
0 0 1
1 0 1
1 1 0
0 0 1 -
2 1 1 2
1 2 4 5
=
1
2
4
5
RS
0 1 0
0 1 1
1 0 2
1 1 0 -
2 1 2 3
1 2 4 5
=
1
2
4
5
RS
1 0 0
1 1 0
0 0 1
1 0 0 -
2 1 1 1
4.3. T o u r n a m e n t 3
/'3=
1
2
3
5
6
RS
1 2 1
0 -
1 1
0 1
1 0
2 3
3 0 5 1 6 0
0 0 I 1 0
0 0
1 1 -
1 3 1
1
2
3
5
1
-
1
0
21
-2 0 1 0 1 1 1 0 1
T]=3 5 6
6
RS
0
1
2
0 0 1
3 0 2 -
6 1 4 3
I 2
1
2
3
5
6
RS
0
1 -
1 1
1 0
0 3
3 4
0 2 1 02
1 0
0 1 -
2 5 3
7"33=3 1 5 1 61
4.4. T o u r n a m e n t 4
1
T4=
4 5 6
1
4
5
6
RS
-
1
0
0
1
0 1 1
0 0
1 0
1 1 -
1
2 2 1
1
4
5
6
RS
-
0
1
1
2
1
3 2 1
4 5 6
7.42=42-0 5 1 6 0
1 1
0
1 0 -
1
T~=
4
5
6
RS
-
0
1 0 0
1 0
0
1
1
0 1
0 1 -
1 2 1
N o t e that T o u r n a m e n t 1 corresponds to the single t o u r n a m e n t example presented above. 4.5. T o u r n a m e n t - s p e c i f i c weights case
Problem ( P T K L ) has been solved for each player and for each o f various values o f ~6. F o r purposes o f this example e = 0.01 was used. The results are displayed in Table 2. Clearly from the example it can be seen that the choice o f r/0 does not influence the relative ratings for the players until it reaches a significantly large value in c o m p a r i s o n to 1/e. In the presence o f small values for rio (e.g. rio ~< 1) the players would be ranked: P1 ,,~ P 5 ,v P 2 > P 4 > P 6 > P3.
(Note that P2 at 0.99 can be regarded as having a rating o f ~ 1.00). G o i n g to higher levels o f % we can discriminate m o r e clearly a m o n g the players. F o r example, using rio = 5, the tie a m o n g the 3 players P1, P5 and P2 is broken, and we get P5 > P1 > P2 > P4 > P6 > P3. 4.6. C o m m o n weight case
I f c o m m o n t o u r n a m e n t weights a ( m ) are desired (as opposed to tournament-specific weights Table 2.'Ratings R6 for the players ~ 0.01 0.1 1.0 5.0 10.0
1
2
3
4
5
6
1.00 1.00 1.00 0.94 0.89
0.99 0.99 0.98 0.92 0.84
0.50 0.50 0.49 0.48 0.46
0.77 0.77 0.77 0.75 0.72
1.00 1.00 1.00 1.00 0.99
0.52 0.52 0.52 0.52 0.52
Ranking players in multiple tournaments at(m)),
879
problem (PTKL) reduces to the set of quadratic problems: M
R~ = max ~
ut ~ a(m)a~.(m)
tEKio
m=l
Subject to:
~ vt = 1 t E Ki o
M
Z
vt <~ 0,
for all i
a(m)a~.~< 1,
for alli,
vt Z °~(m)a~o.(m) -- Z
tEK i
m=l
tEK~ o
M
tEKi
ra=l
m=l,...,M-I
a(m)-a(m+l)>le,
a(M) >i e V t - - Vt+ I - - eT"~ ~ 0 ,
t = 1,..., T -
1
vr - e% >10. Solving this problem for Player 1, for example, we get the following variable values. T *~-- 1
Vl =0.967,
u2 =0.021,
u3 =0.011,
u4 =0.001
wI
w~ = 0.021,
w3 = 0.011,
w~ = 0.001
=
0.967,
a*(1) =0.28,
a*(2) =0.02,
a*(3) =0.01
The aggregate ratings for the six players are: R~=0.63,
R~=0.64,
R~=0.30,
R~=0.33,
R~=0.54,
R~=0.18.
Thus, the final ranking of the players under the common tournament weight scenario is P2>PI
> Ps > P4 > P3 > P6,
which differs quite significantly from the rankings above under the tournament-specific weight case. Acknowledgement--This paper was supported under Grant No. A8966.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
D. H. Bent, Score Problems of Round Robin Tournaments, M.Sc. Thesis, U. Alberta (1964). J. W. Moon, Topics on Tournaments. Holt, Rinehard and Winston, New York (1968). S.T. Goddard, Ranking tournaments and group decision making, Mgmt Sci. 29, 1384-1392 (1983). M. Kendall, Rank Correlation Methods. Hafner, New York (1962). W.D. Cook, I. Golan and M. Kress, Heuristics for ranking players in a round robin tournament. Computers Ops Res. 15, 135-144 (1988). I. Ali, W. D. Cook and M. Kress, On the minimum violations ranking of a tournament, Mgmt Sci. 32, 660-672 (1986). E. Zermelo, Die Berechnung der Twinier--Ergebnisse als ein maximum Problem der Wahrscheinlichkeitsrechnung. Mathe. Zeitschrift 29, 436-460 (1926). R. A. Bradley and M. E. Terry, The rank analysis of incomplete block designs: the method of paired comparisons. Biometrika 39, 324-345 (1952). T. H. Wei, The Algebraic Foundations of Ranking Theory, PhD Thesis, Cambridge University (1952). W. D. Cook and M. Kress, Ranking players in a tournament: an ruth generation approach. In Systems and Management Science by External Methods. Kluwer Academic Publishers, Boston (1992). A. Charnes, W. W. Cooper and E. Rhodes. Measuring the efficiency of decision making units. Fur. J. Oper. Res. 2, 429444 (1978). W. D. Cook and M. Kress, Ordinal Information and Preference Structures: Decision Models and Applications. PrenticeHall, Englewood Cliffs, New Jersey (1992). M. Roubens, Preference relations on actions and criteria in multicriteria decision making. Eur. J. Oper. Res. 10, 51-55 (1982).
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14. W. D. Cook and M. Kress, An mth generation model for weak ranking of players in a tournament. J. Oper. Res. Soc. 41, 1111-1119 (1990). 15. W.D. Cook and M. Kress, Partial and multiple match tournaments. Math. Soc. Sci. 15, 303-306 (1988). 16. W. D. Cook and M. Kress, A multiple criteria decision model with ordinal preference data. Eur..I. Oper. Res. 54, 191198 (1991). 17. A. Charncs and W. W. Cooper, An explicit general solution in linear fractional programming. Naval Res. Logist. Quart. 20, 449-467 (1973).
R e a d e r s m a y a d d r e s s i n q u i r i e s t o D r W a d e D . C o o k at: a s 0 0 0 0 4 2 @ o r i o n . y o r k u . c a
Pergamon
RANKING
0305-0M8(95)00082-8
PLAYERS
IN MULTIPLE
Copyright © 1996 ElsevierScienceLtd Printed in Great Britain. All rights reserved 0305-0548/96 $15.00+ 0.00
TOURNAMENTS
Wade D. Cook, lt:~ John Doyle,2§ Rodney Green2¶ and Moshe Kress 311 1Faculty of Administrative Studies, York University, 4700 Keele Street, North York, Toronto, Ontario, Canada M3J 1P3, 2School of Management, University of Bath, Bath BA2 7AY, U.K. and 3Center for Military Analyses, P.O. Box 2250, Haifa, Israel
(Received December 1994; in revisedform August 1995) Scope and Purpose--In a round robin tournament each of a set of n players competes exactly once with every other player, with the outcome from each match being a win of player i over player j, a loss or a tie. The point of much of the previous research in this area has been to derive a ranking of the players based on their performance in the matches. An example of related work appears in a W. D. Cook, I. Golan and M. Kress, Heuristics for ranking players in a round robin tournament. Computers Ops Res. 15, 135-144 (1988). The tournament structure is not unique to the sports environment, but rather appears in a variety of other application areas. Consumer responses regarding preferences for a set of products, for example, are often expressed via pairwise comparisons--"product i is preferred to product j." Thus, any situation in which elements (players, products, projects. . . . ) are compared in a pairwise sense, may give rise to such a "tournament" structure. In the present paper the single competition setup is extended to a multiple comparison environment. Consider a situation in which a set of products are to be evaluated in terms of K criteria or attributes. For criterion no. 1 (e.g. flavor), suppose that only a subset of the full set of products can be compared (e.g. flavor is not an attribute of some of the products). For criterion No. 2, a different subset of the products are compared, although some members of this second subset may have been in the first comparison as well. So each product is involved in various comparisons, with some products in more sets than others. In addition, some tournaments (subsets) are more difficult than others. The purpose of the research here is to develop models for deriving an overall ranking of the products given the performance in the various tournaments, and taking account of the relative difficulty of the tournaments. Abstract--This paper examines a generalization of the standard round robin tournament. First we consider a set of tournaments wherein each player competes in each member of the set, and the tournaments can be ordinally ranked in order of difficulty. The data envelopment analysis method is used to obtain a ranking for each player, while taking account not only of the players' strength within each tournament, but also the differential difficulty of tournaments. Second, we extend this concept to the case where each player may only compete in a subset of the tournaments. Hence, any given player may have fewer matches than other players. Copyright © 1996 Elsevier Science Ltd
1. I N T R O D U C T I O N
Tournament ranking has been the subject of substantial research efforts over the past several decades. In its most basic form, the round robin tournament consists of a set of pair-wise matches among n players in which all pairs compete exactly once, and each match ends in either a strict fWade D. Cook is a Professor of Management Science in the Faculty of Administrative Studies at York University, Toronto, Canada. He received his Ph.D. in Operations Research from Dalhousie University. His research interests are in the areas of ordinal and multiple criteria decision models, preference structures, and consensus formation. His publications have appeared in a number of academic and professional journals including Management Science, EJOR, CORS, Operations Research, and JORS. :~To whom correspondence should be addressed. §John Doyle is lecturer at the School of Management, University of Bath, U.K., where he teaches courses in psychology, quantitative methods, and information systems. He is currently researching haman aspects of the decision process and techniques (both computer-based and noncomputer-based) to support decision making. ¶Rodney Green is a lecturer in the School of Management, University of Bath. After qualifying as a chemical engineer he worked on the design of computer-based process control systems. The inevitable compromises in such systems led to an interest in Operational Research and he now teaches and researches in that area. IIMoshe Kress is a senior researcher at the Center for Military Analyses (CEMA). He received his Ph.D. in Operations Research from the University of Texas at Austin. His research interests are in the areas of ordinal decision models and preference structures and military operations research. His publications have appeared in a number of Management Science/Operations Research journals, including Management Science, European Journal of Operations Research and
Naval Research Logistics Quarterly.
869
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Wade D. Cook et al.
decision (a winner and loser) or a draw (tied game). The central problem is to rank order the players in a manner which best reflects the tournament results. Numerous authors including Bent [1] and Moon [2] have concentrated on the structural problems and properties of tournaments, while Goddard [3], Kendall [4] and others have dealt with computational problems of ranking the players based on the outcomes from the matches. Cook et al. [5] present a number of heuristics generally based on row sums or Kendall scores from the tournament matrix, and Ali et al. [6] present a structure for examining distance on tournaments. Attempts to capture player strength were discussed originally by Zermelo [7] and later by Bradley and Terry [8]. Both of these approaches were based on maximum likelihood methods. Wei [9] re-examined the player strength issue by taking kth powers of the tournament matrix. It can be shown that these kth powers converge to the right eigenvector of the tournament matrix. Building on Wei's ideas, Cook and Kress [10], use an mth generation approach to incorporate player strength. Weights on the m generations are derived using DEA (data envelopment analysis) principles adopted from the efficiency analysis concepts of Charnes et al. [11]. For a full discussion of the various approaches to tournament ranking see Cook and Kress [12]. In the present paper we examine a generalization of the basic tournament idea by looking at a set o f tournaments whereby in each member of the set, a subset of the players compete in a round robin sense. We assume that the tournaments are of unequal and unknown difficulty, but can be ranked in order of difficulty. We develop a model for ranking the players that takes account not only of player strength within a tournament, but also of the differential difficulty of competing in the various tournaments. The model also accounts for the fact that a player i may compete in only a subset of the tournaments, hence participating in a different number of matches than may be the case for another player j. The models presented here have application to problem areas outside of the conventional tournament arena: 1.1. Using consumer preferences to evaluate products
Consider the situation faced by a market researcher who is collecting consumer responses pertaining to preferences among formulations of a product (e.g. a pudding mix). A particularly convenient format for soliciting preferences in such a case is pairwise comparisons ("do you prefer product a to product b?"). The outcome from such comparisons can be presented in the form of a binary tournament matrix. Furthermore, when making comparisons among products on the basis of any given attribute, e.g. flavor, it may be that only a subset of the products can be evaluated. 1.2. Evaluating energy R&D projects
The problem of selecting from a set of projects relating to energy research and development is important when limited funds are available to finance such projects. Comparisons are often easiest when pairs of projects are evaluated relative to one another in an ordinal sense. An evaluation committee is commonly charged with the task of prioritizing the projects, basing judgments on information supplied by the proposer. Here, when projects come from different departments, certain pairs may not be comparable on particular criteria, hence an incomplete or partial tournament arises. 1.3. General multiple criteria problems
Clearly, many multiple criteria problems involve making pairwise comparisons of members of a set of alternatives in terms of several criteria. The required outcome from this exercise is generally to obtain a final ranking of those alternatives. As with the R&D project example above, certain pairs of alternatives may not be comparable on certain criteria. Methods developed previously (e.g. the outranking method of Roubens [13]) generally presume that all pairs of alternatives are compared on all criteria. Such methods could likely also be extended to include partial comparisons. 2. COMPLETE TOURNAMENTS 2.1. The single tournament case
Assume that each of n players participates in T tournaments { T/}r=l, with each match between a pair of players in a tournament resulting in either a decision or a draw. Let A t denote the binary
Ranking players in multiple tournaments
871
incidence matrix for tournament Tt, that is a,~-
j" 1
if player i defeats player j in tournament Tt
/ 0,
otherwise
In Cook and Kress [10, 14] player strength within a single tournament is viewed from the perspective o f mth generation wins. Specifically, if we take the mth power (A) m of any tournament (incidence) matrix A, then the entry a U(m) represents the number of Hamiltonian paths of length m from player i to playerj. That is, each of these paths has the property that there are m - 1 players ii, i2,.., im-l such that i >/il >/i2 >/ • .. >/ira-1 ~>J-t It can then be argued that one way o f accounting for player strength is to compute the row sums a i. (m) = E~=I aij(m) of each of the mth generation matrices A(m)(= (A)m), and take as the score or worth for player i, a weighted total M Ri = ~ o~(m)ai.(m). m:
(1)
I
Here, M denotes the numbers of powers of A (generations) computed, and {a(m)}~=l is a set of weights representing the importance of these generations to the player's total score. As proposed by Cook and Kress [14], it is reasonable to assume that the importance c~(m) attached to an ruth generation win should be at least as great as the importance o~(m + 1) associated with an m + 1st generation win. In this regard it is proposed that any reasonable set of multipliers or weights should satisfy a ( m ) - c~(m+ 1)~> e,
m= 1,...,M-1
and a ( M ) >t e, for some infinitesimal e. Clearly, as will be seen below, one could impose additional restrictions on the a(m), and as well e might be replaced by an actual positive lower bound that is more than an infinitesimal. We will, however, not concern ourselves with such additional restrictions here. In the case of the single tournament, Cook and Kress propose rating the players on the basis of the R i as defined by equation (1), but where the multipliers c~(m) are chosen in accordance with an optimization procedure that gives each player i the best possible opportunity to make Ri as large as possible. Specifically, we solve for each player i0 the linear programming problem: M R~ = max y. a(m)aio.(m) {or(m) } m'~----I
(e)
s.t.
M ~, a(m)ai.(m) ~< 1, m=l
ct(m)-a(m+l)>le
(2)
i= 1,...,n
(3)
m=l,...,M-1
(4)
e.
(5)
In this formulation, constraints (3) are imposed to bound the problem, although the outcomes (at least the ordering of the Ri) are independent of the choice of constant C (= 1 in this case). Clearly, this approach, in the spirit of the D E A methodology as proposed by Charnes et al. [11], is destined to produce a different set of multipliers {a(m)} for each player i0. In that sense, notational correctness would demand that we label the c~(m) as am(m ). We have, however, for simplicity of presentation, omitted the subscript to. In the development below we propose a mechanism for determining a common set of weights if it is desired to evaluate all players according to the same setting. To illustrate model (P) and as a preamble to the multiple tournament case, consider the case of a 4 ?The notation i >t k means player i defeated or played to a draw with player k.
Wade D. Cook et al. tournament in which the outcomes from the matches are as follows:
A=
1
2
3
4
1
0
1
-
1
2
0
-
1
1
3
0
0
-
1
4
1
0
0
-
:h powers for m = 2, 3 are 2
1 1
3
4
-
0
1
2
1
-
0
1
3
1
0
-
4
0
1
1
A2=2
1
2
3
4
1
-
0
0
1
2
1
-
1
0
0
3
0
1
-
0
-
4
0
0
1
-
A3=
1at the row sums for these three matrices are given by:
2 RSA = 3
RSA 2
4
2
2 = 3
R S A3 = 3
4
4
)wing (P) for this particular tournament, Player l's optimization problem becomes: Ri = m a x 2 a l + 3a2 + a3 Subject to:
2al + 2 a 2 + 2c~ 3 ~< 1 2al + 3t~2 + c~3 ~< 1 o~1 -~- o~2 -I--o~3 ~ 1
al + 2 a 2 + a 3 ~< 1
a 2 -- a 3 ~ e
a3 >~e )ptimum Ri ---- 1 or 100%. If this procedure is repeated for each of the other three players the s R 1 = 1,
R 2 = 0.99,
R3 = 0.50,
R4 = 0.67.
the players are ranked 1 > 2 > 4 > 3. lem (P) essentially tries to capture the overall picture from the three different postures for the as represented by the row sums R S A , RSA:, RSA3 for A, A 2 and A 3 respectively. For example, 3 and 4 are tied in the actual tournament A (i.e. the row sums for 3 and 4 are both 1 in RSA). g at the next higher generation, however, Player 4 gets a larger score (2) than is true of Player re of 1). In A 3 they are tied again. So on the basis o f these three postures, Player 4 seems to be roger of the 2 players and should be ranked higher than Player 3. Similar arguments can be ar other pairs of players. ~, problem (P) is aimed at finding the best overall rating for each player. dtiple tournaments
~e case where data from multiple tournaments are available, the problem of rating players ;s more complex. N o t only is there the fact that not all players compete in all tournaments, re is also an issue of tournament difficulty. In the present subsection we will consider only the
Ranking players in multiple tournaments
873
case of complete tournaments, i.e. where all players compete in each tournament. Tournament difficulty (or importance), which we do wish to look at here, can be viewed from at least two perspectives. First, tournament difficulty may be player dependent--competing against better players makes a tournament more diffcult than would be true of one with lesser competitors. Second, the importance of a tournament may in some cases be viewed from a prestige standpoint. Wins in an internationally recognized tournament may be seen as being more important than those in lower profile matches. Thus, the issue of difficulty or importance can involve several considerations. While we do not propose to delve into the matter of how relative importance should be decided, we do wish to look at how one should evaluate players when the relative importance has been expressed. Consider then the case of T tournaments which can be arranged in order of difficulty?f Without loss of generality we assume TI > T2 > " " > Tt > " " T r ,
that is tournament T1 is the most difficult followed by T2,..., and so on. This being the case, we assume that a player is awarded more credit or weight for a win over a given player in a more difficult tournament Tt, than in a less difficult tournament Tt2 (i.e. where TII > Tt2.) In that regard, let wt be a variable (whose value is to be determined) that expresses the level of difficulty of tournament T r In the case of multiple tournaments where the level of difficulty cannot be precisely quantified, there is no clear and definitive approach to finding an overall ordering of the players. A number of approaches are possible:
Approach No. 1: First, evaluate each tournament separately using the DEA model (P), obtaining T different sets of player ratings R~ (and as well T different sets of multipliers {at(m)},n,t .) Second, determine, using DEA concepts again, a best set of weights wt for each player, i.e. a set of weights which give the best overall rating R i for the players. If these weights wt are to express the level of difficulty of the tournaments Tt, then we require, according to the above ordering of the tournaments, that wt > wt+l. Along these lines it is then reasonable to solve for each player the optimization problem: T
Rio = max ~ wtRtio
(6)
t=l T
subject to:
~, wtR: <~ 1,
i= 1,...,n
(7)
t=l,...,T-1
(8)
t=l
w t - w t + l l>e, Wr >t e
(9)
Approach No. 2: If we treat the resulting ratings R[ from each tournament evaluation as providing only a rank ordering r~ of the players i (i.e. replace R: by the implied ordinal rank position r~ that player i occupies in tournament Tt), then an ordinal multi-criteria model along the lines of Cook and Kress [16] could be utilized to rank the players. Specifically, define
da( i) =
1 if player i ranks lth in tournament Tt 0 otherwise.
Let Wtl denote the worth or value (to be determined from the model) associated with being ranked lth in tournament T , and consider the n linear optimization problems: T
n
Rio = max ~ E dtl(io)wtl
(10)
t=l 1=1 t i t should be pointed out that the idea of multiple match tournaments (MMT) was examined earlier by Cook and Kress [15], but under the assumption that all matches between a pair of players were equally difficult or o f equal value.
Wade D. Cook et al.
874 T
n
~. ~" dtt(i)wtt ~< 1,
subject to:
i= 1,...,n
(11)
t = l I=1
Wtl - - Wtl+l /> C,
t = 1 , . . . , T,
wtnl>e, I,V t l - - W t + l l ~ £
l = 1,...,n - 1
t=l,...,T t=l,...,T-1,
,
wrt >>,~,
(12) (13)
l=l,...,n
l= 1,...,n
(14) (15)
For such problems, Cook and Kress [16] suggest that since this creates a different set of weights wit for each player i0, and since it may be desirable in certain circumstances to have a single (or common) set against which all players are measured, equations (10) to (15) might be replaced by max e
(16)
subject to: (11) to (15). In this latter case the n left hand sides of the constraints (11) provide a ranking of the n players, where the common set of weights wa provide maximum discrimination in a certain sense. It is observed that it may in some circumstances be desirable to use a different E between rank positions 1 than between tournaments t. Clearly, a different e could be used in equations (12) to (13) than in (14) to (15). In this way, we could choose to maximize the discrimination between rank positions and not tournaments, for example. With no loss of generality we assume here a common e. Two important observations should be made with regard to these two approaches. First, it is the case that both of these approaches are based on a two-stage, sequential analysis. That is, first the best (highest) rating for each player in each tournament is determined; this is followed by a second stage in which an overall rating for the player across all tournaments is desired. The problem with these approaches is that in determining the best set o f weights w t for a given player i, that player does not benefit from using the most favorable (from his/her standpoint) set of alto I to use to evaluate the other players in the optimization process. In the simultaneous approach given below the aggregate rating for a player will tend to be better or higher, hence giving a truer, best possible evaluation for each player. The second point is that the solution to equations (10) to (15) will trivially force all wt as high as possible, hence making them all separated by e. Thus, it isn't really necessary to solve an optimization problem under Approach No. 2.
Approach No. 3: The above two approaches are based on a two-stage sequential analysis, in that an evaluation is first done on each tournament, followed by a second stage procedure which attaches a worth to the numerical or rank order scores arising out of that first stage. An alternative approach and one that we will adopt herein, (and one that will be more appropriate in the case of partial tournaments to be examined in the next section), involves viewing the aforementioned two stages in a simultaneous rather than sequential manner. Specifically, the approach advocates finding a set of ruth generation multipliers { o ~ t ( m ) ) m for each tournament t = 1 , . . . , T together with a set of tournament difficulty multipliers {wt} t through the following multi-tournament generalization of problem (P): T
Rio = T
subject to:
M
max y ~ wtc~t (m)aio. t (m) {~'(,~)t,{w,t t---'lm=l
(17)
M
~ E wtc~t(m)a~.(m) <~ 1,
i= 1,...,n
(18)
t=l m=l M
at(rn)4.(m)<~ 1,
i= 1,...,n;
t=l,...,T
(19)
rn=l
(IT)
o~t(m)--ctt(m+l)>~e,
m=l,...,M-l;
t=l,...,T
(20)
Ranking players in multiple tournaments
a'(M)>le, W t -- Wt+ I ~ £,
875
t= 1,...,T
(21)
l = 1,..., T - 1
(22)
wr >1 e
(23)
Here, we again provide each player with the opportunity to choose not only the most favorable ctt(m) on the m generations, but also to weigh the importance of the tournaments (while respecting the constraints) in a manner that makes his/her rating R~ as high as possible. It is noted that we require the performance in individual tournaments [equation (19)] also be bounded by 1. In this way, R~ becomes a weighted average of the performances in the tournaments t. Similarly, the left hand side of equation (18) becomes the weighted score being assigned to player i (by player/0). Problem (PT), unlike problem (P), is nonlinear in the presence of products of variables, i.e. wtt~t(m). Generally, such nonlinearities would render the problem very difficult to solve, but in the present case an equivalent linear formulation is at hand.
weights
2.3. A linear representation An equivalent linear representation of problem (PT) can be accomplished through a simple change of variables. Specifically, define (24)
firm = wtolt(m),
and replace equations (19), (20) and (21) by the equivalent constraints M
wt ~ at(m)a~.(m)<...wt,
t=l,...,T
i=l,...,m,
rn=l
m=l,...,M-1,
wtott(m)--wtctt(mq-1)>~wt, wto~t(M) >1 £wt,
t=l,...,T
t = 1,..., T
Now, rewrite problem (PT) in the form T
M
(25)
R;0 = max }-" ~ 3,ma~.(m) {3tin}t=l m=l T
subject to:
M
E Z ,maI.(m)
l,
i = 1,...,n
(26)
t=l m=l M
(PTL)
~ 3tma~.(m)-wt <<,O,
i= l,...,n,
t= l,...,T
(27)
rn=l
]~tm --~tm+l --f-Wt ~ 0 ,
m = 1,...,M-
1,
t= 1,...,T
(28)
3tM-ewt>lO,
t=I,...,T
(29)
wt-wt+l />e,
t= 1,...,T-1
(30)
WT >1 e
(31)
Since all wt are strictly positive in view of equation (31), then an optimal solution to the LP problem (PTL) immediately yields an optimum to (PT) as given by the following theorem: Theorem 1: If (/3t*,w~) is an optimal solution to the linear problem (PTL), then (at*(m),w~) where at*(m) = ~tm/W~, is an optimal solution to (PT).
2.4. Common weights across tournaments In the above model a different set of multipliers at(m) arises for each tournament Tt. If it is desired to obtain a single (common) set a(m) that applies to all tournaments, then one would need to solve the n quadratic problems (PT) where we replace at(m) by a(m). Unfortunately, in this case, the linearization procedure presented above doesn't work. In the section following an example is given for both tournament specific and common weight cases.
Wade D. Cook et al.
876
3. P A R T I A L
TOURNAMENTS
In the previous section it was assumed that each of n players participated in the same set of T tournaments. Here we examine a generalization of this idea. Assume each player ic{l,..., n} competes in some subset Ki C_ {T1, T2,..., Tr} of the T tournaments. Further, it is assumed that any tournament where i competes is complete in the sense that each player in that tournament has exactly one match with each of the other players in the tournament. So, each of these tournaments is round robin in the usual sense.~" Finally, we assume for simplicity that the number of generations M that is used to capture player strength is common across all tournaments. As will be pointed out later, this assumption can be removed without changing the general approach. The problem with attempting to model player performance in the partial tournament setting using equations (17) to (23) is that a player /o who competes in only a small subset Kio of tournaments will tend to be dominated by a player il where Kil is a much larger set; i.e. Rio < Rfi, simply because of the numbers of matches played in the two cases. Hence, this formulation fails to account for the differential numbers of matches Ki played. To accomplish this we propose the following generalization of (PT) M
E
•
W
tEK~ m=l
RIO =
max
{¢~'(m)},{w,}
t t ,a(m)aio.(m)
(32)
E Wt t e Kto
M
E subject to:
E wtat(m)a~o.(m)
t E K i m=l
~ w,
~<1,
i = 1,...,n
l,
i= l,...,n,
(33)
tEKi M
Z
(m)
tEKi
(34)
m=l
(P'm)
at(m) - a'(m + 1) >/e,
m=l,...,M-1,
t=l,...,T
(35)
a'(m) I> E, t = 1 , . . . , T W t -- Wt+ I >1 £~
(36)
t = 1,..., T - 1
(37)
(38)
Wr>~e
Problem (PTK), a fractional linear problem, therefore accounts for tournament participation by way of normalization. In this manner, players can be properly compared regardless of the number of tournaments in which each is involved. In Charnes and Cooper [17] it is shown that through a suitable change of variables such a problem can be converted to a linear programming equivalent. The requirement that the denominator be strictly positive clearly holds in the present case, and the appropriate change of variables here is
I TIO--
(39)
E Wt t E Kio
Following the earlier change of variables 13t,, = a t ( m ) w t , and letting vt = riowt and #tin problem (PTK) is equivalent to the linear programming problem
=
3"iot~tm,
M
RIO = max )-" E
#tmOltio'(m)
(39)
tEKio m=l
? It is noted that this definition o f partial involvement is different than that used by Cook and Kress [15] where in a single tournament each player i may compete with only a subset o f other participants.
Ranking players in multiple toumarnents subject to:
~
877
v, = 1
(40)
i= 1,...,n
(41)
t E Ki0 M
~, E lZtma~.(m)-- E vt <~O, tE Ki ra=l
tE K l M
i= l,...n,
E #tmal. (m) - vt ~ O,
(PTKL)
t E Ki
(42)
m=l
~ t m - - ~trn+l ~ ~.Vt,
Vt-
m=l,...,M-l,
]~tM ~ El~t,
t = 1,...,T
Vt+ I ~ ~Tlo,
t = 1. . . . , T -
t= 1,...,T
(43) (44)
1
(45) (46)
V T >1 Cl'io
In solving problem (PTK) we m a y impose a restriction of the form ~ t e r,0 wt <<,0 for some chosen scaler 0. Since e is an infinitesimal we may with no loss of generality choose 0 = 1. The m a x i m u m value o f R~0 for 0 > 1 is only greater than R/0 at 0 = 1 by an amount o f the order of e. T h a t is, & ( 0 > 1) = & ( 0 ) + 0(~). Thus, from a practical point of view, problem (PTK) is equivalent to a problem with a constraint ~t e rio wt ~< 1 added. Thus, we m a y augment problem ( P T K L ) by the additional restriction r~o >/ 1
(47)
In Section 4 we look at imposing different lower bounds qio on %. Again, given a solution (r~,/~*, v*) to (PTKL), then w; = v;/r~ a n d c/*(m) = to (PTK). It is noted that at the o p t i m u m r$ = ~ . In the following section we illustrate these models with a full scale example.
~;,./v7
is a solution
4. EXAMPLE
T o demonstrate model (PTKL), consider the case of 6 players competing in 4 tournaments. The involvement of the players is shown in Table 1. Table 1. Players assigned to tournaments
Players Tournament
1
2
3
4
l
X
2 3 4
x x x
X
X
X
x x
x
x x
5
6
x x x
x x
Assume the tournaments are already arranged in order of importance (strength), i.e.
TI>T2>T3>T4. Because only 4 players are involved in each of the tournaments, except for T o u r n a m e n t 3, we work with only 3 generations o f wins, i.e. M = 3. The four tournaments and their ruth powers are as follows. The row sums for each ruth power matrix are shown as well:
4.1. Tournament 1 1
TI
2
3
4
RS
1
-
1
1
0
2
2 3 4
0 0 1
0 0
1 0
1 1 -
2 1 1
1 .,.2 "/1
2
3
4
RS
1
-
o
1
2
3
2 3 4
1 1 0
0 1
0 1
1 0 -
2 1 2
1 .,.3 "/1
2
3
4
RS
1
-
0
0
1
1
2 3 4
1 0 0
1 0
1 1
0 0 -
2 1 1
Wade D. Cook et al.
878 4.2. T o u r n a m e n t 2
"2 =
1 2 4 5
1
2
4
5
RS
0 0 1
1 0 1
1 1 0
0 0 1 -
2 1 1 2
1 2 4 5
=
1
2
4
5
RS
0 1 0
0 1 1
1 0 2
1 1 0 -
2 1 2 3
1 2 4 5
=
1
2
4
5
RS
1 0 0
1 1 0
0 0 1
1 0 0 -
2 1 1 1
4.3. T o u r n a m e n t 3
/'3=
1
2
3
5
6
RS
1 2 1
0 -
1 1
0 1
1 0
2 3
3 0 5 1 6 0
0 0 I 1 0
0 0
1 1 -
1 3 1
1
2
3
5
1
-
1
0
21
-2 0 1 0 1 1 1 0 1
T]=3 5 6
6
RS
0
1
2
0 0 1
3 0 2 -
6 1 4 3
I 2
1
2
3
5
6
RS
0
1 -
1 1
1 0
0 3
3 4
0 2 1 02
1 0
0 1 -
2 5 3
7"33=3 1 5 1 61
4.4. T o u r n a m e n t 4
1
T4=
4 5 6
1
4
5
6
RS
-
1
0
0
1
0 1 1
0 0
1 0
1 1 -
1
2 2 1
1
4
5
6
RS
-
0
1
1
2
1
3 2 1
4 5 6
7.42=42-0 5 1 6 0
1 1
0
1 0 -
1
T~=
4
5
6
RS
-
0
1 0 0
1 0
0
1
1
0 1
0 1 -
1 2 1
N o t e that T o u r n a m e n t 1 corresponds to the single t o u r n a m e n t example presented above. 4.5. T o u r n a m e n t - s p e c i f i c weights case
Problem ( P T K L ) has been solved for each player and for each o f various values o f ~6. F o r purposes o f this example e = 0.01 was used. The results are displayed in Table 2. Clearly from the example it can be seen that the choice o f r/0 does not influence the relative ratings for the players until it reaches a significantly large value in c o m p a r i s o n to 1/e. In the presence o f small values for rio (e.g. rio ~< 1) the players would be ranked: P1 ,,~ P 5 ,v P 2 > P 4 > P 6 > P3.
(Note that P2 at 0.99 can be regarded as having a rating o f ~ 1.00). G o i n g to higher levels o f % we can discriminate m o r e clearly a m o n g the players. F o r example, using rio = 5, the tie a m o n g the 3 players P1, P5 and P2 is broken, and we get P5 > P1 > P2 > P4 > P6 > P3. 4.6. C o m m o n weight case
I f c o m m o n t o u r n a m e n t weights a ( m ) are desired (as opposed to tournament-specific weights Table 2.'Ratings R6 for the players ~ 0.01 0.1 1.0 5.0 10.0
1
2
3
4
5
6
1.00 1.00 1.00 0.94 0.89
0.99 0.99 0.98 0.92 0.84
0.50 0.50 0.49 0.48 0.46
0.77 0.77 0.77 0.75 0.72
1.00 1.00 1.00 1.00 0.99
0.52 0.52 0.52 0.52 0.52
Ranking players in multiple tournaments at(m)),
879
problem (PTKL) reduces to the set of quadratic problems: M
R~ = max ~
ut ~ a(m)a~.(m)
tEKio
m=l
Subject to:
~ vt = 1 t E Ki o
M
Z
vt <~ 0,
for all i
a(m)a~.~< 1,
for alli,
vt Z °~(m)a~o.(m) -- Z
tEK i
m=l
tEK~ o
M
tEKi
ra=l
m=l,...,M-I
a(m)-a(m+l)>le,
a(M) >i e V t - - Vt+ I - - eT"~ ~ 0 ,
t = 1,..., T -
1
vr - e% >10. Solving this problem for Player 1, for example, we get the following variable values. T *~-- 1
Vl =0.967,
u2 =0.021,
u3 =0.011,
u4 =0.001
wI
w~ = 0.021,
w3 = 0.011,
w~ = 0.001
=
0.967,
a*(1) =0.28,
a*(2) =0.02,
a*(3) =0.01
The aggregate ratings for the six players are: R~=0.63,
R~=0.64,
R~=0.30,
R~=0.33,
R~=0.54,
R~=0.18.
Thus, the final ranking of the players under the common tournament weight scenario is P2>PI
> Ps > P4 > P3 > P6,
which differs quite significantly from the rankings above under the tournament-specific weight case. Acknowledgement--This paper was supported under Grant No. A8966.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
D. H. Bent, Score Problems of Round Robin Tournaments, M.Sc. Thesis, U. Alberta (1964). J. W. Moon, Topics on Tournaments. Holt, Rinehard and Winston, New York (1968). S.T. Goddard, Ranking tournaments and group decision making, Mgmt Sci. 29, 1384-1392 (1983). M. Kendall, Rank Correlation Methods. Hafner, New York (1962). W.D. Cook, I. Golan and M. Kress, Heuristics for ranking players in a round robin tournament. Computers Ops Res. 15, 135-144 (1988). I. Ali, W. D. Cook and M. Kress, On the minimum violations ranking of a tournament, Mgmt Sci. 32, 660-672 (1986). E. Zermelo, Die Berechnung der Twinier--Ergebnisse als ein maximum Problem der Wahrscheinlichkeitsrechnung. Mathe. Zeitschrift 29, 436-460 (1926). R. A. Bradley and M. E. Terry, The rank analysis of incomplete block designs: the method of paired comparisons. Biometrika 39, 324-345 (1952). T. H. Wei, The Algebraic Foundations of Ranking Theory, PhD Thesis, Cambridge University (1952). W. D. Cook and M. Kress, Ranking players in a tournament: an ruth generation approach. In Systems and Management Science by External Methods. Kluwer Academic Publishers, Boston (1992). A. Charnes, W. W. Cooper and E. Rhodes. Measuring the efficiency of decision making units. Fur. J. Oper. Res. 2, 429444 (1978). W. D. Cook and M. Kress, Ordinal Information and Preference Structures: Decision Models and Applications. PrenticeHall, Englewood Cliffs, New Jersey (1992). M. Roubens, Preference relations on actions and criteria in multicriteria decision making. Eur. J. Oper. Res. 10, 51-55 (1982).
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14. W. D. Cook and M. Kress, An mth generation model for weak ranking of players in a tournament. J. Oper. Res. Soc. 41, 1111-1119 (1990). 15. W.D. Cook and M. Kress, Partial and multiple match tournaments. Math. Soc. Sci. 15, 303-306 (1988). 16. W. D. Cook and M. Kress, A multiple criteria decision model with ordinal preference data. Eur..I. Oper. Res. 54, 191198 (1991). 17. A. Charncs and W. W. Cooper, An explicit general solution in linear fractional programming. Naval Res. Logist. Quart. 20, 449-467 (1973).
R e a d e r s m a y a d d r e s s i n q u i r i e s t o D r W a d e D . C o o k at: a s 0 0 0 0 4 2 @ o r i o n . y o r k u . c a