Equitable birthdate categorization systems for organized minor sports competition

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European Journal of Operational Research 192 (2009) 253–264 www.elsevier.com/locate/ejor

O.R. Applications

Equitable birthdate categorization systems for organized minor sports competition W.J. Hurley

*

Department of Business Administration, Royal Military College of Canada, PO Box 17000, Station Forces, Kingston, Ont., Canada K7K 7B4 Received 28 June 2006; accepted 3 September 2007 Available online 11 September 2007

Abstract In some organized minor sports programs where there is early competitive streaming, players born early in the year are more likely to reach elite levels than those born late in the year. This is generally attributed to the calendar year system most minor sports programs use to group players for the purposes of competition. In this paper I show how to devise more equitable systems based only on player ages. These systems rotate the relative age advantage so that those players born late in the calendar year are not always the youngest players in their age division. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Societal problem analysis; Scheduling

1. Introduction In almost all minor sports programs where there is early competitive streaming, including hockey, soccer, and baseball, players born early in the year are more likely to reach elite levels than those born late in the year. The literature terms this effect the Relative Age Effect (RAE) and its discovery dates to the work of Gini (1912), Pinter and Forlano (1933), and Huntington (1938). Grondin et al. (1984) and Barnsley et al. (1985) were the first to identify this phenomenon in hockey. Other contributions include Daniel and Janssen (1987), Barnsley and Thompson (1988), and Hurley et al. (2001). The interested reader is referred to Musch and Grondin (2001) for a comprehensive review of the RAE in all sports. One of the contributing factors to the RAE is the use of the calendar year to group participants for the purposes of competition. Supposing the start of the year to be January 1,1 participants born early in a year have, on average, a size *

Tel.: +1 613 541 6000x6468; fax: +1 613 541 6315. E-mail address: [email protected] 1 Some minor sports do not define the start of the year to be January 1. For example, Little League Baseball uses August 1. 0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.09.005

and maturity advantage over those born late in the same year. Moreover this happens season after season as these players progress through the system. Consequently, there is an argument for trying to find a more equitable categorization system. Such a system would distribute this age advantage so that those born late in the calendar year would not always be the youngest. In this paper I propose a new set of systems, termed Relative Age Fair Systems (RAF Systems), that are based only on participant birthdates. The important characteristic of these systems is that they are dynamically fair in a sense I will describe more fully herein. Suffice it to say here that, over time, no group of children will have a strict relative age advantage. RAF Systems are a generalization of the Novem System proposed by Boucher and Halliwell (1991). Their system has the characteristic that all players in a given calendar year get to be the oldest player on their team in at least one year over their minor hockey career. It is essentially a calender rotation system for a limited number of age groupings. The difference is that RAF Systems are based on an optimization which finds a rotation scheme that is as ‘‘close’’ to the Calendar Year System as possible. This notion of proximity has the following meaning. Under

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W.J. Hurley / European Journal of Operational Research 192 (2009) 253–264

the Calendar Year System, a minor sports program will typically allow a year’s worth of players to enter each season (and the oldest age grouping the previous season will leave). In order to get a system that is more equitable, it is not possible to allow a year’s worth of players to enter every year. Hence these systems have the characteristic that flow into the system will vary. Some years it will be less than a year’s worth of players; in other years, it will be more. The good news is that RAF Systems have the lowest variability relative to the Calendar Year System and it is the optimization that identifies these systems. The basic problem is to assign participants to age groupings so that, over time, the advantage of being the oldest in an age grouping is spread evenly over all participants. In this sense, the problem has the flavour of a job shop scheduling problem where jobs (participants) must be scheduled on machines (age groupings) over time. But that is as far as the similarity goes. The interested reader is referred to Pinedo (2006) for an excellent introduction to the scheduling literature. Finally, I remark that this paper is another in a growing body of work that applies operations research to problems in sport. This literature is now large. It effectively begins with two books edited by Ladany and Machol (1976, 1977). Gerchak (1994) provided a comprehensive review although now it is slightly dated. More recently, Albert et al. (2005) have produced an excellent anthology of papers relating probability, statistics and sports. In addition there have been three special issues of journals: Hurley (2006), Koning et al. (2003), and Wadkins and Wostenholme (2002). The paper is organized as follows. In the next section I give a brief overview of the RAE and present some new evidence based on some Canadian minor hockey data. In Section 3, I develop the details of the RAF System. Section 4 discusses the implementation issues and a final section summarizes.

Table 1 Distribution of birthdays for the AAA Level (Ontario Minor Hockey Association, registration data for the year 2000)

Minor Novice Novice Minor Atom Atom Minor Peewee Peewee Minor Bantam Bantam Midget

Ages

Q1

Q2

Q3

Q4

Ratio

8 9 10 11 12 13 14 15 16, 17

123 125 142 145 154 129 162 150 130

103 105 139 121 104 124 124 130 138

68 45 57 83 71 87 78 74 97

30 49 40 34 57 32 29 44 64

2.31 2.45 2.90 2.27 2.02 2.13 2.67 2.37 1.66

1260

1088

660

379

2.26

Totals

for the Ontario Minor Hockey Association for the year 2000. Q1 is the first quarter (January, February, and March), Q2 is the second quarter (April, May, and June), and so on. The last column in the table, labelled ‘‘Ratio’’, is the ratio of the number of first half births (Q1 + Q2) to second half births (Q3 + Q4). Note that this ratio exceeds 2 for all but the Midget age division. Given that male births in Canada are uniformly distributed, this is a significant Relative Age Effect. Now consider the House League data shown in Table 2. Looking at the ‘‘Ratio’’ column, note that ratio of first-half births to second-half births is about equal except for the Bantam and Midget age divisions. The overall ratio for House League is 0.949. All in, this dataset strongly supports the conclusion that the RAE has persisted in Canadian minor hockey. For those readers not familiar with the RAE, there is evidence of the effect right up to the National Hockey League (NHL) level. I counted the number of NHL players with birthdays in the first half of year (January, February, . . . , June) and those with birthdays in the second half (July, August, . . . , December) for selected years. These result are shown in Table 3. Note that there are roughly

2. Some new evidence from minor hockey in Canada There have been a number of studies of the RAE in Canadian minor hockey. All of these have found that the older players within an age division have a significant advantage. However they are now slightly dated. Consequently, I present RAE measurements from a relatively recent minor hockey dataset for two reasons. One is to give the reader an appreciation for its magnitude. The other is to provide evidence that the effect has persisted. In Canada, minor hockey participants are categorized along a number of criteria including age, geography, sex, and skill. The most highly skilled players play at the AAA level. The lowest skill level is termed House League and it is at this level that most participants play. There are intervening skill levels (labelled AA, A, BB, B, etc.) but here I focus on the two extremes. In Table 1, I present registration counts aggregated by quarter of birth for each age division at the AAA level

Table 2 Distribution of birthdays for House League (Ontario Minor Hockey Association, registration data for the year 2000)

Novice Atom Peewee Bantam Midget Totals

Ages

Q1

Q2

Q3

Q4

Ratio

8, 9 10, 11 12, 13 14, 15 16, 17

3108 2990 2333 1582 640

3598 3423 2836 1862 835

3573 3463 2763 1925 906

3158 3270 2683 1879 842

0.99 0.95 0.95 0.91 0.84

10,653

12,554

12,630

11,832

0.949

Table 3 Distribution of NHL birthdays for selected years 1982

1996

2000

#Players with first half birthdays #Players with second half birthdays

442 273

664 427

711 442

Ratio

1.62

1.56

1.61

W.J. Hurley / European Journal of Operational Research 192 (2009) 253–264

60% more players with birthdays in the first half of the year. 3. Constructing RAF systems 3.1. Definitions and problem setup I conceive a minor sports career in three parts: preCORE seasons, CORE seasons, and post-CORE seasons. The pre-CORE seasons, perhaps 1 or 2 years, are those seasons where players are introduced to the sport; the postCORE seasons, again 1 or 2 years, are the final seasons of play where registration numbers are dwindling. The CORE seasons, then, represent the bulk of a player’s participation in a minor sport. What I want to do is make the system fair over the CORE seasons. I assume that there are n CORE seasons. To define fairness, I break a calendar year of births into m parts. Each of these m parts is called a birth-period. For example, if m = 4, each calendar year of births is broken into four parts: 1. 2. 3. 4.

birthdates birthdates birthdates birthdates

between between between between

January 1 and March 31; April 1 and June 30; July 1 and September 30; and October 1 and December 31.

More generally, I define y(j) to be birth-period j within calendar year y. The index j will take the values 1, 2, . . . , m where 1 is assigned to the period beginning January 1 and m is assigned to the period ending December 31. Hence players from birth-period y(j) are older than those in y(k) when j < k. I am going to assume that m is large enough (and consequently the birth-periods are short enough) that those players born in a particular birth-period are equivalent in terms of development and maturity. The CORE age divisions are labelled D1, D2, . . . , Dn where players in D1 are younger than those playing in D2, those playing in D2 are younger than those in D3, and so on. I define a Birthdate Categorization System (BCS) to be an assignment of birth-periods to age divisions for all birth-periods and all future seasons. One such BCS is the Calendar Year System currently in use for most minor sports. The mechanics of age determination for the Calendar Year System depends on a specific date usually called the ‘‘date of age determination.’’ For instance, in Canadian minor hockey, a child’s age for the purposes of classification in a season is his or her age on December 31 of that season. For instance suppose that a particular division is to include all participants aged 10. For those participants born in say, December, they will start the season in September at age 9 since their age at December 31 will be 10. All age divisions are assumed to be made up of exactly M contiguous birth-periods. In Canadian minor hockey, House League uses 2-year age divisions so that M = 2m,

255

whereas the higher levels of representative hockey tend to use 1-year age divisions, or M = m. In summary then, there are three parameters which enable the definition of a Birthdate Categorization System: 1. n, the number of seasons in the CORE; 2. m, the number of birth-periods making up a calendar year of births; and 3. M, the number of birth-periods making up a CORE age division. 3.2. Fairness I define relative age in the following way. In each season, a CORE age division will be made up of participants from M contiguous birth-periods. I assign a relative age 1 to the oldest birth-period making up the age division, a 2 to the second oldest, and so on. Let rT(y(j)) denote the relative age of birth-period y(j) in season T. For an age division with M birth-periods, rT(y(j)) 2 {1, 2, . . . , M}. To define my notion of fairness, I employ the concept of a Relative Age List: Definition 1. The Relative Age List for birth-period y(j), L(y(j)), is a list of the relative ages that birth-period y(j) experiences over n CORE seasons ordered from smallest to largest. Denote these ordered relative ages r(1), r(2), . . . , r(n) where rð1Þ 6 rð2Þ 6    6 rðnÞ ð1Þ and therefore L = [r(1),r(2), . . . , r(n)]. Definition 2. A Birthdate Categorization System is said to be Relative Age Fair (RA-Fair) if every birth-period experiences the same Relative Age List over its CORE seasons. That is LðyðjÞÞ ¼ L0

ð2Þ

for all y(j). Such categorization systems are called RAF Systems. I make no attempt to measure the value of different relative age vectors. For example, consider two birth-periods y(j) and y(k), and their respective relative age vectors for a particular BCS: Season y(j) y(k)

s1

s2

s3

s4

s5

s6

s7

s8

1 4

1 4

2 3

2 3

3 2

3 2

4 1

4 1

The temptation is to compare these two vectors, perhaps with some sort of weighted average. However, given that there is no consensus among minor sports experts on when it is best to have an age advantage, I make no such attempt. Hence, my notion of fairness is the simple requirement that all birth-periods have the same Relative Age List.

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It is worth pointing out that the issue of whether a RAF System will reduce the Relative Age Effect is an empirical question. The only way to really find out is to introduce a system and monitor what happens. But whether such systems will solve the problem is not what I tackle in this paper. The issue is to design an age categorization system that is more equitable than the Calendar Year System. 3.3. RAF systems for the case M = m = n

in season T. The problem is to find a sequence of these birth-period additions, ð8Þ

x1 ; x2 ; . . . ; xT ; . . .

so that each birth-period gets the same Relative Age List over its CORE seasons. I term this sequence an Entry Sequence. As it turns out, the search space of sequences can be narrowed by looking at special cases. For this reason, I make the following definition:

Consider the case where the CORE age divisions are 1year divisions and the number of birth-periods in a year, m, is equal to the number of CORE seasons, n. I make these assumptions for two reasons. First, they simplify the analysis. And second, the extension to other RAF Systems using other parameter assumptions is straightforward once this initial problem is solved. To begin, I need to define a slightly different form of modulo arithmetic. For integers a and q, a mod q is defined to be the integer remainder when a is divided by q. For my purposes I need to make a simple change. Normally, if q divides a then a mod q = 0. However, I am going to set a mod q = q. Hence 24 mod 6 = 6, not 0. Under this revised definition, any integer a can always be written in the form

Definition 3. A Cyclic Entry Sequence is one with the property

a ¼ mq þ r;

Definition 4. Consider a Cyclic Entry Sequence of period n as defined in (10). The vector x = (x1, x2, . . . , xn) is termed an entry vector of order n.

ð3Þ

where the remainder, r, satisfies ð4Þ

1 6 r 6 q:

Note that, for normal modulo arithmetic, this last set of inequalities would require that 0 6 r 6 q  1. Assuming this revised definition, I employ the notation haiq ¼ a mod q:

ð5Þ

In the case where it is clear what the modulus q is, I will supress the q and use only hai. The following results are straightforward to prove. Lemma 1. For modulus q > 0 and any integers a1, a2, . . . , an *

X

+

* ¼

ai

i

q

X i

+ hai iq

ð6Þ

: q

Later in the paper we will need the following result, a direct implication of Lemma 1: Lemma 2. For modulus q > 0 and any integers A, a1, a2, . . . ,an * + * * ++ X X A ¼ A ai : ð7Þ h ai i q q

i

q

q

Now to the construction of RAF Systems. A RAF System can be initiated in any season so, without loss in generality, suppose it is initiated in season 1. Let xT be the number of birth-periods that play their first CORE season

ð9Þ

for all T

and we say that the period of this sequence is n. With Definition 3, the Entry Sequence becomes x1 ; x2 ; . . . ; xn ; x1 ; x2 ; . . . ; xn ; . . . |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} subsequence 1

ð10Þ

subsequence 2

or equivalently, a sequence of the vector x ¼ ðx1 ; x2 ; . . . ; xn Þ:

ð11Þ

I reserve a special name for this vector as the following definition makes clear.

I require that the number of birth-periods added over any n-season subsequence of an entry sequence be equivalent to n years worth of players, or n1 X

xT þs ¼ nm ¼ n2

for any T :

ð12Þ

s¼0

This constraint has the flavour of a conservation of flow constraint. If it is not imposed, players the system Pn1 entering 2 x < n Þ or young will become arbitrarily old ð s¼0 T þs Pn1 ð s¼1 xT þs > n2 Þ as the system passes through successive cycles. This leads to the following definition: Definition 5. An entry vector is balanced for the case m = n if n1 X

Proof. See Appendix. h

i

xT ¼ xT þn

xT þs ¼ n2

ð13Þ

s¼0

for all T. Given an entry vector of order n, x = (x1,x2, . . . , xn), I define the (s,t)th contiguous partial sum to be /s;t ðxÞ ¼ xs þ xsþ1 þ    þ xt

ð14Þ

for s 6 t. In the case where s = t, /s,s(x) = xs. Definition 6. The partial sum set, SðxÞ, of an entry vector, x, is the set of all such partial sums except /1,n, or SðxÞ ¼ f/s;t ðxÞ; 1 6 t 6 s 6 ng  f/1;n ðxÞg:

ð15Þ

W.J. Hurley / European Journal of Operational Research 192 (2009) 253–264

257

For instance, if the entry vector is of order 4, then SðxÞ consists of the partial sums

assigned to particular age divisions over seasons T and T + 1:

x1 ; x2 ; x3 ; x4 x1 þ x2 ; x2 þ x3 ; x3 þ x4 ;

Birth-period .. .

Season T .. .

Season T + 1 .. .

y0(1) y0(2) y0(3) y1(1) y1(2) y1(3) y2(1) y2(2) y2(3)

D2 D2 D2 D1 D1 D1

D3 D3 D2 D2 D2 D1 D1 D1

ð16Þ

x1 þ x2 þ x3 ; x2 þ x3 þ x4 : Note that SðxÞ includes the individual components of x but does not include the complete partial sum /1;4 ¼ x1 þ x2 þ x3 þ x4 :

ð17Þ

Definition 7. Consider an entry vector of order n, x, and its associated partial sum set, SðxÞ: SðxÞ is prime if none of its elements are divisible by n. The last concept we need is a special transformation of an entry vector.

Then an exchange partition of x at component t is the vector

Note that birth-periods are shown in the left-hand column. Column 2, labelled ‘‘Season T’’, shows the age divisions, Di, that the various birth-periods are assigned to in season T. Column 3 does the same thing for season T + 1. Looking at the table in detail, note that y1(2) is the middle birth-period in age division D1; hence its relative age in season T is

Ut ðxÞ ¼ ðxtþ1 ; xtþ2 ; . . . ; xn ; x1 ; . . . ; xt Þ:

rT ðy1ð2ÞÞ ¼ 2:

Definition 8. Consider an entry vector x ¼ ðx1 ; . . . ; xt1 ; xt ; xtþ1 ; . . . ; xn Þ:

ð18Þ

ð19Þ

ð22Þ

An exchange partition simply creates a new n-component vector by taking the first t components of x and making them the final t components of the new vector, Ut(x). Note that

Note that 2 birth-periods, y2(1) and y2(2), enter the CORE in Season T + 1. This means that, in season T + 1, y1(3) is now the oldest birth-period in age division D1 and that y1(2) is the youngest in D2 so that

Un ðxÞ ¼ x:

rT þ1 ðy1ð2ÞÞ ¼ 3:

ð20Þ

Hence, the exchange partitions of x for 1 6 t 6 n  1 are termed proper exchange partitions. We will need the following lemma in the proof of the main proposition. Lemma 3. Consider a balanced entry vector of order n, x, and suppose that its associated partial sum set, SðxÞ, is prime. Then SðUt ðxÞÞ is also prime for all t. Proof. See Appendix. h The next set of results show how to determine whether a particular entry vector will generate a RAF System. Lemma 4. Consider an entry vector x = (x1,x2, . . . , xn). For any birth-period y(j), rT ðyðjÞÞ ¼ hrT 1 ðyðjÞÞ  hxT in in

for T ¼ 2; 3; . . .

ð21Þ

ð23Þ

But by Lemma 4 rT þ1 ðyð2ÞÞ ¼ hrT ðy1ð2ÞÞ  h2i3 i3 ¼ h2  2i3 ¼ h0i3 ¼ 3: ð24Þ Hence Lemma 4 returns the same relative age the ADAT does. The following proposition gives a condition for verifying whether a given entry vector generates a RAF System. Proposition 1. For the case M = m = n, suppose that x = (x1,x2, . . . , xn) is a balanced entry vector and that SðxÞ is prime. Then x generates a RAF System. Proof. See Appendix. h Corollary 1. For the case M = m = n, suppose that x = (x1,x2, . . . , xn) generates a RAF System. Then xt 5 n for 1 6 t 6 n.

Proof. The proof is straightforward so will not be presented here. h

Proof. See Appendix. h

I offer the following example to help clarify how this recursion works. Consider the case n = m = 3 and the relationship between rT(y1(2)) and rT+1(y1(2)) when xT+1 = 2 and y1(2) happens to be assigned to age division D1 in season T. I term the following table an Age Division Assignment Table (ADAT). It shows what birth-periods are

In the case of Proposition 1, if any of the partial sums is divisible by n, then x does not generate a RAF System. More particularly, Corollary 1 makes explicit that we cannot set any of the components of x to n. For example, suppose the following entry vector was proposed for the case n = m = 5:

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W.J. Hurley / European Journal of Operational Research 192 (2009) 253–264

x ¼ ð6; 6; 6; 3; 4Þ:

ð25Þ

Note that it is balanced since X xt ¼ 6 þ 6 þ 6 þ 3 þ 4 ¼ 25 ¼ 52 :

ð26Þ

t

However the partial sum /2;4 ¼ x2 þ x3 þ x4 ¼ 15

ð27Þ

is divisible by 5. Therefore r1 = r4 and this sequence cannot generate a RAF System. Here is another example. Consider the case n = m = 4 and the entry vector x ¼ ð3; 3; 3; 7Þ:

ð28Þ

It is balanced since 2

x1 þ x2 þ x3 þ x4 ¼ 16 ¼ 4 :

ð29Þ

In this table, the CORE age divisions are labelled Di; the pre-CORE divisions are denoted e; and the post-CORE divisions are labelled Ej. Now consider the birth-year y1 and its associated birth-periods, y1(1), y1(2), y1(3), y1(4). The first season of the cycle is season T + 1. Note that, in season T + 1, three of the four y1 birth-periods, y1(1), y1(2), y1(3), begin play in their first CORE season in age division D1. Birth-period y1(4) continues to play the preCORE program in season T + 1. Hence the number of birth-periods entering the CORE in season T + 1 is xT+1 = 3. Note as well that xT þ2 ¼ 3; xT þ3 ¼ 3; xT þ4 ¼ 7: The table can be used to determine the relative ages for each of the birth-periods in their CORE seasons: Birth-period

Note, also, that xt 5 4 for all t and that x1 þ x2 ¼ 6; x2 þ x3 ¼ 6; x3 þ x4 ¼ 10;

ð30Þ

x1 þ x2 þ x3 ¼ 9; x2 þ x3 þ x4 ¼ 13: None of these partial sums is divisible by 4. Hence x generates a RAF System. To see this more clearly, here is the corresponding Age Division Assignment Table: Birth-period .. . y0(3) y0(4) y1(1) y1(2) y1(3) y1(4) y2(1) y2(2) y2(3) y2(4) y3(1) y3(2) y3(3) y3(4) y4(1) y4(2) y4(3) y4(4) y5(1) y5(2) y5(3) y5(4)

Season T .. .

T+1 .. .

T+2 .. .

T+3 .. .

T+4 .. .

D1 D1 e e e e

D2 D1 D1 D1 D1 e e e e e

D2 D2 D2 D2 D1 D1 D1 D1 e e e e e e

D4 D4 D4 D2 D2 D2 D2 D1 D1 D1 D1 e e e e e e e

E5 E5 D4 D4 D4 D4 D3 D3 D3 D3 D2 D2 D2 D2 D1 D1 D1 D1 e e e e

T+5 .. . E6 E5 E5 E5 E5 D4 D4 D4 D4 D3 D3 D3 D3 D2 D2 D2 D2 D1 D1 D1 D1 e

ð31Þ

y1(1) y1(2) y1(3) y1(4)

Season T+1

T+2

T+3

T+4

T+5

2 3 4

3 4 1 2

4 1 2 3

1 2 3 4

1

Each birth-period gets the Relative Age List [1, 2, 3, 4]. In fact, if the table were extended down and to the right (by adding more birth-periods and seasons), it could be verified that all other birth-periods would get the same Relative Age List. It is worth pointing out that the partial sum set is not independent in the sense that if particular partial sums are not divisible by n then related partial sums are also not divisible by n. For instance, if x is balanced and /1,2 is not divisible by n, then /3,n is also not divisible by n. Hence it is not necessary to check that /3,n is divisible by n if /1,2 has already been checked. The complete set of these dependencies is easily specified. They are based on a partition of the complete partial sum as follows. Suppose x is balanced and /1,t is not divisible by n. Since x is balanced, /1;n ¼ n2 :

ð32Þ

Note that /1;n ¼ /1;t þ /tþ1;n

ð33Þ

for 1 6 t 6 n  1 and therefore /tþ1;n ¼ /1;n  /1;t ¼ n2  ðm1;t n þ r1;t Þ;

ð34Þ

where m1,t and r1,t are integers and r1,t 5 0. Hence /tþ1;n ¼ nðn  m1;t Þ  r1;t

ð35Þ

and therefore /t+1,n is not divisible by n. For a particular order n, the RAF System entry vector is not unique. For instance, above I identified the entry vector x ¼ ð3; 3; 3; 7Þ

ð36Þ

W.J. Hurley / European Journal of Operational Research 192 (2009) 253–264

for the case m = n = 4. Another is 0

x ¼ ð6; 5; 2; 3Þ:

ð37Þ

The question arises, then, which is best. The nice thing about the Calendar Year System entry vector, (4, 4, 4, 4), is that it gives a uniform flow of new participants to the system every season. Hence it would be nice to find an entry vector which differs from the Calendar Year System entry vector by as little as possible. One way to do this would be to solve the following optimization problem: minx

n X

k¼1 n X

s:t:

x k ¼ n2 ;

x t ¼ n þ ht

ð39Þ

ðn þ hk  nÞ2 ¼

k¼1

n X

h2k :

ð40Þ

k¼1

The problem is to find a vector h ¼ ðh1 ; h2 ; . . . ; hn Þ ð41Þ Pn 2 which makes k¼1 hk as low as possible. The requirement that x is balanced gives n2 ¼

n X

xk ¼

k¼1

n X

ðn þ hk Þ ¼ n2 þ

k¼1

n X

hk

ð42Þ

k¼1

and the implication is n X

hk ¼ 0:

ð48Þ

SðhÞ is prime; jhk j P 1 and integer for k ¼ 1; 2; . . . ; n:

ð43Þ

As for partial sums and their divisibility by n, note that /s;t ¼ xs þ xsþ1 þ    þ xt ¼ n þ hs þ n þ hsþ1 þ    þ n þ ht ¼ ðt  s þ 1Þn þ hs þ hsþ1 þ    þ ht :

ð44Þ

for k = 1, 2, . . . , n. To solve this transformed program, first note that a lower bound Pn on the optimal value of the objective function is n since k¼1 h2k is minimized when each hk is set to either 1 or +1. However such solutions are not feasible for n P 3. To see this, note that if n is odd, there is no vector h ¼ ðh1 ; h2 ; h3 Þ

Pn with hk 2 {1, 1} which satisfies k¼1 hk ¼ 0: And if n is even, there is no way to make sure that /s,t(h) 5 0 since there must be at least two adjacent components of h, say hp and hp+1, for which hp + hp+1 = 0. Hence the question is whether there are solutions to the program which set |ht| to at most 2 for all t? Consider the case n = 3. Based on the argument above, at least one of the components of h must be 2 or +2 and hence, assuming that one of the components of h is set to 2 or +2, a lower bound on the objective function value is 3 X

2

2

2

h2k ¼ ð2Þ þ ð1Þ þ ð1Þ ¼ 6:

ð50Þ

Now consider h = (1, 1, 2). This vector satisfies the constraints of the program and has an objective function value of 6. Hence it is an optimal solution. More generally, it is possible to show that if xt = n + ht generates a RAF System, then so too does yt = n  ht. That is, for any entry vector generating a RAF System, there is a second that turns out to be a reflection about the Calendar Year entry vector c ¼ ðn; n; . . . ; nÞ:

Defining /s;t ðhÞ ¼ hs þ hsþ1 þ    þ ht

ð49Þ

k¼1

k¼1

ð45Þ

ð51Þ

Hence I offer the following lemma which is easy to prove: Lemma 5. If x = c + h generates a RAF System, then so too does y = c  h.

for 1 6 s 6 t 6 n, /s,t can be rewritten as /s;t ¼ ðt  s þ 1Þn þ /s;t ðhÞ

hk ¼ 0;

x k ¼ n þ hk

for t = 1, 2, . . . , n. Then the objective function is transformed to

k¼1

k¼1 n X

Any solution to this program can be converted to a solution for (38) using

Note that the objective function uses a standard Euclidean norm to measure distance. The constraints guarantee that the optimal entry vector will generate a RAF System. However, for reasons that will become clearer later, I am going to solve a variation of this program. Define

n X

ð47Þ

Moreover, as above, SðhÞ is said to be prime if none of its elements are divisible by n meaning that /s,t(h) 5 0 for all such elements. Hence an equivalent program to the one in (38) is n X h2k minh

ð38Þ

k¼1

ðxk  nÞ2 ¼

SðhÞ ¼ f/s;t ðhÞ; 1 6 s 6 t 6 ng  f/1;n ðhÞg:

k¼1

SðxÞ is prime; xk P 1 and integer for k ¼ 1; 2; . . . ; n:

n X

an entry vector, x, to be the set of all partial sums /s,t(h) except /1,n(h), or

s:t:

ðxk  nÞ2

259

ð46Þ

and therefore, if /s,t(h) 5 0, then /s,t is not divisible by n. Proceeding as above, I define the partial sum set, SðhÞ, of

I refer to this pair of entry vectors as dual partners. Now consider the program in (48) for the case n = 4. Moreover suppose there is a h which generates a RAF

260

W.J. Hurley / European Journal of Operational Research 192 (2009) 253–264

System with at most one component ±2. By the preceding lemma, one of the dual partners will have a component for which hk = 2. For h to be feasible, the other three elements, all of which are either +1 or 1, must sum to 2. Clearly there is no way that this can happen. Hence a feasible solution for (48) must have at least two components for which |hk| = 2. P Therefore a lower bound for the objective function value is h2k ¼ 6: Now consider the vector h ¼ ð2; 1; 2; 1Þ:

As for solving program (48) for higher values of n, I have been able to show (in much the same way as for n = 3 and n = 4) that the following dual partners are based on optimal solutions: Pn 2 Order Entry vectors k¼1 ðxk  nÞ n=2

(1, 3) (3, 1)

2 2

ð52Þ

n=3

(2, 2, 5) (4, 4, 1)

6 6

ð53Þ

n=4

(2, 3, 6, 5) (6, 5, 2, 3)

10 10

n=5

(3, 3, 6, 7, 6) (7, 7, 4, 3, 4)

14 14

n=6

(4, 4, 5, 8, 8, 7) (8, 8, 7, 4, 4, 5)

18 18

n=7

(5, 5, 8, 9, 9, 8, 5) (9, 9, 6, 5, 5, 6, 9)

22 22

n=8

(6, 6, 6, 7, 10, 10, 10, 9) (10, 10, 10, 9, 6, 6, 6, 7)

26 26

n=9

(7, 7, 7, 8, 11, 11, 11, 11, 8) (11, 11, 11, 10, 7, 7, 7, 7, 10)

30 30

n = 10

(8, 8, 8, 8, 9, 12, 12, 12, 12, 11) (12, 12, 12, 12, 11, 8, 8, 8, 8, 9)

34 34

Note that h1 þ h 2 þ h 3 þ h 4 ¼ 0 and that h1 þ h2 ¼ 3; h2 þ h3 ¼ 1; h3 þ h4 ¼ 3 h1 þ h2 þ h3 ¼ 1; h2 þ h3 þ h4 ¼ 2:

ð54Þ

Hence this vector satisfies the constraints of (48). Moreover, it has an objective function value of 6 so it must be an optimal solution. The Age Division Assignment Table for h = (2,1,  2,  1) (equivalently x = (6, 5, 2, 3)) is shown below: Birth-period .. . y0(3) y0(4) y1(1) y1(2) y1(3) y1(4) y2(1) y2(2) y2(3) y2(4) y3(1) y3(2) y3(3) y3(4) y4(1) y4(2) y4(3) y4(4) y5(1) y5(2) y5(3) y5(4)

Season 0 .. .

1 .. .

2 .. .

3 .. .

D1 D1

D2 D2 D2 D2 D1 D1 D1 D1

D4 D3 D3 D3 D3 D2 D2 D2 D2 D1 D1 D1 D1

D4 D4 D4 D3 D3 D3 D3 D2 D2 D2 D2 D1 D1 D1 D1

4 .. .

D4 D4 D4 D4 D3 D3 D3 D3 D2 D2 D2 D2 D1 D1 D1 D1

5 .. .

D4 D4 D4 D4 D3 D3 D3 D3 D2 D2 D2 D2 D1 D1

Note, first, that the pre- and post-CORE age divisions have not been shown. Second, 6 birth-periods have been added in Season 1 (y1(1), y1(2), y1(3), y1(4), y2(1), y2(2)), 5 birth-periods in Season 2 (y2(3), y2(4), y3(1), y3(2), y3(3)), 2 birth-periods in Season 3, and 3 in Season 4. In addition, note that the Relative Age List for each birth-period beginning with y1(1) is [1,2,3,4].

Relaxing M = m = n When constructing a RAF System, it is usually the case that the size of an age division, M, is fixed. That is, it is usually 1 year (m birth-periods) or 2 years (2m birth-periods). For now, consider the case where age divisions are 1 year (M = m) and the number of CORE age divisions, n, is an integral multiple of the number of birth-periods in a year, m: n ¼ pm; ð55Þ where p P 2 is an integer. We say that there are p cycles, each of length m seasons. For instance suppose m = 4 and n = 8. Hence there would be eight seasons of play in the CORE for each birth-period and each age division would comprise four quarters of participants. Let xðkÞ ¼ ðxk1 ; xk2 ; . . . ; xkm Þ be the entry vector for cycle k. Let

ð56Þ

ð57Þ xp ¼ ðxð1Þ ; xð2Þ ; . . . ; xðpÞ Þ be the complete entry vector for all p cycles. The following result shows how entry vectors for the case n = m can be used to specify entry vectors for the case n = pm. Proposition 2. Suppose the entry vector x* generates a RAF System for the case m = n. Then xp ¼ ðxð1Þ ; xð2Þ ; . . . ; xðpÞ Þ ¼ ðx ; x ; . . . ; x Þ generates a RAF System for the case n = pm.

ð58Þ

W.J. Hurley / European Journal of Operational Research 192 (2009) 253–264 Table 4 The age division assignment table for nine age divisions and three subperiods Birth-period

y1(1) y1(2) y1(3) y2(1) y2(2) y2(3) y3(1) y3(2) y3(3) y4(1) y4(2) y4(3) y5(1) y5(2) y5(3) y6(1) y6(2) y6(3) y7(1) y7(2) y7(3) y8(1) y8(2) y8(3) y9(1) y9(2) y9(3)

Season s1

s2

s3

s4

s5

s6

s7

s8

s9

D2 D2 D1 D1 D1

D3 D2 D2 D2 D1 D1 D1

D3 D3 D3 D2 D2 D2 D1 D1 D1

D5 D5 D4 D4 D4 D3 D3 D3 D2 D2 D2 D1 D1 D1

D6 D5 D5 D5 D4 D4 D4 D3 D3 D3 D2 D2 D2 D1 D1 D1

D6 D6 D6 D5 D5 D5 D4 D4 D4 D3 D3 D3 D2 D2 D2 D1 D1 D1

D8 D8 D7 D7 D7 D6 D6 D6 D5 D5 D5 D4 D4 D4 D3 D3 D3 D2 D2 D2 D1 D1 D1

D9 D8 D8 D8 D7 D7 D7 D6 D6 D6 D5 D5 D5 D4 D4 D4 D3 D3 D3 D2 D2 D2 D1 D1 D1

D9 D9 D9 D8 D8 D8 D7 D7 D7 D6 D6 D6 D5 D5 D5 D4 D4 D4 D3 D3 D3 D2 D2 D2 D1 D1 D1

first deal with a number of minor implementation issues. And I’ll do this in the context of minor hockey in Canada, a system I have some familiarity with since I played in it and, at this writing, both of my boys are largely through it. RAF Systems do not require any additional resource. The primary resource requirements for minor hockey are ice-time and people to coach and administer. A RAF System simply redistributes players among the existing teams. There would be no additional requirement for ice-time or people. That said, if a new system were to be implemented, there would be a one-time cost associated with making the change. This cost would include the dollars and volunteer time required to inform and educate parents and children on how a new system would work. One of the main criticisms of RAF Systems put to me is that they may be too complicated for parents and participants to understand. Based on my experience with two parental focus groups, I do not believe this to be true. For each of these groups, I first instructed participants on how an Age Division Assignment Table would work in the context of Canadian minor hockey. A section of one of the Age Division Assignment Tables I used is shown below: Birth year

Birth period Season

1994

January– March April–June July– September October– December January– March April–June July– September October– December January– March April–June July– September October– ..December .

Proof. See Appendix. h

1994 1994

Here is an example. Suppose n = 9 and m = 3. That is, the CORE is 9 seasons, there are 9 age divisions (D1, D2, . . . , D9), and each year of births is divided into three birth-periods, each 4 months. Consider the entry vector

1994

xp ¼ ð5; 2; 2; 5; 2; 2; 5; 2; 2Þ:

Lðy1ð3ÞÞ ¼ ½1; 1; 1; 2; 2; 2; 3; 3; 3:

1995

ð59Þ

Note that each of the three cycles has a corresponding entry vector x* = (5, 2, 2) and this vector generates a RAF System for the case n = m = 3. The ADAT for the above example is shown in Table 4. Note that all birth-periods receive the Relative Age List

1995 1995 1995 1996

ð60Þ

4. Implementation issues I have shown that RAF Systems are more equitable than the Calendar Year System. While the focus has been minor hockey in Canada, there are other minor sports where relative age is a problem. The Musch and Grondin (2001) review indicates that the Relative Age Effect is a problem in soccer and Little League Baseball worldwide. In my view, RAF Systems could easily be applied in the minor systems for these sports. I believe there is only one significant implementation problem with these systems. But before discussing it, I will

261

1996 1996 1996 .. .

2002 2003 2004 2005 2006    NOV atm

ATM pw

PW

NOV atm NOV atm

ATM ATM pw atm ATM pw

  

NOV NOV atm

ATM pw



nov

NOV atm

ATM pw



nov nov

NOV atm atm NOV NOV atm

ATM    ATM   

nov

nov

NOV atm

ATM   

nov

NOV atm

ATM   

nov nov

NOV NOV atm nov NOV atm

 

nov

NOV atm



.. .

.. .

.. .

.. .

.. .

..

.

This is a simple lookup table. For instance if a parent’s child was born on July 13, 1995, the parent would first look down the initial column (labelled Birth Year) and locate 1995. Next, the second column (labelled Birth Period) would be examined to locate the appropriate birth-period interval. In this case, it would be the ‘‘July–September’’

262

W.J. Hurley / European Journal of Operational Research 192 (2009) 253–264

row of Birth Year 1995 since the child’s birthdate is July 13, 1995. This then is the row that specifies exactly what age divisions the child would play in over all seasons from 2002 to 2006. For instance in season 2006, the child would play in the ‘‘ATM’’ (Major Atom) age division. Once I had explained this ADAT to parents, I then asked them to identify what age division their child would play in for three specific seasons using a one-page questionnaire. All parents were able to answer these questions correctly. While it is true that this exercise does not prove that all parents are capable of understanding the new system, it’s certainly a pretty good indication that a majority would. If a RAF System were introduced, there would be a reference birth-year and for that birth-year and subsequent birth-years, each birth-period would experience the same relative age competition. For those birth-years already in the system, the ones with players older than the reference birth-year, the relative ages of participants would vary. Hence, the introduction of a RAF System would result in a more equitable allocation of relative age for these birthyears, albeit these allocations will not be completely fair in the sense described above for CORE seasons. Therefore the introduction of a RAF System results in a more equitable system for these older players. For RAF Systems, some birth-periods must begin CORE play in age division D2. Parents and administrators might be concerned that the jump directly to D2 might be problematic for some children. The reality is that there is no jump. It is always true that a RAF System has a participant competing among the same number of contiguous birth-periods. At worst, a participant will be in the youngest birth-period in age division D2. But how is this worse than the Calendar Year System for those participants with late year births? They will always be, on average, the youngest in every age division they play in over their minor hockey careers. RAF Systems can be devised for systems with other criteria besides age. One such system is the Pop Warner minor football program in the United States. It uses age and weight. The following table outlines how children are classified: Division

Age

Weight

Flag

6–7

40–70

Mighty Mites Older/Lighter

8 9 10

45–90 45–85 45–65

Junior Pee Wee Older/Lighter

8–9–10 11

55–95 55–75

Pee Wee Older/Lighter

9–10–11 12

70–110 70–95

Junior Midget Older/Lighter

80–130 13

80–130 80–110

Midget Older/Lighter

11–12–13–14 15

95–145 95–130

Note the choice this table gives participants who are small for their age. For instance, consider an 11-year-old who is 72 pounds. This child can play either in the Junior Pee Wee division or the Pee Wee division. Given the importance of size in some minor sports such as hockey, such a categorization might be quite useful. It is straightforward to overlay a RAF System on such systems. The major implementation challenge would be political. Minor sport programs tend to put considerable weight on local concerns. If too many local organizations object to changes in the age categorization system, they will not be implemented nationally. Good evidence for this comes from Canadian minor hockey. The structure of Canadian minor hockey requires that any change to age classifications be approved by Hockey Canada. Certainly Hockey Canada is aware of the problem. Some 7 years ago, in the summer of 1999, they organized and sponsored a conference, the Open Ice Hockey Summit, to examine the state of hockey in Canada. Participants included players, coaches, and managers from all levels of Canadian hockey including the NHL. The conference produced eleven recommendations to improve hockey in Canada. Recommendation 3 was this: Examine the date of age determination. (The cut-off date is currently December 31st, but some thought has been given to rotating it throughout the calendar year. The objective is for a player to not always be the youngest or oldest in a given division.)2 But to date, nothing has been done and there is no plan to do anything. I believe there are two reasons for this. First, Hockey Canada is not sure exactly how such a rotation would work. And second, the organizational effort required to implement the change would be significant. There is some evidence on this last point. At their 2001 Annual General Meeting, Hockey Canada implemented a very simple change in age categories. The old and new age categorizations are shown in the following table:

Novice Atom Peewee Bantam Midget

Old Ages

New Ages

8, 9 10, 11 12, 13 14, 15 16, 17

7, 8 9, 10 11, 12 13, 14 15, 16, 17

The new categorization still has two-year age divisions with the exception of Midget. The primary difference is that the new age divisions begin with players one year younger. Hockey Canada took the position that the new age categories would reduce size disparities within age divisions: ‘‘The overwhelming scientific evidence showed that pairing of male athletes age 13 and 14 was best. The male 2 See the CHA website http://nt.canadianhockey.ca/openice/e/index/ html.

W.J. Hurley / European Journal of Operational Research 192 (2009) 253–264

growth spurt, usually takes place in this age group. Although the 13 and 14 years old grouping does not develop a perfect coupling, it minimizes many characteristic differences that are evident with other possible pairings.’’3

263

Proof of Lemma 3. The lemma is true for t = n since SðUn ðxÞÞ ¼ SðxÞ. So consider any exchange partition Ut ðxÞ ¼ ðxtþ1 ; . . . ; xn ; x1 ; . . . ; xt Þ;

ð64Þ

This proposal generated significant discussion and emotion across the country for the better part of 2 years. Suffice it to say that this change, a minimal change at best, was very difficult to implement politically. There is little doubt that Hockey Canada currently feels that its organizational energies would best be spent on other priorities.

where t 6 n  1 (a proper exchange partition). First note that any contiguous partial sum of the first n  t components of Ut(x) is not divisible by n since SðxÞ is prime. By the same reasoning, any contiguous partial sum of the last t components of Ut(x) is not divisible by n. Therefore it remains to check contiguous partial sums that include the elements xn and x1. Denote the vector Ut(x) with

5. Conclusion

Ut ðxÞ ¼ ðy 1 ; . . . ; y i1 ; y i ; . . . ; y nt ; y ntþ1 ; . . . ; y j ; y jþ1 ; . . . ; y n Þ; ð65Þ

This paper begins by summarizing the evidence that the categorization systems used in organized minor sports programs are unfair. The first step in a solution to this problem is to design a new system. Here I have presented an operations research approach to the problem. The main contribution has been to put the age categorization problem on a sound mathematical footing in the case where the categorization is based only on age. In addition I have made the meaning of fairness explicit. For minor sports where size is an advantage, RAF Systems are not likely to be the complete answer. They are demonstrably more equitable than existing systems. But they do not really take care of children who are small relative to the other children. One approach would be to use both age and weight as classification criteria much like the Pop Warner minor football program does in the United States. Future research will examine how a weight criterion can be incorporated into a RAF System.

Proof of Lemma 1. Note that the ai can be written ai ¼ m i q þ r i ;

ð66Þ

y n ¼ xt Then the contiguous partial sum Y i;j ¼ y i þ    þ y nt þ y ntþ1 þ    þ y j

ð67Þ

includes the components xn and x1. Also define Y 1;i1 ¼ y 1 þ    þ y i1 ; Y jþ1;n ¼ y jþ1 þ    þ y n :

ð68Þ ð69Þ

Note that Y 1;i1 þ Y i;j þ Y jþ1;n ¼ n2

Y i;j ¼ ai;j n; ð61Þ

where mi and ri are integers and ri, the remainder, satisfies ð62Þ

1 6 ri 6 q: Hence, * + * + X X ai ¼ ðmi q þ ri Þ q

i

* ¼

q *

¼

ð70Þ

X

mi þ

i

X

X i

+ hai iq

and the result is proved.

+

* ¼

ri q

X i

ð71Þ

where ai,j is an integer. From (70), we have that Y 1;i1 þ Y jþ1;n ¼ n2  ai;j n ¼ nðn  ai;j Þ:

ð72Þ

Taking the modulus with respect to n of both sides, we have hY 1;i1 þ Y jþ1;n in ¼ hnðn  ai;j Þin ¼ n:

q

i

3

y 1 ¼ xtþ1 ; y nt ¼ xn ; y ntþ1 ¼ x1 ;

since x is balanced. Now suppose that Yi,j is divisible by n. In particular, suppose

Appendix

i

where

ð73Þ

But note that Y1,i1 + Yj+1,n is a contiguous partial sum contained in SðxÞ: Hence, it is not divisble by n and therefore

+ ri

hY 1;i1 þ Y jþ1;n in < n:

q

ð63Þ q

h

Taken from ‘‘The New Age of Hockey: A Minor Hockey Age Change Information Guide’’ which can be found at the CHA’s website: www.canadianhockey.ca.

ð74Þ

This is a contradiction since it is impossible that h Y1,i1 + Yj+1,nin is simultaneously equal to n and less than n. Therefore Yi,j is not divisible by n for all feasible i and j. Hence SðUt ðxÞÞ is prime for all t and the proof is complete. h Proof of Proposition 1. Suppose players in birth-period y(k) are scheduled to begin CORE play in season T1. By Lemma 3, the entry vector corresponding to seasons T1,

264

W.J. Hurley / European Journal of Operational Research 192 (2009) 253–264

T2, . . . , Tn is prime. Hence, without loss in generality, let the corresponding entry vector be x. If the entry vector x is to generate a RAF System, every birth-period must have the Relative Age List [1, 2, . . . , n]. Using (21), note that: rsþ1 ðyðkÞÞ ¼ hrs ðyðkÞÞ  hxsþ1 ii

ð75Þ

and rs ðyðkÞÞ ¼ hrs1 ðyðkÞÞ  hxs ii:

ð76Þ

Substituting the second into the first for rs(y(k)) gives

ð77Þ

by Lemma 2. Continuing this composition,

ð78Þ

where n P t > s P 1. But SðxÞ is prime. Therefore /s+1,t is not divisible by n and hence ð79Þ

for all values of t and s that satisfy n P t > s P 1. Therefore the relative ages r1 ðyðkÞÞ; r2 ðyðkÞÞ; . . . ; rn ðyðkÞÞ

ð80Þ

are distinct. And since 1 6 rt ðyðkÞÞ 6 n

ð81Þ

for all t, y(k) must have the Relative Age List [1, 2, . . . , n]. But the selection of birth-period y(k) was arbitrary. Hence all birth-periods must receive the Relative Age List [1, 2, . . . , n]. Therefore x generates a RAF System and the proof is complete. h Proof of Corollary 1. Suppose that x generates a RAF System and that xt = n. From Lemma 4, we have that rt ðyðjÞÞ ¼ hrt1 ðyðjÞÞ  nin ¼ rt1 ðyðjÞÞ for 2 6 t 6 n; ð82Þ which is a contradiction. Now suppose x1 = n. Then rn ðyðkÞÞ ¼ hr1 ðyðkÞÞ  hx2 þ x3 þ    þ xn ii:

ð83Þ

But if the entry vector is balanced, x1 þ x2 þ    þ xn ¼ n2

ð84Þ

or x2 þ    þ xn ¼ n2  x1 ¼ n2  n:

ð85Þ

Note that the right-hand side of this last equation is divisible by n. Hence x2 + x3 + . . . + xn is also divisible by n. Therefore r1 ðyðkÞÞ ¼ rn ðyðkÞÞ

0

p2 s

ð87Þ

pn0 s

Therefore all birth-periods get the same Relative Age List and therefore x generates a RAF System and the proof is complete. h References

rt ðyðkÞÞ ¼ hrs ðyðkÞÞ  hxsþ1 þ xsþ2 þ    þ xt ii

rt ðyðkÞÞ 6¼ rs ðyðkÞÞ

6 7 41; 1; . . . ; 1; 2; 2; . . . ; 2; . . . ; n; n; . . . ; n5: |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} p1 s

¼ hrs1 ðyðkÞÞ  hxs i  hxsþ1 ii

¼ hrs ðyðkÞÞ  h/sþ1;t ii;

Proof of Proposition 2. Based on Proposition 1, the Relative Age List for any cycle and all birth-periods is [1, 2, . . . , m]. Hence the Relative Age List for each birth-period over all p cycles is 2 3

0

rsþ1 ðyðkÞÞ ¼ hhrs1 ðyðkÞÞ  hxs ii  hxsþ1 ii ¼ hrs1 ðyðkÞÞ  hxs þ xsþ1 ii

which is a contradiction to the assumption that x generated a RAF System and the proof is complete. h

ð86Þ

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