VOL. 88, No. 2282

the rank method, when applied to the academic easehere considered, fails to satisfy the following postulate,which may be regarded by many people as a funda-mental principle:

Postulate of relevancy. The relative position of anytwo teams-or the relative position of any three teamsamong themselves-ought to be determinable on thebasis of the actual examination-grades obtained bythe members of these teams alone. In other words, indetermining the relative standing of any teams amongthemselves, it ought not to be neecssary to take intoaccount the performance of any contestants who donot belong to these teams.To show that the rank method fails to satisfy this

postulate, we construct the six following examples,each of which is a hypothetical, but entirely possible,case. There are four teams of three men each: "teamA" has members A1, A2, A3; "team B" has membersB1, B2, B3; and similarly for "team C" and "team D."The actual scores obtained by each of the twelve con-

TABLE II

CaseVII

C1 99 1 CtAi 98 2

AiB1

Bi 94 3A2 93 4

B2D1C2

B3D2CsA3

D3

a89188

87

85a84

8180

78

B 16A 17

A2DiB2

D2C2B3

C3

A3D3

56789

1011

12

CaseVIII

99

97L95

1 929190

888786

81

I79C 78 :

A 17B 18

1

23

456

789

10

11112

testants, in each of the six eases I, II, III, IV, V, VI,and shown in the table; and the question is, how shallwe determine the relative merits of teams A, B, C, ineach of the six cases ?

If we accept the postulate above, the relative stand-ing of the teams A, B, C, whatever it may be, mustobviously be the same in all six cases, since the actualgrades obtained by the members of these teams do notvary from case to case. On the other hand, if weadopt the method of ranks, we see from the table thateach of the six possible orders ABC, ACB, BAC, BCA,CAB, CBA, is represented by one of the six cases.The actual scores recorded in the six eases differ onlyin the scores obtained by two men, D1 and D2; and yetthe variation in these two supposedly irrelevant itemsis enough to upset completely the order of merit amongthe three teams A, B, C.No such ambiguity arises if the sum-of-the-grades

method is employed. This method gives A 268, B 267,C 266, thus determining the order ABC.As an even more striking illustration of the para-

doxes which may result from the rank method, we con-struct two further examples, VII and VIII, as shownin the second table. Here there are four teams A, B,C, D, and the question raised concerns the order ofthe two teams A and B. In passing from case VII tocase VIII every member of team B has improved hisgrade, while every member of team A has lowered hisgrade. Yet if the method of ranks is adopted, B startsout higher than A in case VII and then falls below Ain case VIII. The change which has brought about thisreversal of order is due entirely to changes in thegrades of two men, D1 and D2, who do not belong toeither of the teams A or B.

This paradoxical situation would not occur if thesum-of-the-grades method were used. According tothis method we would have A 271 and B 268 in caseVII, and B 271 and A 268 in case VIII, so that thechange in the final order of merit in passing froincase VII to case VIII would reflect in a natural mani-ner the improvement in B's grade and the deteriorationin A's grade.

These examples may suffice to show that if the pos-tulate of relevancy is accepted, some method other thanthe method of ranks must be adopted.

E. V. HUNTINGTONHARVARD UNIVERSITY

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