Optimal Ranking of Tournaments
J. Spencer The Rand Corporation Santa Monica, California
Dedicated to the Memory of Jon Hal Folkman December 8,1938 - January 23,1969
1. INTRODUCTION
Let us be given a tournament Tn (i-e., a complete direct-
ed graph) on n players. It is natural to look for a ranking (linear order) L on the n players that best reflects the tour- nament result. Thinking of L as a complete directed transitive graph we may use IE(T) fl E(L) I, where E(T) = edge set of T, as a measure of how good a fit L is. We ask, what is the worst case. Set
f (n) = min max IE(T ) nE(L) 1 . n Tn
We wish to find bounds of f(n).
such that every T contains a set of k consistent arcs. The
arcs in a set S are here said to be consistent if we may re-
label the nodes so that if ij is in S then i > j. Erdbs and Moon fll moved
It is easy to show that f(n) is the greatest integer k
n
+
cl, c ... represent positive absolute constants. The value
of c was improved by Reid [21 and Jung (unpublished) showed 2’
1
Any views expressed i n t h i s paper are those of the author. They should not be interpreted as r e f l e c t i n g the vieus of The RAND Corporation or the o f f i c i a l opinion o r policy of any of i t s governmental or pyiuate research sponsors.
Networks, 1: 135-138 3 1971 John Wiley &i Sons, Inc. 135
136 SPENCER n
f ( n ) - > 1 / 2 ( 2 ) + c2n log n. Our main r e s u l t i s
Theorem A: f f n ) > 1 / 2 f : ) + c3n 3 / 2 . -
W e also show
Theorem B : f f n ) - < 1 / 2 f i l + ( f i / 4 + ~ f l I l n ~ ’ ~ ~
2. A LOWER BOUND FOR f (n)
I n t h i s s e c t i o n w e prove Theorem A. W e show it f o r c =
f o r m < n .
(1 < i < n) win si games.
p l ac ing p l aye r i on top and us ing induct ion t o g e t t h e o rde r on t h e o t h e r (n - 1) p laye r s . Then
Suppose by induct ion w e have f ( m ) 1_ m/4 + c m 3/2 3 3
I f s
F i x the tournament T = Tn. L e t p l aye r i
> n/2 + & create an L by i - -
2 3 /2 + J;; > n /4 + c3n - and w e are done.
similar r e s u l t s .
f o r a22 i. Set
For s
Thus w e may assume 1-42 - 6 < s
< n/2-& w e p l a c e i a t t h e bottom with i < n/2 + 6 i -
U = { ( x , B) : x E TI B C T , - x 4 B,IBI = n/2,
I { y E B : x + y)I > n/4 + 6 1 .
For f i x e d x s e t S ( x ) = {y : x + y}. Then
I{B : (x, B) E U}l =
n (n/2)
n
Prob [x 6 B, ( S ( x ) fl B I - > n/4 c J;; : IBI = n/2]
- 1 / 2 (n /2) Prob [ I S ( x ) n BI as x 4 B i s p r a c t i c a l l y independent of I S ( x ) fl BI .
t i o n 13, p. 2191. ( S ( x ) I = n/2 - &. Deviat ion = J1/2 n/2 1 / 2 1 / 2 = &/4 so t h e
n/4 + J;; : IBI = 11/21
This P r o b a b i l i t y i s g iven by a hypergeometric d i s t r i b u -
Then E [ I S ( x ) f-7 B ( 3 = n/4 -&2, Standard I t i s smallest when I S ( x ) \ i s minimal,
OPTIMAL RANKING OF TOURNAMENTS 137
Prob > 1 - O(6) where - X
J -m
i s t h e cumulative normal.
n n/2
n L e t t i n g x range over T we see I u I > n(n/2 1 (1 - @(6)).
A s t h e number of B is ( t h e r e e x i s t s a f i x e d B so t h a t
I{x : ( x , B ) E U)l> - n ( l - @ ( 6 ) ) .
F ix t h a t B and X = {x : (x , B) E U ) and R = T - B - X. Define l i n e a r o r d e r s L ( B ) , L ( X ) , L ( R ) t h a t agree with T I B , T I X , T I R r e s p e c t i v e l y on a t least h a l f t h e edges [by induct ion - or by j u s t t r y i n g L and Lc] . X < B < R or R < X < B so t h a t a t least h a l f of t h e edges with one node i n R and one no t agree wi th T.
Then de f ine L by e i t h e r
Then w e have
Q.E.D. 2 I E ( L ) n E ( T ) I 2 n /4 + n 3 j 2 ( l - Q(6)).
By more c a r e f u l manipulations w e have been able t o prove Theorem A f o r c3 = .1577989...
3. AN UPPER BOUND FOR f (n)
I n t h i s s e c t i o n w e prove Theorem B. Think of l i n e a r o rde r s L as permutat ions, and t h e r e f o r e func t ions
L : E l , ..., n) -41, ..., n ) . Set t = [&I and
L ( t + 1)< ... < L ( 2 t )
L ( k t + 1) < L ( k t + 2 ) < ... < L ( k t + t)
1
I t can e a s i l y be shown t h a t < Ln. Set
V = ( ( T , L) : L €2, I E ( T ) n E ( L ) I 1/2(;)+ a)
Following t h e argument of Erdt)s and Moon [ l l w e can show
138 SPENCER
when a = ( f i / 4 + o(1) G. Thus t h e r e e x i s t s T such t h a t
L € 2 But given any L , w e rearrange L ( k t + 11, ..., L ( k t + t) t o
form L* E 2 where IE(L)AE(L*) I 5 t ( 2 ) - 1/2n
Thus for a l l L ,
n t 3/2 .
I E ( T ) n E ( L ) I - < I E ( T ) n E(L*) I + I E ( L * ) A E ( L ) (
= 1 / 2 ( ~ ) + ( 6 / 4 + b ( l ) ) n 3 / 2
REFERENCES
1. E r d o s , P. and J. W. Moon, "On S e t s of C o n s i s t e n t Arcs i n a Tournament," Canad. Math. BUZZ. 8 , 1965, pp. 269- 271.
2. Reid, K. B. , " S t r u c t u r e i n F i n i t e Graphs," Ph. D. Thesis, U n i v e r s i t y of I l l i n o i s , 1968.
3 . Wilks, s. s., MathematicaZ Statistics, John Wiley & Sons, N.Y. , 1962.
Paper received May 20, 1970.
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