Do Stronger Players Win More Knockout Tournaments?

Do Stronger Players Win More Knockout Tournaments?Author(s): F. R. K. Chung and F. K. HwangSource: Journal of the American Statistical Association, Vol. 73, No. 363 (Sep., 1978), pp. 593-596Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2286606 .

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Do Stronger Players Win More Knockout Tournaments? F. R. K. CHUNG and F. K. HWANG*

We study knockout tournaments using a set of players with which a (pairwise) preference scheme is associated. We show that if the preference scheme is of the Bradley-Terry type, then for any krnock- out tournament, if player i has a larger merit value than player j, ha also has a greater probability of winning the tournament. When the preference scheme is not of the Bradley-Terry type, counterexamples are given for several of the main results.

KEY WORDS: Bradley-Terry model; Knockout tournaments; Paired comparisons; Rooted binary trees.

1. INTRODUCTION A knockout tournament (David 1963, Hartigan 1966,

Hwang 1977, Moon 1968) among n players can be defined as a partially ordered set of games, each involving two players, which satisfies the following properties:

(i) Each game has a winner and a loser; a loser of a game is not involved in any further game.

(ii) The tournament ends when all players but one have lost a game; the player left is the winner of the tournament.

(iii) The subset of games involving any one player are linearly ordered.

It is easy to verify that a knockout tournament among n players will end after exactly n - 1 games.

It is convenient to describe this problem in the termi- nology of graph theory. For our purpose, we represent a knockout tournament among n players by a rooted binary tree with n terminal nodes (a terminal node is a node of indegree one and outdegree zero) each labeled with a distinct player. Thus the set of terminal- nodes represents the starting positions of the players in the tournament. The tournament starts with any pair of labeled nodes which have the same father-node (a node whose two outgoing links connect the pair). This pair of nodes and the links which go to them are then elimi- nated from the binary tree, while the winner of the game now labels the father-node. Thus the tournament among n players is reduced to a tournament among n - 1 players, and we proceed inductively until the only node with indegree zero (called root) of the rooted binary tree is labeled by the winner of the tournament.

The rooted binary tree, ignoring the assignment of players to nodes, is called a knockout tournament plan. Thus a knockout tournament plan for n players can generate n! knockout tournaments, each with a distinct assignment.

* F.R.K. Chung and F.K. Hwang are both members of the technical staff in the Department of Discrete Mathematics, Bell Laboratories, Murray Hill, NJ 07974. The authors wish to thank A.M. Odlyzko, the Editor, and the referees for useful comments.

Let 7rij denote the constant probability that player C, beats player Cj in a game, so that 7rij + 7rji = 1. Then r = I ri3j} is known as a preference scheme. David (1963) considers the following types of preference schemes:

(i) If irij > l2 t rk > 2 1 ik> is said to satisfy stochastic transitivity.

(ii) If irj > 24, 7jk 2 2 => rik > maxQ(ir, 7rijk), X iS said to satisfy strong stochastic transitivity.

(iii) Suppose that player Ci has true merit Mi when judged on some characteristic, and 7rij can be expressed for all i, j as H (Mi - Mj), where H (x) increases monotonically from H (-oo ) = 0 to H(oo) = 1, and H(-x) = 1 - H(x). Then 7r is said to be a linear preference scheme.

It is clear that strong stochastic transitivity implies stochastic transitivity, and a linear preference scheme always satisfies strong stochastic transitivity. When a linear preference scheme satisfies

1 XO H(Mi - Mj) = - sech2 4ydy

4 _(Mi-Mj )

we obtain the important subease (known as the Bradley- Terry model (1952)),

rij = ri,/(7ri + 7rj), where 7ri = emi

which has been extensively studied in the literature (see Bradley 1976, David 1963, Davidson and Farquhar 1976 for some references). We will call such a 7r a B-T preference scheme.

Let Tn denote a knockout tournament plan among n players and A an assignment of players to the starting positions. Let Pi(Tn, A; 7r) be the probability that player Ci wins the tournament (Tn, A) under the preference scheme ir. Presumably, PFi(Tn, A; 7r) depends on the starting position of C, in the tournament. So if we only want to compare the players, we can neutralize the effect of the starting positions by averaging over all n! assignments. Thus we define

Pi(T., 7r) - ;Pi(Tn, A; 7r). n! A

If Xr is such that the players can be naturally ordered, such as in the three cases David (1963) has considered, then a stronger player by that ordering should have a

? Journal of the American Statistical Association September 1978, Volume 73, Number 363

Theory and Methods Section

593



594 Journal of the American Statistical Association, September 1978

larger Pi (T, ir) for any T,. However, it is easy to see that the property of stochastic transitivity cannot guarantee this result-counterexamples are readily found; e.g., 7r12 = 713 - .5 + E, and 723 = 1 - f for small e in a three-player tournament. We believe that the property of strong stochastic transitivity for r is sufficient to preserve the ordering in {Pi(TX, 7r) }. But this is a surprisingly difficult conjecture to prove. The main result of this note is a proof that if 7r is a B-T preference scheme, then Ci stronger than Cj in ir implies Pi(Tnn, r) > Pj(T,n, r) for any Tn.

2. THE BRADLEY-TERRY PREFERENCE SCHEME If T,n is of the form shown in Figure A; i.e., the corre-

spondence of T. to a sequence of games is unique, then Tn is called a ladder tree and is denoted by Ln. We name the highest terminal node vn, the second highest v,,1, and so on; we name the internal nodes un, . . ., u2, where ui is the father-node of vi. Consider the assignment A to Ln: C- vi for i = 1, . .., n.

A. A Ladder Tree Ln

n

vn-m2' n .A: VI

Lemma 1: Let 7r be a B-T preference scheme and 7r' another B-T preference scheme obtained from 7r by changing 7r, to rl'. Then we have

irn + Ti' Pn(Ln, A; in) irn-t1

irn + Ti PnL A; in') 2inr + Til xl2 r

The proof is givenl in the Appendix.

Only the left inequality is needed in the rest of this article (chiefly in Lemma 2). However, the right inequality is necessary in our induction proof of the left inequality.

Lemma 2: Consider the ladder tree Ln with assignment A: C, -* vi, i = 1, . . ., n, and a B-T preference scheme 7r. Let A j be the assignment obtained from A by inter- changing Ci and Cj. Then 7rn > 7i implies

(i) Pn(Lny Ai; 71) > Pi(Ln, A; ir); (ii) Pi(Ln, Aln; 7) > Pi(Ln, A; 1r) for i = 2,

n - 1; (iii) PI(Ln, Aln; 7r) < Pn(Ln, A; yr).

Proof: Lemma 2 is trivially true for n = 2. Assume n > 3. Let Ai be the assignment A restricted to the terminal nodes of Li.

(i) P. (L n A n; 7r) n-i 71n n r nrIl lrn >11 = PI(Ln, A; 7r) j=1 71n + 71j j=2 71r + 7rj

(ii) Pi(Ln, Aln; 7r)

Pi(Li, Alni; 7r) - III 71ri + 71i j=i+l 7ri + 7rj

Ti n-I 71-

Pi(Li, Ai; ) H __ 7ri + 7rn j=i+l 7ri + 7rj

= Pi(Ln, A; 7r) by Lemma 1 n

(iii) Pi(Lny Aln; r) = 1 - E Pi(Ln, Aln; 7) I=2 n-i

< 1 - Pi(Ln,A;i7r) i=i

-Pn(Ln, A; 7r) by using (i) and (ii)

Theorem: Let 7r be a B-T preference scheme and Tn a knockout tournament plan for n players. Then

71-i 7rj X= Pi(Tn, A; 7r) > Pj(Tn, Aij; 7r)

for any A. Proof: The theorem is trivially true for n = 2. We

prove the general case by induction.

Case 1: There exists a first-round game involving neither Ci nor Cj.

Let Ch and Ck be the two players involved in a first- round game, where I h, k} I\ { i, j} = 0. Let Tn_I be the tournament plan obtained from Tn by deleting the two terminal nodes labeled Ch and Ck under A. Let A (h) (A (k))

be the assignment on Tni1 obtained from A by assigning Ch (Ck) to the new terminal node. Then

Pi(Tn, A; 7r) = + 7rk 1 i A(^);h A

+ P A Pi(Tn-1- A)(k) 7+h + 71k

> Pj(Tn-1 A+ P(h); 7r) lRa~~~~7- + 71-k

+ k Pj(Tni,y A13(k); xr) 71h + 71k

- P,(Tn, Ai1; sr) by induction.



Chung and Hwang: Knockout Tournaments 595

Case 2: Every first-round game involves Ci or Cj. There are two possible subeases. In the first, there is

exactly one first-round game involving either Ci or C (or both). In the second, there are exactly two first- round games that involve Ci and Cj separately. These two subeases can be treated together.

Let a and b be the two terminal nodes in Tn labeled by Ci and Cj, respectively. Suppose M is the root of the smallest subtree of Tn containing both nodes a and b. Let Tm denote this tree rooted at M1, and Ta (Tb) the largest proper subtree of Tm containing node a (b). Then Ta and Tb are both ladder trees. Furthermore, the tree L obtained from Tm by replacing Tb by a simple node is also a ladder tree. Finally, let T* be the tree obtained from T by replacing Tm by a single node.

Suppose A is an assignment on T. Then let Am, AL, and A* be the assignments on Tm, L, and T* obtained by restricting A to the terminal nodes on these respective subtrees. (In the cases of AL and A*, the new node is assumed to be labeled by Ci.) The set of players assigned to Ta by A, except C,, is denoted by Ca. Then clearly,

Pi(TU, A; 7r) Pi(Tm, Am; 7r) Pi(T*, A*; 7r)

Pj (Tn I A ij; 7r) Pi(Tmy Aijm; 7r) Pj(T*, Aij*; 7r)

It is easy to show by induction that

Pi (T*, A*; 7r) > Pj (T*, A ij*; 7r)

when ir satisfies strong stochastic transitivity. Further- more,

7r, Pi(Tm, Am; 7r) = 1 i

Pi(L, AL; 7r) CkECa 7i + rk

II Pj(L, Ai,jL; r) CkEC a 71rj + 7rk

Pj(Trn, Aijm; 7r) by Lemma 2(iii)

By averaging over all possible assignments, we obtain:

Corollary: 7ri 2 rj > Pi (Tn 7r) > Pj(TTn 7r).

3. CONCLUSIONS We proved the theorem and its corollary when the

preference scheme is of the Bradley-Terry type, and we showed that they do not hold if only stochastic transitivity is assumed. We conjecture that the corollary to the theorem is still true if strong stochastic transitivity is assumed. However, the approach used in this paper does not extend easily to this case because:

(i) Lemma 2 (iii) and the theorem are not true if only strong stochastic transitivity is assumed.

(ii) The left inequality of Lemma 1 does not hold even if ir is assumed to be a linear preference scheme.

To see (i), consider the ladder tree L5 and the assignment A in Figure B. Suppose ir is the following matrix

B. The Tree L5 and an Assignment

C4

C3 C5

{ Tij}: 1 2 3 4 5

1 .5+E 1 1 1 2 .5 + E 1E- 1-E 3 .5+E 1-E 4 .5 + e

where e < .25. Then it can easily be shown that

P4(L5, A; ir) - P5(L5, A45; 7r) = E(.5 - E)(.5 - 2e)2 < 0

However, this is not a counterexample to the corollary. To see (ii), consider the following counterexample

supplied by a referee. Let H(x) = 2 + x for x E [2, 1]; i.e., a uniform probability distribution over the pre- scribed interval. Then when n = 3 with (M1, M1', M2, M3) = (I, 2, 0, 4), it follows that

H(M3-M) 2 9 P3(L3, A; r) H(M3-M1') 4 P3(L3, A;ir')'

where A assigns Ci to vi, i = 1, 2, 3. It is easily verified that this is not a counterexample to Lemma 2.

We summarize our results in the accompanying table.

A Summary of Proven Results

Left inequality Property of Lemma 1 Lemma 2(iii) Theorem Corollary

STa false false false false SSTa false false false unknown LPSb false unknown unknown unknown B-T true true true true

'(S)ST = (strong) stochastic transitivity. bLPS = linear preference scheme.



596 Journal of the American Statistical Association, September 1978

The two counterexamples indicate the difficulty of the question posed in the title to this article. A fresh approach is needed to attack the problem in its full generality.

APPENDIX

Proof of Lemma 1: Lemma 1 is easily verified for n = 2. We prove the general case by induction on n.

Let Ti denote the tournament obtained from (La, A) by replacing L2 with root u2 by a single terminal node labeled Ci, where i = 1 or 2. Then by induction,

Pn(Ln, A; ,r)

P.(Ln, A; ir')

-Pn(Tl; 7r) + - ;Pn(7T2; lr) 71 + 71r2 irf + 7r2

In 7r2

?r 1' 2 Pn('rl; 7r') ? -rl + in2 Pn (T2; 7-') 71 ,n +rl 7

P n (r I; 7r') + Pn -(2; 7r) 71 + 7r2 inn ? 7 tn1+72

I Pn+(r (71; 7n') +

72 Pn(T2; 7r)

< T1' + 72n + rl + ?r2 ini +7n2 ini +in2

(A.1)

Therefore it suffices to show that the right side of (A.1) is less than or equal to (7nr + ir i)/(irn + r1). Since Pn(T2; 7n) = Pn(r2; 7n'), this is equivalent to

(inn 7r2)P(r2; in) < (On + in1l)Pn(rl; in') . (A.2)

When in> ?2, or 7r,' < 72, then by induction

Pn(T2; ir) inn + i' Por (rP ; 'rl) 7 n -> n2 <

-- ~or

Pn(r1; ir') - in + in2 Pn(r2; n) - 7rn + rl'

respectively. In either case, (A.2) follows immediately. To prove the other inequality in Lemma 1, we use

induction to obtain

Pn(Ln, A; 7r)

P.(Lr,, A; 7!')

p.. (ri; t) + 72 Pn(72; 7r) > 7ri + 7r' 7rn + 7rl 7rl + 7r2

7!17!-Tn7!7217-

P?(T1; 7r') + P,- P( 72; 7-') 7r1 + 72 71' + 7r2 (A.3)

Therefore it suffices to prove that the right side of (A.3) is greater than or equal to (r, - 7r)/(T + 7r,), which is equivalent to

IT 71(7r - 7r- ) + 27r,irl' + n2i(7rl + inl')}Pn(72, 7r)

> (inn - rl') (7ri' - 7rj)Pn (Ti, 7n') . (A.4)

When 7r i' ? in2 or 7rl' < 72,

Pn(T2; in) inn - ml1 Pn(rl; 7r') rn + m2 -- >- or < Pn(r1; 7in) in + i2 P,(T2; 7r) in7rn + in

respectively. Since

n (7I' - in1) + 2r 7r ' + in2(inl + in1')

2 (7rn + in2) (7I - m1)

(A.4) follows immediately in either case.

[Received December 1976. Revised February 1978.]

REFERENCES Bradley, Ralph A. (1976), "Science, Statistics and Paired Com-

parisons," Biometrics, 32, 213-232. and Terry, Milton E. (1952), "Rank Analysis of Incomplete

Block Designs I. The Method of Paired Comparisons," Biometrika, 39, 324-345.

David, H.A. (1963), The Method of Paired Comparisons, New York: Hafner Press.

Davidson, 'R.R., and Farquhar, P.H. (1976), "A Bibliography on the Method of Paired Comparisons," Biometrics, 32, 241-252.

Hartigan, J.A. (1966), "Probabilistic Completion of a Knockout Tournament," Annals of Mathematical Statistics, 37, 495-503.

Hwang, Frank K. (1977), "Several Problems on Knockout Tourna- ments," Proceedings of the 8th Southeastern Conference on Com- binatorics, Graph Theory and Computing, Baton Rouge, 363-380.

Moon, John W. (1968), Topics on Tournaments, New York: Holt, Rinehart & Winston.



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Do Stronger Players Win More Knockout Tournaments?