Graphs and Combinatorics 4, 95-99 (1988)

Graphs and Combinatorics © Springer-Verlag 1988

Stronger Players Win More Balanced Knockout Tournaments

Robert Chen 1. and F.K. Hwang 2

1 University of Miami, Coral Gables, FL 33124, USA a Bell Laboratories, Murray Hill, NJ 07974, USA

AbStract. A player is said to be stronger than another player if he has a better chance of beating the other player than vice versa and his chance of beating any third player is at least as good as that of the other player. Recently, Israel gave an example which shows that a stronger player can have a smaller probability Of winning a knockout tournament than a weaker one when players are randomly assigned to starting positions. In this paper we prove that this anomaly cannot happen if the tournament plan is a balanced one.

1. Introduction

Let T denote a binary tree with n terminal nodes. Then T can serve as a knockout tournament plan (KTP) for n players by having each of the terminal nodes labeled by a distinct player and by having each pair of labeled nodes having a common father node constituting a match, where the father node is to be labeled by the winner of the match. Clearly, the player whose name labels the root of T is the winner of the tournament.

A preference scheme P = {p~} is a set of probabilities where p~j denotes the constant probability that player i beats playerj in a match and pj~ = 1 - p~j. Define W~(T, P, L) to be the probability that player i wins the tournament given T, P, and L where L is a labeling of the terminal nodes. Define W~(T, P) to be the expectation of W~(T, P, L) where L is randomly selected from all possible labelings. Finally, let W~(P) denote the expectation of W~(T, P) where T is randomly selected from all binary trees with n terminal nodes.

A preference scheme P is strongly stochastic transitive (SST) if there is an indexing Cranking) for the players such that i < j implies P~k > Pjk for all k (Pkk is defined to be ½). Without loss of generality we may assume that in any SST scheme P the players are indexed as above. It was proved by Hwang and Hsuan [4] that i fP is SST, then i < j implies that W~(P) > Wj(P). However, the stronger conjecture of Chung and Hwang [1] that if P is SST, t h e n / < j implies that W~(T, P) > Wj(T, P) for all T, turns out to be false.

* The work by this author was done while consulting at Bell Laboratories.

96 R. Chen, F.K. Hwang

Recently, Israel [5] gave a counterexample to the conjecture of Chung and Hwang. The existence of such an anomaly is quite disturbing since on the one hand the notion of "stronger player" is a natural one under the assumption of an SST scheme, yet on the other hand it is meaningless to call a player stronger if his chance of winning the tournament can be less than that of a weaker player with randomized starting positions. Thus it is important to understand how far the anomaly can go. For practical considerations we note: (i) The KTP is usually considered to be a good plan for selecting a strongest player.

Does Israel's counter-example refute this claim? (ii) In practice most KTP's reduce the impact of starting positions by making them

as balanced as possible in terms of the maximum number of matches (nodes) between the starting positions and the championship (root). To answer (i), one of the authors [2] has recently proved that the anomaly

cannot occur to a strongest player. In this paper we answer (ii) by proving that the anomaly cannot occur in a balanced KTP.

I

2. The Main Results

A binary tree T is balanced if there exists a positive integer k such that each terminal node is distance k away from the root. Clearly, to have a balanced KTP, the number of players must be a power of two. We first give some lemmas.

Lemma 1. Let P denote an SST preference scheme and let P' be obtained from P by substituting P~k for Pik for a fixed i such that P;k > Pik for all k ~ i. Then W~( T, P', L) 2 Wi(T, P, L)for every T and L.

Proof. A straightforward induction on the number of matches works. []

Lemma 2. Suppose that i and j are players labeling brother nodes for given T and L. Then

i < j implies

Wi(T, P, L) > Wj(T, P, L) for every SST P.

Proof. Let T' be the tournament obtained from T by eliminating the (i,j) match and let Li(L ~) be the labeling obtained from L by eliminating i(j) and placing j(i) at the father node. Then

W/(T, P, L) = Pli W/( T', P, L J) j > pjiWi(T , n , L )

> pjiWj(T',P,L i) by Lemma 1

= Wj(T,P,L). []

For given i and j, let S(u, v) denote the set of all labelings such that player i labels the terminal node u, and player j labels the terminal node v.

Stronger Players Win More Balanced Knockout Tournaments 97

Theorem. Suppose that P is SST, i < j, and T is balanced. Then

Z W~(T,P,L)>_ Z Wj(T,P,L). LES(u,v) L~S(u,v)

Proof. Suppose that T has 2 k terminal nodes. The Theorem is trivially true for k = 1. We prove the general case by induction on k. If u and v are brother nodes, then the Theorem follows from Lemma 2. Therefore we assume that u and v are not brother nodes.

For {a, b} fq {i,j} = ~, a # b, let Sab(u, v) ~ S(u, v)

denote the set of all labelings such that player a labels the brother node of u, and player b labels the brother node of v. Then the inequality in the Theorem can also be written as

a<b LESa, b(U,V) L¢Sb,a(u,v)

a<b L¢Sa, b(u,v) LeSb,a(U,V )

For given i, j, a, b, let M denote a matching of all players except i, j, a, b into pairs. Let C(M) denote the set of all possible choices of 2 k-1 - 2 players with one player from each pair of M. For c ~ C(M) let Pc denote the probability that in each pair of M, the player in c beats the player not in c. Finally, let L(M) denote the set of all labelings of the terminal nodes by players in M satisfying the following two conditions: (i) No player labels u, v and their brother nodes. (ii) Each pair in M must label two brother nodes.

For given i, j, a, b,

E Wi(T,P,L)+ E Wi(T,P,L) L ¢ S~, b(u, v) L ~ Sb,~(u,v)

=X 2 y 2 M c~C(M) [LeL(M)OSa, b(u,v ) LEL(M)[')Sb, a(U,V )

= E Z Pc ~(PiaPjb + PibPja) Z Wi(T',P,L') M c¢C(M) {. L'¢S(u',v')

÷ PiaPbj E W/(T', P, L~j) ÷ PibPaj ~, Wi(T', P, L:j)~, L" ~S(u',v') L" ¢S(u',v') )

where T' is a K T P for 2 k-1 players obtained from T by eliminating the 2 k terminal nodes, L' is a labeling of i, j and players in c to the terminal nodes of T', u' and v' are the father nodes of u and v, respectively, and L'j is obtained by L by interchanging x and j. But

Z Wi(T',P,L') > ~_, Wj(T',P,L') L" ¢ S(u', v') L" ~ S(u', v')

by our induction hypothesis. Furthermore, for x = a, b,

98 R. Chen, F.K. Hwang

WI(T',P,L'j) >_ Wj(T',P,L~I )

by Lemma 1. Finally, Pl, Pbi >- Pj, Pbj and PabP,~ >-- PjbP, i. Therefore

Y', Wi(T,P,L) + ~ Wi(T,P,L) > ~ Wj(T,P,L) L e Sa, b(u, v) L E Sb, a(U, v) L ~ Sa, b(u, v)

+ ~" Wj(T,P,L). L ~ Sb,,,(u, v)

Since the above inequality is true for all a < b, {a, b} N {i,j} = ~, the Theorem is proved. []

Corollary. Suppose that P is SST, i < j, and T is balanced. Then

W~(T,P) > Wj(T,P) for any T and P.

3. When The Number of Players Is not a Power of Two

There are two methods to approximate a balanced K T P when n is not a power of two. The first method is to add dummy players until n = 2 k for some k. If we enlarge the preference scheme to include the dummy players such that a dummy player loses to any real player with certainty and beats another dummy player with probability one half, then the SST property is preserved. So the Theorem and its Corollary remain true. However, since the randomization of labeling is over all players we may have to live with the following unhappy situation D: At one of the k rounds there exist two matches each engaging a real player with a dummy player (instead of the two real players engaged with each other).

Since for a given labeling L it is straightforward to check whether D can occur, we can eliminate those labelings from the set of random labelings (so that they never come up). Since interchanging the opponents of player i and player j does not affect the occurrence of D, the proof of the Theorem (hence its Corollary) comes through.

There are other reasonable conditions we may wish to impose, for example, the condition that no player should play against a dummy player more than once. However, interchanging the opponents does affect the satisfiability of this condition. So it requires a new proof to reestablish the Theorem.

A second method is to use an almost balanced K T P for the n players, one in which the distances of any two terminal nodes from the root can differ by at most one (this difference is zero for balanced KTP). We conjecture that the Theorem and its Corollary remain true but no proof is available.

4. Some Concluding Remarks

There are various theories about what is the objective function of a KTP. We argue that at least there are some situations, for examples, in Olympic trials and taste experiments, the objective function is to select the strongest player with as fair and as economical a way as possible. Even in situations where entertainment and

Stronger Players Win More Balanced Knockout Tournaments 99

making money is the dominant issue, it is still important to keep the image of K T P as a fair way of producing a champion (or support may erode). Thus how fair a K T P is is a topic relevant to the real world.

Next we discuss the role of a knockout tournament to select a champion when a priori rankings of players already exist. We believe that the role of a knockout tournament is, like any other statistical experiment, in updating the information such that the/ t priori information can be modified. This brings up the question of whether the ~ priori ranking should be used in designing a KTP. "Seeding" is one such method which scatters the stronger players into remote starting positions such that they don't play against each other in early rounds. While seeding may be recommended based on other considerations (for example, a final between the two strongest players brings a dramatic finish), it was shown [ ) ] that no ordinary seeding can avoid the anomaly as discussed in this paper.

References

1. Chung, F.R.K., Hwang, F.K.: Do stronger players win more knockout tournaments? J. Amer. Stat. Assoc. 73 593-596 (1978)

2. Hwang, F.K.: An anomaly in knockout tournaments. Congr. Numerantium 35, 379-386 (1982)

3. Hwang, F.K.: New concepts for seeding knockout tournaments. Amer. Math. Mon. 89, 235- 238 (1982)

4. Hwang, F.K., Hsuan, F.: Stronger players win more knockout tournaments on average. Commun. Stat. Theory Methods A9, 107-113 (1980)

5. Israel, R.: Stronger players need not win more knockout tournaments. J. Amer. Stat. Assoc. 76, 950-951 (1982)

Received: November 10, 1985