A Theory of Tournament Representations

√ d ) must be a forbidden configuration . To answer a dual question , using a celebrated result of Forster & Simon ( 2006 ) , we show a lower bound of O ( √ n ) on the minimum dimension needed to represent all tournaments on n nodes . For any given tournament , we show a novel upper bound on the smallest representation dimension that depends on the least size of the number of unique nodes in any feedback arc set of the flip class associated with a tournament . We show how our results also shed light on upper bound of sign-rank of matrices . 1 INTRODUCTION . In this work , we lay the the foundations for a theory of tournament representations . A tournament is a complete directed graph and arises naturally in several applications including ranking from pairwise preferences , sports modeling , social choice , etc . We say that a tournament T on n nodes can be represented in d dimensions if there exists a skew symmetric matrix M ∈ Rn×n of rank d such that a directed edge from i to j is present in T if and only if Mij > 0 . Real world tournaments are almost always intransitive ( Tversky ( 1969 ) ; Klimenko ( 2015 ) ) and it is not known what type of tournaments can be represented in how many dimensions . This is important to understand because of the following reason : As a modeler of preference relations using tournaments , it is often more natural to have structural domain knowledge such as ‘ The tournaments under consideration do not have long cycles ’ as opposed to algebraic domain knowledge such as ‘ The rank of the skew symmetric matrix associated with the tournaments of interest is at most k ’ . However , algorithms that learn rankings from pairwise comparison data typically need as input the algebraic quantity - the rank of the skew symmetric matrices associated with tournaments or equivalently the dimension where they are represented ( Rajkumar & Agarwal ( 2016 ) ) . To bridge the gap between the structural and the algebraic world , we ask and answer two fundamental questions regarding the representations of tournaments . 1 ) What structurally characterizes the class of tournaments that can be represented in d dimensions ? . 2 ) Given a tournament T on n nodes , what is the minimum dimension d needed to represent it ? . We answer the first question by investigating the intricate structure of the rank d tournament class via the notion of forbidden configurations . Specifically , we show that the set of forbidden config- uration for the rank d tournament class must necessarily be a union of flip classes , a novel way to partition the set of all tournaments into equivalence classes . We explicitly characterize the forbidden configurations for the rank 2 tournament class and exhibit a forbidden configuration for the general rank d tournament class . Specifically , we show that the rank 2 tournaments are equivalent to locally transitive tournaments , a previously studied class of tournaments ( Cohen et al . ( 2004 ) ) . Our results throw light on the connections between transitive and locally transitive tournaments and also lets us develop a classic Quicksort based algorithm to solve that the minimum feedback arc set problem on rank 2 tournaments with O ( n2 ) time complexity . Our results for the general rank d tournament class have connections to the classic long standing Hadamard conjecture and we discuss this as well . Figure 1 gives a glimpse of some of the main results . We answer the second question by proving lower and upper bounds on the smallest dimension needed to represent a tournament on n nodes . We exhibit a lower bound of Ω ( √ n ) using a variation of the celebrated dimension complexity result of Forster & Simon ( 2006 ) for sign matrices . To show upper bounds , we introduce a novel parameter associated to a tournament called the Flip Feedback Node set of a Tournament . This quantity depends on the least number of unique nodes in any feedback arc set of an associated tournament class for the tournament of interest and upper bounds linearly the representation dimension of any tournament . We show how our results can be used to provide upper bounds on the classic notion of sign-rank of a matrix . Previously known upper bounds for sign rank depended on the VC dimension of the associated binary function class ( Alon et al . ( 2016 ) ) . Our upper bounds on the other hand have a graph theoretic flavour . Organization of the Paper : We discuss briefly in Section 2 the foundational works this paper builds upon . We introduce necessary preliminaries in Section 3 . The answer to the first question about the structural characterization of d dimensional tournament classes span Sections 4 , 5 and 6 . We devote Section 7 to answer the second question about the number of dimensions needed to represent a tournament . Section 8.1 explores connections of our results to upper bounds on the sign rank of a sign matrix . Finally , we conclude in Section 10 . All proofs are defered to the Appendix . 2 RELATED WORK . The work in this paper builds on several pieces of work across different domains . We summarize below the most important related works under different categories : Intransitive Pairwise Preference Models : One of the main reasons to study representations of tournaments is to model pairwise preferences . Parametric pairwise preference models that can model intransitivity have gained recent interest . Rajkumar & Agarwal ( 2016 ) develop a low rank pairwise ranking model that can model intransitivity . However , their study and results were restricted to just the transitive tournaments in these classes . A generalization of the classical Bradley-Terry-Luce model ( Bradley & Terry ( 1952 ) , Luce ( 2012 ) ) was studied in Causeur & Husson ( 2005 ) . However , no structural characterization is known . Same holds for the more recent models studied in Chen & Joachims ( 2016 ) , Makhijani & Ugander ( 2019 ) and Bower & Balzano ( 2020 ) . Flip Classes : The notion of flip classes , a novel way to partition the set of all tournaments on n nodes , was first introduced in Fisher & Ryan ( 1995 ) . The goal however was completely different and was on studying equilibrium on certain generalized rock-paper-scissors games on tournaments . Interestingly , and perhaps surprisingly , the notion of flip classes turn out to be fundamental to our study of understanding forbidden configurations of d dimensional tournament classes . Dimension Complexity and Sign Rank : Dimension complexity and sign rank of sign pattern matrices were studied in Forster ( 2002 ) . These results have found significant applications in learning theory and lower bounds in computational complexity ( Forster & Simon ( 2006 ) ) . More recently , Alon et al . ( 2016 ) study the sign rank for function classes with fixed Vapnik-Chervonenkis ( VC ) dimension and show upper bounds . Our upper bounds however depend on certain graph theoretic properties . 3 PRELIMINARIES . Tournaments : A tournament T is a complete directed graph i.e. , a graph on n nodes where for every pair of nodes ( i , j ) either an edge is oriented from i to j or j to i . The number of nodes n in T will be usually clear from the context or will be explicitly specified . For nodes i and j in T , we say i T j if there is a directed edge from i to j . Given a node i , we define the out and in neighbours of i as T+i = { j : i T j } and T − i = { j : j T i } respectively . Given a set of nodes S , we denote by T ( S ) the induced sub-tournament of T on the nodes in S. Feedback Arc Set and Pairwise Disagreement Error : Given a permutation σ on n nodes , the feedback arc set of σ w.r.t the tournament T is defined as Eσ ( T ) = { ( i , j ) : σ ( i ) > σ ( j ) , i > T j } . The pairwise diagreement error of σ w.r.t T is given by |E σ ( T ) | ( n2 ) . It is known that finding the σ that minimizes the pairwise disagreement error w.r.t a general tournament T is a NP-hard problem ( Charbit et al . ( 2007 ) ) . Skew Symmetric Tournament Classes : A square matrix M ∈ Rn×n is skew symmetric if Mij = −Mji∀i , j . In this paper , whenever we refer to a skew symmetric matrix M ∈ Rn×n , we always assume Mij 6= 0 ∀i 6= j and Mii = 0 ∀i . Given such an M , we denote by T { M } the tournament on n nodes induced by M where i T j ⇐⇒ Mij > 0 . We refer to class of tournaments induced by rank d skew symmetric matrices as the rank d tournament class . Forbidden Configurations : A tournament class T is a collection of tournaments . T is said to forbid a tournament T if no tournament in T has a sub-tournament that is isomorphic to T. We call T a forbidden configuration for T if T forbids T but does not forbid any sub-tournament of T. For example , the class of all acyclic/transitive tournaments has the 3-cycle as a forbidden configuration i.e , the tournament T on three nodes i , j , k where i T j T k T i . Representations and Tournaments : The matrix Arot ∈ Rd×d is defined for every even d and is a block diagonal matrix which consists of d/2 blocks of [ 0−1 ; 1 0 ] . This is the canonical representation of a non-degenerate skew symmetric matrix i.e. , any non-singular skew symmetric matrix can be brought to this form by a suitable basis transformation . Given a set of vectors in H = { h1 , . . . , hn } where each hi ∈ Rd for some even d , we refer to the set H as a tournament inducing representation if hTi A rothj 6= 0 for all i 6= j . Furthermore , we refer to the tournament induced by H as T [ H ] where a directed edge exits from node i to j if and only if hTi A rothj > 0 . It is easy to verify that hTAroth = 0 for any h ∈ Rd . Any skew symmetric matrix M ∈ Rn×n of rank d can be written as M = HTArotH for some H ∈ Rd×n . It follows that if M = HTArotH , then T { M } = T [ H ] . Positive Spans : A set of vectors H = { h1 , . . . , hn } where each hi ∈ Rd is said to positively span Rd if for any w ∈ Rd , there exists non-negative constants c1 , . . . , cn ≥ 0 such that ∑ i cihi = w. If H positively spans Rd , then there does not exist a w ∈ Rd such that wThi > 0 ∀i . This is a easy consequence of Farkas ’ Lemma . Remark on Notation : We reiterate that we use T ( · ) , T { · } and T [ · ] to mean different objects - the tournament induced by a subset of nodes , the tournament induced by a skew symmetric matrix and the tournament induced by a representation of a set of vectors respectively . These will be usually clear from the context .

Summary: This paper studies the theory of tournament representations, i.e. low-rank matrices $M$ whose sign agrees with the sign matrix of a tournament $T$.  The authors show several properties of such representations, reducing the study to so called $R$-cones, i.e. tournaments where one vertex beats all others. They also characterize completely rank 2 tournaments, and provide a forbiden class for rank $d$ tournaments.  Finally, they provide an upper bound on the minimum dimension of a presentation of any tournament $T$, in the for of a bound involving minimum feedback arc sets of $T$, and show how this can be extended to sign matrices.