On the complexity of schedule control problems for knockout tournaments

Knockout tournaments constitute a common format of sporting events, and also model a specific type of election scheme (namely, sequential pairwise elimination election). In such tournaments the designer controls the shape of the tournament (a binary tree) and the seeding of the players (their assignment to the tree leaves). In this paper we investigate the computational complexity of tournament schedule control, i.e., designing a tournament that maximizes the winning probability a target player. We start with a generic probabilistic model consisting of a matrix of pairwise winning probabilities, and then investigate the problem under two types of constraint: constraints on the probability matrix, and constraints on the allowable tournament structure. While the complexity of the general problem is as yet unknown, these various constraints -- all naturally occurring in practice -- serve to push to the problem to one side or the other: easy (polynomial) or hard (NP-complete).

[1]  Jean-François Laslier,et al.  Tournament Solutions And Majority Voting , 1997 .

[2]  Emerson M. S. Niou,et al.  A Problem with Referendums , 2000 .

[3]  Lirong Xia,et al.  Sequential voting rules and multiple elections paradoxes , 2007, TARK '07.

[4]  Michael A. Trick,et al.  How hard is it to control an election? Math , 1992 .

[5]  Jeff Horen,et al.  Comparing Draws for Single Elimination Tournaments , 1985, Oper. Res..

[6]  J. Moon.  Topics on tournaments , 1968 .

[7]  Vincent Conitzer,et al.  When are elections with few candidates hard to manipulate? , 2007, J. ACM.

[8]  R. Tollison,et al.  Toward a theory of the rent-seeking society , 1982 .

[9]  Peter C. Fishburn,et al.  Chapter 4 Voting procedures , 2002 .

[10]  Edith Elkind,et al.  Hybrid Voting Protocols and Hardness of Manipulation , 2005, ISAAC.

[11]  J W Moon.  A Problem on Rankings by Committees , 1976 .

[12]  Glenn C. Loury,et al.  Market Structure and Innovation , 1979 .

[13]  P. Fishburn,et al.  Voting Procedures , 2022 .

[14]  M. Trick,et al.  The computational difficulty of manipulating an election , 1989 .

[15]  R. Varga,et al.  Proof of Theorem 4 , 1983 .

[16]  Yoav Shoham,et al.  Optimal seeding in knockout tournaments , 2010, AAMAS.

[17]  Sarit Kraus,et al.  How to Rig Elections and Competitions , 2008 .

[18]  Frank K. Hwang,et al.  New Concepts in Seeding Knockout Tournaments , 1982 .

[19]  Hitoshi Matsushima,et al.  Handbook of Social Choice and Welfare, Volume 1 (Book) , 2004 .

[20]  Dmitry Ryvkin.  The Predictive Power of Noisy Elimination Tournaments , 2005 .

[21]  Jörg Rothe,et al.  Anyone but him: The complexity of precluding an alternative , 2005, Artif. Intell..

[22]  Ariel D. Procaccia,et al.  Junta Distributions and the Average-Case Complexity of Manipulating Elections , 2007, J. Artif. Intell. Res..

[23]  Toby Walsh,et al.  Winner Determination in Sequential Majority Voting , 2007, IJCAI.

[24]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[25]  David R. Appleton,et al.  May the best man win , 1995 .

[26]  J. Moon,et al.  On Generalized Tournament Matrices , 1970 .

[27]  Allen J. Schwenk,et al.  What Is the Correct Way to Seed a Knockout Tournament? , 2000, Am. Math. Mon..