Balancing Act: Optimizing Knockout Tournament Design

1. On the Agenda Control Problem for Knockout Tournaments Thuc Vu, Alon Altman, Yoav Shoham {thucvu, epsalon, shoham}@stanford.edu COMSOC’08, Liverpool, UK

2. 1 1 4 1 2 4 5 3 4 5 6 1 2 3 4 5 6 Knockout Tournament • One of the most popular formats • Players placed at leaf-nodes of a binary tree • Winner of pairwise matches moving up the tree

3. Knockout Tournament Design Space Very rich space with several dimensions: • Objective functions • Predictive power vs. Fairness vs. Interestingness etc… • Structures of the tournament • Unconstrained vs. Balanced vs. Limited matches • Models of the players/ Information available • Unconstrained vs. Monotonic vs. Deterministic etc… • Sizes of the problem • Exact small cases vs. Unbounded cases • Type of results • Theoretical vs. Experimental

4. Related Works: Axiomatic Approaches • Objectives: Set of axioms • “Delayed Confrontation”, “Sincerity Rewarded”, and “Favoritism Minimized” in [Schwenk’00] • “Monotonicity” in [Hwang’82] • Structure: Balanced knockout tournament • Model: Monotonic • The players are ordered based on certain intrinsic abilities • The winning probabilities reflect this ordering • Size: Unbounded number of players

5. Related Works: Quantitative Approaches • Objective function: Maximizing the predictive power • Probability of the strongest player winning the tournament • Structure: Balanced knockout tournament • Model: Monotonic • Size: Focus on small cases such as 4 or 8 players [Appleton’95, Horen&Riezman’85, and Ryvkin’05]

6. Related Works: Under Voting Context • Election with sequential pairwise comparisons • Model: • Deterministic comparison results [Lang et al. ’07] • Probabilistic comparison results [Hazon et al. ’07] • Structure: • Consider general, balanced, and linear order • Objective function: control the election • Show that with balanced voting tree, some modified versions are NP-complete • Computational aspects of other control methods [Bartholdi et al. ’92][Hemaspaandra et al. ’07]

7. Our Work We focus on the following space: • Structure: Knockout tournament with • Unconstrained general structure • Balanced structure • Tournament with round placements • Model of players: • Unconstrained general model • Deterministic • Monotonic • Objective function: • Maximizing the winning probability of a target player

8. The General Model • Given input: • Set N of players • Matrix P of winning probabilities • Pi,j – probability i win against j • 0  Pi,j=1- Pj,i  1 • No transitivity required • A general knockout tournament K defined by: • Tournament structure T – binary tree • Seeding S – a mapping from N to leaf nodes of T Probability p(j,K) of player j winning tournament K can be calculated efficiently

9. k KT1 KT2 The General Problem Objective function: Find (T,S) thatmaximizes the winning probability of a given player k With the general model: • Open problem • Optimal structure must be biased

10. New result with structure constraint • Balanced knockout tournament (BKT) • Tournament structure is a balanced binary tree • Can only change the seeding Theorem: Given N and P, it is NP-complete to decide whether there exists a BKT such that p(k,BKT)≥δ for a given k in N and δ≥0

11. How about deterministic model? • Win-Lose match tournament • Winning probabilities can be either 0 or 1 • Analogous to sequential pairwise eliminations • Question: Find (T,S) that allows k to win • Complexity of this problem • Without structure constraints, it is in P [Lang’07] • For a balanced tournament, it is an open problem

12. NP-hard with round placements • Knockout tournament with round placements • Each player j has to start from round Rj • The tournament is balanced if Rj=1 for all j • Certain types of matches can be prohibited Theorem: Given N, win-lose P, and feasible R, it is NP-complete to decide whether there exists a tournament K with round placement R such that a given player k will win K

13. Complexity Results

14. Sketch of Proof Reduction from Vertex Cover Vertex Cover: Given G={V,E} and k, is there a subset C of V such that |C|≤k and C covers E? Reduction Method: Construct a tournament K with player o such that o wins K <=> C exists K contains the following players: • Objective player o • n vertex players vi • m edge players ei • Filler players fr for o • Holder players hrj for v

15. Sketch of Proof (cont.) • Winning probabilities

16. (n-1) o v1 vn o vi1 v1 h11 vn h1n Round 2 Round 1 Three phases of the tournament Phase 1: (n-k) rounds • o and vi start at round 1 • At each round r, there are (n-r) new holders hri • o eliminates v’ not in C at each round

17. (n-2) o v1 vn o vi2 v1 h11 vn h1n Three phases of the tournament Phase 1: (n-k) rounds • o and vi start at round 1 • At each round r, there are (n-r) new holders hri • o eliminates v’ not in C at each round Round 3 Round 2

18. Three phases of the tournament Phase 1: (n-k) rounds • o and vi start at round 1 • At each round r, there are (n-r) new holders hri • o eliminates v’ not in C at each round (k) Round (n-k) o vj1 vjk At most k vertex players remain

19. Three phases of the tournament Phase 2: m rounds • o plays against fr • ej starts at round j and plays against the covering v • The (k-1) remaining vi play against holders hri k vertex players Round 2 o vj1 vjk v’ Round 1 o fr vj1 h11 vjk h1k v’ e1 (k-1) vertex players

20. Three phases of the tournament Phase 2: m rounds • o plays against fr • ej starts at round j and plays against the covering v • The (k-1) remaining vi play against holders hri k vertex players remain iff all e’s eliminated by v’s Round m o vj1 vjk v’ Round (m-1) o fr vj1 h11 vjk h1k v’ em (k-1) vertex players

21. (k-1) o vj2 vjk o vj1 vj2 h12 vjk h1k Round 2 Round 1 Three phases of the tournament Phase 3: k rounds • o eliminates the remaining v’s • At each round r, there are (k-r) new holders hri • o wins the tournament iff all edge players were eliminated by one of the k vertex players

22. Three phases of the tournament Phase 3: k rounds • o eliminates the remaining v’s • At each round r, there are (k-r) new holders hri • o wins the tournament iff all edge players were eliminated by one of the k vertex players o wins the tournament iff there are k vertex players at the beginning of phase 3 Round k o o vjk Round (k-1)

23. Win-Lose-Tie Constraint • Win-Lose-Tie (WLT) match tournament • Winning probabilities can be 0, 1, or 0.5 Question: Find (T,S) that maximizes the winning probability of a given player k • Complexity of this problem • Without structure constraints, it is in P • For a balanced tournament, it is an NP-complete problem

24. Complexity Results

25. Balanced WLT Tournaments Theorem: Given N, and win-lose-tie P, it is NP-complete to decide whether there exists a balanced WLT tournament K such that p(k,K)≥δ for a given k in N and δ≥0 Sketch of Proof: Similar to hardness proof for round placement tournament • Need gadgets to simulate round placements • Make sure any round placement at most O(log(n)) • Possible since the players can have ties

26. How about Monotonic Model? • Tournament with monotonic winning prob. • Very common model in the literature • The winning probability matrix P satisfies • Pi,j+Pj,i=1 • Pi,j≥Pj,i for all (i,j): i≤j • Pi,j≤Pi,j+1 for all (i,j) • Open problem for both cases: • Balanced knockout tournament • Without structure constraints

27. NP-hard with Relaxed Constraint • ε-monotonic: relax one of the requirements • Pi,j≤Pi,j+1 + ε for all (i,j) with ε > 0 Theorem: Given N, and ε-monotonic P, it is NP-complete to decide whether there exists a balanced tournament K such that p(k,K)≥δ for a given k in N and δ≥0

28. Complexity Results

29. Conclusions and Future Works • Addressed the tournament design space • Showed that for balanced tournament, the agenda control problem is NP-hard • Even for win-lose-tie or ε-monotonic probabilities • Future directions: • Balanced tournament with deterministic results • Approximation methods • Other objective functions such as fairness or “interestingness”

30. Thank you! Questions?