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COMSOC’08, Liverpool, UK On the Agenda Control Problem for Knockout Tournaments Thuc Vu, Alon Altman, Yoav Shoham {thucvu, epsalon,

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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

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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

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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

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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]

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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]

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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

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The General Model Given input: Set N of players Matrix P of winning probabilities P i,j – probability i win against j 0 P i,j =1- P j,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

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The General Problem Objective function: Find (T,S) that maximizes the winning probability of a given player k With the general model: Open problem Optimal structure must be biased k KT 1 KT 2

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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

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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

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NP-hard with round placements Knockout tournament with round placements Each player j has to start from round R j The tournament is balanced if R j =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

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Complexity Results GeneralWin-Lose General Open (Biased) O(n 2 ) [Lang’07] Balanced NP-hardOpen Round- placements NP-hard

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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 v i m edge players e i Filler players f r for o Holder players h r j for v

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Sketch of Proof (cont.) Winning probabilities vjvj ejej frfr hrthrt o1010 vivi arbitrary1 if v i covers e j, 0 o.w.01 eiei --11 frfr --arb.1 hrthrt --

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Three phases of the tournament Phase 1: (n-k) rounds o and v i start at round 1 At each round r, there are (n-r) new holders h r i o eliminates v’ not in C at each round ov i1 o v1v1 h11h11 v1v1 vnvn h1nh1n vnvn (n-1) Round 1 Round 2

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Three phases of the tournament Phase 1: (n-k) rounds o and v i start at round 1 At each round r, there are (n-r) new holders h r i o eliminates v’ not in C at each round ov i2 o v1v1 h11h11 v1v1 vnvn h1nh1n vnvn (n-2) Round 2 Round 3

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Three phases of the tournament Phase 1: (n-k) rounds o and v i start at round 1 At each round r, there are (n-r) new holders h r i o eliminates v’ not in C at each round ov j1 v jk (k) Round (n-k) At most k vertex players remain

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Three phases of the tournament Phase 2: m rounds o plays against f r e j starts at round j and plays against the covering v The (k-1) remaining v i play against holders h r i ofrfr o v j1 h11h11 v jk h1kh1k (k-1) vertex players v’e1e1 Round 1 Round 2 k vertex players

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Three phases of the tournament Phase 2: m rounds o plays against f r e j starts at round j and plays against the covering v The (k-1) remaining v i play against holders h r i ofrfr o v j1 h11h11 v jk h1kh1k (k-1) vertex players v’emem Round (m-1) Round m k vertex players remain iff all e’s eliminated by v’s

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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 h r i o wins the tournament iff all edge players were eliminated by one of the k vertex players ov j1 o v j2 h12h12 v jk h1kh1k (k-1) Round 1 Round 2

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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 h r i o wins the tournament iff all edge players were eliminated by one of the k vertex players ov jk o Round (k-1) Round k o wins the tournament iff there are k vertex players at the beginning of phase 3

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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

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Complexity Results General Model Win- Lose-Tie Win-Lose General Structure Open (Biased) O(n 2 ) [Lang’07] Balanced Structure NP-hard Open Round- placements NP-hard

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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

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

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NP-hard with Relaxed Constraint ε-monotonic: relax one of the requirements P i,j ≤P i,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

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Complexity Results GeneralWin- Lose-Tie Win-Loseε-monoMono General Structure Open (Biased) O(n 2 )O(n 2 ) [Lang’07] Open Balanced Structure NP-hard OpenNP-hardOpen Round- placements NP-hard Open

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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”

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Thank you! Questions? GeneralWin- Lose-Tie Win-Loseε-monoMono General Structure Open (Biased) O(n 2 )O(n 2 ) [Lang’07] Open Balanced Structure NP-hard OpenNP-hardOpen Round- placements NP-hard Open

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