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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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Presentation on theme: "COMSOC’08, Liverpool, UK On the Agenda Control Problem for Knockout Tournaments Thuc Vu, Alon Altman, Yoav Shoham {thucvu, epsalon,"— Presentation transcript:

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

2 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 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

9 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

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 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

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

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

15 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 --

16 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

17 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

18 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

19 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

20 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

21 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

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 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

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 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

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 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

27 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

28 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

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? 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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