On the Agenda Control Problem for Knockout Tournaments

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

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

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

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

Related Works: Quantitative Approaches n Objective function: Maximizing the predictive power n n 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]

Related Works: Under Voting Context n Election with sequential pairwise comparisons n Model: n n n Structure: n n Consider general, balanced, and linear order Objective function: control the election n n Deterministic comparison results [Lang et al. ’ 07] Probabilistic comparison results [Hazon et al. ’ 07] 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]

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

The General Model n Given input: n n Set N of players Matrix P of winning probabilities n n Pi, j – probability i win against j 0 Pi, j=1 - Pj, i 1 No transitivity required A general knockout tournament K defined by: n n 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

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

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

How about deterministic model? n Win-Lose match tournament n n 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 n n Without structure constraints, it is in P [Lang’ 07] For a balanced tournament, it is an open problem

NP-hard with round placements n Knockout tournament with round placements n n n 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 whethere exists a tournament K with round placement R such that a given player k will win K

Complexity Results General Win-Lose General Open O(n 2) (Biased) [Lang’ 07] Balanced NP-hard Open NP-hard Roundplacements

Sketch of Proof Reduction from 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: n n n Objective player o n vertex players vi m edge players ei § Filler players fr for o § Holder players hrj for v

Sketch of Proof (cont. ) n Winning probabilities vj ej fr h rt o 1 0 vi arbitrary 1 if vi covers ej, 0 o. w. 0 1 ei - - 1 1 fr - - arb. 1 h rt - - arb.

Three phases of the tournament Phase 1: (n-k) rounds n n n 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 (n-1) Round 2 Round 1 o o v 1 vi 1 vn h 1 n

Three phases of the tournament Phase 1: (n-k) rounds n n n 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 (n-2) Round 3 Round 2 o o v 1 vi 2 v 1 vn h 1 n

Three phases of the tournament Phase 1: (n-k) rounds n n n 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 vj 1 vjk At most k vertex players remain

Three phases of the tournament Phase 2: m rounds n n n 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 Round 1 o o vj 1 fr vj 1 vjk h 1 1 vjk (k-1) vertex players v’ h 1 k v’ e 1

Three phases of the tournament Phase 2: m rounds n n n 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 Round (m-1) o o vj 1 fr vj 1 vjk h 1 1 vjk (k-1) vertex players v’ h 1 k v’ em

Three phases of the tournament Phase 3: k rounds n n n 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 (k-1) Round 2 Round 1 o o vj 2 vj 1 vj 2 vjk h 1 k

Three phases of the tournament Phase 3: k rounds n n n 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 Round k Round (k-1) o o iff vjk there are k vertex players at the beginning of phase 3

Win-Lose-Tie Constraint n Win-Lose-Tie (WLT) match tournament n Winning probabilities can be 0, 1, or 0. 5 Question: Find (T, S) that maximizes the winning probability of a given player k n Complexity of this problem n n Without structure constraints, it is in P For a balanced tournament, it is an NP-complete problem

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

Balanced WLT Tournaments Theorem: Given N, and win-lose-tie P, it is NP-complete to decide whethere 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 n n Need gadgets to simulate round placements Make sure any round placement at most O(log(n)) n Possible since the players can have ties

How about Monotonic Model? n Tournament with monotonic winning prob. n n Very common model in the literature The winning probability matrix P satisfies n n 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: n n Balanced knockout tournament Without structure constraints

NP-hard with Relaxed Constraint n ε-monotonic: relax one of the requirements n 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 whethere exists a balanced tournament K such that p(k, K)≥δ for a given k in N and δ≥ 0

Complexity Results General Win-Lose ε-mono Mono -Tie General Structure Open O(n 2) (Biased) NP-hard Balanced Structure NP-hard Roundplacements O(n 2) Open [Lang’ 07] Open NP-hard Open

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

Thank you! Questions? General Win-Lose ε-mono Mono -Tie General Structure Open O(n 2) (Biased) NP-hard Balanced Structure NP-hard Roundplacements O(n 2) Open [Lang’ 07] Open NP-hard Open