COMSOC’08, Liverpool, UK
On the Agenda Control Problem for Knockout
Tournaments
Thuc Vu, Alon Altman, Yoav Shoham
{thucvu, epsalon, shoham}@stanford.edu
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
1 2
1
3 4
4
5 6
5
4
1
1 2
3 4 5 6
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
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
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]
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]
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
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
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
KT1 KT2
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
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
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
Complexity Results
General
Win-Lose
General Open
(Biased)
O(n2)[Lang’07]
Balanced NP-hard Open
Round-placements
NP-hard NP-hard
Sketch of Proof
Reduction from Vertex CoverVertex 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
Sketch of Proof (cont.)
Winning probabilities
vj ej fr hrt
o 1 0 1 0
vi arbitrary 1 if vi covers ej, 0 o.w.
0 1
ei - - 1 1
fr - - arb.
1
hrt - - arb.
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 hr
i
o eliminates v’ not in C at each round
o vi1
o
v1 h11
v1
vn h1n
vn
(n-1)
Round 1
Round 2
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 hr
i
o eliminates v’ not in C at each round
o vi2
o
v1 h11
v1
vn h1n
vn
(n-2)
Round 2
Round 3
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 hr
i
o eliminates v’ not in C at each round
o vj1 vjk
(k)
Round (n-k)
At most k vertex players remain
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 hr
i
o fr
o
vj1 h11
vj1
vjk h1k
vjk
(k-1) vertex players
v’ e1
v’
Round 1
Round 2
k vertex players
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 hr
i
o fr
o
vj1 h11
vj1
vjk h1k
vjk
(k-1) vertex players
v’ em
v’
Round (m-1)
Round m
k vertex players remain iff all e’s eliminated by v’s
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 hr
i
o wins the tournament iff all edge players were eliminated by one of the k vertex players
o vj1
o
vj2 h12
vj2
vjk h1k
vjk
(k-1)
Round 1
Round 2
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 hr
i
o wins the tournament iff all edge players were eliminated by one of the k vertex players
o vjk
o
Round (k-1)
Round k
o wins the tournament
iff
there are k vertex players at the beginning of phase 3
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
Complexity Results
GeneralModel
Win-Lose-Tie
Win-Lose
General Structure
Open
(Biased)
O(n2) O(n2)[Lang’07]
Balanced Structure
NP-hard NP-hard Open
Round-placements
NP-hard NP-hard NP-hard
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
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
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
Complexity Results
General
Win-Lose-Tie
Win-Lose
ε-mono
Mono
General Structure
Open
(Biased)
O(n2) O(n2) [Lang’07]
Open Open
Balanced Structure
NP-hard NP-hard Open NP-hard Open
Round-placements
NP-hard NP-hard NP-hard NP-hard Open
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”
Thank you! Questions?
General
Win-Lose-Tie
Win-Lose
ε-mono
Mono
General Structure
Open
(Biased)
O(n2) O(n2) [Lang’07]
Open Open
Balanced Structure
NP-hard NP-hard Open NP-hard Open
Round-placements
NP-hard NP-hard NP-hard NP-hard Open
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