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
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One of the most popular formats
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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:
 Objective functions
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Structures of the tournament
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Unconstrained vs. Monotonic vs. Deterministic etc…
Sizes of the problem
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Unconstrained vs. Balanced vs. Limited matches
Models of the players/ Information available
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Predictive power vs. Fairness vs. Interestingness etc…
Exact small cases vs. Unbounded cases
Type of results

Theoretical vs. Experimental
Related Works: Axiomatic Approaches
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Objectives: Set of axioms
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Structure: Balanced knockout tournament
Model: Monotonic
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“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

Objective function: Maximizing the predictive power


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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
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Model:
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Structure:
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Consider general, balanced, and linear order
Objective function: control the election
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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:
 Structure: Knockout tournament with
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Model of players:
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Unconstrained general structure
Balanced structure
Tournament with round placements
Unconstrained general model
Deterministic
Monotonic
Objective function:

Maximizing the winning probability of a target player
The General Model

Given input:

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Set N of players
Matrix P of winning probabilities
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
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Pi,j – probability i win against j
0  Pi,j=1- Pj,i  1
No transitivity required
A general knockout tournament K defined by:

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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
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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
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
Without structure constraints, it is in P [Lang’07]
For a balanced tournament, it is an open problem
NP-hard with round placements
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Knockout tournament with round placements
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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
O(n2)
(Biased) [Lang’07]
Balanced
NP-hard Open
NP-hard NP-hard
Roundplacements
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:
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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.)
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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
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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
v1
vi1
v1
vn
h11
vn
h1 n
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
(n-2)
Round 3
Round 2
o
o
v1
vi2
v1
vn
h11
vn
h1 n
Three phases of the tournament
Phase 1: (n-k) rounds


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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
Three phases of the tournament
Phase 2: m rounds

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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
vj1
fr
vj1
v’
vjk
h11
vjk
(k-1) vertex players
h1k
v’
e1
Three phases of the tournament
Phase 2: m rounds


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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
vj1
fr
vj1
v’
vjk
h11
vjk
(k-1) vertex players
h1k
v’
em
Three phases of the tournament
Phase 3: k rounds

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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
vj2
vj1
vj2
vjk
h12
vjk
h1k
Three phases of the tournament
Phase 3: k rounds


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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
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Win-Lose-Tie (WLT) match tournament
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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
General
Structure
General WinModel
Lose-Tie
Win-Lose
Open
O(n2)
(Biased)
O(n2)
NP-hard NP-hard
Balanced
Structure
NP-hard NP-hard
Roundplacements
[Lang’07]
Open
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.

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Very common model in the literature
The winning probability matrix P satisfies

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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 WinLose-Tie
Win-Lose ε-mono Mono
General
Structure
Open
O(n2)
(Biased)
O(n2)
Balanced
Structure
NP-hard NP-hard
Open
NP-hard Open
NP-hard
NP-hard Open
NP-hard NP-hard
Roundplacements
Open
Open
[Lang’07]
Conclusions and Future Works


Addressed the tournament design space
Showed that for balanced tournament, the
agenda control problem is NP-hard
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Even for win-lose-tie or ε-monotonic probabilities
Future directions:
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Balanced tournament with deterministic results
Approximation methods
Other objective functions such as fairness or
“interestingness”
Thank you! Questions?
General WinLose-Tie
Win-Lose ε-mono Mono
General
Structure
Open
O(n2)
(Biased)
O(n2)
Balanced
Structure
NP-hard NP-hard
Open
NP-hard Open
NP-hard
NP-hard Open
NP-hard NP-hard
Roundplacements
Open
Open
[Lang’07]