On the Agenda Control Problem for Knockout Tournaments

1
On the Agenda Control Problem for Knockout
Tournaments

• Thuc Vu, Alon Altman, Yoav Shoham
• thucvu, epsalon, shoham_at_stanford.edu

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 Schwenk00
• Monotonicity in Hwang82
• 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
• Appleton95, HorenRiezman85, and Ryvkin05

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. 92Hemaspaandra 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
• Pi,j probability i win against j
• 0 ? Pi,j1- 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

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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

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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)d for a given k in N and d0

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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
  Lang07
• 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 Rj
• The tournament is balanced if Rj1 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
General Win-Lose
General Open (Biased) O(n2) Lang07
Balanced NP-hard Open
Round-placements NP-hard NP-hard
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Sketch of Proof

• Reduction from Vertex Cover
• Vertex Cover Given GV,E and k, is there a
  subset C of V such that Ck and C covers E?
• Reduction Method Construct a tournament K with
  player o such that o wins K ltgt 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

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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.
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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

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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

Round 3
Round 2
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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

(k)
Round (n-k)
o
vj1
vjk
At most k vertex players remain
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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 hri

k vertex players
o
vj1
vjk
v
Round 2
o
fr
vj1
h11
vjk
h1k
v
e1
Round 1
(k-1) vertex players
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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 hri

k vertex players remain iff all es eliminated
by vs
o
vj1
vjk
v
Round m
o
fr
vj1
h11
vjk
h1k
v
em
Round (m-1)
(k-1) vertex players
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Three phases of the tournament

• Phase 3 k rounds
• o eliminates the remaining vs
• 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

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Three phases of the tournament

• Phase 3 k rounds
• o eliminates the remaining vs
• 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 iff there are k vertex
players at the beginning of phase 3
Round k
o
o
vjk
Round (k-1)
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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
GeneralModel Win-Lose-Tie Win-Lose
General Structure Open (Biased) O(n2) O(n2) Lang07
Balanced Structure NP-hard NP-hard Open
Round-placements NP-hard NP-hard 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)d for
  a given k in N and d0
• 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
• Pi,jPj,i1
• Pi,jPj,i for all (i,j) ij
• Pi,jPi,j1 for all (i,j)
• Open problem for both cases
• Balanced knockout tournament
• Without structure constraints

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NP-hard with Relaxed Constraint

• e-monotonic relax one of the requirements
• Pi,jPi,j1 e for all (i,j) with e gt 0
• Theorem Given N, and e-monotonic P, it is
  NP-complete to decide whether there exists a
  balanced tournament K such that p(k,K)d for a
  given k in N and d0

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Complexity Results
General Win-Lose-Tie Win-Lose e-mono Mono
General Structure Open (Biased) O(n2) O(n2) Lang07 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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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 e-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?
General Win-Lose-Tie Win-Lose e-mono Mono
General Structure Open (Biased) O(n2) O(n2) Lang07 Open Open
Balanced Structure NP-hard NP-hard Open NP-hard Open
Round-placements NP-hard NP-hard NP-hard NP-hard Open