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Ensuring your favorite player wins: Tournament Rigging and Bribery - - PowerPoint PPT Presentation

Mar 15, 2023 105 likes •1.41k views

Ensuring your favorite player wins: Tournament Rigging and Bribery Sushmita Gupta, Sanjukta Roy, Saket Saurabh, Meirav Zehavi (Knockout)Tournaments How can we ensure our favorite player/team wins the tournament? How to ensure that favorite

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Ensuring your favorite player wins: Tournament Rigging and Bribery

Sushmita Gupta, Sanjukta Roy, Saket Saurabh, Meirav Zehavi

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(Knockout)Tournaments

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How can we ensure our favorite player/team wins the tournament?

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How to ensure that favorite wins the tournament?

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How to ensure that favorite wins the tournament?

We have predictive information about various match-ups

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How to ensure that favorite wins the tournament?

We have predictive information about various match-ups

What if favorite didn’t have to play those it can’t beat….?

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How to ensure that favorite wins the tournament?

We have predictive information about various match-ups

What if favorite didn’t have to play those it can’t beat….?

A n e x a m p l e …

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FIFA WORLD CUP’18 Knockout Stages

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SLIDE 12
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Seeding in a Tournament

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Seeding in a Tournament

Preliminary ranking for the purpose of draw/ bracket. Originally used in Tennis

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Seeding in a Tournament

Preliminary ranking for the purpose of draw/ bracket. Originally used in Tennis It describes a player’s path to the final and potential opponents in each round. Specific to the tournament.

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Seeding in a Tournament

Preliminary ranking for the purpose of draw/ bracket. Originally used in Tennis It describes a player’s path to the final and potential opponents in each round. Specific to the tournament.

F

  • r

m a l l y w e s a y …

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 v1 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 v1 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 v1 v4 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 v1 v4 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 v1 v4 v5 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 v1 v4 v5 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 v1 v4 v5 v8 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 v1 v4 v5 v8 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 v1 v4 v5 v8 v4 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 v1 v4 v5 v8 v4 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 v1 v4 v5 v8 v4 v5 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 v1 v4 v5 v8 v4 v5 seeding

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Seeding

v1 v2 v3 v4 v5 v6 v7 v8 v1 v4 v5 v8 v4 v5 v4 seeding w i n n e r !

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Is there a seeding to ensure that favorite wins the tournament ?

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Our problem: TOURNAMENT FIXING

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Our problem: TOURNAMENT FIXING

INPUT: Win-lose graph

A C E F G H B

✶

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Our problem: TOURNAMENT FIXING

INPUT: Win-lose graph

QUESTION: Does there exist a seeding that ensures that favorite wins?

A C E F G H B

✶

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Our problem: TOURNAMENT FIXING

INPUT: Win-lose graph

…introduced by Vu,Altman,Shoham AAMAS’09

QUESTION: Does there exist a seeding that ensures that favorite wins?

A C E F G H B

✶

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How hard is TOURNAMENT FIXING ?

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How hard is TOURNAMENT FIXING ?

If there is a player who beats all

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How hard is TOURNAMENT FIXING ?

If there is a player who beats all

ie win-lose graph is acyclic

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How hard is TOURNAMENT FIXING ?

If there is a player who beats all

ie win-lose graph is acyclic

Easy to decide

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How hard is TOURNAMENT FIXING ?

In general: NP-hard

If there is a player who beats all

ie win-lose graph is acyclic

Easy to decide

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How hard is TOURNAMENT FIXING ?

In general: NP-hard [Aziz et al. AAAI’14]

If there is a player who beats all

ie win-lose graph is acyclic

Easy to decide

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TOURNAMENT FIXING is NP-Hard.

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TOURNAMENT FIXING is NP-Hard.

  • Existence of cycles makes it interesting!
  • i.e no dominant player exists.
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TOURNAMENT FIXING is NP-Hard.

  • Existence of cycles makes it interesting!
  • i.e no dominant player exists.

Intuitively, let k = Number of ``upsets’’

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TOURNAMENT FIXING is NP-Hard.

  • Existence of cycles makes it interesting!
  • i.e no dominant player exists.

Parameter k = Number of pairings in which a player will beat a player of higher rank.

Intuitively, let k = Number of ``upsets’’

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TOURNAMENT FIXING is NP-Hard.

  • Existence of cycles makes it interesting!
  • i.e no dominant player exists.

Parameter k = Number of pairings in which a player will beat a player of higher rank.

Intuitively, let k = Number of ``upsets’’

….But, how do we define rank ?

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A C

✶

E F G H B

Ranking the players

We are given feedback arc set of the win-lose graph

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A C

✶

E F G H B

Ranking the players

We are given feedback arc set of the win-lose graph

We obtain a ranking of the players

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A B C

✶

E F G H A C

✶

E F G H B

Ranking the players

We are given feedback arc set of the win-lose graph

We obtain a ranking of the players

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A B C

✶

E F G H A C

✶

E F G H B

Ranking the players

We are given feedback arc set of the win-lose graph

We obtain a ranking of the players

Back to our problem

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TOURNAMENT FIXING,

k arcs away from acyclic

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TOURNAMENT FIXING,

k arcs away from acyclic

INPUT: Win-lose graph QUESTION: Does there exist a seeding that ensures that favorite wins the tournament?

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TOURNAMENT FIXING,

k arcs away from acyclic

INPUT: Win-lose graph QUESTION: Does there exist a seeding that ensures that favorite wins the tournament?

KNOWN RESULT: (1) Solvable in O(nk) [Aziz et al.

AAAI’14]

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TOURNAMENT FIXING,

k arcs away from acyclic

INPUT: Win-lose graph QUESTION: Does there exist a seeding that ensures that favorite wins the tournament?

(2) 2O(k2log k)poly(n) using ILP [Ramanujam and

Szeider AAAI’17]

KNOWN RESULT: (1) Solvable in O(nk) [Aziz et al.

AAAI’14]

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OUR WORK: PARAM TOURNAMENT FIXING

Gupta,Roy,Saurabh & Zehavi IJCAI’18

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OUR WORK: PARAM TOURNAMENT FIXING

Algorithm runs in time 2O(k log k)poly(n)

Gupta,Roy,Saurabh & Zehavi IJCAI’18

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OUR WORK: PARAM TOURNAMENT FIXING

Algorithm runs in time 2O(k log k)poly(n) Combinatorial algorithm using a greedy strategy

Gupta,Roy,Saurabh & Zehavi IJCAI’18

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OUR WORK: PARAM TOURNAMENT FIXING

Algorithm runs in time 2O(k log k)poly(n) Combinatorial algorithm using a greedy strategy Reveals structural properties

Gupta,Roy,Saurabh & Zehavi IJCAI’18

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Spanning Binomial Arborescences (SBA)

A C

✶

E F G H B

Win-lose graph

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Spanning Binomial Arborescences (SBA)

E B F G H

✶

A C A C

✶

E F G H B

Win-lose graph

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Spanning Binomial Arborescences (SBA)

E B F G H

✶

A C A C

✶

E F G H B

Win-lose graph

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Spanning Binomial Arborescences (SBA)

E B F G H

✶

A C A C

✶

E F G H B

Win-lose graph

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Spanning Binomial Arborescences (SBA)

E B F G H

✶

A C A C

✶

E F G H B

Win-lose graph

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Spanning Binomial Arborescences (SBA)

E B F G H

✶

A C A C

✶

E F G H B

Win-lose graph

SBA

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Binomial Arborescences (BA)

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Binomial Arborescences (BA)

A unlabeled BA T rooted at v is defined recursively

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Binomial Arborescences (BA)

A unlabeled BA T rooted at v is defined recursively A single node v is a BA rooted at v

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Binomial Arborescences (BA)

A unlabeled BA T rooted at v is defined recursively A single node v is a BA rooted at v

Given 2 vertex disjoint BA of equal size, Tv rooted at v and Tu rooted at u, adding arc v ➞ u gives a BA Tvu rooted at v

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Binomial Arborescences (BA)

A unlabeled BA T rooted at v is defined recursively A single node v is a BA rooted at v

Given 2 vertex disjoint BA of equal size, Tv rooted at v and Tu rooted at u, adding arc v ➞ u gives a BA Tvu rooted at v

If T ⊆ D (a directed graph) and V(T) = V(D), then T is labeled spanning BA (SBA)

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Tournament Fixing ⟷ SBA

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Tournament Fixing ⟷ SBA

Theorem: Let D be a win-lose graph where favorite is a vertex.

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Tournament Fixing ⟷ SBA

Theorem: Let D be a win-lose graph where favorite is a vertex. There is a seeding of the vertices in D s.t. the resulting tournament is won by favorite ⟺

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Tournament Fixing ⟷ SBA

Theorem: Let D be a win-lose graph where favorite is a vertex. There is a seeding of the vertices in D s.t. the resulting tournament is won by favorite ⟺ D has an SBA s.t. favorite is the root.

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Tournament Fixing ⟷ SBA

Theorem: Let D be a win-lose graph where favorite is a vertex. There is a seeding of the vertices in D s.t. the resulting tournament is won by favorite ⟺ D has an SBA s.t. favorite is the root.

[ Williams AAAI’10]

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TOURNAMENT FIXING In terms of an SBA

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TOURNAMENT FIXING In terms of an SBA

INPUT: Win-lose graph

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TOURNAMENT FIXING In terms of an SBA

INPUT: Win-lose graph

(OLD) QUESTION: Does there exist a seeding ensuring that favorite wins the tournament?

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TOURNAMENT FIXING In terms of an SBA

INPUT: Win-lose graph

(OLD) QUESTION: Does there exist a seeding ensuring that favorite wins the tournament?

(NEW) QUESTION: Does the win-lose graph have a subgraph that is an SBA with favorite as the root ?

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TOURNAMENT FIXING In terms of an SBA

INPUT: Win-lose graph

(OLD) QUESTION: Does there exist a seeding ensuring that favorite wins the tournament?

(NEW) QUESTION: Does the win-lose graph have a subgraph that is an SBA with favorite as the root ?

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Sketch of the Algorithm

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Sketch of the Algorithm

  • I. GUESS:
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Sketch of the Algorithm

  • I. GUESS:
  • II. VERIFY:
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Sketch of the Algorithm

  • I. GUESS:
  • II. VERIFY:
  • III. GREEDY:
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Sketch of the Algorithm

  • I. GUESS:
  • II. VERIFY:
  • III. GREEDY:

(i) A template — a partial structure of some SBA where certain paths and subtrees are compressed. (ii) We know the position of the affected vertices & position of their Least common ancestor. (iii) Length of those paths and sizes of subtrees

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Sketch of the Algorithm

  • I. GUESS:
  • II. VERIFY:
  • III. GREEDY:

(i) A template — a partial structure of some SBA where certain paths and subtrees are compressed. (ii) We know the position of the affected vertices & position of their Least common ancestor. (iii) Length of those paths and sizes of subtrees (i) Our guess is ``realizable” in terms of an SBA

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Sketch of the Algorithm

  • I. GUESS:
  • II. VERIFY:
  • III. GREEDY:

(i) A template — a partial structure of some SBA where certain paths and subtrees are compressed. (ii) We know the position of the affected vertices & position of their Least common ancestor. (iii) Length of those paths and sizes of subtrees (i) Our guess is ``realizable” in terms of an SBA (i) Fill up the paths & subtrees of the template (ii) If final outcome is an SBA, then done. (iii) Or else, guess again

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What if no favorable seeding exists for favorite ?

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Can favorite win with bribery?

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Can favorite win with bribery?

INPUT: Win-lose graph

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Can favorite win with bribery?

QUESTION: Is it possible to fix some (say l) matches so that there is a seeding that enables favorite to win ? INPUT: Win-lose graph

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Can favorite win with bribery?

QUESTION: Is it possible to fix some (say l) matches so that there is a seeding that enables favorite to win ? INPUT: Win-lose graph

In terms of an SBA ….

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Can favorite win with bribery?

QUESTION: Is it possible to fix some (say l) matches so that there is a seeding that enables favorite to win ? INPUT: Win-lose graph Can we reverse l arcs in the win-lose graph so that there will be an SBA with favorite as the root ?

In terms of an SBA ….

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Can favorite win with bribery?

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Can favorite win with bribery?

Answered in 2npoly(n) time & poly(n) space, n := number of players.

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Can favorite win with bribery?

Answered in 2npoly(n) time & poly(n) space, n := number of players. Answered in 2O(k2log k)poly(n) time & poly(n) space, k := FAS of win-lose graph

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Can favorite win with bribery?

Answered in 2npoly(n) time & poly(n) space, n := number of players. Answered in 2O(k2log k)poly(n) time & poly(n) space, k := FAS of win-lose graph Uses our algorithm for TOURNAMENT FIXING

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ELITE (PLAYERS) CLUB

A B C D

✶

F G H

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ELITE (PLAYERS) CLUB

A B C D

✶

F G H

Type 1

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ELITE (PLAYERS) CLUB

A B C D

✶

F G H

Type 1 2

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ELITE (PLAYERS) CLUB

A B C D

✶

F G H

Type 1 2 3 4

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ELITE (PLAYERS) CLUB

A B C D

✶

F G H

Type 1 2 3 4 5

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ELITE (PLAYERS) CLUB

A B C D

✶

F G H

Type 1 2 3 4 5 Type 6

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ELITE (PLAYERS) CLUB

S is an ELITE CLUB if

A B C D

✶

F G H

Type 1 2 3 4 5 Type 6

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ELITE (PLAYERS) CLUB

S is an ELITE CLUB if

  • 1. every player in S beats favorite

A B C D

✶

F G H

Type 1 2 3 4 5 Type 6

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ELITE (PLAYERS) CLUB

S is an ELITE CLUB if

  • 1. every player in S beats favorite
  • 2. If a player v of type i belongs to S, then all other

players of type i that beat v also belong to S

A B C D

✶

F G H

Type 1 2 3 4 5 Type 6

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ELITE (PLAYERS) CLUB

S is an ELITE CLUB if

  • 1. every player in S beats favorite
  • 2. If a player v of type i belongs to S, then all other

players of type i that beat v also belong to S

A B C D

✶

F G H

Type 1 2 3 4 5 Type 6

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ELITE (PLAYERS) CLUB

S is an ELITE CLUB if

  • 1. every player in S beats favorite
  • 2. If a player v of type i belongs to S, then all other

players of type i that beat v also belong to S

ELITE CLUB={A,B, C,D }

A B C D

✶

F G H

Type 1 2 3 4 5 Type 6

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How can favorite win with bribery?

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How can favorite win with bribery?

Sufficient to fix matches that feature favorite

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How can favorite win with bribery?

Sufficient to fix matches that feature favorite Sufficient to fix matches that feature favorite and someone from ELITE CLUB

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How can favorite win with bribery?

Sufficient to fix matches that feature favorite Sufficient to fix matches that feature favorite and someone from ELITE CLUB Once we know which matches to fix, find the seeding using our earlier algorithm

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How can favorite win with bribery?

Sufficient to fix matches that feature favorite Sufficient to fix matches that feature favorite and someone from ELITE CLUB Once we know which matches to fix, find the seeding using our earlier algorithm

Properties used by

  • ur algorithm.
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IN CONCLUSION

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IN CONCLUSION

Interesting class of problems

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IN CONCLUSION

Interesting class of problems Interesting structural properties

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IN CONCLUSION

Interesting class of problems Interesting structural properties Many secondary and tertiary parameters to explore

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IN CONCLUSION

Interesting class of problems Interesting structural properties Many secondary and tertiary parameters to explore

Are these problems solvable in time f(k)poly(n), k := FVS in win-lose graph

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Thank You!

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When can favorite win ?

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When can favorite win ?

Win lose graph is acyclic favorite can win ⟺ #players beaten by favorite is ≥ n

2l − 1

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When can favorite win ?

Win lose graph is acyclic favorite can win ⟺ #players beaten by favorite is ≥ Win lose graph is not acyclic favorite can win ⟺ there exists players U s.t. there is a seeding on U ∪ { favorite } that makes favorite win favorite wins if it beats players

n 2l − 1 n 2l − 1 n 2l − 1

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(Knockout)Tournaments

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(Knockout)Tournaments

SPORTS Tennis Tournaments Last four rounds of FIFA World Cup Olympic heats

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(Knockout)Tournaments

SPORTS Tennis Tournaments Last four rounds of FIFA World Cup Olympic heats POLITICS Multi-level Elections

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(Knockout)Tournaments

SPORTS Tennis Tournaments Last four rounds of FIFA World Cup Olympic heats POLITICS Multi-level Elections INDUSTRY/LIFE etc Decision making