Nominating Representatives in Single-Elimination Tournaments - PowerPoint PPT Presentation

Basic Notions Static scenario Dynamic Scenario Future work Nominating Representatives in Single-Elimination Tournaments Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini University of Warwick April 24 th 2019 PhDs in Logic, Bern Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Introduction Tournaments : competitive environments, widely used in practice. A method of selecting a winner, based on pairwise comparisons. Knockout (single-elimination) tournaments: played in rounds. Connections with the social choice theory : tournaments are social choice functions (e.g pairwise majority contests)! Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Contributions We lift the setting of knockout tournaments to competitions between coalitions . Study of algorithmic aspects of game-theoretic problems in this scenario. Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Contributions We lift the setting of knockout tournaments to competitions between coalitions . Study of algorithmic aspects of game-theoretic problems in this scenario. Applications: real-life tournaments (sports), social choice: selection of candidates for elections. Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Knockout Tournaments A knockout-tournament ( SE π, C ) is based on: A set of players C A seeding π A round-robin tournament D A winner in a round advances forward! SE k π, C denotes k th round of the tournament. Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Coalitional Knockout Tournaments We consider a seeding of coalitions ( C = { C 1 , . . . , C l } ), selecting representatives . A digraph on the coalitions. The winning coalition: The one whose representative wins the tournament. Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Exploration of the setting We will focus on algorithmic properties of game-theoretic solution concepts. Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Exploration of the setting We will focus on algorithmic properties of game-theoretic solution concepts. Settings Dynamic Static win/lose b win/lose win/lose b win/lose Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Static Strategies A strategy profile in the static scenario is a choice of players: Definition Given a set of coalitions C = { C 1 , . . . , C l } , a strategy-profile is a tuple ( p 1 , . . . , p l ) , such that for any p i , p i ∈ C i . Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Static Solution Concepts Win/lose scenario : Fix a seeding π , a set of coalitions C and a digraph D . A profile s = ( p 1 , . . . , p ℓ ) is a: Nash equilibrium: if for every coalition C i and a player p ′ i ∈ C i , if C i wins under s − i , p ′ i , it wins under s . Dominant Strategy equilibrium: if for any profile s’ , if C i wins under s’ , C i wins under s’ , C i wins under s’ − i , s [ i ] . Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Results Any DSE is a NE. Checking if s is a NE or DSE is polynomial. NE doesn’t always exist. Finding a NE is quasi-polynomial. text Finding a DSE is polynomial. a 1 , a 2 b 1 , b 2 a 1 ≻ b 2 , b 1 ≻ a 1 , a 2 ≻ b 1 , b 2 ≻ a 2 . Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Beyond Win/Lose Static Solution Concepts Beyond win/lose scenario : Coalitions care about how high they advance! A profile s = ( p 1 , . . . , p ℓ ) is a: Nash equilibrium: if for every coalition C i and a player p ′ i , if C i is represented in SE k i , it is in SE k π, s . π, s ı , p ′ Dominant Strategy equilibrium: if for any profile s’ , if C i is represented in SE k π, s’ , it is in SE k π, s’ − i , s [ i ] . Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Results NE doesn’t always exist: by the same example! Recognizing both concepts is polynomial. Finding a NE is quasi-polynomial. Finding a DSE is polynomial. Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Dynamic Strategies In this scenario, we allow coalitions to pick a representative at every round! A strategy is a specification which player should be chosen when a particular coalition is encountered. Definition Let C = { C 1 , . . . , C l } be a set of coalitions. Then, a strategy of a coalition C i is a function σ i : C → C i . a strategy profile is a tuple ( σ 1 , . . . , σ l ) Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Dynamic Solution Concepts - Win/Lose Let us adapt the static solution concepts to the new setting! A profile σ = ( σ 1 , . . . , σ ℓ ) is a: Nash equilibrium: if for every coalition C i and a strategy σ ′ i , if C i wins under σ , it wins under σ ′ − i ,σ [ i ] . Dominant Strategy equilibrium: if for any profile σ ′ , if C i wins under σ ′ , C i wins under σ ′ − i ,σ [ i ] . Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Results NE doesn’t always exist: by the same example! Recognizing both concepts is polynomial. Finding a NE is quasi-polynomial. Finding a DSE is polynomial. Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Dynamic Solution Concepts - Beyond Win/Lose A profile σ = ( σ 1 , . . . , σ ℓ ) is a: Nash equilibrium: if for every coalition C i and a strategy σ ′ i if C i gets to the round SE k i it does under σ . π,σ − i ,σ ′ Dominant Strategy equilibrium: if for any profile σ ′ , if C i gets to the round SE k under σ ′ , it does under σ ′ − i ,σ [ i ] . Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Results NE doesn’t always exist: by the same example! Recognizing both concepts is polynomial. Finding a DSE and NE is polynomial. Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Summary of Results We considered algorithmic properties of solution concepts in several types of the considered setting. Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Summary of Results We considered algorithmic properties of solution concepts in several types of the considered setting. STATIC DYNAMIC CHECK FIND CHECK FIND W / L NEaaa P Q-P P Q-P W / L DSEaa P P P P B W / L NEa P Q-P P P B W / L DSE P P P P Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Future Work Check if we can provide polynomial time algorithms for finding the solution concepts. Extend the setting with probabilities : Non-deterministic round robin tournament. Mixed strategies. Further connections with social choice theory: Voting theory: How to choose a candidate for a president? Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work Binomial arboresences 1 17 9 3 2 5 10 4 25 21 7 19 18 11 13 6 29 27 26 20 8 15 23 22 12 14 28 24 16 31 33 30 32 Example of a spanning binomial arborescence Grzegorz Lisowski , Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments