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Basic Notions Static scenario Dynamic Scenario Future work

Nominating Representatives in Single-Elimination Tournaments

Grzegorz Lisowski, Ramanujan Sridharan, Paolo Turrini

University of Warwick

April 24th 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 (SEπ,C) is based on:

A set of players C A seeding π A round-robin tournament D

A winner in a round advances forward! SEk

π,C denotes kth

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

• f coalitions

(C = {C1, . . . , Cl}), 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 Static Dynamic 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 = {C1, . . . , Cl}, a strategy-profile is a tuple (p1, . . . , pl), such that for any pi, pi ∈ Ci.

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 = (p1, . . . , pℓ) is a:

Nash equilibrium:if for every coalition Ci and a player p′

i ∈ Ci,

if Ci wins under s−i,p′

i , it wins under s.

Dominant Strategy equilibrium: if for any profile s’, if Ci wins under s’, Ci wins under s’, Ci wins under s’−i,s[i].

Grzegorz Lisowski, Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments

Basic Notions Static scenario Dynamic Scenario Future work

Results

Checking if s is a NE or DSE is polynomial. Finding a NE is quasi-polynomial. Finding a DSE is polynomial. Any DSE is a NE. NE doesn’t always exist.

text a1, a2 b1, b2

a1 ≻ b2, b1 ≻ a1, a2 ≻ b1, b2 ≻ a2.

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 = (p1, . . . , pℓ) is a:

Nash equilibrium: if for every coalition Ci and a player p′

i , if

Ci is represented in SEk

π,sı,p′

i , it is in SEk

π,s.

Dominant Strategy equilibrium: if for any profile s’, if Ci is represented in SEk

π,s’, it is in SEk 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 = {C1, . . . , Cl} be a set of coalitions. Then, a strategy of a coalition Ci is a function σi : C → Ci. a strategy profile is a tuple (σ1, . . . , σl)

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 = (σ1, . . . , σℓ) is a:

Nash equilibrium: if for every coalition Ci and a strategy σ′

i , if

Ci wins under σ, it wins under σ′

−i,σ[i].

Dominant Strategy equilibrium: if for any profile σ′, if Ci wins under σ′, Ci 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 = (σ1, . . . , σℓ) is a:

Nash equilibrium: if for every coalition Ci and a strategy σ′

i if

Ci gets to the round SEk

π,σ−i,σ′

i it does under σ.

Dominant Strategy equilibrium: if for any profile σ′, if Ci gets to the round SEk 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 2 3 4 5 7 8 6 9 10 11 12 13 14 15 16 33 17 18 19 21 20 22 23 24 25 26 27 28 29 30 31 32

Example of a spanning binomial arborescence

Grzegorz Lisowski, Ramanujan Sridharan, Paolo Turrini Coalitional Tournaments