The Tale of the Tournament Equilibrium Set. Felix Brandt Dagstuhl, March 2012
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2 The Tale of the Tournament Equilibrium Set Felix Brandt Dagstuhl, March 2012
3 Rational Choice Theory Amartya K. Sen 2
4 Rational Choice Theory Let U be a universe of at least three alternatives. Amartya K. Sen 2
5 Rational Choice Theory Let U be a universe of at least three alternatives. Amartya K. Sen Alternatives are chosen from feasible subsets. 2
6 Rational Choice Theory Let U be a universe of at least three alternatives. Amartya K. Sen Alternatives are chosen from feasible subsets. Throughout this talk, the set of feasible sets F(U) contains all finite and non-empty subsets of U. 2
7 Rational Choice Theory Let U be a universe of at least three alternatives. Amartya K. Sen Alternatives are chosen from feasible subsets. Throughout this talk, the set of feasible sets F(U) contains all finite and non-empty subsets of U. A choice function is a function S:F(U) F(U) such that S(A) A. A S(A) 2
8 Rational Choice Theory Let U be a universe of at least three alternatives. Amartya K. Sen Alternatives are chosen from feasible subsets. Throughout this talk, the set of feasible sets F(U) contains all finite and non-empty subsets of U. A choice function is a function S:F(U) F(U) such that S(A) A. Two standard consistency conditions: Let A,B F(U) with B A and S(A) B. A S(A) B 2
9 Rational Choice Theory Let U be a universe of at least three alternatives. Amartya K. Sen Alternatives are chosen from feasible subsets. Throughout this talk, the set of feasible sets F(U) contains all finite and non-empty subsets of U. A choice function is a function S:F(U) F(U) such that S(A) A. Two standard consistency conditions: Let A,B F(U) with B A and S(A) B. - Contraction (α): S(A) B S(B) A S(A) B 2
10 Rational Choice Theory Let U be a universe of at least three alternatives. Amartya K. Sen Alternatives are chosen from feasible subsets. Throughout this talk, the set of feasible sets F(U) contains all finite and non-empty subsets of U. A choice function is a function S:F(U) F(U) such that S(A) A. Two standard consistency conditions: Let A,B F(U) with B A and S(A) B. - Contraction (α): S(A) B S(B) - Expansion (β + ): S(A) B S(B) A S(A) B 2
11 Rational Choice Theory Let U be a universe of at least three alternatives. Amartya K. Sen Alternatives are chosen from feasible subsets. Throughout this talk, the set of feasible sets F(U) contains all finite and non-empty subsets of U. A choice function is a function S:F(U) F(U) such that S(A) A. Two standard consistency conditions: Let A,B F(U) with B A and S(A) B. - Contraction (α): S(A) B S(B) - Expansion (β + ): S(A) B S(B) The conjunction of both properties is equivalent to Samuelson s weak axiom of revealed preference (Sen, 1969; Bordes, 1976). A S(A) B 2
12 From Choice to Social Choice Let N be a finite set of voters and R(U) the set of all transitive and complete relations over U. A social choice function (SCF) is a function S:R(U) N F(U) F(U) such that S(R,A) A. Useful conditions on SCFs IIA (Independence of Irrelevant Alternatives): Choice only depends on preferences over alternatives within the feasible set. Pareto-optimality: Alternative y is not chosen if there exists some x that is unanimously strictly preferred to y. Non-dictatorship: There should be no voter whose most preferred alternative is always uniquely chosen. 3
13 Arrow s Impossibility Kenneth J. Arrow 4
14 Arrow s Impossibility Theorem (Arrow, 1951, 1959): There exists no SCF that simultaneously satisfies IIA, Pareto-optimality, nondictatorship, α, and β +. Kenneth J. Arrow 4
15 Arrow s Impossibility Kenneth J. Arrow Theorem (Arrow, 1951, 1959): There exists no SCF that simultaneously satisfies IIA, Pareto-optimality, nondictatorship, α, and β +. In the context of SCFs, IIA is only a mild framework requirement (Bordes and Tideman, 1991) and dropping it offers little relief (Banks, 1995). 4
16 Arrow s Impossibility Kenneth J. Arrow Theorem (Arrow, 1951, 1959): There exists no SCF that simultaneously satisfies IIA, Pareto-optimality, nondictatorship, α, and β +. In the context of SCFs, IIA is only a mild framework requirement (Bordes and Tideman, 1991) and dropping it offers little relief (Banks, 1995). Dropping Pareto-optimality offers little relief (Wilson, 1972). 4
17 Arrow s Impossibility Kenneth J. Arrow Theorem (Arrow, 1951, 1959): There exists no SCF that simultaneously satisfies IIA, Pareto-optimality, nondictatorship, α, and β +. In the context of SCFs, IIA is only a mild framework requirement (Bordes and Tideman, 1991) and dropping it offers little relief (Banks, 1995). Dropping Pareto-optimality offers little relief (Wilson, 1972). Dropping non-dictatorship is unacceptable. 4
18 Arrow s Impossibility Kenneth J. Arrow Theorem (Arrow, 1951, 1959): There exists no SCF that simultaneously satisfies IIA, Pareto-optimality, nondictatorship, α, and β +. In the context of SCFs, IIA is only a mild framework requirement (Bordes and Tideman, 1991) and dropping it offers little relief (Banks, 1995). Dropping Pareto-optimality offers little relief (Wilson, 1972). Dropping non-dictatorship is unacceptable. Dropping β + offers little relief (Sen, 1977). 4
19 Arrow s Impossibility Theorem (Arrow, 1951, 1959): There exists no SCF that simultaneously satisfies IIA, Pareto-optimality, nondictatorship, α, and β +. In the context of SCFs, IIA is only a mild framework requirement (Bordes and Tideman, 1991) and dropping it offers little relief (Banks, 1995). Dropping Pareto-optimality offers little relief (Wilson, 1972). Dropping non-dictatorship is unacceptable. Dropping β + offers little relief (Sen, 1977). Dropping α allows for reasonable SCFs! Kenneth J. Arrow 4
20 Majoritarian SCFs An SCF is majoritarian if its outcome only depends on the pairwise majority relation within the feasible set. Majoritarianism implies all Arrovian conditions except α and β +. We assume for convenience that individual preferences are strict and there is an odd number of voters. Hence, the pairwise majority relation is asymmetric and complete, i.e., it can be represented by a tournament graph. Let D (x)={y y x} denote the dominators of x. b d a c e 5
21 Positive Results 6
22 Positive Results Theorem (Bordes, 1976): The top cycle is the smallest majoritarian SCF satisfying β +. John I. Good 6
23 Positive Results Theorem (Bordes, 1976): The top cycle is the smallest majoritarian SCF satisfying β +. - β + can be weakened to γ: S(A) S(B) S(A B) for all A,B. John I. Good 6
24 Positive Results Theorem (Bordes, 1976): The top cycle is the smallest majoritarian SCF satisfying β +. - β + can be weakened to γ: S(A) S(B) S(A B) for all A,B. John I. Good Theorem (Moulin, 1986): The uncovered set is the smallest majoritarian SCF satisfying γ. Peter C. Fishburn 6
25 Positive Results Theorem (Bordes, 1976): The top cycle is the smallest majoritarian SCF satisfying β +. - β + can be weakened to γ: S(A) S(B) S(A B) for all A,B. John I. Good Theorem (Moulin, 1986): The uncovered set is the smallest majoritarian SCF satisfying γ. - γ can be weakened to ρ + : S(D (a)) S(A) for all a A. Peter C. Fishburn 6
26 Positive Results Theorem (Bordes, 1976): The top cycle is the smallest majoritarian SCF satisfying β +. - β + can be weakened to γ: S(A) S(B) S(A B) for all A,B. John I. Good Theorem (Moulin, 1986): The uncovered set is the smallest majoritarian SCF satisfying γ. - γ can be weakened to ρ + : S(D (a)) S(A) for all a A. Theorem (B., 2011): The Banks set is the smallest majoritarian SCF satisfying ρ +. Peter C. Fishburn Jeffrey S. Banks 6
27 Positive Results Theorem (Bordes, 1976): The top cycle is the smallest majoritarian SCF satisfying β +. - β + can be weakened to γ: S(A) S(B) S(A B) for all A,B. John I. Good Theorem (Moulin, 1986): The uncovered set is the smallest majoritarian SCF satisfying γ. - γ can be weakened to ρ + : S(D (a)) S(A) for all a A. Theorem (B., 2011): The Banks set is the smallest majoritarian SCF satisfying ρ +. - ρ + can be weakened to ρ: S(D (a)) S(A) for all a S(A). Peter C. Fishburn Jeffrey S. Banks 6
28 Positive Results Theorem (Bordes, 1976): The top cycle is the smallest majoritarian SCF satisfying β +. - β + can be weakened to γ: S(A) S(B) S(A B) for all A,B. Theorem (Moulin, 1986): The uncovered set is the smallest majoritarian SCF satisfying γ. - γ can be weakened to ρ + : S(D (a)) S(A) for all a A. Theorem (B., 2011): The Banks set is the smallest majoritarian SCF satisfying ρ +. - ρ + can be weakened to ρ: S(D (a)) S(A) for all a S(A). Conjecture (Schwartz, 1990): The tournament equilibrium set (TEQ) is the smallest majoritarian SCF satisfying ρ. John I. Good Peter C. Fishburn Jeffrey S. Banks 6
29 Tournament Equilibrium Set Thomas Schwartz 7
30 Tournament Equilibrium Set Let S be a choice function and define S( )=. Thomas Schwartz 7
31 Tournament Equilibrium Set Let S be a choice function and define S( )=. A non-empty set of alternatives B is S-retentive if S(D (x)) B for all x B. x B Thomas Schwartz 7
32 Tournament Equilibrium Set Let S be a choice function and define S( )=. A non-empty set of alternatives B is S-retentive if S(D (x)) B for all x B. Idea: No alternative in the set should be properly dominated by an outside alternative. x B Thomas Schwartz 7
33 Tournament Equilibrium Set Let S be a choice function and define S( )=. A non-empty set of alternatives B is S-retentive if S(D (x)) B for all x B. Idea: No alternative in the set should be properly dominated by an outside alternative. x B Thomas Schwartz S is a new choice function that yields the union of all inclusion-minimal S-retentive sets. 7
34 Tournament Equilibrium Set Let S be a choice function and define S( )=. A non-empty set of alternatives B is S-retentive if S(D (x)) B for all x B. Idea: No alternative in the set should be properly dominated by an outside alternative. S is a new choice function that yields the union of all inclusion-minimal S-retentive sets. x B Thomas Schwartz The tournament equilibrium set (TEQ) of a tournament is defined as TEQ=TEQ. 7
35 Tournament Equilibrium Set Let S be a choice function and define S( )=. A non-empty set of alternatives B is S-retentive if S(D (x)) B for all x B. Idea: No alternative in the set should be properly dominated by an outside alternative. S is a new choice function that yields the union of all inclusion-minimal S-retentive sets. x B Thomas Schwartz The tournament equilibrium set (TEQ) of a tournament is defined as TEQ=TEQ. Recursive definition (unique fixed point of ring-operator) 7
36 Tournament Equilibrium Set Let S be a choice function and define S( )=. A non-empty set of alternatives B is S-retentive if S(D (x)) B for all x B. Idea: No alternative in the set should be properly dominated by an outside alternative. S is a new choice function that yields the union of all inclusion-minimal S-retentive sets. x B Thomas Schwartz The tournament equilibrium set (TEQ) of a tournament is defined as TEQ=TEQ. Recursive definition (unique fixed point of ring-operator) Conjecture (Schwartz, 1990): Every tournament contains a unique inclusion-minimal TEQ-retentive set. 7
37 The Mystery of TEQ 8
38 The Mystery of TEQ Theorem (Laffond et al., 1993; Houy 2009; B., 2011; B. and Harrenstein, 2011): The following statements are equivalent: 8
39 The Mystery of TEQ Theorem (Laffond et al., 1993; Houy 2009; B., 2011; B. and Harrenstein, 2011): The following statements are equivalent: Every tournament contains a unique minimal TEQ-retentive set. 8
40 The Mystery of TEQ Theorem (Laffond et al., 1993; Houy 2009; B., 2011; B. and Harrenstein, 2011): The following statements are equivalent: Every tournament contains a unique minimal TEQ-retentive set. TEQ is the smallest majoritarian SCF satisfying ρ. 8
41 The Mystery of TEQ Theorem (Laffond et al., 1993; Houy 2009; B., 2011; B. and Harrenstein, 2011): The following statements are equivalent: Every tournament contains a unique minimal TEQ-retentive set. TEQ is the smallest majoritarian SCF satisfying ρ. TEQ satisfies monotonicity (and numerous other desirable properties). 8
42 The Mystery of TEQ Theorem (Laffond et al., 1993; Houy 2009; B., 2011; B. and Harrenstein, 2011): The following statements are equivalent: Every tournament contains a unique minimal TEQ-retentive set. TEQ is the smallest majoritarian SCF satisfying ρ. TEQ satisfies monotonicity (and numerous other desirable properties). TEQ satisfies α and γ (and is set-rationalizable and self-stable). 8
43 The Mystery of TEQ Theorem (Laffond et al., 1993; Houy 2009; B., 2011; B. and Harrenstein, 2011): The following statements are equivalent: Every tournament contains a unique minimal TEQ-retentive set. TEQ is the smallest majoritarian SCF satisfying ρ. TEQ satisfies monotonicity (and numerous other desirable properties). TEQ satisfies α and γ (and is set-rationalizable and self-stable). TEQ is group-strategyproof (for Kelly s preference extension). 8
44 The Mystery of TEQ Theorem (Laffond et al., 1993; Houy 2009; B., 2011; B. and Harrenstein, 2011): The following statements are equivalent: Every tournament contains a unique minimal TEQ-retentive set. TEQ is the smallest majoritarian SCF satisfying ρ. TEQ satisfies monotonicity (and numerous other desirable properties). TEQ satisfies α and γ (and is set-rationalizable and self-stable). TEQ is group-strategyproof (for Kelly s preference extension). Furthermore, a simple and very efficient heuristic for computing TEQ (which is NP-hard in general) relying on Schwartz s conjecture exists (B. et al., 2010). 8
45 The Mystery of TEQ Theorem (Laffond et al., 1993; Houy 2009; B., 2011; B. and Harrenstein, 2011): The following statements are equivalent: Every tournament contains a unique minimal TEQ-retentive set. TEQ is the smallest majoritarian SCF satisfying ρ. TEQ satisfies monotonicity All or (and nothing: numerous other desirable Either properties). TEQ is a most appealing SCF or it is severely flawed. TEQ satisfies α and γ (and is set-rationalizable and self-stable). TEQ is group-strategyproof (for Kelly s preference extension). Furthermore, a simple and very efficient heuristic for computing TEQ (which is NP-hard in general) relying on Schwartz s conjecture exists (B. et al., 2010). 8
46 Schwartz s Conjecture 9
47 Schwartz s Conjecture There is no counterexample with less than 13 alternatives; checked 154 billion tournaments (B. et al., 2010). 9
48 Schwartz s Conjecture There is no counterexample with less than 13 alternatives; checked 154 billion tournaments (B. et al., 2010). TEQ satisfies all nice properties when there are less than 13 alternatives. 9
49 Schwartz s Conjecture There is no counterexample with less than 13 alternatives; checked 154 billion tournaments (B. et al., 2010). TEQ satisfies all nice properties when there are less than 13 alternatives. No counterexample was found by searching billions of random tournaments with up to 50 alternatives. 9
50 Schwartz s Conjecture There is no counterexample with less than 13 alternatives; checked 154 billion tournaments (B. et al., 2010). TEQ satisfies all nice properties when there are less than 13 alternatives. No counterexample was found by searching billions of random tournaments with up to 50 alternatives. Checking significantly larger tournaments is computationally intractable. 9
51 Schwartz s Conjecture There is no counterexample with less than 13 alternatives; checked 154 billion tournaments (B. et al., 2010). TEQ satisfies all nice properties when there are less than 13 alternatives. No counterexample was found by searching billions of random tournaments with up to 50 alternatives. Checking significantly larger tournaments is computationally intractable. Over the years, various incorrect proof attempts of Schwartz s conjecture by ourselves and other researchers were discarded. 9
52 Weaker Statements TEQ Conjecture Schwartz (1990) TRIV (k) Conjecture Brandt et al. (2010) ME Conjecture Brandt (2011) TRIV (2) Conjecture Brandt et al. (2010) MC Conjecture Dutta (1988) bs M Conjecture k Brandt (2011) CF Conjecture Brandt et al. (2012) TRIV (1) Conjecture Good (1971) bs M 4 Conjecture Brandt (2011) 10
53 Weaker Statements TEQ Conjecture Schwartz (1990) TRIV (k) Conjecture Brandt et al. (2010) ME Conjecture Brandt (2011) TRIV (2) Conjecture Brandt et al. (2010) MC Conjecture Dutta (1988) bs M Conjecture k Brandt (2011) CF Conjecture Brandt et al. (2012) TRIV (1) Conjecture Good (1971) bs M 4 Conjecture Brandt (2011) 10
54 Weaker Statements TEQ Conjecture Schwartz (1990) TRIV (k) Conjecture Brandt et al. (2010) ME Conjecture Brandt (2011) TRIV (2) Conjecture Brandt et al. (2010) MC Conjecture Dutta (1988) bs M Conjecture k Brandt (2011) CF Conjecture Brandt et al. (2012) TRIV (1) Conjecture Good (1971) bs M 4 Conjecture Brandt (2011) 10
55 Weaker Statements TEQ Conjecture Schwartz (1990) TRIV (k) Conjecture Brandt et al. (2010) ME Conjecture Brandt (2011) TRIV (2) Conjecture Brandt et al. (2010) MC Conjecture Dutta (1988) bs M Conjecture k Brandt (2011) CF Conjecture Brandt et al. (2012) TRIV (1) Conjecture Good (1971) bs M 4 Conjecture Brandt (2011) 10
56 Weaker Statements TEQ Conjecture Schwartz (1990) TRIV (k) Conjecture Brandt et al. (2010) ME Conjecture Brandt (2011) TRIV (2) Conjecture Brandt et al. (2010) MC Conjecture Dutta (1988) bs M Conjecture k Brandt (2011) CF Conjecture Brandt et al. (2012) TRIV (1) Conjecture Good (1971) bs M 4 Conjecture Brandt (2011) 10
57 Weaker Statements TEQ Conjecture Schwartz (1990) TRIV (k) Conjecture Brandt et al. (2010) ME Conjecture Brandt (2011) TRIV (2) Conjecture Brandt et al. (2010) MC Conjecture Dutta (1988) bs M Conjecture k Brandt (2011) CF Conjecture Brandt et al. (2012) TRIV (1) Conjecture Good (1971) bs M 4 Conjecture Brandt (2011) CF Conjecture: Let (A,B) be a partition of the vertex set of a tournament T. Then A or B contains a transitive subtournament that is undominated in T. 10
58 Weaker Statements TEQ Conjecture Schwartz (1990) TRIV (k) Conjecture Brandt et al. (2010) ME Conjecture Brandt (2011) TRIV (2) Conjecture Brandt et al. (2010) MC Conjecture Dutta (1988) bs M Conjecture k Brandt (2011) CF Conjecture Brandt et al. (2012) TRIV (1) Conjecture Good (1971) bs M 4 Conjecture Brandt (2011) Theorem (B., Chudnovsky, Kim, Liu, Norin, Scott, Seymour, CF Conjecture: Let (A,B) be a partition of the vertex set of a tournament T. and Thomassé; 2012): The CF conjecture is false. Then A or B contains a transitive subtournament that is undominated in T. 10
59 Weaker Statements TEQ Conjecture Schwartz (1990) TRIV (k) Conjecture Brandt et al. (2010) ME Conjecture Brandt (2011) TRIV (2) Conjecture Brandt et al. (2010) MC Conjecture Dutta (1988) bs M Conjecture k Brandt (2011) CF Conjecture Brandt et al. (2012) TRIV (1) Conjecture Good (1971) bs M 4 Conjecture Brandt (2011) Theorem (B., Chudnovsky, Kim, Liu, Norin, Scott, Seymour, CF Conjecture: Let (A,B) be a partition of the vertex set of a tournament T. and Thomassé; 2012): The CF conjecture is false. Then A or B contains a transitive subtournament that is undominated in T. 10
60 11
61 The proof is non-constructive and relies on a probabilistic argument by Erdös and Moser (1964). 11
62 The proof is non-constructive and relies on a probabilistic argument by Erdös and Moser (1964). Neither the counter-example nor its size can be deduced from proof. 11
63 The proof is non-constructive and relies on a probabilistic argument by Erdös and Moser (1964). Neither the counter-example nor its size can be deduced from proof. Smallest counter-example of this type requires about vertices. 11
64 The proof is non-constructive and relies on a probabilistic argument by Erdös and Moser (1964). Neither the counter-example nor its size can be deduced from proof. Smallest counter-example of this type requires about vertices. The estimated number of atoms in the universe is approx
65 Moral of the Tale? 12
66 Moral of the Tale? In principle, TEQ is severely flawed. 12
67 Moral of the Tale? In principle, TEQ is severely flawed. If there does not exist a substantially smaller counterexample, this has no practical consequences. 12
68 Moral of the Tale? In principle, TEQ is severely flawed. If there does not exist a substantially smaller counterexample, this has no practical consequences. The 22-year-old conjecture of a political scientist has been refuted using extremal graph theory. 12
69 Moral of the Tale? In principle, TEQ is severely flawed. If there does not exist a substantially smaller counterexample, this has no practical consequences. The 22-year-old conjecture of a political scientist has been refuted using extremal graph theory. Politics shouldn t be some mind-bending exercise. It s about what you feel in your gut (British PM David Cameron, April 2011) 12
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