Minimal Retentive Sets in Tournaments
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1 Minimal Retentive Sets in Tournaments From Anywhere to TEQ Felix Brandt Markus Brill Felix Fischer Paul Harrenstein Ludwig-Maximilians-Universität München Estoril, April 12, / 17
2 The Trouble with Tournaments Tournaments are oriented complete graphs Many applications: social choice theory, sports tournaments, game theory, argumentation theory, webpage and journal ranking, etc. Question: How to select the winner(s) of a tournament in the absence of transitivity? a b c d e 2 / 17
3 The Trouble with Tournaments Tournaments are oriented complete graphs Many applications: social choice theory, sports tournaments, game theory, argumentation theory, webpage and journal ranking, etc. Question: How to select the winner(s) of a tournament in the absence of transitivity? a b c d e 2 / 17
4 Overview Tournament solutions Retentiveness and Schwartz s Tournament Equilibrium Set (TEQ) Properties of minimal retentive sets Approximating TEQ A new tournament solution 3 / 17
5 Tournament Solutions a A tournament T = (A, ) consists of: a finite set A of alternatives a complete and asymmetric relation on A b d c e 4 / 17
6 Tournament Solutions a A tournament T = (A, ) consists of: a finite set A of alternatives a complete and asymmetric relation on A b d c e A tournament solution S maps each tournament T = (A, ) to a set S(T) such that S(T) A and S(T) contains the Condorcet winner if it exists S is called proper if a Condordet winner is always selected as only alternative 4 / 17
7 Tournament Solutions a A tournament T = (A, ) consists of: a finite set A of alternatives a complete and asymmetric relation on A b d c e A tournament solution S maps each tournament T = (A, ) to a set S(T) such that S(T) A and S(T) contains the Condorcet winner if it exists S is called proper if a Condordet winner is always selected as only alternative Examples: Trivial Solution (TRIV), Top Cycle (TC), Uncovered Set, Slater Set, Copeland Set, Banks Set, Minimal Covering Set (MC), Tournament Equilibrium Set (TEQ),... 4 / 17
8 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) a b 5 / 17
9 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) a b 5 / 17
10 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) a b 5 / 17
11 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) a b 5 / 17
12 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) 5 / 17
13 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) 5 / 17
14 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) 5 / 17
15 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) 5 / 17
16 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) 5 / 17
17 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) 5 / 17
18 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) 5 / 17
19 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) 5 / 17
20 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) b a 5 / 17
21 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) b a 5 / 17
22 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) b a 5 / 17
23 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) b a 5 / 17
24 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) b a 5 / 17
25 Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) Note: SSP is equivalent to ˆα (see Felix s lecture) (SSP MON) implies WSP and IUA b a 5 / 17
26 Examples Definition: TRIV returns the set A for each tournament T = (A, ) 6 / 17
27 Examples Definition: TRIV returns the set A for each tournament T = (A, ) Definition: TC returns the smallest dominating set, i.e. the smallest set B A with B A \ B Intuition: No winner should be dominated by a loser 6 / 17
28 Examples Definition: TRIV returns the set A for each tournament T = (A, ) Definition: TC returns the smallest dominating set, i.e. the smallest set B A with B A \ B Intuition: No winner should be dominated by a loser B A \ B 6 / 17
29 Examples Definition: TRIV returns the set A for each tournament T = (A, ) Definition: TC returns the smallest dominating set, i.e. the smallest set B A with B A \ B Intuition: No winner should be dominated by a loser Define D(b) = {a A : a b} TC is the smallest set B satisfying D(b) B for all b B B D(b) A \ B b B 6 / 17
30 Examples Definition: TRIV returns the set A for each tournament T = (A, ) Definition: TC returns the smallest dominating set, i.e. the smallest set B A with B A \ B Intuition: No winner should be dominated by a loser Define D(b) = {a A : a b} TC is the smallest set B satisfying D(b) B for all b B Both TRIV and TC satisfy all four basic properties B D(b) A \ B b B 6 / 17
31 Retentiveness Intuition: An alternative a is only properly dominated by a good alternatives Thomas Schwartz 7 / 17
32 Retentiveness Intuition: An alternative a is only properly dominated by a good alternatives, i.e., alternatives selected by S from the dominators of a Thomas Schwartz 7 / 17
33 Retentiveness Intuition: An alternative a is only properly dominated by a good alternatives, i.e., alternatives selected by S from the dominators of a No winner should be properly dominated by a loser Thomas Schwartz 7 / 17
34 Retentiveness Definition: all b B B is S-retentive if B and S(D(b)) B for D(b) b Thomas Schwartz B 7 / 17
35 Retentiveness Definition: all b B B is S-retentive if B and S(D(b)) B for D(b) b Thomas Schwartz B 7 / 17
36 Retentiveness Definition: all b B B is S-retentive if B and S(D(b)) B for S(D(b)) D(b) b Thomas Schwartz B 7 / 17
37 Retentiveness Definition: all b B B is S-retentive if B and S(D(b)) B for S(D(b)) D(b) b Thomas Schwartz B Definition: S returns the union of all minimal S-retentive sets 7 / 17
38 Retentiveness Definition: all b B B is S-retentive if B and S(D(b)) B for S(D(b)) D(b) b Thomas Schwartz B Definition: S returns the union of all minimal S-retentive sets Call S unique if there always exists a unique minimal S-retentive set 7 / 17
39 Retentiveness Definition: all b B B is S-retentive if B and S(D(b)) B for S(D(b)) D(b) b Thomas Schwartz B Definition: S returns the union of all minimal S-retentive sets Call S unique if there always exists a unique minimal S-retentive set Minimal S-retentive sets exist for each tournament S is unique if and only if there do not exist two disjoint S-retentive sets 7 / 17
40 Example Proposition: TRIV = TC 8 / 17
41 Example Proposition: TRIV = TC Proof: A set is TRIV-retentive if and only if it is dominating D(b) TRIV(D(b)) = D(b) b B 8 / 17
42 The Tournament Equilibrium Set The tournament equilibrium set (TEQ) is defined recursively as TEQ = TEQ 9 / 17
43 The Tournament Equilibrium Set The tournament equilibrium set (TEQ) is defined recursively as TEQ = TEQ well-defined because D(a) < A for each a A 9 / 17
44 Example a b d c e x D(x) a {c} b {a, e} c {b, d} d {a, b} e {a, c, d} 10 / 17
45 Example a b d c e x D(x) TEQ(D(x)) a {c} {c} b {a, e} {a} c {b, d} {b} d {a, b} {a} e {a, c, d} {a, c, d} 10 / 17
46 Example a b d c e x D(x) TEQ(D(x)) a {c} {c} b {a, e} {a} c {b, d} {b} d {a, b} {a} e {a, c, d} {a, c, d} 10 / 17
47 Example a b d c e x D(x) TEQ(D(x)) a {c} {c} b {a, e} {a} c {b, d} {b} d {a, b} {a} e {a, c, d} {a, c, d} TEQ-retentive sets: {a, b, c, d, e}, {a, b, c, d}, {a, b, c} 10 / 17
48 Example a b d c e x D(x) TEQ(D(x)) a {c} {c} b {a, e} {a} c {b, d} {b} d {a, b} {a} e {a, c, d} {a, c, d} TEQ-retentive sets: {a, b, c, d, e}, {a, b, c, d}, {a, b, c} TEQ(T) = {a, b, c} 10 / 17
49 The Tournament Equilibrium Set The tournament equilibrium set (TEQ) is defined recursively as TEQ = TEQ well-defined because D(a) < A for each a A 11 / 17
50 The Tournament Equilibrium Set The tournament equilibrium set (TEQ) is defined recursively as TEQ = TEQ well-defined because D(a) < A for each a A Schwartz s Conjecture: TEQ is unique, i.e., each tournament admits a unique minimal TEQ-retentive set. 11 / 17
51 The Tournament Equilibrium Set The tournament equilibrium set (TEQ) is defined recursively as TEQ = TEQ well-defined because D(a) < A for each a A Schwartz s Conjecture: TEQ is unique, i.e., each tournament admits a unique minimal TEQ-retentive set. Theorem (Laffond et al., 1993, Houy, 2009): satisfies any of MON, WSP, SSP, and IUA. TEQ is unique if and only if TEQ 11 / 17
52 Inheritance of Basic Properties Recall: S returns the union of all minimal S-retentive sets 12 / 17
53 Inheritance of Basic Properties Recall: S returns the union of all minimal S-retentive sets Theorem: If S satisfies MON, WSP, SSP, or IUA, so does S. 12 / 17
54 Inheritance of Basic Properties Recall: S returns the union of all minimal S-retentive sets Theorem: If S satisfies MON, WSP, SSP, or IUA, so does S. Theorem: If S satisfies (MON SSP), WSP, SSP, or IUA, so does S 12 / 17
55 Inheritance of Basic Properties Recall: S returns the union of all minimal S-retentive sets Theorem: If S satisfies MON, WSP, SSP, or IUA, so does S. Theorem: If S satisfies (MON SSP), WSP, SSP, or IUA, so does S if S is unique. 12 / 17
56 Convergence Define S (0) = S and S (k+1) = S(k). Thus, we obtain sequences like: TRIV, TC, TC, TC (2), TC (3),... MC, MC, MC (2), MC (3), MC (4), / 17
57 Convergence Define S (0) = S and S (k+1) = S(k). Thus, we obtain sequences like: TRIV, TC, TC, TC (2), TC (3),... MC, MC, MC (2), MC (3), MC (4),... Definition: S converges to S if for each T there is some k T N such that S (k T ) (T) = S (n) (T) = S (T) for all n k T 13 / 17
58 Convergence Define S (0) = S and S (k+1) = S(k). Thus, we obtain sequences like: TRIV, TC, TC, TC (2), TC (3),... MC, MC, MC (2), MC (3), MC (4),... Definition: S converges to S if for each T there is some k T N such that S (k T ) (T) = S (n) (T) = S (T) for all n k T Theorem: Every tournament solution converges to TEQ. 13 / 17
59 Convergence Define S (0) = S and S (k+1) = S(k). Thus, we obtain sequences like: TRIV, TC, TC, TC (2), TC (3),... MC, MC, MC (2), MC (3), MC (4),... Definition: S converges to S if for each T there is some k T N such that S (k T ) (T) = S (n) (T) = S (T) for all n k T Theorem: Every tournament solution converges to TEQ. Proof: S (n 1) (T) = TEQ(T) for all tournaments T of order n 13 / 17
60 Reaching the Limit Theorem: If S TEQ, then S (k) TEQ for all k 0. b i S(T) TEQ(T) a i 14 / 17
61 Reaching the Limit Theorem: If S TEQ, then S (k) TEQ for all k 0. b i S(T) b 1 TEQ(T) a 1 a i 14 / 17
62 Reaching the Limit Theorem: If S TEQ, then S (k) TEQ for all k 0. b i S(T) TEQ(T) b 1 a 1 b 2 a 2 a i 14 / 17
63 Reaching the Limit Theorem: If S TEQ, then S (k) TEQ for all k 0. b i S(T) TEQ(T) b 1 a 1 b 2 a 2... a i 14 / 17
64 Reaching the Limit Theorem: If S TEQ, then S (k) TEQ for all k 0. b i S(T) TEQ(T) b 1 a 1 b 2 a 2... a i 14 / 17
65 Approximating TEQ Theorem (Brandt et al. 2008): Computing TEQ is NP-hard. 15 / 17
66 Approximating TEQ Theorem (Brandt et al. 2008): Computing TEQ is NP-hard. Theorem: S is efficiently computable if and only if S is. S, S, S (2), S (3),... TEQ 15 / 17
67 Approximating TEQ Theorem (Brandt et al. 2008): Computing TEQ is NP-hard. Theorem: S is efficiently computable if and only if S is. S, S, S (2), S (3),... TEQ We would like to have nice convergence / 17
68 Approximating TEQ Theorem (Brandt et al. 2008): Computing TEQ is NP-hard. Theorem: S is efficiently computable if and only if S is. S, S, S (2), S (3),... TEQ We would like to have nice convergence... Theorem: all k 0. If S S, TEQ S and TEQ is unique, then TEQ S (k+1) S (k) for 15 / 17
69 Approximating TEQ Theorem (Brandt et al. 2008): Computing TEQ is NP-hard. Theorem: S is efficiently computable if and only if S is. S, S, S (2), S (3),... TEQ We would like to have nice convergence... Theorem: all k 0. In particular, If S S, TEQ S and TEQ is unique, then TEQ S (k+1) S (k) for TRIV TC TC TC (2) TEQ. Thus, TEQ can be approximated by an anytime algorithm. 15 / 17
70 Approximating TEQ Theorem (Brandt et al. 2008): Computing TEQ is NP-hard. Theorem: S is efficiently computable if and only if S is. S, S, S (2), S (3),... TEQ We would like to have nice convergence... Theorem: all k 0. In particular, If S S, TEQ S and TEQ is unique, then TEQ S (k+1) S (k) for TRIV TC TC TC (2) TEQ. Thus, TEQ can be approximated by an anytime algorithm. As uniqueness of TC (k) implies uniqueness of TC (k 1), we have an infinite sequence of increasingly difficult conjectures. 15 / 17
71 The Minimal Top Cycle Retentive Set TRIV, TC, TC, TC (2), TC (3),... TEQ 16 / 17
72 The Minimal Top Cycle Retentive Set TRIV, TC, TC, TC (2), TC (3),... TEQ Theorem: TC is unique. c 2i+1... c 2i. b 1 c 1 b 0 c 0 16 / 17
73 The Minimal Top Cycle Retentive Set TRIV, TC, TC, TC (2), TC (3),... TEQ Theorem: TC is unique. c 2i+1... Consequence: TC satisfies MON, SSP, WSP, and IUA TC lies between TC and TEQ TC is efficiently computable b 1 c 2i c 1. b 0 c 0 16 / 17
74 Conclusion Retentiveness as an operation on tournament solutions Inheritance of basic properties by minimal retentive sets Convergence and approximating TEQ TC first new concept in sequence with desirable properties Future work: Prove (or disprove) uniqueness of TC (2), MC,..., TEQ 17 / 17
75 Conclusion Retentiveness as an operation on tournament solutions Inheritance of basic properties by minimal retentive sets Convergence and approximating TEQ TC first new concept in sequence with desirable properties Future work: Prove (or disprove) uniqueness of TC (2), MC,..., TEQ Thank you! 17 / 17
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