Porsche Families, Qatar, Lower Saxony indirectly control. Volkswagen AG. We
will consider (indirect) control rather than ownership. Hans Peters, Dominik Karos
...
Mutual control structures Hans Peters, Dominik Karos
June 2013
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
0 / 23
An example: the Porsche-Volkswagen case (≥ 2007) Porsche Fam. Qatar Lower Saxony
90% 10% 17% 20%
Porsche SE 50.7% Volkswagen AG
100%
Porsche AG
9.9% Others
Porsche Families controls Porsche SE Qatar, Lower Saxony, Porsche SE control Volkswagen AG (...) Volkswagen AG controls Porsche AG ⇒ Porsche Families, Qatar, Lower Saxony indirectly control Volkswagen AG We will consider (indirect) control rather than ownership. Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
1 / 23
Main related literature • Gambarelli and Owen (1994); distinguish between investors and firms. Propose an updating procedure to capture indirect control. • Work in game theory on (cooperative) games and graphs; e.g., conjunctive permission structure in Gilles, Owen, and van den Brink (1992) can be seen as a special case of our mutual control structure. • Crama and Leruth (2011) present an overview of theoretical and empirical literature (corporate governance, corporate finance); Crama and Leruth (2007) also present a procedure to capture control and voting power in corporate networks. • Work on power indices: Shapley (1953), Shapley and Shubik (1954); Dubey (1975), Einy and Haimanko (2011). • This presentation is based on the forthcoming working paper “Indirect control and power in mutual control structures” by Dominik Karos (Saarbr¨ ucken University) and Hans Peters (Maastricht University), June 2013. Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
2 / 23
Outline of the presentation 1. Mutual control structures, indirect control 2. Mutual control structures and simple games 3. Power indices for invariant mutual control structures 4. Concluding remark: ownership versus control
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
3 / 23
1. Mutual control structures, indirect control
Throughout, there is a set of players N = {1, . . . , n}. By P(N) we denote the set of all subsets of N, i.e., of all coalitions.
Definition: Mutual control structure A mutual control structure (mcs) is a map C : P(N) → P(N) satisfying (i) C (∅) = ∅, (ii) monotonicity: C (S) ⊆ C (T ) for all S, T ∈ P(N) with S ⊆ T . The set of all mutual control structures is denoted by C. If i ∈ C (S) then we say that i is controlled by S, or that S controls player i.
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
4 / 23
The Porsche-Volkswagen case revisited (1) 1-Porsche Fam. 2-Qatar 3-Lower Saxony
90%
4-Porsche SE 50.7%
10% 17%
5-Volkswagen
20%
100%
6-Porsche AG
9.9% Others
Can be represented by a mutual control structure C with: 4 ∈ C (S) ⇔ 1 ∈ S, 5 ∈ C (S) ⇔ {2, 3, 4} ⊆ S, 6 ∈ C (S) ⇔ 5 ∈ S for all S ∈ P(N), where N = {1, . . . , 6}. We would like to include, for instance: {1, 2, 3} ⊆ S ⇒ 5 ∈ C (S), for all S ∈ P(N). Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
5 / 23
Indirect control
Definition: Indirect control The mutual control structure C is invariant if it satisfies the following condition. Indirect control: For all S, T , R ∈ P(N) with T ⊆ C (S) and R ⊆ C (S ∪ T ) we have R ⊆ C (S). The set of all invariant mutual control structures is denoted by C ∗ .
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
6 / 23
Indirect control
Definition: Indirect control The mutual control structure C is invariant if it satisfies the following condition. Indirect control: For all S, T , R ∈ P(N) with T ⊆ C (S) and R ⊆ C (S ∪ T ) we have R ⊆ C (S). The set of all invariant mutual control structures is denoted by C ∗ . • An invariant mutual control structure C is transitive: for all S, T , R ∈ P(N) with T ⊆ C (S) and R ⊆ C (T ) we have R ⊆ C (S). (This follows since R ⊆ C (T ) implies R ⊆ C (S ∪ T ) by monotonicity of C .)
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
6 / 23
Indirect control
Definition: Indirect control The mutual control structure C is invariant if it satisfies the following condition. Indirect control: For all S, T , R ∈ P(N) with T ⊆ C (S) and R ⊆ C (S ∪ T ) we have R ⊆ C (S). The set of all invariant mutual control structures is denoted by C ∗ . • An invariant mutual control structure C is transitive: for all S, T , R ∈ P(N) with T ⊆ C (S) and R ⊆ C (T ) we have R ⊆ C (S). (This follows since R ⊆ C (T ) implies R ⊆ C (S ∪ T ) by monotonicity of C .) • The mutual control structure C in the Porsche-Volkswagen case violates indirect control with S = {1, 2, 3}, T = {4}, and R = {5}.
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
6 / 23
How to make a mutual control structure invariant? • Let C be an arbitrary mutual control structure. We define C 1 , C 2 , . . . recursively by C (S) if k = 1 k C (S) = for each S ∈ P(N). k−1 C (S ∪ C (S)) if k > 1
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
7 / 23
How to make a mutual control structure invariant? • Let C be an arbitrary mutual control structure. We define C 1 , C 2 , . . . recursively by C (S) if k = 1 k C (S) = for each S ∈ P(N). k−1 C (S ∪ C (S)) if k > 1 • Since, by monotonicity, C (S) ⊆ C 2 (S) ⊆ . . . ⊆ C k (S) ⊆ . . . for every S ∈ P(N), there must be a natural number p ≥ 1 such that for each S ∈ P(N) we have C (S) = C 1 (S) ⊆ C 2 (S) ⊆ . . . ⊆ C p (S) = C p+1 (S) = C p+2 (S) = . . .
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
7 / 23
How to make a mutual control structure invariant? • Let C be an arbitrary mutual control structure. We define C 1 , C 2 , . . . recursively by C (S) if k = 1 k C (S) = for each S ∈ P(N). k−1 C (S ∪ C (S)) if k > 1 • Since, by monotonicity, C (S) ⊆ C 2 (S) ⊆ . . . ⊆ C k (S) ⊆ . . . for every S ∈ P(N), there must be a natural number p ≥ 1 such that for each S ∈ P(N) we have C (S) = C 1 (S) ⊆ C 2 (S) ⊆ . . . ⊆ C p (S) = C p+1 (S) = C p+2 (S) = . . . We denote C p by C ∗ . We have
Proposition: invariance of C ∗ Let C be a mutual control structure. Then C ∗ is an invariant mutual control structure. Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
7 / 23
The Porsche-Volkswagen case revisited (2) 1-Porsche Fam.
90% 10% 17%
2-Qatar 3-Lower Saxony
20%
4-Porsche SE 50.7% 5-Volkswagen
100%
6-Porsche AG
9.9% Others
The mutual control structure C ∗ is given by: 4 ∈ C ∗ (S) ⇔ 1 ∈ S 5 ∈ C ∗ (S) ⇔ {2, 3, 4} ⊆ S or {1, 2, 3} ⊆ S 6 ∈ C ∗ (S) ⇔ 5 ∈ S or {2, 3, 4} ⊆ S or {1, 2, 3} ⊆ S . for all S ∈ P(N).
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
8 / 23
Minimal invariant extensions • D ∈ C ∗ is an invariant extension of C ∈ C if C ⊆ D, i.e., C (S) ⊆ D(S) for all S ∈ P(N).
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
9 / 23
Minimal invariant extensions • D ∈ C ∗ is an invariant extension of C ∈ C if C ⊆ D, i.e., C (S) ⊆ D(S) for all S ∈ P(N). • Clearly, C ∗ is an invariant extension of C , but also D ∈ C ∗ defined by D(S) = N for all S ∈ P(N), is an invariant extension of any mutual control structure.
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
9 / 23
Minimal invariant extensions • D ∈ C ∗ is an invariant extension of C ∈ C if C ⊆ D, i.e., C (S) ⊆ D(S) for all S ∈ P(N). • Clearly, C ∗ is an invariant extension of C , but also D ∈ C ∗ defined by D(S) = N for all S ∈ P(N), is an invariant extension of any mutual control structure. • An invariant extension D of C is minimal if D ⊆ D ′ for every invariant extension D ′ of C . Obviously, minimal invariant extensions are unique. We have the following result.
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
9 / 23
Minimal invariant extensions • D ∈ C ∗ is an invariant extension of C ∈ C if C ⊆ D, i.e., C (S) ⊆ D(S) for all S ∈ P(N). • Clearly, C ∗ is an invariant extension of C , but also D ∈ C ∗ defined by D(S) = N for all S ∈ P(N), is an invariant extension of any mutual control structure. • An invariant extension D of C is minimal if D ⊆ D ′ for every invariant extension D ′ of C . Obviously, minimal invariant extensions are unique. We have the following result.
Proposition: minimal invariant extension Let C ∈ C. Then C ∗ is the (unique) minimal invariant extension of C .
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
9 / 23
2. Mutual control structures and simple games • A simple game is a map v : P(N) → {0, 1} such that v (∅) = 0 and S ⊆ T ⇒ v (S) ≤ v (T ) for all S, T ∈ P(N). If v (S) = 1 coalition S is winning, otherwise it is loosing. The set of all simple games is denoted by Σ.
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
10 / 23
2. Mutual control structures and simple games • A simple game is a map v : P(N) → {0, 1} such that v (∅) = 0 and S ⊆ T ⇒ v (S) ≤ v (T ) for all S, T ∈ P(N). If v (S) = 1 coalition S is winning, otherwise it is loosing. The set of all simple games is denoted by Σ. • Given a mutual control structure C , we define for each player i ∈ N a simple game wiC by wiC (S) = 1 ⇔ i ∈ C (S), for all S ∈ P(N).
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
10 / 23
2. Mutual control structures and simple games • A simple game is a map v : P(N) → {0, 1} such that v (∅) = 0 and S ⊆ T ⇒ v (S) ≤ v (T ) for all S, T ∈ P(N). If v (S) = 1 coalition S is winning, otherwise it is loosing. The set of all simple games is denoted by Σ. • Given a mutual control structure C , we define for each player i ∈ N a simple game wiC by wiC (S) = 1 ⇔ i ∈ C (S), for all S ∈ P(N). • We denote w C = (w1C , . . . , wnC ) ∈ ΣN . An element of ΣN is a simple game structure.
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
10 / 23
2. Mutual control structures and simple games • A simple game is a map v : P(N) → {0, 1} such that v (∅) = 0 and S ⊆ T ⇒ v (S) ≤ v (T ) for all S, T ∈ P(N). If v (S) = 1 coalition S is winning, otherwise it is loosing. The set of all simple games is denoted by Σ. • Given a mutual control structure C , we define for each player i ∈ N a simple game wiC by wiC (S) = 1 ⇔ i ∈ C (S), for all S ∈ P(N). • We denote w C = (w1C , . . . , wnC ) ∈ ΣN . An element of ΣN is a simple game structure. • There is a 1-1 correspondence between ΣN and C. For w ∈ ΣN we denote by C w the associated mutual control correspondence: i ∈ C w (S) ⇔ wi (S) = 1, for all i ∈ N and S ∈ P(N). Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
10 / 23
The Porsche-Volkswagen case revisited (3) • The mutual control structure C associated with the Porsche-Volkswagen case was given by: 4 ∈ C (S) ⇔ 1 ∈ S, 5 ∈ C (S) ⇔ {2, 3, 4} ⊆ S, 6 ∈ C (S) ⇔ 5 ∈ S for all S ∈ P(N), where N = {1, . . . , 6}. • The associated simple game structure w C has for all S ∈ P(N): w1C (S) = w2C (S) = w3C (S) = 0, w4C (S) = 1 ⇔ {1} ⊆ S, w5C (S) = 1 ⇔ {2, 3, 4} ⊆ S, w6C (S) = 1 ⇔ {5} ⊆ S. Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
11 / 23
How to make a simple game structure invariant? • For an arbitrary w ∈ ΣN it is not necessarily the case that C w is invariant, i.e., C w ∈ C ∗ .
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
12 / 23
How to make a simple game structure invariant? • For an arbitrary w ∈ ΣN it is not necessarily the case that C w is invariant, i.e., C w ∈ C ∗ . • Let w ∈ ΣN . For each pair i , j ∈ N with i = 6 j we define a map N N ti ,j : Σ → Σ as follows: wk (S) if k 6= i ti ,j (wk )(S) = ei (S) if k = i w for each k ∈ N, where 1 if wi (S) = 1 1 if S = (S1 \ {j}) ∪ S2 for some S1 , S2 ei (S) = w such that wi (S1 ) = 1, wj (S2 ) = 1 0 otherwise
Such a map ti ,j is called an elementary substitution: if player j is in a winning coalition S1 in wi , then he may be replaced by a winning coalition S2 in the game wj . Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
12 / 23
The Porsche-Volkswagen case revisited (4) • The with C associated simple game structure w was: w4 (S) = 1 ⇔ {1} ⊆ S, w5 (S) = 1 ⇔ {2, 3, 4} ⊆ S, w6 (S) = 1 ⇔ {5} ⊆ S.
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
13 / 23
The Porsche-Volkswagen case revisited (4) • The with C associated simple game structure w was: w4 (S) = 1 ⇔ {1} ⊆ S, w5 (S) = 1 ⇔ {2, 3, 4} ⊆ S, w6 (S) = 1 ⇔ {5} ⊆ S. Applying the elementary substitution t5,4 yields w4 (S) = 1 ⇔ {1} ⊆ S, w5 (S) = 1 ⇔ {2, 3, 4} or {1, 2, 3} ⊆ S, w6 (S) = 1 ⇔ {5} ⊆ S.
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
13 / 23
The Porsche-Volkswagen case revisited (4) • The with C associated simple game structure w was: w4 (S) = 1 ⇔ {1} ⊆ S, w5 (S) = 1 ⇔ {2, 3, 4} ⊆ S, w6 (S) = 1 ⇔ {5} ⊆ S. Applying the elementary substitution t5,4 yields w4 (S) = 1 ⇔ {1} ⊆ S, w5 (S) = 1 ⇔ {2, 3, 4} or {1, 2, 3} ⊆ S, w6 (S) = 1 ⇔ {5} ⊆ S. Applying, next, the elementary substitution t6,5 yields w4 (S) = 1 ⇔ {1} ⊆ S, w5 (S) = 1 ⇔ {2, 3, 4} or {1, 2, 3} ⊆ S, w6 (S) = 1 ⇔ {5} or {1, 2, 3} or {2, 3, 4} ⊆ S. Call the last structure w ∗ , then C w = C ∗ is invariant. ∗
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
13 / 23
Minimal invariant extensions • We call w ∈ ΣN invariant if it does not change under elementary substitutions. By (ΣN )∗ we denote the set of all invariant simple game structures. •w ˜ is an invariant extension of w if w ˜ ≥ w and w ˜ ∈ (ΣN )∗ . •w ˜ is a minimal invariant extension of w if it is an invariant extension of w and if w ˜ ≤ w ′ for any other invariant extension w ′ of w .
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
14 / 23
Minimal invariant extensions • We call w ∈ ΣN invariant if it does not change under elementary substitutions. By (ΣN )∗ we denote the set of all invariant simple game structures. •w ˜ is an invariant extension of w if w ˜ ≥ w and w ˜ ∈ (ΣN )∗ . •w ˜ is a minimal invariant extension of w if it is an invariant extension of w and if w ˜ ≤ w ′ for any other invariant extension w ′ of w .
Theorem: minimal invariant extension Let w ∈ ΣN . Then applying elementary substitutions results in the unique minimal invariant extension of w , denoted by w ∗ . Moreover, we have ∗ C w = (C w )∗ .
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
14 / 23
Minimal invariant extensions • We call w ∈ ΣN invariant if it does not change under elementary substitutions. By (ΣN )∗ we denote the set of all invariant simple game structures. •w ˜ is an invariant extension of w if w ˜ ≥ w and w ˜ ∈ (ΣN )∗ . •w ˜ is a minimal invariant extension of w if it is an invariant extension of w and if w ˜ ≤ w ′ for any other invariant extension w ′ of w .
Theorem: minimal invariant extension Let w ∈ ΣN . Then applying elementary substitutions results in the unique minimal invariant extension of w , denoted by w ∗ . Moreover, we have ∗ C w = (C w )∗ . Summary
w ∈ ΣN −→ Cw ∈ C ↓ ↓ ∗ w ∗ ∈ (ΣN )∗ −→ C w = (C w )∗ ∈ C ∗
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
14 / 23
3. Power indices for invariant mutual control structures • We now consider only invariant mutual control structures and associated invariant simple game structures.
Definition: power index A power index is a map ϕ : C ∗ → RN . • We will axiomatically derive a class of power indices.
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
15 / 23
3. Power indices for invariant mutual control structures • We now consider only invariant mutual control structures and associated invariant simple game structures.
Definition: power index A power index is a map ϕ : C ∗ → RN . • We will axiomatically derive a class of power indices.
Definition: marginal control contribution, null player, permutation Let C ∈ C ∗ , i ∈ N. ∆Ci (S) = C (S) \ C (S \ {i }) is the marginal control contribution of player i to a coalition S. Player i is a null player in C if ∆Ci (S) = ∅ and i ∈ / C (S) for all S ∈ P(N). For a permutation π : N → N, πC ∈ C ∗ is defined by (πC )(S) = π(C (π −1 (S))) for all S ∈ P(N).
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
15 / 23
The axioms Let ϕ : C ∗ → RN be a power index.
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
16 / 23
The axioms Let ϕ : C ∗ → RN be a power index. Null-Player (NP) ϕi (C ) = 0 for every null player i with respect to C , for every C ∈ C ∗ .
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
16 / 23
The axioms Let ϕ : C ∗ → RN be a power index. Null-Player (NP) ϕi (C ) = 0 for every null player i with respect to C , for every C ∈ C ∗ . P Zerosum (ZS) ϕi (C ) = 0 for every C ∈ C ∗ . i ∈N
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
16 / 23
The axioms Let ϕ : C ∗ → RN be a power index. Null-Player (NP) ϕi (C ) = 0 for every null player i with respect to C , for every C ∈ C ∗ . P Zerosum (ZS) ϕi (C ) = 0 for every C ∈ C ∗ . i ∈N
Anonymity (AN) ϕπ(i ) (πC ) = ϕi (C ) for every player i ∈ N, every permutation π of N, and every C ∈ C ∗ .
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
16 / 23
The axioms Let ϕ : C ∗ → RN be a power index. Null-Player (NP) ϕi (C ) = 0 for every null player i with respect to C , for every C ∈ C ∗ . P Zerosum (ZS) ϕi (C ) = 0 for every C ∈ C ∗ . i ∈N
Anonymity (AN) ϕπ(i ) (πC ) = ϕi (C ) for every player i ∈ N, every permutation π of N, and every C ∈ C ∗ . Transfer Property (TP) ϕ(C ) − ϕ(C ′ ) = ϕ(D) − ϕ(D ′ ) for all C , C ′ , D, D ′ ∈ C ∗ such that C ′ ⊆ C , D ′ ⊆ D, and C (S) \ C ′ (S) = D(S) \ D ′ (S) for every S ⊆ N.
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
16 / 23
A family of power indices • We introduce a family of power indices, defined in terms of the associated simple games. • For a simple game wi , the dividend di (S) of coalition S is defined, recursively, by ( 0 if S = ∅ P di (S) = di (T ) otherwise. wi (S) − T S
• The Shapley value of player k ∈ N in the game wi is Shk (wi ) =
X di (S) . |S|
S:k∈S
• For C ∈ C ∗ and i ∈ N, we write diC for the dividends of wiC . Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
17 / 23
For every weight vector ω = (α1 , . . . , αn−1 , β2 , . . . , βn ) ∈ R2n−2 , we define the power index Φω by X X d C (S) X d C (S) k k Φωi (C ) = α + β |S| |S| |S| |S| S:i ∈S,k∈S k∈N\{i } S:i ∈S,k ∈S / X X d C (S) X d C (S) i i − α + β |S| |S| |S| |S| k∈N\{i }
S:i ∈S,k∈S /
S:i ∈S,k∈S
for all C ∈ C ∗ and i ∈ N.
First line: dividends for player i from controlling other players. Second line: dividends for other players from controlling player i .
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
18 / 23
Theorem: Characterization of family Φω Let ϕ be a power index. Then ϕ satisfies NP, ZS, AN, and TP if and only if there is a weight vector ω = (α1 , . . . , αn−1 , β2 , . . . , βn ) ∈ R2n−2 such that ϕ(C ) = Φω (C ) for every C ∈ C ∗ .
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
19 / 23
Theorem: Characterization of family Φω Let ϕ be a power index. Then ϕ satisfies NP, ZS, AN, and TP if and only if there is a weight vector ω = (α1 , . . . , αn−1 , β2 , . . . , βn ) ∈ R2n−2 such that ϕ(C ) = Φω (C ) for every C ∈ C ∗ . • Weights are completely arbitrary. Consider the following axiom. Monotonicity (MO) ϕi (C ) ≥ ϕi (D) for all C , D ∈ C ∗ and i ∈ N such C that (i) i ∈ C (S) ⇒ i ∈ D(S) and (ii) ∆D i (S) ⊆ ∆i (S) for all S ⊆ N.
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
19 / 23
Theorem: Characterization of family Φω Let ϕ be a power index. Then ϕ satisfies NP, ZS, AN, and TP if and only if there is a weight vector ω = (α1 , . . . , αn−1 , β2 , . . . , βn ) ∈ R2n−2 such that ϕ(C ) = Φω (C ) for every C ∈ C ∗ . • Weights are completely arbitrary. Consider the following axiom. Monotonicity (MO) ϕi (C ) ≥ ϕi (D) for all C , D ∈ C ∗ and i ∈ N such C that (i) i ∈ C (S) ⇒ i ∈ D(S) and (ii) ∆D i (S) ⊆ ∆i (S) for all S ⊆ N.
Corollary: monotonicity and nonnegative weights Let ϕ be a power index. Then ϕ satisfies NP, ZS, AN, TP, and MO if and only if there is a weight vector ω = (α1 , . . . , αn−1 , β2 , . . . , βn ) ∈ R2n−2 + such that ϕ(C ) = Φω (C ) for every C ∈ C ∗ .
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
19 / 23
A further normalization • Consider the following axiom. Controlled player (CP) For all C ∈ C ∗ , j ∈ C (N), and i ∈ N \ C (N), −1 if ∆Cj (S) = ∅ for all S ⊆ N ϕj (C ) = ϕi (C ) − 1 if ∆Ci (S) = ∆Cj (S) for all S ⊆ N. • Note: if ϕ also satisfies NP and i is a null player, then the second consequence in CP implies the first.
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
20 / 23
A further normalization • Consider the following axiom. Controlled player (CP) For all C ∈ C ∗ , j ∈ C (N), and i ∈ N \ C (N), −1 if ∆Cj (S) = ∅ for all S ⊆ N ϕj (C ) = ϕi (C ) − 1 if ∆Ci (S) = ∆Cj (S) for all S ⊆ N. • Note: if ϕ also satisfies NP and i is a null player, then the second consequence in CP implies the first.
Corollary: weights equal to one There is a unique power index satisfying NP, ZS, AN, TP, and CP, namely the power index Φω with ω = (1, . . . , 1) ∈ R2n−2 . Hence: each player collects his Shapley values from controlling other players, but “pays” the Shapley values to the other players from controlling him. Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
20 / 23
The Porsche-Volkswagen case revisited (5) 1-Porsche Fam.
90% 10% 17%
2-Qatar 3-Lower Saxony
20%
4-Porsche SE 50.7% 5-Volkswagen
100%
6-Porsche AG
9.9% Others
We take ω = (1, . . . , 1) and compute: Φω1 (C ∗ ) = Φω4 (C ∗ ) =
14 12 , − 56 ,
Φω2 (C ∗ ) = Φω5 (C ∗ ) =
7 12 , − 12 ,
Φω3 (C ∗ ) =
7 12 ,
Φω6 (C ∗ ) = −1.
Before Porsche Families started buying (so before 2007) they had larger power, namely, equal to 2: is elaborated in the paper. Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
21 / 23
4. Concluding remark: ownership versus control • We consider control rather than ownership. Example: 80%
a
60%
d
b
40%
20%
c
Control: C ({a}) = {b}, C ({d}) = {a}, C ({c}) = ∅, C ({b}) = ∅. Hence C ∗ ({d}) = {a, b}, C ∗ ({c}) = ∅. Ownership: d owns 48% of b indirectly via a, and c owns 20% of b directly and 32% via a, in total: 52%. Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
22 / 23
THANK YOU FOR YOUR ATTENTION
Hans Peters, Dominik Karos ()
Mutual control structures
June 2013
23 / 23
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