Applied cooperative game theory:

Transcription

1 Applied cooperative game theory: Dividends Harald Wiese University of Leipzig April 2 Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 / 22

2 Overview Dividends De nition and interpretation oalition functions as vectors Spanning and linear independence The basis of unanimity games Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 2 / 22

3 De nition and interpretation of dividends I De nition (Harsanyi dividend) Let v 2 V N be a coalition function. The dividend (also called Harsanyi dividend) is a coalition function d v on N de ned by Theorem (Harsanyi dividend) d v (T ) = ( ) jt j jk j v (K ). K T For any coalition function v 2 V N, its Harsanyi dividends are de ned by the induction formula d v (S) = v (S) for jsj =, d v (S) = v (S) d v (K ) for jsj > K S Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 3 / 22

4 De nition and interpretation of dividends II onsider a player i who is a member of 2 n coalitions T N. Player i owns coalition T together with the other players from T where his ownership fraction is jt j. Each coalition T brings forth a dividend d v (T ). Player i should obtain d v (T ) = Sh i (v). i2t N jt j Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 4 / 22

5 oalition functions as vectors I We need 2 n entries to describe any game v 2 V N. (The worth of is always zero!) For example, u f,2g 2 V (f, 2, 3g) can be identi ed with the vector from R {z}, {z}, {z}, fg f2g f3g {z} f,2g, {z} f,3g, {z} f2,3g, {z} f,2,3g A. Problem Write down the vector that describes the Maschler game 8 <, jk j = v (K ) = 6, jk j = 2 : 72, jk j = 3 Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 5 / 22

6 oalition functions as vectors II You know how to sum vectors. We can also multiply a vector by a real number (scalar multiplication). oth operations proceed entry by entry: Problem onsider v = (, 3, 3), w = (2, 7, 8) and α = 2 αw. and determine v + w and Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 6 / 22

7 Spanning and linear independence I Spanning R m, m, is a prominent class of vector spaces some of which obey m = 2 n. De nition (linear combination, spanning) A vector w 2 R m is called a linear combination of vectors v,..., v k 2 R m if there exist scalars (also called coe cients) α,..., α k 2 R such that w = k α`v` `= holds. The set of vectors fv,..., v k g is said to span R m if every vector from R m is a linear combinations of the vectors v,..., v k. Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 7 / 22

8 Spanning and linear independence II Spanning onsider, for example, R 2 and the set of vectors f(, 2), (, ), (, )g. Any vector (x, x 2 ) is a linear combination of these vectors by 2x (, 2) (3x x 2 ) (, ) x (, ) = (2x x, 4x (3x x 2 ) x ) = (x, x 2 ). Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 8 / 22

9 Spanning and linear independence III Spanning Problem Show that (, ) is a linear combination of the other two vectors, (, 2) and (, )! Using the result of the above exercise, we have 2x (, 2) (3x x 2 ) (, ) x (, ) = 2x (, 2) (3x x 2 ) [(, 2) (, )] x (, ) = [2x (3x x 2 )] (, 2) [x + (3x x 2 )] (, ) so that any vector from R 2 is a linear combination of just (, 2) and (, ). Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 9 / 22

10 Spanning and linear independence IV Linear independence If we want to span R 2 (or any R m ), we try to nd a minimal way to do so. Any vector in a spanning set that is a linear combination of other vectors in that set, can be eliminated. De nition (linear independence) A set of vectors fv,..., v k g is called linearly independent if no vector from that set is a linear combination of other vectors from that set. Problem Are the vectors (, 3, 3), (2,, ) and (8, 9, 9) linearly independent? Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 / 22

11 Spanning and linear independence V asis De nition (basis) A set of vectors fv,..., v k g is called a basis for R m if it spans R m and is linearly independent. An obvious basis for R m consists of the m unit vectors (,,..., ), (,,,..., ),..., (,...,, ). Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 / 22

12 Spanning and linear independence VI asis Any x = (x,..., x m ) is a linear combination of these vectors by x (,,..., ) + x 2 (,,,..., ) x m (,...,, ) = (x,,..., ) + (, x 2,,..., ) (,...,, x m ) = (x,..., x m ). This proves that the unit vectors do indeed span R m. In order to show linear independence, consider any linear combination of m unit vectors, for example α (,,..., ) + α 2 (,,,..., ) α m (,...,,, ) which is equal to (α,..., α m coe cients α,..., α m., ) and unequal to (,...,, ) for any Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 2 / 22

13 Spanning and linear independence VII asis Theorem (basis criterion) Every basis of the vector space R m has m elements. Any set of m elements of the vector space R m that span R m form a basis. Any set of m elements of the vector space R m that are linearly independent form a basis. Theorem (uniquely determined coe cients) Let fv,..., v m g be a basis of R m and let x be any vector such that x = Then α i = β i for all i =,.., m. m m α i v i = β i v i. i= i= Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 3 / 22

14 The basis of unit games Unit vectors form a basis. For the vector space V N, this means that the 2 n coalition functions v T, T 6=, given by, S = T v T (S) =, S 6= T form a basis. Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 4 / 22

15 The basis of unanimity games I Lemma (unanimity games form basis) The 2 n unanimity games u T, T 6=, form a basis of the vector space V N. Thus, there exist uniquely determined coe cients λ v (T ) such that or holds. v (S) = v = λ v (T ) u T T 22 N nf g λ v (T ) u T (S), S N. T 22 N nf g Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 5 / 22

16 The basis of unanimity games II Lemma The coe cients for the linear combination of v are λ v (T ) := d v (T ). Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 6 / 22

17 The coe cients: an example I onsider N := f, 2, 3g and the coalition function v given by 8, jsj = >< 6, S = f, 2g v (S) = 48, S = f, 3g 3, S = f2, 3g >: 72, S = N This coalition function can also be expressed by the vector (fg) (f2g) (f3g) 6 (f, 2g) 48 (f, 3 (f2, 3g) A 72 (f, 2, 3g) Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 7 / 22

18 The coe cients: an example II d v (fg) = d v (f2g) = d v (f3g) =, d v (f, 2g) = ( ) jf,2gj jk j v (K ) K 22 f,2g nf g = ( ) 2 v (fg) + ( ) 2 v (f2g) + ( ) 2 2 v (f, 2g) = + + 6, d v (f, 3g) = ( ) 2 v (fg) + ( ) 2 v (f3g) + ( ) 2 2 v (f, 3g) = 48, d v (f2, 3g) = ( ) 2 v (f2g) + ( ) 2 v (f3g) + ( ) 2 2 v (f2, 3g) = 3 and d v (f, 2, 3g) = K 22 N nf g ( ) jn j jk j v (K ) =... = 66 Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 8 / 22

19 The coe cients: an example III d v (f, 2g) u f,2g + d v (f, 3g) u f,3g + d v (f2, 3g) u f2,3g + d v (N) u N = A + A + A A A Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 9 / 22

20 The coe cients: exercises Problem alculate the coe cients for the following games on N = f, 2, 3g : v 2 V N is de ned by v (f, 2g) = v (f2, 3g) = v (f, 2, 3g) = and v (fg) = v (f2g) = v (f3g) = v (f, 3g) =. v 2 V N is de ned by 8 <, jsj v (S) = 8, jsj = 2 : 9, S = N Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 2 / 22

21 The proof of the lemma I Lemma (unanimity games form basis) The 2 n V N. unanimity games u T, T 6=, form a basis of the vector space It is su cient to show that the unanimity games are linearly independent. We use a proof by contradiction and assume that there is a unanimity game u T that is a linear combination of the others: where u T = k. β`u T` `= the coalitions T, T,..., T k are all pairwise di erent, k 2 n 2 holds and β` 6= holds for all ` =,..., k. Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 2 / 22

22 The proof of the lemma II Let us assume jt j jt`j for all ` =,.., k. If not, rearrange and rename. Using the coalition T as an argument, we now obtain Problem = u T (T ) and hence the desired contradiction. k = β`u T` (T ) `= k = β` `= = In the above proof, do you see why u T` (T ) = holds for all ` =,..., k? Harald Wiese (hair of Microeconomics) Applied cooperative game theory: April 2 22 / 22