A DEA Game II. Juha Saloheimo S ysteemianalyysin. Laboratorio. Teknillinen korkeakoulu

Transkriptio

1 A DEA Game II Juha Salohemo

2 Content Recap of the Example The Shapley Value Margnal Contrbuton, Ordered Coaltons, Soluton to the Example DEA Mn Game Summary Home Assgnment

3 Recap of the Example Table 1. Normalzed Score Matrx Teammate A B C D Sum Bran 0,45 0,25 0,2 0,1 1 Bravery 0,25 0,4167 0,2083 0,125 1 Labor 0,1154 0,2308 0,2692 0, Total Score 0,8154 0,8975 0,6775 0, We are tryng to fnd a way to dvde a prze pool of $10000 between teammates A, B, C and D.

4 Table 2. Coalton and Characterstc Functon Coalton {A,B} {A,C} {A,D} {B,C} {B,D} {C,D} Bran 0,7 0,65 0,55 0,45 0,35 0,3 Bravery 0,67 0,46 0,38 0,625 0,54 0,33 Labor 0,35 0,38 0,5 0,5 0,62 0,65 Coalton {A,B,C} {A,B,D} {A,C,D} {B,C,D} {A,B,C,D} Bran 0,9 0,8 0,75 0,55 1 Bravery 0,86 0,79 0,58 0,75 1 Labor 0,62 0,73 0,77 0, Table 2 shows all the possble coaltons and ther scores. -The bold numbers ndcate the hghest score wthn the crtera for coaltons and to maxmze coalton s proporton of the prze money, t should choose weght 1 for ths crteron. - In ths case the optmal proporton of the prze money c(s) for a coalton S equals to the hghest crteron score for ths coalton

5 The Shapley Value Margnal contrbuton One soluton to the problem of dvdng prze money between teammates can be reached wth help of Shapley values The margnal contrbuton of a player k to a coalton S that ncludes player k, can be defned as c(s) c(s {k}) Wth help of the Tables 1 and 2 we can calculate for each k and S the margnal contrbutons

6 -For example C s margnal contrbuton to coalton {A,B,C} can be calculated wth help of prevous tables as c({a,b,c}) c({a,b}) = 0,9 0,7 = 0,2 Table 3. Each Member s Margnal Contrbuton to Coalton Coalton A B C D {A} 0, {B} 0 0, {C} 0 0 0,27 0 {D} ,38 {A,B} 0,28 0, {A,C} 0,38 0 0,2 0 {A,D} 0, ,1 {B,C} 0 0,36 0,21 0 {B,D} 0 0,23 0 0,2 {C,D} 0 0 0,27 0,38 {A,B,C} 0,28 0,25 0,2 0 {A,B,D} 0,18 0,25 0 0,1 {A,C,D} 0,12 0 0,22 0,12 {B,C,D} 0 0,23 0,27 0,26 {A,B,C,D} 0,12 0,23 0,2 0,1

7 The Shapley Value Ordered Coaltons We ntroduce orderng n formng a coalton If we suppose that players jon the coalton n order A,B,C,D, the formed ordered coalton s denoted by ABCD. In ths case the margnal contrbuton for the player D s calculated as c{({a,b,c,d}) c({a,b,c}), for the player C as c{({a,b,c}) c({a,b}),

8 for the player B as c{({a,b}) c({a}) and for the player A as c{({a}) c({0}) The Shapley Value Orderng n Coaltons

9 The Shapley Value Defnton The Shapley value k (c) of player k by means of the characterstc functon c(s) s defned as ( c) k 1 n! S: ks N ( s 1)!( n s)! { c( S) c( S { k})}, (1) where s s the number of members n the coalton S and n the total number of players n the game. Shapley value calculates player k s contrbuton to the entre game as the average of margnal contrbutons to coaltons whch nclude k.

10 Soluton The Shapley Value The Shapley value counts the player k s margnal contrbuton n all permutatons of the players ncludng k. If we assume that the ordered coalton S ncludes player k as a last member of S so that t s preceded by 1,..., s 1, and k s successors are 1,..., n s. Ths pattern appears (s 1)!(n s)! tmes n the all permutatons n! If the all permutatons occur wth equal probablty, k s Shapley value can be calculated as stated

11 The Shapley Value The Soluton to the Example - On the table 4 all the margnal contrbutons are calculated for each 4! = 24 ordered coaltons. - Shapley values are calculated as an average over the margnal contrbutons for each ordered coalton. - Calculated Shapley values recommend that for A should be gven $2600, B $2900, C $2300 and D $2200 of the amount of $ Table 4. The Shapley Value Coalton A B C D Sum A<-B<-C<-D 0,45 0,25 0,2 0,1 1 A<-B<-D<-C 0,45 0,25 0,2 0,1 1 A<-C<-B<-D 0,45 0,25 0,2 0,1 1 A<-C<-D<-B 0,45 0,23 0,2 0,12 1 A<-D<-B<-C 0,45 0,25 0,2 0,1 1 A<-D<-C<-B 0,45 0,23 0,22 0,1 1 B<-A<-C<-D 0,28 0,42 0,2 0,1 1 B<-A<-D<-C 0,28 0,42 0,2 0,1 1 B<-C<-A<-D 0,28 0,42 0,21 0,1 1 B<-C<-D<-A 0,12 0,42 0,21 0,26 1 B<-D<-A<-C 0,18 0,42 0,2 0,2 1 B<-D<-C<-A 0,12 0,42 0,27 0,2 1 C<-A<-B<-D 0,38 0,25 0,27 0,1 1 C<-A<-D<-B 0,38 0,23 0,27 0,12 1 C<-B<-A<-D 0,28 0,36 0,27 0,1 1 C<-B<-D<-A 0,12 0,36 0,27 0,26 1 C<-D<-A<-C 0,12 0,23 0,27 0,38 1 C<-D<-C<-A 0,12 0,23 0,27 0,38 1 D<-A<-B<-C 0,17 0,25 0,2 0,38 1 D<-A<-C<-B 0,17 0,23 0,22 0,38 1 D<-B<-A<-C 0,18 0,23 0,2 0,38 1 D<-B<-C<-A 0,12 0,23 0,27 0,38 1 D<-C<-A<-B 0,12 0,23 0,27 0,38 1 D<-C<-B<-A 0,12 0,23 0,27 0,38 1 Shapley Value 0,26 0,29 0,23 0,22 1

12 DEA Mn Game In DEA mn game (N,d) the coalton S s mnmzng the dvson that t can expect from the game d( S) mn s. t. m 1 w w, m 1 w w x ( S) 0 (2)

13 DEA Mn Game Super-addtvty DEA mn game s super-addtve.e., d( S T) d( S) d( T). S, T N wth S T 0 Thus the gans ncrease by enlargng the coalton and for the grand coalton N we get d(n) = 1.

14 DEA Mn Game Mn Games versus Max Games Between the games (N,d) and (N,c) we have the followng relatonshp. d( S) c( N S) 1. S N Proof. By renumberng the ndexes, N = {1,.n}, we assume that S = {1,,h} and N S = {h+1,,n} For these two sets, t holds that

15 DEA Mn Game Mn Games versus Max Games d( S) c( N S) mn mn( mn(1 1 max jh1 max j j1 jh1 h n x x j j1 jh1 n jh1 n x x j j n x j ) max max n ) max x j jh1 n n jh1 x x j jh1 j n x 1. j

16 DEA Mn Game Mn Games versus Max Games Theorem 15.1 (Nakabayash and Tone (2006)) The Shapley values of the DEA max and mn games (N,c) and (N,d) are the same. Proof. For all ( c) k 1 n! 1 n! 1 n! S: ks N S: ks N S: ks N k N we have ( s 1)!( n s)! { c( S) c( S ( s 1)!( n s)! [{1 d( N ( s 1)!( n s)! { d( N { k})} S)} {1 d( N S { k}) d( N S)} S { k})}]

17 DEA Mn Game Mn Games versus Max Games By replacng S = N S + {k} and defnng s as the number of members n the coallton S the prevous formula becomes: 1 n! S ': ks ' N ( s' 1)!( n s')! { d( S') d( S' { k})} ( d), where (d) s the Shapley value defned n (1). k Although the max game represents selfsh and the mn game self-sacrfcng behavor, the Shapley solutons are the same n ths cooperatve game,.e., k d) k ( k ( d)

18 Concluson Shapley values can be used to dvson gans between the players n DEA max and mn games. The Shapley values are the same for both max and mn games.

19 Home assgnment Teammates Crteron A B C X Y Z How much should each player be pad of $100 accordng to Shapley values, f ther performance s evaluated by three crtera X, Y and Z. (10 ponts for formulaton and answer)