A DEA Game I Chapters

Transkriptio

1 A DEA Game I Chapters Saara Tuurala

2 Agenda Introducton General Formulaton Assumpton on the Game and Far Dvson Coalton and Characterstc Functon Summary Home Assgnment

3 Introducton /5 A DEA game deals wth consensus-makng among ndvduals or organzatons under multple crtera envronment An example: How should a prze pool of $0,000 be shared among four teammates? Teammate A B C D Sum Bran Bravery Labor Total Score

4 Introducton 2/5 A smple soluton s to dvde the reward n proportons to each member s total score: Teammate Reward A 8/70 x $0,000 = $2,77 B 2/70 x $0,000 = $3,000 C 6/70 x $0,000 = $2,286 D 8/5 x $0,000 = $2,43

5 Introducton 3/5 The mportance of the crtera can be taken nto account wth fxed weghts: Teammate Weghts A B C D Sum Bran vs. Bravery vs. Labor Reward $2,57 $3,000 $2,286 $2,43 $0,000 : : Reward $3,34 $3,060 $2,64 $,642 $0,000 3 : 2 : Reward $2,055 $2,945 $2,397 $2,603 $0,000 : 2 : 3 Ths knd of fxed weght scheme can pose a dffcult problem on the way we choose the weghts

6 Introducton 4/5 In order to resolve ths problem, we apply a varable weght scheme: ω Suppose that each member can choose the weghts to get the bggest reward. Let ω, ω 2 and ω 3 be wegths to the three crtera Then A wll solve a problem: = ω + ω + ω R A max9 ω st. 20ω + 24ω + 26ω = 0, = 500, ω = ω = 0, R A = 9ω = $4,500

7 Introducton 5/5 Wth optmal weghts for each teammate we get: A B C D Sum ω * ω 2 * ω 3 * Reward $4,500 $4,67 $2,692 $3,846 $5,206 (> $0,000) Notce that the total reward exceeds the prze pool Next we generalze the problem

8 General Formulaton /4 Suppose n players and m crtera wth scores x j and weghts w k (that are most preferable for player k) Let s defne player k s relatve score and m maxmze t: k max ω k st. ω k = ω x m n k ω = j= 0 ( ) k x j

9 General Formulaton 2/4 Before contnung, let s normalze the scores and the total prze and use Charnes-Cooper transformaton cheme m k to express the nonlnear c( k) = max x k ω k ω problem as lnear: = m k k st. ω =, ω ( ) The c(k) ndcates the hghest relatve score for player k wth hs optmal weghts =

10 General Formulaton 3/4 As we normalze the prevous example: The lnear problem for A s: ( A) Teammate A B C D Sum Bran Bravery Labor Total Score c st. ω + ω + ω =, ω ω = max 0.45ω ω ω 2 =, ω 2 3 = ω 3 = 0, c 2 0( =,2,3) ( A) =

11 General Formulaton 4/4 The total gan due to ths prncple amounts to =.5205 whch exceeds the total purse of by 52.05% (equals to the non-normalzed case) The man objectve s now to fnd a far dvson by cooperatve game theory n the way that the total gan s equal to the total purse

12 Assumpton on the Game and Far Dvson Let s assume the followng agreements (each player stll stcks to hs most preferable weghts) (A) All players agree not to break off the game (A2) All players are wllng to negotate wth each other to attan a reasonable and far dvson

13 Coalton and Characterstc Functon /3 Any subset S of the palyer set N = (,, n) s called a coalton (here {A,B}, {B,C,D} etc.) Let s defne the score for coalton S: ( S ) x, ( =, m) x = j..., j S

14 Coalton and Characterstc Functon 2/3 The maxmum outcome s obtaned by solvng the followng LP: c ( S ) = max ω x ( S ) st. m = ω m = ω =, ω 0 ( ) The c(s) defnes a characterstc functon of the coalton S

15 Coalton and Characterstc Functon 3/3 Let s enumerate all coaltons (except for the sngle player case) and ther characterstc functons of the example: Coalton {A,B} {A,C} {A,D} {B,C} {B,D} {C,D} Bran Bravery Labor Coalton {A,B,C} {A,B,D} {A,C D} {B,C,D} {A,B,C,D} Bran Bravery Labor

16 Summary A DEA game deals wth consensus-makng among ndvduals or organzatons under multple crtera envronment We learned how to formulate a lnear problem from a non-lnear DEA game We learned the concepts of: coalton characterstc functon

17 Home Assgnment The team won 2000 n a competton Teammates Annu Jore Sakke Ptu Sum Swmmng Runnng Cyclng Rock Clmbng Total Score Normalze the scores and the total prze. (2 p.) How much s the total gan? (2 p.) Evaluate all coaltons (3 p.) and ther characterstc functons (3 p.)