1 / 16

Section 2.4 The Shapley-Shubik Power Index

Section 2.4 The Shapley-Shubik Power Index. Sequential Coalition Every coalition starts with a first player, who may then be joined by a second player, then a third, and so on. ~ the order in which the players joined the coalition. Banzhaf Shapley-Shubik

veda-cote
Télécharger la présentation

Section 2.4 The Shapley-Shubik Power Index

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 2.4The Shapley-Shubik Power Index

  2. Sequential Coalition • Every coalition starts with a first player, who may then be joined by a second player, then a third, and so on.

  3. ~ the order in which the players joined the coalition

  4. Banzhaf Shapley-Shubik {P1,P2,P3}< P1,P2,P3> < P1,P3,P2> < P2,P1,P3> < P2,P3,P1> < P3,P1,P2> < P3,P2,P1>

  5. < > notation indicates a sequential coalition ~ order matters

  6. In each sequential coalition, there is one player that “tips the scales” – the moment that player joins the coalition, the coalition changes from a losing to a winning coalition. • The players get added from left to right, one at a time until tally is bigger than quota.

  7. We call such a player a pivotal player for the sequential coalition. • There is only one pivotal player per coalition. • The number of times a player is pivotal is known as “S”

  8. General Method to find Shapley—Shubik Power Index • Step 1 – Make a list of all sequential coalitions (call it T) • Step 2 – Determine the pivotal player in each sequential coalition • Step 3 – Count the total times P is pivotal (call it S) • Step 4 – Find the Shapley Shubik Power Index by =s/t

  9. The Shapley-Shubik Power Index of Player P is S/T.

  10. How many sequential coalitions???? • The number of sequential coalitions with N players is … • N! = N (N-1) (N-2) ··· 1

  11. Ex 2-14 Cones and flavors of ice cream 2 choices of cones and 3 flavors of ice cream (multiply 2*3, the number of choices or options)

  12. Ex. 2-15 Ice cream has cones, flavors, and toppings 5 cones, 31 flavors, 8 toppings 5*31*8=1240 ways

  13. Ex. 2-16 How many sequential coalitions do we have? 5 choices….120 sequential coalitions 5! In calculator OR5(4)(3)(2)(1)= 120

  14. Ex. 2-17 [4: 3, 2, 1] Step 1, write all the sequential coalitions (how do you know how many? 3!=6) <p1,p2,p3> <p1,p3,p2> <p2,p1,p3> <p2,p3,p1> <p3,p1,p2> <p3,p2,p1>

  15. Step2 underline pivotal player you need to add left to right until you reach quota. When you reach quota player is pivotal • Step 3 add up the pivotal players • p1=SS1=4 • p2=SS2=1 • p3=SS3=1 T=6 • Step 4 divide each ss/t to get shapleyshubik power index

  16. You try: NBA Draft [6: 4,3,2,1] Follow the steps, 1 through 4 to get the Shapley Shubik Power Index

More Related