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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

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Section 2.4 The Shapley-Shubik Power Index

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  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

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