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

► BanzhafShapley-Shubik {P 1, P 2, P 3 } {P 1, P 2, P 3 }

► notation indicates a sequential coalition ~ order matters

► 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.

- 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”

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

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

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

► 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)

Ex ► Ice cream has cones, flavors, and toppings ► 5 cones, 31 flavors, 8 toppings ► 5*31*8=1240 ways

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

Ex ► [4: 3, 2, 1] ► Step 1, write all the sequential coalitions (how do you know how many? 3!=6) ► ►

► 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 T=6  Step 4 divide each ss/t to get shapley shubik power index

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