Section 2.4Section 2.4The Shapley-The Shapley-Shubik Power Shubik Power

IndexIndex

►Sequential CoalitionSequential Coalition

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

~ the ~ the orderorder in which the in which the players joined the coalitionplayers joined the coalition

►BanzhafBanzhaf Shapley-ShubikShapley-Shubik

{P{P1,1,PP2,2,PP33}} < P< P1,1,PP2,2,PP33>>< P< P1,1,PP3,3,PP22>>< P< P2,2,PP1,1,PP33>>< P< P2,2,PP3,3,PP11>>< P< P3,3,PP1,1,PP22>>< P< P3,3,PP2,2,PP11>>

►< > notation indicates a < > notation indicates a sequential coalition ~ order sequential coalition ~ order mattersmatters

►In In eacheach sequential coalition, there sequential coalition, there is is oneone player that “tips the scales” player that “tips the scales” – the moment that player joins the – the moment that player joins the coalition, the coalition changes coalition, the coalition changes from a losing to a winning coalition.from a losing to a winning coalition.

►The players get added from left to The players get added from left to right, one at a time until tally is right, one at a time until tally is bigger than quota.bigger than quota.

- We call such a player a We call such a player a pivotal pivotal playerplayer for the sequential coalition. for the sequential coalition.

- There is only one pivotal player There is only one pivotal player per coalition.per coalition.

- The number of times a player is The number of times a player is pivotal is known as “S”pivotal is known as “S”

General Method to find General Method to find Shapley—Shubik Power IndexShapley—Shubik Power Index

►Step 1 – Make a list of all sequential Step 1 – Make a list of all sequential coalitions (call it T)coalitions (call it T)

►Step 2 – Determine the pivotal player Step 2 – Determine the pivotal player in each sequential coalitionin each sequential coalition

►Step 3 – Count the total times P is Step 3 – Count the total times P is pivotal (call it S)pivotal (call it S)

►Step 4 – Find the Shapley Shubik Power Step 4 – Find the Shapley Shubik Power Index by =s/t Index by =s/t

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

►How many sequential How many sequential coalitions????coalitions????

►The number of sequential The number of sequential coalitions with N players is …coalitions with N players is …

►N! = N (N-1) (N-2) N! = N (N-1) (N-2) ··· 1··· 1

►Ex 2-14Ex 2-14►Cones and flavors of ice creamCones and flavors of ice cream►2 choices of cones and 3 flavors of ice 2 choices of cones and 3 flavors of ice

cream (multiply 2*3, the number of cream (multiply 2*3, the number of choices or options)choices or options)

Ex. 2-15Ex. 2-15

► Ice cream has cones, flavors, and Ice cream has cones, flavors, and toppingstoppings

►5 cones, 31 flavors, 8 toppings5 cones, 31 flavors, 8 toppings►5*31*8=1240 ways 5*31*8=1240 ways

Ex. 2-16Ex. 2-16

►How many sequential coalitions do we How many sequential coalitions do we have?have?

►5 choices….120 sequential coalitions5 choices….120 sequential coalitions

5! In calculator OR5(4)(3)(2)(1)= 1205! In calculator OR5(4)(3)(2)(1)= 120

Ex. 2-17Ex. 2-17► [4: 3, 2, 1][4: 3, 2, 1]► Step 1, write all the sequential coalitions Step 1, write all the sequential coalitions

(how do you know how many? 3!=6)(how do you know how many? 3!=6)

►<p1,<p1,p2p2,p3>,p3>►<p1,<p1,p3p3,p2>,p2>►<p2,<p2,p1p1,p3>,p3>►<p2,p3,<p2,p3,p1p1>>►<p3,<p3,p1p1,p2>,p2>►<p3,p2,<p3,p2,p1p1>>

►Step2 underline pivotal player you need to add Step2 underline pivotal player you need to add left to right until you reach quota. When you left to right until you reach quota. When you reach quota player is pivotalreach quota player is pivotal

►Step 3 add up the pivotal playersStep 3 add up the pivotal players►p1=SS1=4p1=SS1=4►p2=SS2=1p2=SS2=1►p3=SS3=1p3=SS3=1

T=6T=6 Step 4 divide each ss/t to get shapley shubik Step 4 divide each ss/t to get shapley shubik

power indexpower index

You try: NBA DraftYou try: NBA Draft

►[6: 4,3,2,1][6: 4,3,2,1]►Follow the steps, 1 through 4 to get Follow the steps, 1 through 4 to get

the Shapley Shubik Power Indexthe Shapley Shubik Power Index