Chapter 13 – Weighted Voting Lecture Part 2

Chapter 13 – Lecture Part 2 The Banzhaf Power Index –Counting the number of subsets of a set –Listing winning coalitions and blocking coalitions –Listing a coalition’s critical voters

Counting Subsets – Finding Banzhaf Power Index A set of n elements will have 2 n subsets and will have 2 n – 1 proper subsets. Thus, for a weighted voting system with n voters, there are 2 n possible coalitions. We count both the set of all voters and the empty set as coalitions. The set of all voters is sometimes called the grand coalition. The subset consisting of no voters is called the empty coalition. To determine the Banzhaf power index for a given system we list all of a voting system’s coalitions (or sometimes to save time we list only the winning coalitions.)

Banzhaf Power Index – Finding Winning Coalitions We determine the weight of each of the winning coalitions and the number of extra votes that coalition has above the quota. That is, we determine the number of votes the coalition could lose and still be a winning coalition. A particular voter is critical in a given coalition if it’s weight is greater than the extra votes of that coalition. A certain voter would not be critical if its weight was equal or less than the extra votes of the coalition.

Banzhaf Power Index – An Example Consider the voting system [ 51: 26, 26, 26, 22 ]. There are 2 4 = 16 different coalitions including both the grand and empty coalitions. We are interested in those coalitions that are winning (and the ones that are blocking). We will create a list of all coalitions (later we might try to be quicker by listing only winning coalitions). We list the coalition of all voters, coalitions of 3 voters, coalitions of 2 voters, coalitions of only 1 voter, and the empty coalition.

Banzhaf Power Index – An Example coalitioncoalition weight (w)extra votes ( w – q )winning ?critical voters A, B, C, D10049yesnone A, B, C7827yesnone A, B, D7423yesA, B A, C, D7423yesA, C B, C, D7423yesB, C A, B521yesA, B A, C521yesA, C A, D48no B, C521yesB, C B, D48no C, D48no A26no B26no C26no D22no empty0no A = 26, B = 26 C = 26, D = 22 q = 51

Banzhaf Power Index – An Example coalitionweight (w)extra votes: blocking (w – bq)blocking ?critical voters A, B, C, D10050yesnone A, B, C7828yesnone A, B, D7424yesA, B A, C, D7424yesA, C B, C, D7424yesB, C A, B522yesA, B A, C522yesA, C A, D48no B, C522yesB, C B, D48no C, D48no A26no B26no C26no D22no empty0no A = 26, B = 26 C = 26, D = 22 q = 51 bq = 50

Banzhaf Power Index – An Example coalitioncoalition weight (w)extra votes ( w – q )winning ?critical voters A, B, C, D10049yesnone A, B, C7827yesnone A, B, D7423yesA, B A, C, D7423yesA, C B, C, D7423yesB, C A, B521yesA, B A, C521yesA, C B, C521yesB, C A = 26, B = 26 C = 26, D = 22 q = 51 Here we list only the winning coalitions. Note that A is critical in 4 of the winning coalitions, B is also critical in 4 winning and likewise, C is critical in exactly 4 winning coalitions. Notice that D is not critical in any winning coalitions.

Banzhaf Power Index – An Example coalitionweight (w)extra votes: blocking (w – bq)blocking ?critical voters A, B, C, D10050yesnone A, B, C7828yesnone A, B, D7424yesA, B A, C, D7424yesA, C B, C, D7424yesB, C A, B522yesA, B A, C522yesA, C B, C522yesB, C A = 26, B = 26 C = 26, D = 22 q = 51 bq = 50 Here we list only the blocking coalitions. Notice that A, B and C are each critical in exactly 4 blocking coalitions. Also, notice that D is not critical to any blocking coalition.

The Banzhaf Power Index – An Example A is critical in 4 winning and blocking coalitions. Thus, we say the Banzhaf Power of voter A is 8. This is the same for voters B and C. We found that D was critical in 0 winning and 0 blocking coalitions. D’s Banzhaf Power is 0. The Banzhaf Power Index for the system [ 51: 26, 26, 26, 22 ] is given by the numbers ( 8, 8, 8, 0 ). Sometimes the Banzhaf Power Index is written so that each term represents the fraction of times each voter is critical out of the total number of times all voters are critical. In this case, the total number of times all voters are critical is = 24. Thus, we could write the Banzhaf index as ( 8/24, 8/24, 8/24, 0/24 ) or as, after reducing, ( 1/3, 1/3, 1/3, 0 ). In a sense, voters A, B and C each share 1/3 of the power while D really has no power in this system.

Banzhaf Power Index - Shortcuts The number of blocking coalitions to which a voter is critical equals the number of winning coalitions to which that voter is critical. –If a voting member is critical to a winning coalition, that member is therefore also critical to the opposing coalition – that is, to the coalition of voters who’d hope to block that measure. –And likewise, for any blocking coalition to which a voter is critical, that voter is also critical to the opposing coalition – that is, to the coalition of voters who’d hope to pass that measure. –For each coalition that a voter is critical there is exactly one opposing coalition to which that voter is also critical. Thus, once a voter is critical to a coalition, that voter is critical to an equal number of winning and blocking coalitions.

Banzhaf Power Index - Shortcuts The number of blocking coalitions to which a voter is critical equals the number of winning coalitions to which that voter is critical. To determine the Banzhaf power index for a weighted voting system, we list winning coalitions, count the number of times each voter is critical to each winning coalition and double that number. We can begin by considering all coalitions of each possible size and determine which of those are winning.

Banzhaf Power Index – An example Let’s return to the voting system given by [ 16 : 9, 9, 7, 3, 1, 1 ]. There are 2 6 = 64 coalitions, however not all of them are winning coalitions. There are = 1 coalition of 6 = 6 coalitions of 5 = 15 coalitions of 2 = 20 coalitions of 3 = 15 coalitions of 4 = 6 coalitions of 1

Banzhaf Power Index – An example Coalitions of 1: none are winning Coalitions of 2: s1, s2182s1, s2 s1, m160s1, m s2, m160s2, m Coalitions of 3: s1, s2, m259none s1, s2, I215s1, s2 s1, s2, L193s1, s2 s1, s2, e193s1, s2 s1, m, I193s1, m s1, m, L171s1, m s1, m, e171s1, m s2, m, I193s2, m s2, m, L171s2, m s2, m, e171s2, m Note: tables list winning coalitions, weight, extra votes, and critical voters, in that order, from left to right. Let’s refer to the Smith’s as s1 and s2 and the others by the first letter of their last name.

Banzhaf Power Index – An example Coalitions of 4: s2, m, L, e182s2, m s2, m, I, e204s2, m s2, m, I, L204s2, m s1, m, L, e182s1, m s1, m, I, e204s1, m s1, m, I, L204s1, m s1, s2, L, e204s1, s2 s1, s2, I, e226s1, s2 s1, s2, I, L226s1, s2 s1, s2, m, e2610none s1,s2, m, L2610none s1, s2, m, I2812none

Banzhaf Power Index – An example Coalitions of 5: s2, m, I, L, e215s2, m s1, m, I, L, e215s1, m s1,s2, I, L, e237s1, s2 s1,s2, m, L, e2711none s1,s2, m, I, e2913none s1,s2, m, I, L2913none Coalitions of 6: There is only one (all voters) and no voter is critical to that coalition.

Banzhaf Power Index – An example s2, m, L, e182s2, m s2, m, I, e204s2, m s2, m, I, L204s2, m s1, m, L, e182s1, m s1, m, I, e204s1, m s1, m, I, L204s1, m s1, s2, L, e204s1, s2 s1, s2, I, e226s1, s2 s1, s2, I, L226s1, s2 s1, s2, m, e2610none s1,s2, m, L2610none s1, s2, m, I2812none s1, s2, m259none s1, s2, I215s1, s2 s1, s2, L193s1, s2 s1, s2, e193s1, s2 s1, m, I193s1, m s1, m, L171s1, m s1, m, e171s1, m s2, m, I193s2, m s2, m, L171s2, m s2, m, e171s2, m s1, s2182s1, s2 s1, m160s1, m s2, m160s2, m s2, m, I, L, e215s2, m s1, m, I, L, e215s1, m s1,s2, I, L, e237s1, s2 s1,s2, m, L, e2711none s1,s2, m, I, e2913none s1,s2, m, I, L2913none For each voter, we count the number of times that voter is critical to some coalition ….

Banzhaf Power Index – An example Results: S1 is critical 16 times, S2 is critical 16 times, and M is critical 16 times. Neither I, L or E are ever critical. Conclusion: For the voting system given by [ 16 : 9, 9, 7, 3, 1, 1 ] we have determined that the Banzhaf power index for this system is given by ( 32, 32, 32, 0, 0, 0 ). Significance: This implies that Dr.’s Ruth and Ralph Smith and Dr. Mansfield share power equally in this system while neither of Dr.’s Ide, Lambert or Edwards have any power. Even though the nominal power of Dr. Mansfield is less than the Smiths, he has equal Banzhaf power in this system. Dr. Ide, Dr. Lambert and Dr. Edwards are dummy voters ( their vote is of no significance.)