Coalitions. would say that the set with zero players also counts as a coalition.
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1 Coalitions Coalition: Any set of players. 1 Weight of a coalition: The total number of votes controlled by the players in the coalition; that is, the sum of the weights of individual players in the coalition. Winning coalition: A coalition whose weight is at least q (enough to pass a motion). Losing coalition: A coalition whose weight is less than q (not enough to pass a motion). Grand coalition: The coalition consisting of all players. (This is always a winning coalition.) 1 Tannenbaum (p.48) requires that a coalition contain at least one player. I (JLM) would say that the set with zero players also counts as a coalition.
2 Coalitions For example, in the WVS [58; 31, 31, 28, 21, 2, 2]: {P 2, P 3, P 5 } and {P 1, P 4, P 5, P 6 } are coalitions (sets of players). {P 2, P 3, P 5 } is a winning coalition because w 2 + w 3 + w 5 = = {P 1, P 4, P 5, P 6 } is a losing coalition because w 1 + w 4 + w 5 + w 6 = = 56 < 58.
3 Counting Coalitions How many different possible coalitions are there?
4 Counting Coalitions How many different possible coalitions are there? N = 1: 2 coalitions: {} and {P 1 }.
5 Counting Coalitions How many different possible coalitions are there? N = 1: 2 coalitions: {} and {P 1 }. N = 2: 4 coalitions: {}, {P 1 }, {P 2 }, {P 1, P 2 }
6 Counting Coalitions How many different possible coalitions are there? N = 1: 2 coalitions: {} and {P 1 }. N = 2: 4 coalitions: {}, {P 1 }, {P 2 }, {P 1, P 2 } N = 3: 8 coalitions: {}, {P 1 }, {P 2 }, {P 1, P 2 }, {P 3 }, {P 1, P 3 }, {P 2, P 3 }, {P 1, P 2, P 3 }
7 Counting Coalitions How many different possible coalitions are there? 2 If there are N players, there are 2 N coalitions. N 2 N N 2 N This is if you count the coalition {}. If you follow Tannenbaum and require that a coalition contain at least one player, then the formula would be 2 N 1 instead of 2 N.
8 Coalition Tables Coalition table: A list of all possible coalitions that indicates which are winning and which are losing. Example 1: [8; 5, 5, 4]. Coalition Total votes Winning/losing {} 0 Losing {P 1 } 5 Losing {P 2 } 5 Losing {P 3 } 4 Losing {P 1, P 2 } 10 Winning {P 1, P 3 } 9 Winning {P 2, P 3 } 9 Winning {P 1, P 2, P 3 } 14 Winning
9 Coalition Tables Coalition table: A list of all possible coalitions that indicates which ones are winning and which ones are losing. Example 2: [10; 5, 5, 4]. Coalition Total votes Winning/losing {} 0 Losing {P 1 } 5 Losing {P 2 } 5 Losing {P 3 } 4 Losing {P 1, P 2 } 10 Winning {P 1, P 3 } 9 Losing {P 2, P 3 } 9 Losing {P 1, P 2, P 3 } 14 Winning
10 Coalition Tables Coalition table: A list of all possible coalitions that indicates which ones are winning and which ones are losing. Example 3: [12; 5, 5, 4]. Coalition Total votes Winning/losing {} 0 Losing {P 1 } 5 Losing {P 2 } 5 Losing {P 3 } 4 Losing {P 1, P 2 } 10 Losing {P 1, P 3 } 9 Losing {P 2, P 3 } 9 Losing {P 1, P 2, P 3 } 14 Winning
11 Equivalence of Weighted Voting Systems We can now tell when two weighted voting systems represent the same allocation of power: when the same coalitions are winning in both systems. For example, let s compare two apparently different WVS s: [10; 5, 5, 4] and [60; 44, 22, 11].
12 Equivalence of Weighted Voting Systems [10; 5, 5, 4] [60; 44, 22, 11] Coalition Total W/L Total W/L {} {P 1 } {P 2 } {P 3 } {P 1, P 2 } {P 1, P 3 } {P 2, P 3 } {P 1, P 2, P 3 }
13 Equivalence of Weighted Voting Systems [10; 5, 5, 4] [60; 44, 22, 11] Coalition Total W/L Total W/L {} 0 Losing {P 1 } 5 Losing {P 2 } 5 Losing {P 3 } 4 Losing {P 1, P 2 } 10 Winning {P 1, P 3 } 9 Losing {P 2, P 3 } 9 Losing {P 1, P 2, P 3 } 14 Winning
14 Equivalence of Weighted Voting Systems [10; 5, 5, 4] [60; 44, 22, 11] Coalition Total W/L Total W/L {} 0 Losing 0 Losing {P 1 } 5 Losing 44 Losing {P 2 } 5 Losing 22 Losing {P 3 } 4 Losing 11 Losing {P 1, P 2 } 10 Winning 66 Winning {P 1, P 3 } 9 Losing 55 Losing {P 2, P 3 } 9 Losing 33 Losing {P 1, P 2, P 3 } 14 Winning 77 Winning
15 Equivalence of Weighted Voting Systems [10; 5, 5, 4] [60; 44, 22, 11] Coalition Total W/L Total W/L {} 0 Losing 0 Losing {P 1 } 5 Losing 44 Losing {P 2 } 5 Losing 22 Losing {P 3 } 4 Losing 11 Losing {P 1, P 2 } 10 Winning 66 Winning {P 1, P 3 } 9 Losing 55 Losing {P 2, P 3 } 9 Losing 33 Losing {P 1, P 2, P 3 } 14 Winning 77 Winning These two WVS s are equivalent!
16 Dummies Question: How can we detect dummies mathematically?
17 Dummies Question: How can we detect dummies mathematically? Answer: Player P is a dummy if there is no winning coalition that P can turn into a losing coalition by deserting. Equivalently, P is a dummy if there is no losing coalition that P can turn into a winning coalition by joining. (A dummy can still be part of a winning coalition! But a dummy s presence or absence makes no difference to whether or not the coalition is winning or losing.)
18 Dummies Example: [10; 5, 5, 4]. The only winning coalitions are {P 1, P 2 } and {P 1, P 2, P 3 } (the grand coalition) The only winning coalition that includes P 3 is the grand coalition and removing P 3 still results in a winning coalition, namely {P 1, P 2 }. Therefore, P 3 is a dummy.
19 Dummies Example: [12; 5, 5, 4]. Now the only winning coalition is {P 1, P 2, P 3 } (the grand coalition). Removing P 3 from the grand coalition results in a losing coalition, namely {P 1, P 2 }. Therefore, P 3 is not a dummy.
20 Dummies Example: [8; 5, 5, 4]. There are four winning coalitions: {P 1, P 2 }, {P 1, P 3 }, {P 2, P 3 }, and the grand coalition {P 1, P 2, P 3 }. Removing P 3 from the winning coalition {P 1, P 3 } results in the losing coalition {P 1 }. Therefore, P 3 is not a dummy.
21 Measuring Power Question: How do we measure power using mathematics? Idea: Measure how often each player in a WVS controls the balance of power. That is, for each player P, what is the likelihood that (a) P is part of the winning coalition and that (b) P s support is necessary for that coalition to be a winning one? Such a player is called critical for that winning coalition.
22 Critical Players Example 1: [11; 7, 5, 4]. The winning coalitions and their weights are: {P 1, P 2 } } {{ } weight 12 {P 1, P 3 } } {{ } weight 11 {P 1, P 2, P 3 } } {{ } weight 16 In {P 1, P 2 }, the participation of P 2 is necessary: if P 2 leaves the coalition, it becomes a losing one. In {P 1, P 2, P 3 }, the participation of P 2 is not necessary: if P 2 leaves, then the coalition is still winning. P 2 is critical for {P 1, P 2 }, but not for {P 1, P 2, P 3 }.
23 Critical Counts and the Banzhaf Power Index Example 1: [11; 7, 5, 4]. Winning Coalition Weight Critical Players {P 1, P 2 } = 12 P 1, P 2 {P 1, P 3 } = 11 P 1, P 3 {P 1, P 2, P 3 } = 16 P 1 Critical count = # of coalitions for which a player is critical. The critical count for P 1 is 3; the critical counts for P 2 and P 3 are both 1.
24 Critical Counts and the Banzhaf Power Index Winning Coalition Weight Critical Players {P 1, P 2 } = 12 P 1, P 2 {P 1, P 3 } = 11 P 1, P 3 {P 1, P 2, P 3 } = 16 P 1 Player Critical Count P 1 3 P 2 1 P 3 1 Total 5
25 Critical Counts and the Banzhaf Power Index Winning Coalition Weight Critical Players {P 1, P 2 } = 12 P 1, P 2 {P 1, P 3 } = 11 P 1, P 3 {P 1, P 2, P 3 } = 16 P 1 Player Critical Count Banzhaf power index P 1 3 3/5 P 2 1 1/5 P 3 1 1/5 Total 5
26 Measuring Power, Again Idea: We can measure a player s power by determining how often that player is critical for some coalition. I.e., how often does each player s name appear in the last column of the table? Lionel Penrose (1946): The Elementary Statistics of Majority Voting John Banzhaf (1965): Weighted Voting Doesn t Work James Coleman (1971): Control of Collectives and the Power of a Collectivity to Act
27 Banzhaf s Article Banzhaf s article Weighted Voting Doesn t Work (Rutgers Law Review, vol. 19, 1965) In almost all cases weighted voting does not do the one thing which both its supporters and opponents assume that it does... voting power is not proportional to the number of votes a legislator may cast. The purpose of this paper is neither to attack nor defend weighted voting per se. As with any objective mathematical analysis, its intent is only to explain the effects which necessarily follow once the mathematical model and the rules of its operation are established.
28 Banzhaf s Article Banzhaf analyzed the Nassau County Board of Supervisors the weighted voting system [58; 31, 31, 28, 21, 2, 2]. Banzhaf showed that, as we have seen, players P 4, P 5 and P 6 are dummies. The article Weighted Voting Doesn t Work became the basis for a series of lawsuits (not a frequent achievement for an article on mathematics!)
29 Calculating Banzhaf Power Indices For any weighted voting system with N players: 1. Find all the winning coalitions. 2. For each one, determine the critical players. 3. Count the number of times that each player P i is critical. Call this number B i. 4. The Banzhaf power index of P i is β i = B i B 1 + B B N The Banzhaf power distribution of the weighted voting system is the complete list (β 1, β 2,..., β N ).
30 Banzhaf Power: Example 1 Example 1: The weighted voting system [8; 5, 5, 4]. Winning Coalition Weight Critical Players {P 1, P 2 } = 10 P 1, P 2 {P 1, P 3 } = 9 P 1, P 3 {P 2, P 3 } = 9 P 2, P 3 {P 1, P 2, P 3 } = 14 None
31 Banzhaf Power: Example 1 Each of the three players has the same critical count: B 1 = B 2 = B 3 = 2. Thus every player has the same Banzhaf power index. The sum of all critical counts is T = 6, so β 1 = β 2 = β 3 = 2 6 = 1 3. Therefore, the Banzhaf power distribution is ( ) 1 3, 1 3, 1. 3 This confirms that all players have equal power!
32 Banzhaf Power: Example 2 Example 2: The weighted voting system [10; 5, 5, 4]. Winning Coalition Weight Critical Players {P 1, P 2 } 10 {P 1, P 2, P 3 } 14
33 Banzhaf Power: Example 2 Example 2: The weighted voting system [10; 5, 5, 4]. Winning Coalition Weight Critical Players {P 1, P 2 } 10 P 1, P 2 {P 1, P 2, P 3 } 14 P 1, P 2
34 Banzhaf Power: Example 2 Example 2: The weighted voting system [10; 5, 5, 4]. Winning Coalition Weight Critical Players {P 1, P 2 } 10 P 1, P 2 {P 1, P 2, P 3 } 14 P 1, P 2 Player B i Banzhaf power index P 1 2 2/4 = 1/2 P 2 2 2/4 = 1/2 P Total T = 4
35 Banzhaf Power: Example 2 Example 2: The weighted voting system [10; 5, 5, 4]. Winning Coalition Weight Critical Players {P 1, P 2 } 10 P 1, P 2 {P 1, P 2, P 3 } 14 P 1, P 2 Player B i Banzhaf power index P 1 2 2/4 = 1/2 P 2 2 2/4 = 1/2 P Dummy! Total T = 4
36 Banzhaf Power: Example 2 Conclusion: In the WVS [10; 5, 5, 4], the Banzhaf power distribution is (β 1, β 2, β 3 ) = (1/2, 1/2, 0). Therefore, player P 3 is a dummy. A player is a dummy if and only if his/her Banzhaf power index is 0.
37 Banzhaf Power: Example 3 Example 3: The weighted voting system [12; 5, 5, 4] Winning Coalition Weight Critical Players {P 1, P 2, P 3 } = 14 P 1, P 2, P 3 All players have the same critical count: B 1 = B 2 = B 3 = 1. So all players have the same Banzhaf power index: β 1 = β 2 = β 3 = 1/3.
38 Banzhaf Power: Examples 1 and 3 In both [8; 5, 5, 4] and [12; 5, 5, 4], the Banzhaf power distribution is (β 1, β 2, β 3 ) = (1/3, 1/3, 1/3). Therefore, in each system, all players have the same power. But that does not necessarily mean that the voting systems are equivalent. (In the first case, a motion requires two players to agree; in the second case, unanimous agreement is required.)
39 Banzhaf Power: Example 4 Example 4: Weighted voting system [22; 26, 5, 5, 4].
40 Banzhaf Power: Example 4 Example 4: Weighted voting system [22; 26, 5, 5, 4]. There are lots of winning coalitions namely, those (and only those) coalitions that contain player P 1 ( Mom ).
41 Banzhaf Power: Example 4 Example 4: Weighted voting system [22; 26, 5, 5, 4]. There are lots of winning coalitions namely, those (and only those) coalitions that contain player P 1 ( Mom ). In each winning coalition, Mom is a critical player, but no one else is.
42 Banzhaf Power: Example 4 Example 4: Weighted voting system [22; 26, 5, 5, 4]. There are lots of winning coalitions namely, those (and only those) coalitions that contain player P 1 ( Mom ). In each winning coalition, Mom is a critical player, but no one else is. Therefore, Mom s Banzhaf power index is 1 (= 100%) and everyone else s Banzhaf power index is 0.
43 Banzhaf Power: Example 4 Example 4: Weighted voting system [22; 26, 5, 5, 4]. There are lots of winning coalitions namely, those (and only those) coalitions that contain player P 1 ( Mom ). In each winning coalition, Mom is a critical player, but no one else is. Therefore, Mom s Banzhaf power index is 1 (= 100%) and everyone else s Banzhaf power index is 0. A player is a dictator if and only if his/her Banzhaf power index is 100%.
44 Banzhaf Power: Summary The Banzhaf power index (BPI) of a player measures how often that player is critical for some winning coalition. β i = B i T = critical count for player P i sum of critical counts for all players The BPI of a dummy is always 0 (= 0%). The BPI of a dictator is always 1 (= 100%). If two players have the same Banzhaf power index, then they really do have the same amount of power.
45 Banzhaf Power: Example 5 Example 5: In the weighted voting system [10; 8, 4, 2, 1], what is the Banzhaf power distribution?
46 Banzhaf Power: Example 5 Example 5: In the weighted voting system [10; 8, 4, 2, 1], what is the Banzhaf power distribution? Winning Coalition Critical Players {P 1, P 2 } P 1, P 2 {P 1, P 3 } P 1, P 3 {P 1, P 2, P 3 } P 1, P 2 {P 1, P 2, P 4 } P 1, P 2 {P 1, P 3, P 4 } P 1 {P 1, P 2, P 3, P 4 } P 1
47 Banzhaf Power: Example 5 Example 5: In the weighted voting system [10; 8, 4, 2, 1], what is the Banzhaf power distribution? Winning Coalition Critical Players {P 1, P 2 } P 1, P 2 {P 1, P 3 } P 1, P 3 {P 1, P 2, P 3 } P 1, P 2 {P 1, P 2, P 4 } P 1, P 2 {P 1, P 3, P 4 } P 1 {P 1, P 2, P 3, P 4 } P 1 B 1 = 6 B 2 = B 3 = 2 B 4 = 0 T = 10
48 Banzhaf Power: Example 5 For the weighted voting system [10; 8, 4, 2, 1], the Banzhaf power distribution is (6/10, 2/10, 2/10, 0) = (3/5, 1/5, 1/5, 0). P 1 has veto power (he is critical for every winning coalition). P 2 has the same amount of power as P 3. P 4 is a dummy (her Banzhaf power index is 0).
49 Example: The UN Security Council The UN Security Council consists of 5 permanent members: China, France, Russia, United Kingdom, USA 10 rotating members (currently: Bosnia/Herzegovina, Brazil, Gabon, Lebanon, Nigeria, Colombia, Germany, India, Portugal, South Africa) Passing a motion requires at least nine votes, including all five of the permanent members. What is the Banzhaf power distribution? (There must be a better way of doing this than having to write out all 2 15 = 32, 768 coalitions!)
50 Example: The UN Security Council Observations: We don t need to assign numerical weights all we need to do is figure out which coalitions are winning and which ones are losing. Each winning coalition must contain all five permanent members, along with at least four rotating members. Every permanent member has the same power as every other permanent member. Every rotating member has the same power as every other rotating member.
51 Example: The UN Security Council Permanent Members Every permanent member (say, Russia) is critical for every winning coalition. Therefore, the critical count for Russia is just the total number of winning coalitions, which is the number of ways to choose at least four of the ten rotating members. Using permutations and combinations (coming soon!), this number can be calculated as 848.
52 Example: The UN Security Council Rotating Members Each rotating member (say, Portugal) is critical for a winning coalition only if there are exactly three other rotating members. Therefore, the critical count for Portugal is the number of ways to choose exactly three of the nine other rotating members. Using permutations and combinations, this number can be calculated as 84.
53 Example: The UN Security Council 5 permanent members, each with critical count rotating members, each with critical count 84
54 Example: The UN Security Council 5 permanent members, each with critical count rotating members, each with critical count 84 Banzhaf power index of each permanent member: 848 (5 848) + (10 84) = = 16.69%
55 Example: The UN Security Council 5 permanent members, each with critical count rotating members, each with critical count 84 Banzhaf power index of each permanent member: 848 (5 848) + (10 84) = = 16.69% Banzhaf power index of each rotating member: 84 (5 848) + (10 84) = = 1.65%
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