Counting Maximal Chains in Weighted Voting Posets

Weighted voting is built around the idea that voters have differing amounts of influence in elections, with familiar examples ranging from company shareholder meetings to the United States Electoral College. We examine the idea that each voter has a uniquely determined weight, paying particular attention to how voters leverage this weight to get their way on a specific yes/no motion (for example, by forming coalitions). After some more background on weighted voting, we describe a natural partial order relation between these coalitions of voters. This ordering can be modeled by a partially ordered set (poset), which we call a coalitions poset. Using this poset, we derive another important poset via a natural ordering on collections of coalitions. Our results begin by detailing a method for counting the number of maximal chains in the derived poset. After employing this method to find the number of maximal chains in the derived poset with 5 voters, we extend our method for use in the coalitions poset. Finally, we conjecture a formula for the number of maximal chains in the coalitions poset with n voters. Acknowledgements: The author would like to thank his thesis advisors, Dr. Jason Parsley and Dr. Sarah Mason, for all of their help on this paper. Page 158 RHIT Undergrad. Math. J., Vol. 14, No. 1 1 An Introduction to Weighted Voting Systems The focus of this paper is the study of weighted voting systems. A weighted voting system is defined as a collection of n voters, v1, v2, v3, ..., vn, who vote on a yes or no motion. Each voter vi has some given weight wi. In order for a motion to pass, the sum of the weights of all voters voting for the motion must meet or exceed some fixed quota q. Otherwise, the motion is said to fail. We define a coalition as a subset of our n voters who all vote the same way on a motion. A coalition can contain any number of voters, from no voters to all voters in the system. The weight of a coalition is the sum of the weights of each of its voters. The interested reader may wish to peruse [1] and [4] as a supplement to our introduction of weighted voting. Now that we have been acquainted with the basic concepts of weighted voting systems, let us introduce more mathematical definitions: Definition 1.1. A coalition A is a collection of j voters from a system with n voters such that every member of A votes the same way on a motion. The coalition A has the following properties: • 0 ≤ j ≤ n • wA = w1 + w2 + w3 + · · ·+ wj, where wA is the weight of coalition A The grand coalition is precisely the coalition containing all voters in the system. In other words, the grand coalition occurs when j = n. Similarly, the empty coalition contains no voters and occurs when j = 0. Definition 1.2. A weighted voting system (q : wn, wn−1, ..., w2, w1) is defined by the following characteristics: • There are n voters, to whom we give the names v1, v2, v3, ..., vn. • There is a weight wi ≥ 0 corresponding to each voter vi. • There is a quota q such that a motion will pass if, given the coalition A containing all voters voting in favor of the motion, we have wA ≥ q. In most literature, voters are referred to by their subscripts. In other words, we would simply call voter vn by the abbreviated name, voter n. As a convention, we enumerate our n voters by increasing weight. That is, voter 1 has the least weight, voter 2 has the second least weight,..., and voter n has the greatest weight. We do not require that the weights be strictly increasing. So, we have: 0 ≤ w1 ≤ w2 ≤ w3 ≤ · · · ≤ wn Contrastingly, we list the voters comprising a coalition by decreasing weight. Consequently, the coalition containing voters 1, 3, and 4 would be written {4, 3, 1} = {431} because RHIT Undergrad. Math. J., Vol. 14, No. 1 Page 159 voter 4 has the greatest weight, voter 3 has the second greatest weight, and voter 1 has the least weight. Similarly, the grand coalition containing all n voters would be written {n, n − 1, n − 2, ..., 2, 1}. We should note that this ordering is contrary to the majority of voting literature, which tends to list the weights of the voters in a coalition in increasing order. The convention of ordering weights in this new way came from a paper by Mason and Parsley [3]. The advantage to this ordering is that it makes it much easier to determine the rank of a coalition in a certain partially ordered set M(n), which is central to the study of weighted voting theory and to this paper. Essentially, weighted voting is predicated on the idea that not all voters are equal. Familiar real world applications of weighted voting range from shareholder meetings to the Electoral College of the United States. In shareholder meetings, some shareholders will have more weight than others by virtue of owning more stock in the company. For example, a shareholder’s weight might be equal to the number of shares he owns. Similarly, some states have more weight than others in the Electoral College because they have a larger population. Let us examine an example of weighted voting in a real-world scenario. Example 1.1. Alice, Bob, and Charlie are the sole shareholders of a small construction company. The distribution of shares is illustrated in the following table: Shareholder Number of Shares Alice 49 Bob 46 Charlie 5 The company has recently fallen on hard times because of the decline in the housing market. Alice proposes expanding the business to other markets, reasoning that this will increase their revenue. Bob opposes the idea, explaining that this proposition will also lead to higher expenditures. Given the share distribution shown above, how can Alice get her motion to pass? We will set the quota q at 51 shares. Because we can pick a quota (namely, q = 51) and assign weights to each voter i (simply take wi = the number of shares held by i), we have a weighted voting system. Using Definition 1.2, we notate this weighted voting system as (51 : 49, 46, 5). If Alice is the only one to vote for the motion, she will form a coalition with weight w = 49. But 49 < q, so Alice’s motion would fail. On the other hand, if Charlie votes with Alice on the motion, they will form a coalition with weight w = 49 + 5 = 54. Since 54 > q, Alice’s motion would pass. Finally, if Alice is able to convince Bob to support her motion, they will form a coalition with weight w = 49 + 46 = 95. Since 95 > q, Alice’s motion would pass. So, we have found that Alice cannot singlehandedly force her motion to pass; she needs the help of either Bob or Charlie. Of course, Alice could also win by teaming up with both Bob and Charlie. However, this observation is trivial. Once Alice forms a coalition with one of the other two shareholders, they already have enough weight between themselves to force her motion to pass. Adding the third shareholder to the coalition would have no effect on the outcome. This idea is important and will come up again when we discuss the notion of minimal winning coalitions. Page 160 RHIT Undergrad. Math. J., Vol. 14, No. 1 Example 1.1 is meant to be a representative example of weighted voting systems. So, we should note that while it was convenient for the weights of our voters to sum to 100, it was not necessary. Furthermore, we did not have to choose q = 51 as our quota; it could have just as easily been taken to be q = 30 or even q = 100. However, to avoid the situation in which two opposing coalitions are both winning, we will disallow quotas set at less than or equal 50 percent of the combined weight of all voters. The reader may also notice that the weights of each voter were different in Example 1.1. This was also not necessary. It is perfectly valid to have a weighted voting system in which the weights of certain (or even all) voters are equal. Consider the following example, which is a variation of Example 1.1: Example 1.2. Alice, Bob, and Charlie are the sole shareholders of the construction company. Their shares are distributed according to the following table. Shareholder Number of Shares Alice 33 Bob 33 Charlie 33 Alice again makes her proposition, and is opposed by Bob as in Example 1.1. In this scenario, how can Alice get her motion to pass? The quota q is set at 50 shares. Analyzing the system as we did before, we find that Alice needs the help of either Bob or Charlie to get her motion to pass. The reader may notice that this result is identical to the result from Example 1.1. This similarity will be even more apparent once the notion of winning coalitions is introduced. 2 Winning Coalitions Before we proceed any further, we will give a formal definition for an idea that we have been taking for granted, winning coalitions. Our intuitive notion of a winning coalition has served us well up to this point, but it will be useful to have an explicit definition. In a natural extension of this idea, we will also define losing coalitions. Definition 2.1. Given a weighted voting system (q : wn, wn−1, ..., w2, w1), define the coalition A as a collection of all voters who vote the same way on some motion. • A is a winning coalition if and only if wA ≥ q • A is a losing coalition if and only if wA < q By Definition 1.1, the weight of any given coalition A must be unique. In other words, no one coalition can have two weights at once. Therefore, the weight of coalition A must satisfy either wA ≥ q or wA < q, but not both. This tells us that a coalition cannot be both RHIT Undergrad. Math. J., Vol. 14, No. 1 Page 161 winning and losing; however, it must be one of the two. To help us see this more explicitly, we will find all of the winning and losing coalitions from Example 1.1. Recall that the quota was set at q = 51. Coalition Weight of Coalition Winning/Losing {∅} 0 Losing {Alice} 49 Losing {Bob} 46 Losing {Charlie} 5 Losing {Alice, Bob} 49 + 46 = 95 Winning {Alice, Charlie} 49 + 5 = 54 Winning {Bob, Charlie} 46 + 5 = 51 Winning {Alice, Bob, Charlie} 49 + 46 + 5 = 100 Winning Next, we will compute all of the winning and losing coalitions from Example 1.2. Upon doing so, the similarity to Example 1.1 will be even more obvious. Coalition Weight of Coalition Winning/Losing {∅} 0 Losing {Alice} 33 Losing {Bob} 33 Losing {Charlie} 33 Losing {Alice, Bob} 33 + 33 = 66 Winning {Alice, Charlie} 33 + 33 = 66 Winning {Bob, Charlie} 33 + 33 = 66 Winning {Alice, Bob, Charlie} 33 + 33 + 33 = 99 Winning Notice that the winning coalitions in Example 1.1 and Example 1.2 are the same. Furthermore, the losing coalitions are also the same. We would like to show that the second observation follows from the first. Because we set our quota at greater than 50 percent of the combined weight of all voters, the set of losing coalitions L is the complement of the set of winning coalitions W . That is, W c = L. In order to generalize our argument, we will assume that two arbitrarily chosen weighted voting systems have the same winning coalitions (as was the case with Example 1.1 and Example 1.2). Restating the assumption, the set of winning coalitions of system 1 must equal the set of winning coalitions of system 2. W1 = W2 (Assumption) (W1) c = (W2) c (Taking the complement preserves equality) L1 = L2 (Since W c = L) Page 162 RHIT Undergrad. Math. J., Vol. 14, No. 1 We have now seen that all information regarding a voting system’s losing coalitions can be obtained from information about the winning coalitions. This is an important observation because it tells us that voting systems are characterized by their winning coalitions. In fact, voting systems are actually characterized by their minimal winning coalitions, but we will get to that in the next section. Essentially, if we know the winning coalitions of a weighted voting system, then the losing coalitions are the leftover coalitions (assuming q is set at greater than 50 percent of the combined weight of all voters). 3 Minimal Winning Coalitions Now that we have been introduced to winning coalitions, let us try to gain a deeper understanding via an example. Namely, consider all four of the winning coalitions in Example 1.1. By definition of a winning coalition A, it must be true that wA ≥ q, or in this case wA ≥ 51. Indeed, one of these coalitions has a weight equal to the quota, while the other three have more weight than is needed to win (wA > q). A good question to ask would be if we could remove a voter from one of these three coalitions and still have a winning coalition. If we could do so, then that voter would not be critical to his coalition winning. This leads us to another definition: Definition 3.1. A critical voter in a coalition is a voter whose removal would cause a winning coalition to become a losing coalition. We can use Definition 3.1 to define minimal winning coalitions: Definition 3.2. A minimal winning coalition is a winning coalition in which every voter is a critical voter. Because every voter in a minimal winning coalition is critical, removing any one would cause the coalition to become a losing coalition. To get a better understanding of this idea, we will identify the minimal winning coalitions in Example 1.1. Recall that the quota was set at 51 shares. We begin by identifying all of the winning coalitions. During our discussion of winning coalitions, we found that there were exactly four. Now, we select one of these four coalitions to start with. We choose {Alice, Bob}. For each voter in the coalition (in this case, Alice and Bob), we must determine if that voter is critical. We will start with Alice. Our first step is to remove Alice from the coalition {Alice, Bob}. We are left with the coalition {Bob}. The weight w of this coalition is equal to Bob’s weight, which is 46. But 46 < q. Alice’s removal has caused the winning coalition to become a losing coalition. Therefore, Alice is a critical voter in the coalition {Alice, Bob}. Similarly, removing Bob from the coalition leaves only {Alice}. But 49 < q. Bob’s removal has also caused the coalition to become a losing coalition. Therefore, Bob must be a critical voter in the coalition {Alice, Bob}. Since every voter in the coalition {Alice, Bob} is critical, it must be a minimal winning coalition. The same process can be repeated for the other three winning coalitions. Doing so, we find: RHIT Undergrad. Math. J., Vol. 14, No. 1 Page 163 Coalition Weight of Coalition Minimal/Not Minimal {Alice, Bob} 49 + 46 = 95 Minimal {Alice, Charlie} 49 + 5 = 54 Minimal {Bob, Charlie} 46 + 5 = 51 Minimal {Alice, Bob, Charlie} 49 + 46 + 5 = 100 Not Minimal We can see that there are no critical voters in the coalition {Alice, Bob, Charlie}. The removal of any one voter still leaves the coalition with a weight of at least 51. In a way, this winning coalition is less important than the other three winning coalitions. Imagine that we only knew the minimal winning coalitions for this weighted voting system. We could use this knowledge to find all of the non-minimal winning coalitions. In this case, we would know that {Alice, Bob} is winning. Adding Charlie to this coalition could only increase its weight, so {Alice, Bob, Charlie} must also be winning. So, as we hinted at in Section 2, weighted voting systems are completely characterized by their minimal winning coalitions. This leads us to the following definition. Definition 3.3. Two distinct weighted voting systems are isomorphic if and only if they have the same minimal winning coalitions. When two distinct weighted voting systems are isomorphic, it means that they are essentially the same. While the two isomorphic weighted voting systems may have different quotas and different weights assigned to their voters, the structure and the power of each voter is the same in each system. We talked about the similarities between Examples 1.1 and 1.2. Now, with this new idea in mind, we might guess that these two weighted voting systems are isomorphic. Indeed, they both have precisely the same minimal winning coalitions, so they must be isomorphic. In fact, we have a name for this type of weighted voting system: majority rule. As we can see from our minimal winning coalitions, any two voters can team up to win. Definition 3.4. A weighted voting system with n voters is called a majority rule system if and only if any coalition containing greater than 50 percent of the n voters is a winning coalition. In the case of Example 1.1 and Example 1.2, n = 3. Since 3 2 = 1.5, any coalition with two or more voters must be winning. Indeed, this was exactly what we observed. In more general terms, when n is odd, any coalitions with n+1 2 or more voters must be winning. Likewise, when n is even, any coalitions with n 2 + 1 or more voters must be winning. We previously noted that the choice of quota q = 51 in Example 1.1 was arbitrary. In the following example, we will change the quota to q = 52. The reader might not expect this to make a significant difference on our example. However, we will see that it changes the entire structure of the system. Example 3.1. Since we are raising our quota, all losing coalitions from Example 1.1 will stay losing. So, we only need to consider the winning coalitions. The only difference between this Page 164 RHIT Undergrad. Math. J., Vol. 14, No. 1 example and Example 1.1 is that the coalition {Bob, Charlie} no longer has enough weight to win with a quota of 52. So, our only minimal winning coalitions are {Alice, Bob} and {Alice, Charlie}. Coalition Weight of Coalition Minimal/Not Minimal {Alice, Bob} 49 + 46 = 95 Minimal {Alice, Charlie} 49 + 5 = 54 Minimal {Bob, Charlie} 46 + 5 = 51 Losing {Alice, Bob, Charlie} 49 + 46 + 5 = 100 Not Minimal We can see that Alice is in every winning coalition. In other words, any coalition that Alice is not a part of is losing. So, in this weighted voting system, Alice can force a motion to fail by voting against it. We define this type of weighted voting system by saying that Alice has veto power. Definition 3.5. A voter in a weighted voting system has veto power if and only if that voter appears in every (minimal) winning coalition. Suppose that we change the quota again, this time to q = 55. In the following example, we will examine the effect that this increase in quota has on our system. Example 3.2. As in Example 3.1, we are raising our quota. So, all coalitions that were losing at lower quotas will still be losing here. The only difference between this example and Example 3.1 is that the coalition {Alice, Charlie} no longer has enough weight to win with a quota of q = 55. Thus, our only minimal winning coalition is {Alice, Bob}. Coalition Weight of Coalition Minimal/Not Minimal {Alice, Bob} 49 + 46 = 95 Minimal {Alice, Charlie} 49 + 5 = 54 Losing {Bob, Charlie} 46 + 5 = 51 Losing {Alice, Bob, Charlie} 49 + 46 + 5 = 100 Not Minimal We can see that Charlie is not in any minimal winning coalitions. The only way he can win is to form a coalition with both Alice and Bob. However, Alice and Bob can win without Charlie. So, Charlie has no power in this system. Formally, we call him a dummy voter. Definition 3.6. A voter in a weighted voting system is called a dummy if and only if that voter does not appear in any minimal winning coalitions. Finally, we will consider the effect of changing the quota to q = 96. Example 3.3. Again, since we are raising the quota, all coalitions that were losing at lower quotas will still be losing here. The only difference between this example and Example 3.2 is that the coalition {Alice, Bob} no longer has enough weight to win with a quota of q = 96. Thus, our only minimal winning coalition is {Alice, Bob, Charlie}. RHIT Undergrad. Math. J., Vol. 14, No. 1 Page 165 Coalition Weight of Coalition Minimal/Not Minimal {Alice, Bob} 49 + 46 = 95 Losing {Alice, Charlie} 49 + 5 = 54 Losing {Bob, Charlie} 46 + 5 = 51 Losing {Alice, Bob, Charlie} 49 + 46 + 5 = 100 Minimal We can see that all voters must vote together in order to win. We call this type of system a consensus system. Definition 3.7. A weighted voting system with n voters is called a consensus system if and only if the only winning coalition is the grand coalition containing all n voters. 4 Partial Order Relations The results of this paper are centered around partially ordered sets (also known as posets). Because of their importance in our research, will we take a step back from voting theory to explain the mathematics of partially ordered sets. We will begin with partial order relations, the building blocks of posets. Definition 4.1. A binary relation R on a set S is a partial order relation if and only if R satisfies the following conditions: 1. R is reflexive: For all x ∈ S, x R x 2. R is antisymmetric: For all x, y ∈ S, x R y and y R x implies x = y 3. R is transitive: For all x, y, z ∈ S, x R y and y R z implies x R z Consider the “subset” relation on the power set of a set S. For readers unfamiliar with this terminology, the power set P (S) is simply the set of all subsets of S. We define the subset relation as follows. Given two subsets U and V in the power set P (S), U R V if and only if U ⊆ V . In Figure 1, we can see a visual representation of U R V . Page 166 RHIT Undergrad. Math. J., Vol. 14, No. 1