Voting Games: Part II

Joseph Malkevitch
York College (CUNY)
malkevitch at york.cuny.edu

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1. Introduction

Democratic societies and international organizations use a wide variety of complex rules to reach decisions. Examples where it is not always easy to understand the consequences of the way voting is done include the Electoral College used to elect the President of the United States, the United Nations Security Council, the governance structure of the World Bank and the International Monetary Fund, and various councils of the European Union.

In attempting to make sense of what goes on in complex voting situations it is tempting to think of the participants as players in a game and investigate those players who can come together to "win." Such groups of winning players are known as winning coalitions. Furthermore, if each of the players in such a game could be assigned a weight, and the winning coalitions could be determined by finding those groups of players whose weight exceeds a fixed quota, then it would appear that it would be easier to see the relative power of the players involved. Unfortunately, this approach turns out to be unsuccessful because many games can not be represented as weighted voting games. Even in weighted voting games a player's influence or power is not always easily related to the weighted vote that the player casts. Thus, a player i can have positive weight but never be a member of a minimal winning coalition (a coalition that wins and the removal of a single player does not allow the coalition to win any longer). Such players are known as dummies. It can also happen that players with unequal weights do not have the same "power." Here I will examine the insights that mathematics has provided into complex voting game systems, thereby providing political scientists, the lay public, and the participants in these games insights which they would not otherwise have had.

2. Voting games

In a situation where there is a voting body where the legislators represent geographical regions, say, of different sizes, it is tempting to believe that equity can be achieved in the following way: Each representative will be assigned a weight proportional to the population of the geographic region that the legislator represents. A quota is established by summing these weights, dividing by 2, and moving up to the next largest integer. When this is done we have an example of a weighted voting game. The winning coalitions are precisely those coalitions whose weights sum to at least the quota. More generally one might set the quota in a more arbitrary manner (typically larger than the quota just defined) and take the winning coalitions to be those whose members have weights that sum to at least the quota. This approach captures the idea of having more than a simple majority for an action to be taken - for example, to approve a treaty or amend a constitution. Games that are weighted voting games seem appealingly simple. Let us look at an actual case history to see what can happen if one proceeds in a naive fashion.

This case history involves Nassau County, New York, which is the geographic region on Long Island just to the east of New York City. Nassau County's government took the form of a Board of Supervisors, one representative for each of various municipalities, who cast a block of votes. Not only did Nassau County use weighted voting, but weighted voting was also used for the counties of New York State other than the 5 counties in New York City.

Here are the weighted voting systems used at various times by Nassau County. The passing quota shown reflects the number of votes needed to pass "ordinary legislation." Some special voting situations required larger quotas than the numbers shown.

The numerical weights were chosen to try to take into account the populations of the different municipalities, which were quite disparate. It is easy to see that in 1958, Oyster Bay, Long Beach and Glen Cove were dummies (!) and that North Hempstead had the same power as Hempstead 1 and Hempstead 2 because any two of these communities formed a minimal winning coalition. You should check for yourself that in 1964 there were also dummies. After 1964 the quota was raised to guarantee that no community was a dummy. However, eventually Nassau County converted to having a county legislature because of the difficulties of using a weighted voting system in a situation where there were so few players and such large differences in the populations of the communities.

Still, this approach to governance has its own problems. District lines cut across what had often been natural historical political entities. Furthermore, though the districts served by each member of the county legislature could be made approximately equal in population, this could not be done in a way that avoided some members of the legislature being tempted to vote in an identical way because, though they are different people, they have identical interests, given the geographical communities that they serve. To the extent that it is desirable that every member of the legislature act in an independent manner, this has been hard to achieve.

3. When are voting games weighted voting games?

When a voting game is described in a complex verbal way it is often difficult for the participants in the game and the people whom the game serves to know what the "influence" or "power" relations of the players are. If a game is given in weighted voting form at least one can relatively easily check if a particular set of players is a winning coalition or not. Thus, it is a natural question to ask whether or not a game can be represented as a weighted voting game. Again, this means that the players cast weights and there is a value Q such that a coalition is winning if and only if the sum of the weights cast by the players in the coalition is Q or greater.

For simple games with few players all simple voting games are weighted. However, for larger simple games this is no longer true.

Suppose we have a weighted voting game G and two winning coalitions of players C₁ and C₂ where C₁ contains a player i who is not in C₂, and C₂ contains a player j not in C₁. Now form two new coalitions by having i leave C₁ and join C₂, and j leave C₂ to join C₁. There are two possibilities: either i and j have equal weight or they do not. In the first case the new coalitions after the switch must still both be winning coalitions since the weights of the players in the two coalitions have not changed after the switch. However, in the case of unequal weights for i and j, after the switch one of the new coalitions has increased in weight. Hence, after the switch it can not be that both of the new coalitions are losing. Therefore, if we can find a voting game having coalitions where switching a pair of players, as described above, changes two winning coalitions to two losing coalitions, the game we started with can not be a weighted voting game! The argument above can be extended in a natural way to exchanging groups of players instead of individual players. Using this idea of trading, it is possible to show that the "game" of passing a bill by the U.S. Congress and the President can not be represented as a weighted voting game. (This game involves 435 members of the House of Representatives, 100 Senators, the Vice-President (who votes when there is a tie in the Senate but not otherwise) and the President!)

Although not every voting game is weighted, Alan Taylor and William Zwicker showed that every simple voting game can be given a "vector" representation. The idea in a vector-representation of a voting game is that one has an n-tuple of numbers and a quota associated with each of the positions of the n-tuple. The winning coalitions are found by vector addition (adding the components) for the players having total weight exceeding the quotas for each component. For example, with the following vector weights:

A member of the House of Representatives:

(0, 1)

A member of the Senate:

(1, 0)

The Vice-President:

(.5, 0)

The President:

(16.5, 72)

and quotas of (67, 290), one successfully captures the winning coalitions of the "passing a law" game. Thus, while the voting game for passing a federal law is not weighted, it can be understood as a relatively simple vector game. These ideas have been used to try to measure the "complexity" of the rules involved in various real-world voting games. Taylor and Zwicker also obtained many fascinating results about the relationship between "trading" and the structure of voting games and weighted voting games.

4. Power indices

While the issue of being able to represent a voting game as a weighted voting game is a convenience for the players in thinking about the game, the advantage is deceptive. The reason is that the weight relations that are involved in a weighted voting game do not reflect the "influence" or "power" relations. We have already seen that a player may have positive weight and yet be a dummy. Thus, it is natural to ask whether there is a way to find the power or influence of the players in a voting game. If this were possible in a way that people could agree on, then one could try to assign weights to players that gave rise to "power," i.e. that was proportional to population as a way of ensuring equity in the weighted voting game setting.

Relatively early in the development of game theory, proposals were put forward to measure the strength, influence, or power of the players in voting games. Among the earliest power indices that were developed for voting games used a game theory idea now known as the Shapley Value, named for Lloyd Shapley. The power index which grows out of this concept is known as the Shapley-Shubik Index which is named for the mathematician Lloyd Shapley (below, left) and the economist Martin Shubik (below, right). Both Shapley and Shubik have had influential careers in creating and applying game theory ideas.

Although the Shapley-Shubik index can be approached axiomatically, let me explain the idea in a more intuitive way. Suppose that in a voting game one imagines all the players being "polled" in a particular order. When a winning coalition forms the first time, the person who changed the situation to a winning coalition is given a "pivot point." Thus, i gets a pivot point if a yes vote of the players who are polled before i do not form a winning coalition, but with i, they do. This means that whether a player gets a pivot or not depends on the player's role in the winning coalition structure and the ordering picked.

Consider all the orderings of the n players in a voting game. Repeating the idea above in slightly different terms, the players vote in a particular order, and it is noted at what point some player i, together with the previous players (if there were any), form a winning coalition. We will give player i in this ordering a pivot point if the players up to that point do not form a winning coalition but the players together with i do form a winning coalition. If there are n players, there are n! orderings and one pivot point per ordering. The Shapley-Shubik power index for player i is the total number of pivots for i divided by n!.

Here is a simple example with four players:

Note that the quota required here (7) is more than a simple majority, which would be 6. To avoid confusion between the weight a player casts and the names of the voters we will use the names of w, x, y, and z for the players who cast votes of weight 4, 3, 2, and 1, respectively.

Since there are 4 players, we have 24 different ways to arrange them. Reading from left to right I have placed a star to the right of that player's name whose vote together with those of the previous players first creates a winning coalition (i.e. weight sum of 7 or more).

wx*yz
wx*zy
wyx*z
wyz*x
wzx*y
wzy*x
xw*yz
xw*zy
xyw*z
xyzw*
xzw*y
xzyw*
ywx*z
ywz*x
yxw*z
yxzw*
yzw*x
yzxw*
zwx*y
zwy*x
zxw*y
zxyw*
zyw*x
zyxw*

The Shapley-Shubik power index (written as a fraction in lowest terms) of each of the players is below:

w power 7/12,

x power 1/4,

y power 1/12, and

z power 1/12.

First, note that the fractions sum to 1, and thus one can interpret these numbers as probabilities. Note also that the power reflects the "symmetries" present in the way that the players enter into the minimal winning coalitions. The minimal winning coalitions are {w,x} and {w,y,z}. Thus, it should not be surprising that y and z have the same power though they have different weights.

More important for political games was a power index proposed by the mathematically trained (but lawyer by profession) John Banzhaf III. Banzhaf realized - and took the issue to the public and the courts - that it was naive to try to get equity for voting games by merely assigning weights proportional to population.

Banzhaf also developed a way of computing the power of the players in voting games that is both intuitively appealing and seems more natural for political situations than that of Shapley and Shubik. Suppose that we have a voting game and consider a "bill" that has been placed before the players in the game. Each of the players can be thought of as voting yes or no on the bill. Now, consider a particular yes-no voting pattern P. We can see if the bill passes or not for pattern P. When the bill does pass, this means that we must have a winning coalition for the players who voted in this pattern. Now, for each player s in P who voted yes we can determine what would happen if s had voted no. If the change from yes to no changes the bill from passing to failing to pass, we assign s a pivot point. Note there may be several pivots for a given pattern P or possibly no pivots assigned for that pattern. (This contrasts with the Shapley-Shubik situation where for each of the n! orderings of the players one gets one pivot per ordering. In the Banzhaf case there are 2ⁿ yes/no patterns for n players, but not all of these patterns result in a pass.) We will define the Banzhaf power index of player i to be the number of pivots that i receives for all patterns P where the bill passes and i favored the bill, and where, if i were to switch his/her vote (the other votes staying fixed), the bill would no longer pass, divided by the total number of pivots.

Shown below is the game for which earlier we computed the Shapley-Shubik power index of the players.

Here is how the calculation goes for the Banzhaf power index. We list below all the winning coalitions (e.g. yes/no patterns which result in a pass), and next to each player (on the right) we put a "^" symbol if that player, by changing a vote from yes to no, would change the winning coalition to a losing one:

w^xyz
w^x^
w^x^y
w^x^z
w^y^z^

There are a total of ten "^" symbols in the list above.

Hence the Banzhaf power indices are:

w has power 1/2,

x has power 3/10,

y has power 1/10, and

z has power 1/10.

Again the equality of power for y and z reflects the symmetry with which these players enter into the minimal winning coalitions. Although it might appear that the Shapley-Shubik and Banzhaf indices give different numerical values but qualitatively the same picture, this is not always the case. (The analysis of a proposed way to amend the Canadian Constitution, which was never adopted, illustrates this point, as does the analysis of the game to pass a bill in the United States (Congress and President interacting)). Note that the powers that we are talking about here do not accrue from the person casting the vote but from the structural properties of the game. It is conceivable that some who structurally would be dummies might have considerable influence in the way that actions actually get taken because of their charisma.

Which is better, the Shapley-Shubik or Banzhaf approach to power indices? Before you decide, you should realize that these are not the only candidates for power indices. There are additional indices, a notable one due to J. Deegan and E. Packel. This index is intriguing for treating coalitions of different sizes in a different way. This might well make sense, since one might think that smaller coalitions are more likely to form than larger coalitions. Yet, one can also imagine scenarios where the dynamics of how coalitions form are very complex, so that it is hard to sort out how power and winning coalition size interact. A variety of scholars have provided interesting axiomatizations of a variety of power indices (e.g. J. Deegan, P. Dubey, E. Packel, L. Shapley). The value of axiomatizations is that if one can characterize different power indices axiomatically, one can analyze a particular real-world situation, thereby helping one decide which power indices make sense.

What is tantalizing is that clever mathematical analysis and the attempts of social scientists (such as Shubik) further both the insights we have into fairness and equity questions as well as intriguing aspects of mathematics. When Taylor and Zwicker collected information for their recent book Simple Games, they came across many independent rediscoveries of ideas related to games in many areas of knowledge. As so often has been the case, mathematics and its applications progress together.

Joseph Malkevitch
York College (CUNY)
malkevitch at york.cuny.edu

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NOTE: Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal, which also provides bibliographic services.