Voting Power Centrality Jeffrey A. Carnegie New York University How much power does a voter have? Given an “influence network” (the ability of voters to influence each other) and a voting rule, what is the chance he can pass a desired proposal? I develop a new centrality measure to estimate this, and use it to analyze the 2000 session of the US Supreme Court. What is power centrality? Power is the ability of an individual to create an outcome that is to his liking in some situation. On famous measure, Banzhaf power, considers an individual’s power to be the proportion of winning coalitions in which he is a critical voter (can make or break the vote). That incorporates the voting rule, but does not allow for differences in the ability of individuals to change the minds of others. Similarly, network centrality measures incorporate a network structure (like measures ability to influence), but they do not consider a voting rule. Power centrality is designed to consider both the influence network structure and the voting rule. Example 1: the power behind the throne [1] is a dictator, and [2], [3], and [4] are ministers. Banzhaf power would have {1] at 1, and the rest at 0. But this ignores the fact that ministers can influence the dictator. Any student of history is aware that people close to the dictator can wield tremendous power. By power centrality, each minister has a 25% chance to get his way by directly influencing the dictator. Ministers [2] and [3] can also influence each other, which is a second route to influence the dictator. 0.25 0.25 0.25 0.5 How is power centrality calculated? Choose a “starter” for which to calculate power centrality. This is the voter who initiates and idea he wants to pass by the voting rule. Simulate the spread of the idea from the starter to other voters using the influence network . Repeat I times. For all the simulations, find the proportion of votes in which the measure passes under the voting rule. This is the starter’s power centrality. Repeat steps 1 to 3, using each voter as the starter in turn (note that you can reuse the same simulations, but must re-analyze the spread of the idea with the new starter). Optional: Repeat steps 1 to 4 many times and calculate the distribution of power centrality to determine a 95% confidence interval. Results: [1]: 1.00 [3]: 0.375 [2]: 0.375 [4]: 0.25 Example 2: Sampson monks How is the influence network determined? There are three possible methods: Assume an influence network based on detailed knowledge of the voting group. For instance, Create an influence network based on an existing social network. For example, if you have a directed friendship network, you can convert it to an influence network. If A is friends with B, B may influence A. With binary ties, assume each person has an equal chance of being influenced by each in-tie as well as by himself, and assume the sum of these probabilities totals to one. Determine an influence network based on observed votes that have already occurred as a result of the influence network. Start with a base influence network, such as everyone having a 10% chance of influencing everyone else. Given the “realizations”, find the chance for each directed pair which maximizes the chance of seeing those realizations. Repeat until the chances seem relatively. Example 3: Florentine families, marriage and business ties This uses the Sampson data, summed for all three time periods, so that ties mentioned all three times are stronger. The numbers are low because the ties are sparse (only nine out-ties per node). The nodes with the most in-ties are the most powerful, but those influencing them are powerful as well. The chart above shows the power centrality for the Sampson network, using a simple majority rule. For comparison, the centrality measures are shown (normalized to the same maximum of 0.1). The numbers in brackets show the first five nodes by each measure. Power centrality is very different in magnitude and order, except for Bonavich power. Bonavich power has the same order for nearly all nodes, and the magnitudes are quite similar when compared (ration is between 1 and 0.63). The power centrality measure remains unique even following a simple majority rule with equal voting weights. This uses the Florentine data, summed for both marriage and business ties. Consider a business proposition that requires a combined wealth of 200. This uses wealth as the voting power, and a q-value of 200. The Medici are the most powerful family by any measure. Note that the Strozzi are 40% richer, but have almost half the power to get an alliance for the business proposition. As before, centrality measures are very different. Even Bonavich power is different because it does not account for voting power. If the q-value is raised to 300, the Medici are still the most powerful, but they dropped more power than anyone else. The order changed as well; the Barbadori, Bischeri, and Castellani families have relatively more influence on the more difficult business proposition. Political Science examples: US Supreme Court, UN Security Council, European Union, US Senate. Extensions: competing ideas, influence probabilities change based on issue type, more complex voting rules.