electoral race
Let the Election Games Begin
The 2012 Presidential Elections, which took place last night, was one of the most exciting moments for many Americans, and especially for those who have voted for the first time in their lives. Each and every viewer was sitting on the edge of their seat as the results from each city, county and state continued to release out to the public. Main candidates Mitt Romney and Barack Obama were racing to win that 270th electoral vote, which would give the candidate the majority, to give them the ultimate win. The suspense continued to build as the polls from each time zone began to close and ballots were rapidly being tallied up. Major television station CNN was feeding America live updates on the status of the votes and also making realistic predictions based on historical data from previous elections. Often, in some states, Romney and Obama would result in differences as little as 0.3%, which is what happened in Florida. Finally after 5 hours of keeping our eyes glues to the TV screen, we were relieved; Obama won.
There is an interesting mathematical concept that is intertwined into the elections and the way votes could be viewed as, Baye’s rule. Baye’s rule is a mathematical model for how information cascades can occur and also is based on conditional probability, or the chance of an event happening given that a previous event has already occurred. We can make a mathematical model for Mitt Romney and Barack Obama to describe the probability of either candidate winning the presidential election. All of the conditional probabilities are based on the results, either a democratic or republican majority, of each state. For example, given that Obama won Florida, the probability of him winning presidency again would be much higher. Vice versa, if Romney won Florida, the probability of him winning presidency would increase significantly. The conditional probability with Florida was significant because more heavily populated states are worth more electoral votes, in this case 27. For even larger states such as California, which is worth 55 electoral votes, significantly increased the probability of winning, for whichever party won that state, or significantly decreased the chance of winning the election for the other party.
Baye’s Rule could also be applied to not only the voting outcome of states individually, but also to different groups of people based on race, ethnicity, age, or any other demographic characteristic. The results of the elections show that Obama won among the younger crowd and minority groups such as Hispanics and African Americans. Since they makeup a large percentage of the voting population, their votes increased Obama’s chances of winning. Given that almost all of the young people voted for the Democratic Party, the probability of Obama being elected increased significantly. The same concept applied to the Hispanics and African Americans.
The outcomes of the “larger” groups such as California, with the highest electoral votes, or young people, which makes up a big population in the United States, were the most influential because they were valued heavily. Winning California would put the candidate, who received points from California, 55 electoral votes closer to the target 270. Consider Alaska, which is a smaller, less significant, and less heavily weighted state. Valued as 3 electoral votes, Alaska made minimal impact on changing the chances of either candidate winning. So even if we were given that Romney or Obama won the 3 electoral votes from Alaska, their probability of gaining presidency would only increase by a trivial amount. With this type of information cascade, the probability approaches 1 as the candidate wins more electoral votes from individual states, or votes in general from different demographic groups of people.
Another interesting principle that the elections demonstrate is directed vs. undirected webs and the strength of connectivity. We can safely assume that both Barack Obama and Mitt Romney do not have any personal relationships or have even heard of most of their voters in America. However, any person voting has obviously heard of both of the candidates and therefore in a web, a directed path would connect the voter to each presidential party. This concept is also known as the global name-recognition network, in which there is a directed path to another person if one has heard of him before. If a web was depicted including Obama, Romney, and several voters, it could look similar to the diagram below. As one can see, the tiny circles on the outsides represent voters and have directed paths towards each party because they know Obama and Romney. However, because it is unlikely that Obama and Romney would have heard of the voters, or their names, there is no directed path that leads back out to them.
By looking at this web example, it is evident that nodes Obama and Romney are reachable from voters A and B. However, nodes A and B are reachable from neither Obama nor Romney. These characteristics prevent this web from having a strongly connected component, which is a subset of nodes in a directed graph where every node in the component has a path to all other nodes in the component. Furthermore, this subset cannot be a part of another strongly connected component. Since Obama and Romney have no paths to any outside voters, a connected component fails to exist; the only connected component is a web consisting solely of the two candidates. This phenomenon is also present in situations with celebrities, where all of their fans love them and have heard of them while the celebrities don’t know one tenth of their fans.
Furthermore, the elections process can sometimes be about decisions being influenced and skewed by the people around you, also known as herding. Often social pressures to conform, imitation reasons, or inferences from others’ decisions, can sway someone in their voting choice. If one is part of a large family, in which all of the members are oscillating between Obama and Romney, and the relatives begin voting for Obama, one will be more tempted to vote in the same manner because his family members’ votes are influencing his decision as well. There also exist voters who are completely oblivious to the aspects of the republican or democratic party, or politics in general. In this case, some voters will simply vote by imitation, copying the way one of their friends or family votes. It is expected that many young voters, who are not yet as educated as the older members in the family, would vote the way their family would. By voting how others vote, there are two benefits, either informational or direct. An example of a direct benefit for a 2 weeks pregnant female voter, for example, supporting Obama would be the ability to legally get an abortion; in this case, if she felt strongly about pro-abortion, she can receive direct benefits by voting for Obama. On the contrary, one may vote for Romney because of informational effects; after listening to many convincing arguments about why Romney would be successful, based on the information inferred from the conversation, one way be swayed to vote for Romney instead.
Information cascades almost seem inevitable in election or voting-like situations. There always exists a decision that would fit the social norms given a specific demographic area or even within groups of friends or family. Because one is always surrounded by thousands of people who feel strongly about their opinions, influence is expected. The elections ultimately resemble important mathematical concepts that one would have never really thought of before. Previously occurring events always affects the future, because everything is conditional upon the past.
Sources:
http://www.cnn.com/2012/11/07/politics/why-romney-lost/index.html?hpt=hp_t1
http://elections.nytimes.com/2012/results/president