VIDEO REPORT BY LURRAINE REES, FALL 2002

For Practical Purposes

Social Choice

Election Theory

 

         In the overview discussion about social choices, the secret of the mathematics of decision-making is presented. The methods of game theory, prisoner’s dilemma, zero sum games, tragedy of-the-commons, and voting theory are explained from a mathematics point of view.

The video opens with a decision most everyone has made at one time or another, namely, “Should I bring an umbrella today?” The mathematician’s first approach to this decision is the statement of the problem. Question: Is it worthwhile to bring an umbrella today? Answer:  It depends upon what the weather could be. At this point a list of possibilities is compiled. Normally statistics are used to estimate the probability of each outcome occurring. The outcomes are listed by what will happen in each case. Sal has two choices to make, take the umbrella or don’t take the umbrella. The weather choices are rain or sunshine. At the outset, there is a 50-50 chance of rain. Once the list of possibilities is compiled, a numerical value according to preference is assigned to each of the outcomes in order. If Sal takes the umbrella and he stays dry while it’s raining, that receives a –2. The expected payoff for this choice is .5 x –2 = -1.5. If it is sunny, he assigns the value of –1 because he does not like having to drag the umbrella around. The expected value is

.5 x –1= -1. If Sal does not take the umbrella and it is sunny, the expected payoff is .5 x –3= -1. The final compilation shows that based on the expected payoff, it doesn’t pay to take the umbrella. The final decision was based on the expected payoff, which is the heart of the mathematical approach to decision making.

         The topic of game theory is concerned with what other people are planning, which then affects decisions involving when to go head-to-head with another. Game theory was one of the most important developments of mathematics in the 20^th century because of its techniques of analysis. In game theory, the decision to make is to decide whether to attack or to retreat. The question to address is: What is the enemy going to do? The choices depend on someone else’s choices. The concept of game theory comes from the game of chess. Zero sum games like chess, are designed in such a way that one player wins and one loses. Robert Axelrod, Professor of Economics, states that real world situations are similar to zero sum games. His recommendation is that we could all do better by cooperating and avoiding an explosive confrontation.

         Prisoner’s dilemma sets up, in a four-part grid, strategies of cooperation and defection with regard to free trade with values given to the expected payoffs. The dilemma is that the decision is caught between the benefits of cooperation and the temptations of defection. Sometimes the best strategy is to defect because it brings a higher return for the individual than would cooperation. If both parties cooperate, both would have a higher return.

         Tragedy of-the-commons addresses the situation of what would happen if everyone pursued the same goal for his own gain. The example given is the favorite fishing hole. It is fine for one individual to fish freely, but what if everyone acted in the same way. What would happen to our resources? This situation is a version of the Prisoner’s Dilemma. The response to this is regulation by organizations that represent all the players, such as the Environmental Protection Agency.

Voting and election theory involves combining the individual preferences into a single decision. If the decision is between two choices, there will be a majority vote that wins the election. However, if there are more than two choices there will not be a majority. The model given was to decide which game to play with a group of thirteen people. Using the mathematical approach, the first step is to rank all three alternatives in order of preference and then throw out the game with the lowest count. An insincere ranking of preferences can take place in order to achieve the intended vote of the last place game choice. This is a major flaw in this voting method.  

         An interview with Kenneth Arrow, who received the Nobel Memorial Prize in Economics Sciences, in 1972, explains that a fair and decisive voting system is impossible to design. In his Impossibility Theorem, he states that there is no voting system that is immune to logical flaws or insincere voting behavior.

         In some cases, a group can agree in advance to reach a decision that best serves all, which is called the Fairness Decision. Each person in the model wanted the Frisbee and decided to write down how much they thought the Frisbee cost, Shelia thought $1.50 but wrote $0.75. While Alice thought the Frisbee was worth $1.00 but wrote $0.50. The result was that both did better that they thought. Decision-making is considered an art. 

         Election theory involves a variety of methods in order to obtain the preferred voting outcome. The theme of the video is an election of candidates for the Replacement Party. The party came up with a new method for this year’s vote and will attempt to have five different ballots and five different methods voted on at the same time. The choice of voting methods can significantly affect or determine the outcome of an election.

A majority rule method is the easiest process that involves two candidates. The candidate garnering the highest count wins the contest. If there are more than two candidates, the candidate with the highest votes will not be the winner based on majority rule.

         The plurality method is the type used in presidential elections. In this case, the candidate with the most votes wins even without receiving the majority of votes.

         Plurality with runoff chooses a small group of top candidates to ensure that the candidate voted last will not run. A major flaw in this method is that one of the top two candidates can lose by getting more votes.

         Sequential runoff is a method of elimination of one candidate at a time. There is a risk with this process that the good alternative will be eliminated early.

         Borda count is a method that lists all preference information at once. An example shows the ranking of top college football teams. The first, second, and third place positions are assigned a point value which then determines the teams overall ranking based on total highest points. The flaw in this method is that insincere votes can manipulate the outcome. If, for example, it is realized who the top winner will be, the insincere vote will rank another team higher in order to throw the outcome. This is known as strategic voting.

         The agenda effect seeks the outcome that is dependent upon the order in which the issue is voted upon. The issues each have a high priority but are paired according to improving the chances of the one issue that was the predetermined to be the winner. By “stacking the agenda,” the alternative issue is brought up for a vote as late as possible to ensure its favorable outcome. All voting methods are vulnerable to strategic voting methods. Due to the influence of public opinion polls, it is best to provide the public with all preferences in advance of voting.

         The Condorset method of voting is the paring of every alternative with every other. The pairing is designed to defeat every other in pair-wise contests. This method is not reliable because it is not decisive.

         In 1953, Kenneth Arrow began analyzing the rules for fair voting methods and wrote his Impossibility Theorem. His proof stated that no voting method can satisfy the fairness criteria. Arrow realized the paradox that majority voting can lead to cycles. Addressing the social choices of society and seeing that each individual holds a preference for each of the alternatives in the set, Arrow continued to write problems, conditions, and other methods, but there was difficulty satisfying all the conditions. He discovered that it could not be satisfied at all.

         For group decision-making, there will always be a drive to seek out a better voting method. Mathematicians will scrutinize for flaws. It is important to know that the voting method we use can determine the decision we make.

         The lesson about social-choice and the mathematics of decision-making was quite a learning experience. The techniques of listing individual preferences and assigning values seemed so simple and elementary yet the final decision can be manipulated by insincere voters. This is amazing. I was struck with the similarity of a child’s decision in choosing which team to play on and how the insincere votes can dramatically influence and throw a decision to another’s intended outcome.

Each of the voting methods was first explained in the context of every day group decisions. Then the process was repeated through the theme of the Replacement Party Election. This format facilitated the explanation of each of the voting methods because under each method a different candidate was chosen the winner. Overall the message was that the most important decision to make is to decide what type of voting method to adopt for the best results.

With our upcoming election, viewing this video was very timely. I now have a more heightened awareness regarding exit polls, party politics, and media manipulation. This new information allows me to analyze the way in which the method of voting affects how my vote will ultimately be counted. I have a better understanding of the Plurality method used in our Presidential elections and why there has been resistance to the third party votes throughout our country’s history. The Agenda Effect and the insincere votes in the Borda Count seem to be the most cunning methods of garnering votes. Both these methods are used widely in the local, state, and federal arenas of government as well as in the sporting field.