VIDEO REPORT BY LURRAINE REES, FALL 2002

For Practical Purposes

Social Choice

Election Theory

         Inthe overview discussion about social choices, the secret of the mathematics ofdecision-making is presented. The methods of game theory, prisoner’sdilemma, zero sum games, tragedy of-the-commons, and voting theory areexplained from a mathematics point of view.

The video opens with adecision most everyone has made at one time or another, namely, “Should Ibring an umbrella today?” The mathematician’s first approach tothis decision is the statement of the problem. Question: Is it worthwhile tobring an umbrella today? Answer: It depends upon what the weather could be. At this point a list ofpossibilities is compiled. Normally statistics are used to estimate theprobability of each outcome occurring. The outcomes are listed by what willhappen in each case. Sal has two choices to make, take the umbrella ordon’t take the umbrella. The weather choices are rain or sunshine. At theoutset, there is a 50-50 chance of rain. Once the list of possibilities iscompiled, a numerical value according to preference is assigned to each of theoutcomes in order. If Sal takes the umbrella and he stays dry while it’sraining, 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 hedoes not like having to drag the umbrella around. The expected value is

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

         Thetopic of game theory is concerned with what other people are planning, whichthen affects decisions involving when to go head-to-head with another. Gametheory was one of the most important developments of mathematics in the 20^thcentury because of its techniques of analysis. In game theory, the decision tomake 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’schoices. The concept of game theory comes from the game of chess. Zero sumgames like chess, are designed in such a way that one player wins and oneloses. Robert Axelrod, Professor of Economics, states that real worldsituations are similar to zero sum games. His recommendation is that we couldall do better by cooperating and avoiding an explosive confrontation.

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

         Tragedyof-the-commons addresses the situation of what would happen if everyone pursuedthe 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 thesame way. What would happen to our resources? This situation is a version ofthe Prisoner’s Dilemma. The response to this is regulation byorganizations that represent all the players, such as the EnvironmentalProtection Agency.

Voting and election theory involves combiningthe individual preferences into a single decision. If the decision is betweentwo choices, there will be a majority vote that wins the election. However, ifthere are more than two choices there will not be a majority. The model givenwas to decide which game to play with a group of thirteen people. Using themathematical approach, the first step is to rank all three alternatives inorder of preference and then throw out the game with the lowest count. Aninsincere ranking of preferences can take place in order to achieve the intendedvote of the last place game choice. This is a major flaw in this votingmethod.  

         Aninterview with Kenneth Arrow, who received the Nobel Memorial Prize inEconomics Sciences, in 1972, explains that a fair and decisive voting system isimpossible to design. In his Impossibility Theorem, he states that there is novoting system that is immune to logical flaws or insincere voting behavior.

         Insome cases, a group can agree in advance to reach a decision that best servesall, which is called the Fairness Decision. Each person in the model wanted theFrisbee 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. 

         Electiontheory involves a variety of methods in order to obtain the preferred votingoutcome. The theme of the video is an election of candidates for theReplacement Party. The party came up with a new method for this year’svote and will attempt to have five different ballots and five different methodsvoted on at the same time. The choice of voting methods can significantlyaffect or determine the outcome of an election.

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

         Theplurality method is the type used in presidential elections. In this case, thecandidate with the most votes wins even without receiving the majority ofvotes.

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

         Sequentialrunoff is a method of elimination of one candidate at a time. There is a riskwith this process that the good alternative will be eliminated early.

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

         Theagenda effect seeks the outcome that is dependent upon the order in which theissue is voted upon. The issues each have a high priority but are pairedaccording to improving the chances of the one issue that was the predeterminedto be the winner. By “stacking the agenda,” the alternative issueis 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 theinfluence of public opinion polls, it is best to provide the public with allpreferences in advance of voting.

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

         In1953, Kenneth Arrow began analyzing the rules for fair voting methods and wrotehis Impossibility Theorem. His proof stated that no voting method can satisfythe fairness criteria. Arrow realized the paradox that majority voting can leadto cycles. Addressing the social choices of society and seeing that eachindividual holds a preference for each of the alternatives in the set, Arrowcontinued to write problems, conditions, and other methods, but there was difficultysatisfying all the conditions. He discovered that it could not be satisfied atall.

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

         Thelesson about social-choice and the mathematics of decision-making was quite alearning experience. The techniques of listing individual preferences andassigning values seemed so simple and elementary yet the final decision can bemanipulated by insincere voters. This is amazing. I was struck with thesimilarity of a child’s decision in choosing which team to play on andhow the insincere votes can dramatically influence and throw a decision toanother’s intended outcome.

Each of the voting methods was first explainedin the context of every day group decisions. Then the process was repeatedthrough the theme of the Replacement Party Election. This format facilitatedthe explanation of each of the voting methods because under each method adifferent candidate was chosen the winner. Overall the message was that themost important decision to make is to decide what type of voting method toadopt for the best results.

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