Brian Mazur
Math 89S
Professor Bray
November 12, 2013
NicolausTideman
Biography of Dr. T. NicolausTideman
Brian Mazur
November 12, 2013
Math 89S
Professor Hubert Bray
Summary
Ever since he was a boy, NicolausTidemanwas transfixed on the election process. When he was just 12 years old, he manipulated a school election and split the votes of the other candidates, which gave him a majority, allowing him to become class treasurer. He did not go on to be a candidate in another election – rather he has spent his life examining collective decision-making and the best methods of voting candidates into office. After receiving his B.A. in mathematics and economics from Reed College and his doctorate in economics from the University of Chicago, he held teaching positions at a number of universities. Tideman has been teaching economics at Virginia Tech for more than 25 years, but has taken a special interest in voting methods and game theory.
Tideman’s most famous discovery in the world of game theory is the Ranked Pairs Method. Tideman stumbled upon Ranked Pairs when he was asked by his colleagues to come up with a way to rank applicants to Virginia Tech’s Economics Department. Because a handful of people were evaluating multiple candidates for a job, they asked Tideman to create a way for them to “vote” for each job-seeker. On a preferential ballot, Ranked Pairs orders candidates by their margins of victory and builds off of “the priority of greatest majors” which also says that the candidate with the largest margin of victory is the winner. Additionally, the winner of a Ranked Pairs election is always a Condorcet winner, which is important because Condorcet winners tend to hail from the center of the political spectrum and are better suited to represent the vast interests of consituents. It has also been found that the winner of Ranked Pairs is almost always the same as the winner of the Schulze method. Tideman’s discoveries have highlighted how successful a preferential ballot election can be, as Ranked Pairs produces that coveted Condorcet winner.
The Condorcet-Hare rule is another method that has been examined by Tideman. He wrote in Quasi-Empirical Evaluation of Voting Rulesthat the Condorcet-Hare method is an excellent repellant against strategic voters or voters who vote dishonestly. This method will give a candidate who garners a majority of the vote an automatic victory. But if no candidate prevails,all candidates who are outside of the Smith Set are eliminated from the race and their votes a redistributed accordingly. This process repeats until a candidate gains a majority of the vote, thus winning the election.
One of Tideman’s early research topicswas the Independence of Clones rule, which says that a winner of an election must be unaffected by the removal of another candidate in an election. He describes two conditions (see later in this paper) that can get rid of clones and spring the Independence of Clones rule to life.
NicolausTideman has made many contributions to the game theory, but most importantly is discovery of Ranked Pairs and his research on the Condorcet-Hare rule. He has an extensive background in economics and also specializes in land value, making him a “renaissance man” in the sphere of numbers.
There are only a few hundred people in the world who have applied mathematical sequences and equations to the voting process. NicolausTideman, who is currently a Professor of Economics at Virginia Polytechnic Institute, has dedicated a large chuck of his research to game theory, collective decision-making and voting. Tideman is probably most famous for his discovery of ranked pairs, but he has also extensively researched the Schulze method and the Condorcet-Hare method. In one of his papers, Quasi-Empirical Evaluation of Voting Rules, Tideman writes, “When the members of a collectivity will be taking a single vote over more than two options, to elect a single officer such as a president or mayor, or to choose a single policy, what voting rule should be used to aggregate the votes and identify the winner?”(Tideman, Green-Armytage and Cosman, 2013). This question seems to be one that he and many other game theorists seek to answer. Although, Tideman writes, it is possible to determine the best way of voting through numbers, we cannot be certain because of the variables in elections (reality) and because every election and its outcome are unique (Tideman, Green-Armytage and Cosman, 2013).
Tideman has spent his life trying the find the perfect method for voting candidates into office. At a young age, Tideman was fascinated with elections, questioning how different elections worked, even taking an interest in local electotionsin San Francisco (Holcombe, 2007). In middle school, Tideman participated in a student council election and pitted his opponents against each other so their votes would offset and he won with a majority (Holcombe, 2007).
One of Tideman’s first works on Game Theory and Voting was in the Journal of Political Economywhen he published the paper “A New and Superior Process for Making Social Changes” with Gordon Tullock in 1976. One interesting element of the voting process that the duo examines is the impact of voting versus not voting. During the 1970s, voting turnout in the United States plummeted to under 60 percent (University of California at Santa Barbara, 2013), making this article very important to those who were considering turning out the polls or not. Tideman and Tullock write, “the only wholly instrumental reason for a person to vote is the possibility that his vote will be decisive” (Tideman and Tullock, 1976).They discuss “truthful” and “non-truthful” votes in this article, something that he would build off in his more recent publications when he discusses strategic voting and which methods protect against them (Tideman and Tullock, 1976). With money playing a considerable part in game theory and collective decision making, Tideman’s knowledge in economics aids him in his studies. He has also authored a number of books, most famously Collective Decisions and Voting: The Potential for Public Choice which examines preferential voting methods and which ones should be used to elect today’s leaders.
Tideman writes that the foundations of voting strategically lie in the Gibbard-Satterhwaite Theorem, which was discovered in 1973.Tideman’s interpretation of the theorem is that it, “shows that no reasonable single-winner voting rule can be free of incentives for strategic voting in all cases” (Tideman, Green-Armytage and Cosman, 2013). In his book, Game Theory over Deomocracy, Professor Hubert Bray of Duke University simplifies this definition, saying that the Gibbard-Satterhwaite Theorem can allow a voter to vote dishonestly to achieve his or her desired result in an election. In Quasi-Empirical Evaluation of Voting Rules, Tideman exhibits how a strategic voter may be happy with the result of an election, but overall strategic voting is not healthy in an election, especially in regards to the long-term dynamic of the region, country or city that election effects. “strategic voting can also lead to a distinctly less efficient result.” Tideman writes. “for example by accentuating the weight placed on the votes of those who behave strategically or by impeding the revelation of voters’ preferences to such a degree that the perceived non-viability of a candidate can become a self-fulfilling prophecy, even in cases where such a candidate would win and provide high social welfare if votes were sincere” (Tideman, Green-Armytage and Cosman, 2013).
Tideman found that the best method to prevent voters from dishonestly influencing voters is the Condorcet-Hare method. The rule will elect a Condorcet winner if there is one available, but should voters try to influence the election so there is not a Condorcet winner, the Hare winner wins (Tideman, Green-Armytage and Cosman, 2013). The Hare method is also known as Instant Runoff Voting. Voters will rank candidates on a ballot, with one being the highest (first choice) and two being (last choice). If no candidate wins a majority of the votes, the last place candidate is taken out of the election (Redford, 2011).
At this point in the with the normal Condorcet method the votes would be redistributed accordingly with the weakest candidate being knocked out of the race. But with the combined Condorcet-Hare method, if no candidate achieves a majority vote, all candidates outside of the Smith Set are removed from the race (Tideman, Green-Armytage and Cosman, 2013). The Smith Set, which was created by Boston College professor John Smith, is definied as, “If all candidates in a set S pairwise-beat all candidate in the complement subset (i.e a voter majority prefers each S-member over each candidate not in S) and S is the smallest nonempty such set, then S is called the "Smith Set." A "Condorcet winner" is a 1-element Smith set” (Rangevoting.org, 2008) Thus, after the first round of tabulation, candidates outside of the Smith Set are knocked out of the race and the votes are recounted accordingly. One should keep in mind, however, if a candidate garners a majority of the vote then that candidate wins outright and the Condorcet-Hare method does not play out to its entirety(Tideman, Green-Armytage and Cosman, 2013).
Tideman and his co-authors ran a model known as the “Politbarometer” which simulated elections that could possibly be manipulated by voters. The experiment affirmed Tideman’s theory that Condorcet-Hare elections were not likely to be influenced dishonestly. Out of the over hundred thousand models run, only two percent could be manipulated by dishonest or strategic voting (Tideman, Green-Armytage and Cosman, 2013). Tideman wrote to me on November 6, 2013 that he thinks this method is superior to Ranked Pairs. Because the weakest candidates who are outside the Smith Set are eliminated from the election and their votes are redistributed to the candidate of the next choice, the Condorcet-Hare rule simply gives voters “more bang for their buck” or the most out of their vote.
One day Tideman was asked by his colleagues to come up with a method of ranking prospective instructors for the Economics Department at Virginia Tech. Tideman was tasked with figuring out how to count the votes, or in this case, evaluations of the hiring committee so that each evaluation was counted to the fairest way possible.Tideman, in order to meet his colleagues’ request devised a method that would differentiate among the job-seekers. Thus, the Ranked Pairs rule was born. But the Ranked Pairs rule is based on another rule, “the priority of greatest majors.” Bray writes that, this is “a natural way to resolve cycles in the preferences of voters.” It also orders candidates by margins of victory, thus resulting directly from votes casted in an election (Bray, 2013). “The priority of greatest majors” factors into Ranked Pairs because in a preferential ballot, the winning candidate is always ranked the highest by the “the priority of greatest majors.” Tideman’s Ranked Pairs will always produce a Condorcet winner. Condorcet methods are considered widely successful because they tend to always produce a winner of an election from the center and best represents the interests of his or her constituents. Additionally, Tideman’s discovery was special because it “was the only single-winner election method known in 1997 that satisfies Condorcet, monotonicity, clone-immunity, majority for solid coalitions, and reversal symmetry” (Schulze and Smith, Unk.).
Ranked Pairs also produces the same outcome as the Schulze method, which is another method of choosing the winner of a preferential ballot. The Schulze rule virtually proclaims a candidate that beats every other candidate becomes the outright winner of an election. The victor has “a chain of victories” (Bray, 2013) that places the winning candidate at the top of the food chain. The winning candidate never loses at head-to-head matchup and therefore will have the greatest margin of victory most of the time. Hence, the Schulze Method and the Ranked Pairs method always deliver us the same winner.
Another rule created by Tideman is the Independence of Clones rule. To understand this rule, we must first examine what a “clone” means in the context of an election. Tideman writes in Independence of Clones as a Criterion for Voting Rulethat a clone is “A proper subset of two or more candidates, S, is a set of clones if no voter ranks any candidate outside of S as either tied with any element of S or between any two elements of S” (Tideman, 1987)
He sets forth two conditions that must be met when clones are present in an election in order for the Independence of Clones to come into play:
1. A candidate that is a member of a set of clones wins if and only if some member of fffffffffffthat set of clones wins after a member of the set is eliminated from the ballot. (Tideman, fffffffffff1987)
2. A candidate that is not a member of a set of clones wins if and only if that candidate vbvvvvvvwins after any clone is eliminated from the ballot. (Tideman, 1987)
Tideman is quick to note that independent clones usually do not exist in Ranked Pairs because t is rare to have a tie between two candidates in Ranked Pairs. If there does happen to be a tie, Tidemanwrites “the outcome is a tie among all the candidates that win under some
way of breaking these ties” (Tideman, 1987). He describes the role of clones, if they exist, in a number of voting rules. For example, plurality voting is not independent of clones, because with multiple candidates it is possible for supporters of a candidate, in this case candidate “A” to categorize another candidate, “C”, as their second choice. If the supporters of C categorize A as their second choice, then A and C are clones. Tideman recalls that his two opponents in his student council election were clones because one of his opponents could have clearly been the dominant candidate and still not win. Thus, plurality voting has clones and is also not a Condorcet method.
Tideman’s discoveries have been invaluable to the world of game theory and voting methods and he has worked tirelessly to ensure that his methods to work in real-life as many countries, cities and regions attempt to find the best way to elect their leaders. His methods are amplified by how applicable they are to real life and how effective they could be if used in an actual election.
Works Cited
- Tideman, Nicolaus, James Green-Armytage, and Rafael Cosman.Quasi-Empirical Evaluation of Voting Rules.Filebox.vt.edu. Virginia Tech University, 23 July 2013. Web. 08 Jan. 2013. <
- Holcombe, Randall G. "NicolausTideman: Collective Decisions and Voting: The Potential for Public Choice." Rev. ofCollective Decisions and Voting: The Potential for Public Choice. n.d.: n. pag. Florida State University, 08 Aug. 2007. Web. 10 Nov. 2013. <
- Tideman, Nicolaus, and Gordon Tullock. "A New and Superior Process for Making Social Choices."Journal of Political Economy84.6 (1976): 1145-159.Jstor.com. Chicago Journals. Web. 10 Nov. 2013. <
- Bray, Hubert.Democracy Over Game Theory: How to Elect Candidates Who Best Represent Voters. N.p.: n.p.,n.d. Print.
- Aredford, Alexander. "Voting Methods."Math.wsu.edu. Washington State University, n.d. Web. 11 Nov. 2013.
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- Schulze, Markus, and Warren Smith. "Schulze's Beatpath Voting Method."Rangevoting.org. Range Voting, n.d. Web. 12 Nov. 2013. <
- "Tideman "Ranked Pairs" Condorcet Voting System."RangeVoting.org. Range Voting, n.d. Web. 12 Nov. 2013. <
- Tideman, Nicolaus. "Independence of Clones as a Criterion for Voting Rules."Social Choice and Welfare4 (1987): 185-206.Springer.com. Springer-Veriag. Web. 12 Nov. 2013. <
- "Dr. NicolausTideman | Economics | Virginia Tech."Economics.virginiatech.edu. Virginia Polytechnic Insitute, n.d. Web. 12 Nov. 2013. <