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The Vexing Mathematics of Democracy

In this discussion, we will explore the mathematical conundrums, inconsistencies, and paradoxes that plague voting systems. For example, Condorcet's paradox shows that the aggregation of rationally voting individuals can lead to irrational group choices. Is democracy mathematically irrational? Arrow's Impossibility Theorem and results from the theory of apportionment will also be discussed.

 The discussion is inspired by George G. Szpiro's book "Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present" which argues that "the only method of government that avoids paradoxes, inconsistencies, and manipulations is a dictatorship". Is the mathematics of voting really this pessimistic? Is democracy mathematically hopeless?

Topics be discussed (emphasis will be placed on participant interests):

• What is Condorcet's Paradox? The Alabama Paradox? The Population Paradox? The New York Paradox? The New State Paradox?

• Is it true "that there are no good or correct methods to allocate seats to Congress or any other parliament"?

• A survey of voting methods including majority rule, qualified two-thirds majority rule, absolute majority rule, pairwise elections, ranked order elections, pairwise knockout voting procedures, the Borda count, the Condorcet method, the Laplace criterion, and the Dodgson method.

• The implications of Arrow's Impossibility Theorem which shows, according to Szpiro, that "No democratic constitution exists that produces a coherent method of social choice; only a dictatorship can fulfill a handful of innocuous sounding conditions."

• A survey of apportionment methods (ways to determine the number of representatives to a parliament or Congress) including the methods of Jefferson, Adams, Webster, Hill, Wilcox, and Huntington, 

• The implications of the Gibbard–Satterthwaite theorem which shows that all elections are subject to strategic manipulation by participants. Does this theorem provide a mathematical basis to explain the nastiness that is American politics today? Is it a mathematical necessity that strategy instead of simple and fair choice dominates democratic institutions?

• The implications of the Balinsky and Young theorem which seems to state that it is mathematically impossible to fairly allocate seats in a parliament or congress.

• The human side of mathematics: Szpiro discusses the biographies of the mathematicians who contributed to the modern mathematical theory of voting and social choice. What can we learn about the culture of mathematics from Szpiro's book?

• Is mathematics a value-free endeavor that resolves questions into black and white clarity (The mathematically right result)? Or is mathematics a messy political affair where paradoxes and preferences delude mathematicians during its development? This point comes from the AMS review of Szpiro's book:

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  • CJ F.

    There is an interesting course on voting that just started on the Coursera platform: "Making Better Group Decisions: Voting, Judgement Aggregation and Fair Division", Eric Pacuit, University of Maryland, College Park

    I'm starting it, we'll see if I finish it ...

    September 2, 2014

  • Sam B.

    Paradoxes kept coming up all through the book, being one of the two problems to voting systems. They were never anything too complicated, but they were the source of major headaches. Condorcet's Paradox creates a loop when people are asked for their preferences for one candidate over another. As you may recall, the example was three people choosing drinks with no clear majority. When the preferences were chosen, it started a loop. In a case like this, paradoxes are simply annoying, since "what to drink" can be solved with the answer to "what makes us drunk?" but their is a second problem presented in the book: People.
    Pliny was the earliest example of people messing with the vote. With three possible outcomes, Pliny was confident his choice would win with a 40, 30, 30 outcome. In this case, an opponent manipulated the vote so his decision would win 60 - 40. And other examples would pop up of people gaming the vote, mostly with people taking advantages of paradoxes in the system.

    August 22, 2014

  • CJ F.

    There is another way in which I think Szpiro exaggerates the unfairness of congressional appropriation. When you carefully read the details in chapter 12, it becomes clear that all divisor methods are immune to the population, Alabama, and new state paradoxes (p. 199). Then on p. 195 he shows that of the divisor methods the Webster-Willcox method is the only divisor method with no bias.

    There is one imperfection that all the divisor methods suffer: they can be "out of quota" (see pp. [masked]). This is the justification for Szpiro's claim that even apportionment is mathematically inherently unfair. But again if you read the text carefully, being "out of quota" means the rounding of some delegations may exceed 0.5 from the fractional allocation the state "deserves". OK, that is a problem. But it happens randomly. It may be possible to try other divisors (the text is unclear here). Finally, the Webster method is out of quota only 0.06% of the time! It seems to be mathematically fairest.

    August 21, 2014

    • CJ F.

      One more point: it should be noted that since 1940 Congress has used the Huntington-Hill method. This method uses the geometric mean which minimizes the relative differences in the number of constituents (or population) per representative. But it is "out of quota" nearly 0.3% of the time and it is biased toward small states. So it is much worse than the Webster-Willcox method.

      However, two distinguished panels (one of which included John von Neumann!) of the National Academy of Sciences declared it superior to Webster-Willcox! And Harvard mathematician Huntinton argued vigorously that it was the best appoportionment method. How can such modern mathematicians be so blinded by a fascinating mathematical proof that distracts one's attention from the bias inherent in the Huntington-Hill method? Is mathematics a game of politics or is it definitive and verifiable?

      August 21, 2014

  • CJ F.

    One of the most detailed parts of George G. Szpiro's book "Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present" concerns apportionment (allocating seats in parliament or Congress).

    Szpiro writes on p. 187: "a fair and true allocation of seats in Congress is also a mathematical impossibility". But a careful reading of the chapter suggests to me that our author is exaggerating!

    The problem is the difficulty in allocating fractional humans or fractional votes to Congress. On p. 196, Szpiro outlines a plan for allocating an extra representative who would have a fractional vote. Congress might grow by 25 congresspeople, but would still have just 435 votes. It seems to me that it would be better to divide the fractional vote among that states other delegates. Wouldn't that solve the problem? And Szpiro reports that a constitutional amendment would not be required. Is it a problem that reps would not all be worth the same number of votes?

    August 20, 2014

  • CJ F.

    One of the monumental results from the mathematics of voting is Arrow's Impossibility Theorem. It shows, according to George Szpiro, that "No democratic constitution exists that produces a coherent method of social choice; only a dictatorship can fulfill a handful of innocuous sounding conditions." (p. 178).

    Szpiro's discussion of the two axioms and five conditions for a good social choice aggregation function is very good. But he provides no proof and no intuition of the necessity of Arrow's theorem. Can anyone explain it?

    Wikipedia attempts to do so:

    Or you could read Arrow's original 1950 paper "A Difficulty in the Concept of Social Welfare" (The Journal of Political Economy, Vol. 58, No. 4., pp. 328-346.

    Or the 2nd edition of Arrow's book "Social Choice and Individual Values":

    Can anyone explain this Sat?

    1 · August 18, 2014

  • CJ F.

    Although there aren't many proofs in George G. Szpiro's book "Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present" , there is a deep one on p. 164. It is the proof that the geometric mean minimizes the relative differences in the number of constituents (or population) per representative.

    There are several steps in Szpiro's proof that I don't understand (namely the ones relating the index i and j on different sides of the inequality. Can you really do that?) I also find it difficult to keep present in my mind the meaning of the theorem with its symbolic notation. Can you explain each line of this proof? If people are interested, we could make this a focus next Saturday. Can you explain easily? Why are the simple arithmetical "proofs" in this book so vexing?

    Huntington's original paper in the Trans. Amer. Math. Soc. 30 (1928),[masked] can be downloaded at

    August 16, 2014

  • CJ F.

    OK, I'm trying to understand Szpiro's parenthetical comment on p. 123: "Actually, any divisor between 28,356 and 28,511 would have worked". I think he got the math wrong. I used the R programming language to analyze this because it is a dataset and R is great with data sets. I wrote this csv file 1790.pop (from Table 9.2 on p. 122):

    Load it into a vector:
    pop <- read.csv("1790.pop",
    [1] 121
    [1] 120
    So the minimal divisor is 28,365 not 28,356!
    > sum(pop%/%28511)
    [1] 120
    > sum(pop%/%28512)
    [1] 119
    So the maximal divisor is 28,511 as Szpiro says. Arrgh, I hate it when typos enter a math text. But it seems to be a simple digit transposition issue.

    Is there any way to determine the minimal and maximal divisors without a trial-by-error binary search?

    August 12, 2014

    • CJ F.

      I need some help with apportionment. On p. 124, Szpiro writes "the number of seats in the House grew from 104 to 240" (by 1830). But on p. 123 and in Wikipedia (http://en.wikipedia.o...­) the correct number is 105. Also on page 124, he says "It remained in force for fifty years, until 1830". But 1830 - 1790 = 40 years!

      He also writes "Delaware, ... with 'raw' numbers of seats of 1.61, 1.78". Checking against the 1800 census which also used a divisor of 33,000 (­) (census data:­), I deduce that Delaware had 58,120 free persons and 6,153 slaves. Article I, Section 2, Clause 3 of the United States Constitution (­) says we count 3/5 of a slave. So I get 58120+(6153*3/5) = 61811.8. [masked]/33000 =[masked] NOT 1.78. Another transposition error?

      August 12, 2014

  • Sam B.

    Would anyone out there be interested in this meetup being broadcasted on Skype or Google hangouts? It's been mentioned to me a couple of times, and I've started thinking about it more and how to actually do it. Most likely, it will be a hangout since I have an android tablet and most of probably have a youtube, google, gmail, or android account. Any questions, comments, concerns, or if you have experience with the sort of thing, message me or reply to this thread.

    August 7, 2014

  • Roy B

    Sorry for the off-topic comment, but I thought the group might be interested in this Coursera offering (which starts today, Aug 1) and/or book: Learning How to Learn: Powerful mental tools to help you master tough subjects (, or "A Mind for Numbers" (

    August 1, 2014

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