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?

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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: http://www.ams.org/notices/201101/rtx110100059p.pdf

CJ Fearnley

Sam Bledsoe

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