In this talk, I will introduce the mathematics and applications of election procedures and apportionment methods.

Elections are easy when there are only two candidates: vote by majority rule.  For three or more candidates, Kenneth Arrow’s Impossibility Theorem shows that no three-candidate election procedure satisfies a set of reasonable axioms, implying that there is no “best” election procedure.  After discussing the axiomatic approach in voting theory, we will review commonly used election procedures, including the use of ranked choice voting in East Pointe, Michigan as part of the resolution of a Voting Rights Act lawsuit.

In the context of the US House of Representatives, the apportionment problem is to determine the number of representatives each state receives in the House.  We will review the history and mathematics of apportioning the House, including its relationship to the Electoral College.  We will conclude with the recent use of apportionment methods to allocate delegates among candidates in the Democratic and Republican presidential primaries.

In this talk, I will introduce the mathematics and applications of bankruptcy problems and matching problems.

When a firm goes bankrupt, the firm’s assets are divided among the firm’s creditors based on how much each creditor is owed.  This is known as the bankruptcy problem.  The history of bankruptcy problems dates back 2000 years to a passage in the Babylonian Talmud.  We will consider the history of the Talmud problem and its relationship to cooperative game theory.  As an application, we will connect bankruptcy problems to the problem of reparations.  Further, we will design a mechanism to apply noncooperative game theory to award travel funds, a problem that is similar to a bankruptcy problem.

David Gale and Lloyd Shapley introduced the stable marriage problem and an algorithm to match spouses in a stable way.  The algorithm had been in use by the National Resident Matching Program (The Match) to match doctors to hospital residency training programs before the Gale-Shapley article. We will review the Gale-Shapley algorithm and discuss the application of algorithms to solve the school choice problem of matching students to public schools (as used in Boston and New York).