# Logical Methods in the Humanities

## The Logic and Mathematics of Voting Procedures

Weekly seminar using logic and mathematics to study voting procedures.

October 17, 2008: Michel Balinski, Ecole Polytechnique and CNRS, Paris
The Majority Judgement: A New Theory and Method of Voting

The traditional model of social choice fails for two reasons: Its logical implications: The Condorcet paradox, Arrow's paradox, ... , more fundamentally of course, Arrow's and Gibbard-Satterthwaite’s impossibility theorems, and aberrations. Its inadequacy as a model of reality: It assumes that the voters of an electorate or judges of a jury have rank-orders in their minds. This is false as practice and experimentation proves.

A more realistic model leads to possibility theorems and the characterization (in many different ways) of the majority judgement as the unique mechanism that satisfies almost all of the desirable properties identified over the years and where it does not, it does the best possible.

A brief exposition of this thesis will be presented, together with a complete description of the mechanism itself (US patent pending).
(Joint work with Rida Laraki, Ecole Polytechnique and CNRS, Paris)

References:
October 23, 2008: Steven J. Brams (NYU)
How Democracy Resolves Conflict in Difficult Games
Steven J. Brams and D. Marc Kilgour

Democracy resolves conflicts in difficult games like Prisoners' Dilemma and Chicken by stabilizing their cooperative outcomes. It does so by transforming these games into games in which voters are presented with a choice between a cooperative outcome and a Pareto-inferior noncooperative outcome. In the transformed game, it is always rational for voters to vote for the cooperative outcome, because cooperation is a weakly dominant strategy independent of the decision rule and the number of voters who choose it. Such games are illustrated by 2-person and n-person public-goods games, in which it is optimal to be a free rider, and a biblical story from the book of Exodus.
November 3, 2008: Rohit Parkh (CUNY)
Belief revision and the logic of campaigning

The talk divides naturally into two halves. Both have to do with belief revision but are otherwise distinct. The first half concerns the AGM theory of belief revision and some results about language splitting, relevance and some generalizations of Craig's lemma. The results are by myself, Korousias and Makinson, and Peppas, Chopra and Foo.

The second part concerns a theory of how voters see a candidate in terms of things she has said in the past and what she might say next to improve her standing in their eyes.

This is joint work with Walter Dean, a CUNY student who is graduating soon.
November 20, 2008: Kenneth Arrow (Stanford)
The Subject-matter of Social Choice Theory

More questions will be asked than answers given. Social choice theory has both normative and descriptive implications. The normative viewpoint is itself complicated. (1) If individuals are rational, "normative" and "descriptive" overlap. (2) Can rationality be ascribed to collectivities? (3) How is social choice theory related to political philosophy?

Descriptively, one can apply social choice thinking to elections (where intransitivities can sometimes be inferred), to legislation (agenda control), or to judicial procedures.
December 4, 2008: Donald G. Saari (UC Irvine)
Mathematics of Voting and Social Choice

Mathematics has been used to understand the source of voting problems probably since the work of the mathematician J.C Borda back in 1770. While progress has been made, so many puzzles remain -- enough so that one should worry whether election outcomes really reflect the views of the voters. In this talk, which assumes no expertise in the area, it will be shown how mathematics explains a wide number of problems including Arrow's famous result, all of those voting paradoxes, and how extensions of these ideas provide insight into concerns ranging from engineering to multiscale analysis issues.
December 5, 2008: Todd Davies (Stanford)
Intuitive Preference Aggregation: Testing Two Criteria in Arrow's Impossibility Theorem

I will report on a series of experiments in which respondents were shown hypothetical preference profiles for seven or fewer voters' rankings of six or fewer alternatives. Aggregation judgments were tested for adherence to two normatively controversial social choice principles underlying Arrow's impossibility theorem (1951, 1963): independence of irrelevant alternatives (IIA) and collective rationality (CR). When pairwise majority winners are large Borda count losers, both naive and sophisticated respondents violate IIA, CR, and a weaker independence principle across rank and pair-comparison format tests, between and within-respondents. Smaller Borda disparities result in presentation effects. Behaviorally-supported criteria include independence of irrelevant voters and positionality-based interpersonal comparisons of utility. I will discuss the meaning of experiments like this for both normative and descriptive social choice theory. (joint work with Raja Shah, Katarina Ling, and Renee Trochet)