Complexity-Theoretic Aspects of Political Science

UR-CS Participating Faculty: Lane Hemaspaandra.

Project Description

This project studies complexity-theoretic aspects of political science--in particular, of voting theory. One focus is an experimental study of Congressional apportionment, and the other is a theoretical study of voting systems.

Regarding the latter, the Condorcet criterion is that an election is won by any candidate who defeats all others in pairwise majority-rule elections. The Condorcet Paradox, dating from 1785, notes that not only is it not always the case that Condorcet winners exist but, far worse, when there are more than two candidates, pairwise majority-rule elections may yield strict cycles in the aggregate preference even if each voter has non-cyclic preferences. (The standard example is an election over candidates a, b, and c in which one third of the voters have preference tex2html_wrap_inline40 , one third of the voters have preference tex2html_wrap_inline42 , and one third of the voters have preference tex2html_wrap_inline44 . In this case, though each voter individually has well-ordered preferences, the aggregate preference of the electorate is that b trounces a, c trounces b, and a trounces c. In short, individually well-ordered preferences do not necessarily aggregate to a well-ordered societal preference.) This is a widely discussed and troubling feature of majority rule.

In 1876, Charles Lutwidge Dodgson--more commonly referred to today by his pen name, Lewis Carroll--proposed an election system that is inspired by the Condorcet criterion (Carroll did not use this term--indeed, Black has shown that Carroll ``almost beyond a doubt'' was unfamiliar with Condorcet's work), yet that sidesteps the abovementioned problem. In particular, a Condorcet winner is a candidate who defeats each other candidate in pairwise majority-rule elections. In Carroll's system, an election is won by the candidate who is ``closest'' to being a Condorcet winner. In particular, each candidate is given a score that is the smallest number of exchanges of adjacent preferences in the voters' preference orders needed to make the candidate a Condorcet winner with respect to the resulting preference orders. Whatever candidate (or candidates, in the case of a tie) has the lowest score is the winner. This system admits ties but, as each candidate is assigned an integer score, no strict-preference cycles are possible.

Bartholdi, Tovey, and Trick, in their paper ``Voting Schemes for which It Can Be Difficult to Tell Who Won the Election,'' raise a difficulty regarding Carroll's election system. Though the notion of winner(s) in Carroll's election system is mathematically well-defined, Bartholdi et al. raise the issue of what the computational complexity is of determining who is the winner. Though most natural election schemes admit obvious polynomial-time algorithms for determining who won, in sharp contrast Bartholdi et al. prove that Carroll's election scheme has the disturbing property that it is NP-hard to determine whether a given candidate has won a given election (a problem they dub DodgsonWinner), and that it is NP-hard even to determine whether a given candidate has tied-or-defeated another given candidate (a problem they dub DodgsonRanking).

Bartholdi, Tovey, and Trick's NP-hardness results establish lower bounds for the complexity of DodgsonRanking and DodgsonWinner. A central focus of this project has been the exact analysis of the complexity, and we have recently achieved this. Ongoing research in this project studies the complexity of other voting systems for which the complexity of determining the winner remains an open issue.

References

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This is a list of selected recent papers, from or related to this project, by University of Rochester authors. Essentially all the papers listed below can be found, in their full technical report versions, in the UR-CS Technical Report Archive's theory section.

HHR
E. Hemaspaandra, L. Hemaspaandra, and J. Rothe. Exact analysis of Dodgson elections: Lewis Carroll's 1876 voting system is complete for parallel access to NP. Journal of the ACM. To appear.

HHR97
E. Hemaspaandra, L. Hemaspaandra, and J. Rothe. Raising NP lower bounds to parallel NP lower bounds. SIGACT News, 28(2):2-13, 1997.

HRSZ
L. Hemaspaandra, K. Rajasethupathy, P. Sethupathy, and M. Zimand. Power balance and Congressional apportionment algorithms. ACM Journal of Experimental Algorithmics. Accepted subject to minor revision.

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Lane A. Hemaspaandra