Coalitional Instability and the Three‐Fifths Compromise

Author Summary by Gordon Ballingrud and Keith L. Dougherty

Were the Constitution’s two methods of legislative apportionment inevitable? This paper determines the coalitional stability of apportionment rules proposed at the Constitutional Convention assuming the Convention limited itself to the rules proposed. An apportionment rule is a criterion for dividing legislative seats, such as the relative populations of each state or the relative values of property within them. Such rules are coalitionally stable if there does not exist another apportionment rule that a majority of voters (in this case, states) prefers to it.

Using each state’s vote share as a measure of state preference, we find that the stability of the apportionment rules proposed at the Constitutional Convention depended upon which states were present. Equal apportionment was in equilibrium (stable) with thirteen states present, as in the Continental Congress, but when Rhode Island and New Hampshire were absent during the first third of the Convention, all of the apportionment rules proposed at the Convention were in a top cycle. That means that if delegates voted to increase their state’s vote share in a pairwise vote, the Convention could move from any one of the proposed rules of apportionment to any other through a series of majority-rule votes.

The Three-Fifths Clause was proposed by James Wilson during this chaotic period, perhaps as an attempt to ground the national legislature on popular rule. With New York departing near the middle of the Convention, equal apportionment and the Three-Fifths Clause became stable -— each of these could not be beaten by any of the other eight rules proposed at the Convention.

This helps us to explain why the Great Compromise was finally reached. We conclude that the Great Compromise was partly the result of historical contingency (i.e., which states participated in in voting and which apportionments were proposed), rather than necessity (as claimed by some historians and legal scholars).

About the Authors: Gordon Ballingrud is Ph.D. Candidate and Keith L. Dougherty is a Professor both in the Department of Political Science at the University of Georgia, Athens. Their research, ”Coalitional Instability and the Three‐Fifths Compromise“ is now available in Early View and will appear in a forthcoming issue of the American Journal of Political Science.