# Balinski–Young theorem

In 1983, two mathematicians, Michel Balinski and Peyton Young, proved that any apportionment method will result in paradoxes for three or more parties (or states).[1][2] The theorem shows that any possible method used to allocate the remaining fraction will necessarily fail to always follow quota. More precisely, their theorem states that there is no apportionment system with the following properties for all house sizes:[3]

• Quota rule: Each party gets one of the two numbers closest to its fair share of seats. For example, if a party's fair share is 7.34 seats, it must get either 7 or 8 seats; any other number will violate the rule.
• Pairwise population monotonicity: If party A gets more votes, with no change in party B's votes, A will not lose a seat to B.

## Specific systems

Methods may have a subset of these properties, but can't have all of them:

• The only methods that are free of the population paradox are the divisor methods.[4] Huntington-Hill, the method currently used to apportion House of Representatives seats, is one of them. However, these methods will fail the quota rule in some elections.
• Such failures can be extremely rare under a good apportionment system--so far, no apportionment of the US congress under Huntington-Hill or Webster would have failed to maintain quota.
• Largest remainder methods like Hamilton obey the quota rule but fail pairwise population monotonicity.
• No method may always follow quota and be free of the population paradox.[4]

Many mathematically analogous situations arise in which quantities are to be divided into discrete equal chunks. The Balinski–Young theorem applies in these situations: it indicates that although very reasonable approximations can be made, there is no mathematically rigorous way in which to reconcile the small remaining fraction while complying with all the competing fairness elements.

## House monotonicity

The Balinski-Young theorem is sometimes quoted with an unnecessary condition called House monotonicity, which says that adding more seats should not cause a party to lose seats. However, this condition turns out to be irrelevant. Balinski and Young have constructed a method that satisfies both quota and house monotonicity, although it is not in common use.[5] It is also possible to satisfy both population and house monotonicity (with divisor methods); thus, house monotonicity is compatible with either of the other two criteria.

However, the quota rule is incompatible with population monotonicity.

## History

The division of seats in an election is a prominent cultural concern. In 1876, the United States presidential election turned on the method by which the remaining fraction was calculated. Rutherford Hayes received 185 electoral college votes, and Samuel Tilden received 184. Tilden won the popular vote. With a different rounding method the final electoral college tally would have reversed.

## Notes

The quota rule is incompatible with immunity to vote management. See Divisor method#Notes for an example.

For proportional representation, the paradox can theoretically be sidestepped by giving party representatives slightly different voting power to correct for any error. For instance, if 13.57% of the voters voted for party A, the total voting power of party A's representatives could be set to 13.57% of the total. However, it makes counting votes much more opaque. This solution is not employed by any current political assembly.