Unique games conjecture

The Huntington–Hill method of apportionment assigns seats by finding a modified divisor D such that each constituency's priority quotient (population divided by D ), using the geometric mean of the lower and upper quota for the divisor, yields the correct number of seats that minimizes the percentage differences in the size of the congressional districts. When envisioned as a proportional voting system, this is effectively a highest averages method in which the divisors are given by $D=\sqrt{n(n+1)}$, n being the number of seats a state is currently allocated in the apportionment process (the lower quota) and n+1 is the number of seats the state would have if it is assigned to this state (the upper quota).

The United States House of Representatives uses this method of apportionment to assign the number of representative seats to each state.

The method is credited to Edward Vermilye Huntington and Joseph Adna Hill.

Examples

The U.S. House of Representatives guarantees one seat for each state. So, each state starts off with a divisor D of 1.41 (the square root of the product of one, the number seats currently assigned, and two, the number of seats that would next be assigned). Each state's census population is divided by its D, to produce a priority value for each state. Since all of the D's are equal for this first seat, the state with the largest population would win (California).

Now, California's D would be changed to 2.45 (the square root of two times three), and we would repeat this process. Unless California has about twice the population of any other state, the next largest state would win this second seat. In this case Texas would win the next seat because its population (24 million) divided by its current D (still 1.41) is greater than California's similarly calculated priority value (37 million/2.45) for priority values of 17 million and 15 million, respectively).

This process is repeated until all empty house seats have been exhausted. If the number of U.S. House of Representatives seats were equal in size to the population of the United States, this method would guarantee that the appointments would equal the populations of each state.

Although the U.S. House of Representatives currently uses the Equal Proportions method, Congress has not always used it. In fact, George Washington used the presidential veto power for the very first time in order to block apportionment legislation less favorable to his home state of Virginia. Had Congress used the Equal Proportionment Method (with a divisor of 34,800) to apportion House seats according to state population following the 1790 census, the 105 seats in the 3rd United States Congress the House of Representatives would have been apportioned as follows:

State	Population	Quotas	Lower	Upper	G. Mean	Rnd. Dir.	Seats
Connecticut	236,841	6.81	6	7	6.48	up	7
Delaware	55,540	1.60	1	2	1.41	up	2
Georgia	70,835	2.04	2	3	2.45	down	2
Kentucky	68,705	1.97	1	2	1.41	up	2
Maryland	278,514	8.00	8	9	8.49	down	8
Massachusetts	475,327	13.66	13	14	13.49	up	14
New Hampshire	141,822	4.08	4	5	4.47	down	4
New Jersey	179,570	5.16	5	6	5.48	down	5
New York	331,589	9.53	9	10	9.49	up	10
North Carolina	353,523	10.16	10	11	10.49	down	10
Pennsylvania	432,879	12.44	12	13	12.49	down	12
Rhode Island	68,446	1.97	1	2	1.41	up	2
South Carolina	206,236	5.93	5	6	5.48	up	6
Vermont	85,533	2.46	2	3	2.45	up	3
Virginia	630,560	18.12	18	19	18.49	down	18

Compared with the actual apportionment, Pennsylvania and Virginia would have lost one seat each, while Delaware and Vermont would have gained one seat each.

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