Werner's Blog — Opinion, Analysis, Commentary
The mathematics of US Senate reform
Seal of the US Senate

As the United States celebrates its Independence Day today, it is perhaps an appropriate day to reflect on the constitutional landscape of the US. After the US Supreme Court recently ruled that federal courts have no jurisdiction over drawing boundaries of electoral districts, the ball is back in the court of individual states to find ways to limit gerrymandering. While state initiatives and referendums may help put the drawing of district boundaries into the hands of politically neutral and independent electoral boundary commissions (which is what we do in Canada), the disparity of power and electors is nowhere more distorted than in the United States Senate, which allocates two seats to each state. The result is that in Wyoming and Vermont each of the two senators represents only about 600,000 people, while each senator in California represents about 40 million people. What could be more unfair?

Eric Orts, a professor colleague at the Wharton School, thinks so to. He penned an article in The Atlantic earlier this year that got me thinking about how to model optimal seat allocation under constraints. What would the US Senate look like if seats were allocated closer to population shares? Some degree of asymmetric representation may well be acceptable or even desirable, although the current immense degree of disproportionality is troubling. The rationale for asymmetric representation is to give smaller states a louder voice. However, the current ratio of asymmetric representation is ultimately overly distorting. (N.B.: Germany's Bundsrat, the upper house that represents the German Lšnder, has each state represented with either 3, 4, 5, or 6 seats. This is not perfectly proportional to population shares either, but a much closer approximation than in the United States.)

There are compromises that retain a degree of asymmetry but reduces the huge disproportionality. Naturally, representing states, a senate with 100 senators should have at least one senator from each state. This allocates 50 of the 100 seats. How should the remaining 50 seats be allocated? This is an optimization problem for which economists have some recipes. For this we need an objective function that we are trying to optimize, and three constraints: one for the total number of seats, one for the minimum number per state (1), and one for the maximum number per state (M). A variation of the the Gallagher Disproportionality Index that is weighted by state populations is a good start for this: \[G^\ast = \frac{10^6}{2} \sum_{i=1}^{50} p_i\max\{0,p_i-s_i\}^2 \] where \(p_i\in(0,1)\) is the population share of state \(i\) and \(s_i\in(0,1)\) is the seat share of that state. The metric above penalizes under-representation, and thus the objective is to minimize \(G^\ast\).

The table below shows the results of maximizing this objective function subject to the three constraints and the resulting number of seats for each share when the maximum number of seats per state (M) is varied from the current 2 to 3, 4, 5, 6 and 15. Population in millions and percent of total are shown as well for each state. Any state that has a population share of more than 2% is currently under-represented (e.g., California, Texas), and any state that has a population share of less than 2% is over-represented (e.g., Wyoming, Vermont).

StatePopulation Maximum Seats Allocation
Mio. [%] 2 3 4 5 6 15
Alabama 4.89  1.50  2 3 2 2 1 1
Alaska 0.74  0.23  2 1 1 1 1 1
Arizona 7.17  2.20  2 3 2 2 2 2
Arkansas 3.01  0.92  2 1 1 1 1 1
California 39.56  12.12  2 3 4 5 6 12
Colorado 5.70  1.74  2 3 2 2 2 1
Connecticut 3.57  1.09  2 1 1 1 1 1
Delaware 0.97  0.30  2 1 1 1 1 1
Florida 21.30  6.52  2 3 4 5 6 6
Georgia 10.52  3.22  2 3 4 3 3 3
Hawaii 1.42  0.44  2 1 1 1 1 1
Idaho 1.75  0.54  2 1 1 1 1 1
Illinois 12.74  3.90  2 3 4 4 4 4
Indiana 6.69  2.05  2 3 2 2 2 2
Iowa 3.16  0.97  2 1 1 1 1 1
Kansas 2.91  0.89  2 1 1 1 1 1
Kentucky 4.47  1.37  2 3 2 2 1 1
Louisiana 4.66  1.43  2 2 2 2 1 1
Maine 1.34  0.41  2 1 1 1 1 1
Maryland 6.04  1.85  2 2 2 2 2 1
Massachusetts 6.90  2.11  2 3 3 2 2 2
Michigan 10.00  3.06  2 3 3 3 3 3
Minnesota 5.61  1.72  2 2 2 2 2 1
Mississippi 2.99  0.91  2 1 1 1 1 1
Missouri 6.13  1.88  2 2 2 2 2 1
Montana 1.06  0.33  2 1 1 1 1 1
Nebraska 1.93  0.59  2 1 1 1 1 1
Nevada 3.03  0.93  2 1 1 1 1 1
New Hampshire 1.36  0.42  2 1 1 1 1 1
New Jersey 8.91  2.73  2 3 3 3 3 2
New Mexico 2.10  0.64  2 1 1 1 1 1
New York 19.54  5.99  2 3 4 5 6 6
North Carolina 10.38  3.18  2 3 4 3 3 3
North Dakota 0.76  0.23  2 1 1 1 1 1
Ohio 11.69  3.58  2 3 4 4 4 3
Oklahoma 3.94  1.21  2 3 1 1 1 1
Oregon 4.19  1.28  2 3 2 1 1 1
Pennsylvania 12.81  3.92  2 3 4 4 4 4
Rhode Island 1.06  0.32  2 1 1 1 1 1
South Carolina 5.08  1.56  2 2 2 2 2 1
South Dakota 0.88  0.27  2 1 1 1 1 1
Tennessee 6.77  2.07  2 3 2 2 2 2
Texas 28.70  8.79  2 3 4 5 6 9
Utah 3.16  0.97  2 1 1 1 1 1
Vermont 0.63  0.19  2 1 1 1 1 1
Virginia 8.52  2.61  2 3 3 3 3 2
Washington 7.54  2.31  2 3 3 3 2 2
West Virginia 1.81  0.55  2 1 1 1 1 1
Wisconsin 5.81  1.78  2 2 2 2 2 1
Wyoming 0.58  0.18  2 1 1 1 1 1
Disparity Index     963.7 722.2 532.8 380.8 262.6 6.7

The number of seats are shaded green if a state is over-represented (by more than about 10%), shaded red if a state is underrepresented (also by more than about 10%), and shaded blue if a state is represented about fairly. Some of the small states (Wyoming, Delaware, Idaho) will always remain over-represented as their population share is significantly below 1%. California, the most populous state, will only gain fair representation when M exceeds 12.

Even a small shift to shuffle seats from small states to large states, limiting large states to 3 seats, brings about more fairness. However, to be fair to the large states requires significant changes and raising M to 12. In that case, California, Texas, Florida, and New York would be represented by 12, 9, 6 and 6 senators each. Even then the populous states are not large enough to form a coalition of the big against the small. To gain 50 seats, the 9 largest states would need to band together against the other 41. If asymmetric representation is meant to limit the possibility of a coalition of the "big" states against the "small" states, then limiting seats per state to 5 or 6 would be defensible. In any case, redistribution of senate seats would create fairer representation.

The discussion above left the number of seats at 100. Another method would involve leaving the minimum number of senators at 2 and adding to the overall number of senators in the Senate. Another solution to the problem would be to double the number of senators from 100 to 200, and allocate the new 100 senators according to population shares. However, a Senate with 100 members may be more effective than a Senate with 200 members.

Naturally, a big question is which political party would gain or lose from such a redistribution of Senate seats. If states with small population shares are disproportionately red and states with large population shares are disproportionately blue, then the shift would create political winners and losers. Even if there was no partisan effect of Senate reform, it would surely put the populous states against the less-populous states. What is fairness to the winners is loss of power to the losers. Fair is, simply put, when each vote counts equally and is not distorted by procedures of the electoral system that maps votes to seats.

What are the chances that the US will ever adopt Senate reform? Probably as small as they are in Canada, where we have had this dicussion for decades as well: close to nil. Canada's appointed Senate is arguably even more problematic than the elected but disproportional United States Senate. Thus there is no reason for Canadian Schadenfreude. Senate reform should be contemplated on both sides of the border—but reform is likely to remain a utopical dream of academics. Senate reform in the United States would certainly be blocked by a coalition of small states which would see their power in Washington diminish if half their seats shifted to larger states. A two-thirds majority in the Seanate for a constitutional amendment of its own reform seems utterly out of the question. There seems to be no feasible route to fairer representation.

The Senate's bias towards less populous states favours the rural states over the urban states. In an increasingly polarized political landscape, it entrenches polarization even further because political preferences align along the urban-rural divide. When the United States formed out of 13 states 243 years ago, disparity was not much of a concern. Nobody envisioned California or Texas at that time. Disproportionality in the US Senate is thus a historic accident that is likely to persist into the indefinite future. Disproportionality would be less of concern in a less partisan world and a more cooperative world.

Happy Independence Day to all of you in the United States. May your democracy flourish and prosper—and perhaps evolve one day to represent states more fairly.

Further readings and sources:

For those interested in how I obtained the results, below is the CPLEX-OPL code. Population shares are based on the 2018 population estimates of the Census Bureau.

int M = 5; int Senators = 100; {string} States = { "AL","AK","AZ","AR","CA","CO","CT","DE","FL","GA", "HI","ID","IL","IN","IA","KS","KY","LA","ME","MT", "NE","NV","NH","NJ","NM","NY","NC","ND","OH","OK", "OR","MD","MA","MI","MN","MS","MO","PA","RI","SC", "SD","TN","TX","UT","VT","VA","WA","WV","WI","WY"}; tuple Population { key string state; float popshare; } {Population} Pop = { <"AL",0.01497211>, <"AK",0.00225886>, <"AZ",0.02196758>, <"AR",0.00923169>, <"CA",0.12116781>, <"CO",0.01744617>, <"CT",0.01094349>, <"DE",0.00296256>, <"FL",0.06524230>, <"GA",0.03222237>, <"HI",0.00435113>, <"ID",0.00537334>, <"IL",0.03902740>, <"IN",0.02049800>, <"IA",0.00966764>, <"KS",0.00891828>, <"KY",0.01368723>, <"LA",0.01427405>, <"ME",0.00409969>, <"MD",0.01850954>, <"MA",0.02114208>, <"MI",0.03061864>, <"MN",0.01718769>, <"MS",0.00914809>, <"MO",0.01876603>, <"MT",0.00325396>, <"NE",0.00590957>, <"NV",0.00929469>, <"NH",0.00415499>, <"NJ",0.02728783>, <"NM",0.00641854>, <"NY",0.05986005>, <"NC",0.03180623>, <"ND",0.00232820>, <"OH",0.03580611>, <"OK",0.01207811>, <"OR",0.01283664>, <"PA",0.03922951>, <"RI",0.00323868>, <"SC",0.01557327>, <"SD",0.00270239>, <"TN",0.02073732>, <"TX",0.08791707>, <"UT",0.00968283>, <"VT",0.00191843>, <"VA",0.02609065>, <"WA",0.02308239>, <"WV",0.00553147>, <"WI",0.01780763>, <"WY",0.00176968> }; // Decision Variables dvar int Seats[States]; // Objective Function minimize sum(p in Pop) (500000*p.popshare*maxl(0,p.popshare-Seats[p.state]/Senators)^2); subject to { sum(s in States) Seats[s] == Senators; forall (s in States) Seats[s]>=1; forall (s in States) Seats[s]<=M; } // Results execute DISPLAY_RESULTS{ for(var p in Pop) writeln("SEATS: ",p.state," ",Seats[p.state]); }
Posted on Thursday, July 4, 2019 at 07:45 — #US | #Politics
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