Allocating Electoral College Seats
I was thinking about the 2016 U.S. Presidential election  unfortunately, that happens a lot these days. The number of electors allocated to each state depends on that state’s population, but the allocation is only updated after a census. I was curious how the allocation of electoral seats would change if the census were performed just before the 2016 election and whether that would have made the Electoral College result closer (spoilers: yes, but not by much).
For this thought experiment, I did the updated allocation using a small mixed integer program. Integer programming is a powerful (and fun) technique, and it’s well suited to modeling different notions of fairness. It’s not, however, the tool used for the real allocation: apparently U.S. House seats (and by extension Electoral College seats) are allocated using the HuntingtonHill method. I know approximately nothing about fair allocation in general or the HuntingtonHill method in particular; feel free to educate me.
Deciding how to allocate seats is a problem more philosophical than mathematical. Is the goal for each elector to represent about the same number of people? To limit under and/or over representation? How do you balance a few people being underrepresented by a lot versus a lot of people being underrepresented by a little? I don’t have more than a rough intuitive answer to these questions, but I think the following is at least defensible:
A particular allocation of seats would seem to be bad to the extent that some people are denied political agency. If, for example, Colorado were given only one seat in the Electoral College, that would seem unfair to the people of Colorado because they have disproportionately little political voice. So for this problem I set the objective as minimizing underrepresentation.
The population data I used is from Wikipedia (onawhim experiments = easiest data available) and represents statelevel population as of July 1st, 2016. For each state [^{1}] $i$ let $p_i$ be the state’s population and $f_i$ be the proportion of the nation’s population living in that state, i.e. $f_i = p_i / \sum_j p_j$. For each state $i$, I denote by $c_i$ the number of seats allocated in the Electoral College. Ideally, $\frac{c_i}{538} = f_i$ for every state, but of course this isn’t possible. For each state, the underrepresentation is $\left(f_i  \frac{c_i}{538}\right)_+$, i.e. 0 if the state has at least as many seats as its population would suggest, otherwise the difference between its population share and its electoral seat share.
Because some states have much larger populations than others, it’s appropriate to weight each state’s underrepresentation by its population. Intuitively, it’s better for 100,000 people to be slightly underrepresented than for 1,000,000 people to be underrepresented by the same amount. So the objective will be to minimize \(\sum\limits_{i=1}^{51} p_i \left(f_i  \frac{c_i}{538}\right)_+\).
That last term is nonlinear in the decision variables $c$, but we can use a standard modeling trick to make everything linear. The formulation is actually quite small:
\[\begin{align*} \underset{c,u}{\text{minimize}}\quad & \sum_{i=1}^{51} p_i u_i \\ \text{such that}\quad & u_i \geq f_i  \frac{c_i}{538} \quad \forall i \\ & c_i \geq 3 \quad \forall i\\ & c_i \text{ integer} \\ & u_i \geq 0 \end{align*}\]Notice that variable $u_i$ is greater than both 0 and the misrepresentation of state $i$. If state $i$ is overrepresented, then $u_i$ will be 0 (since we minimize with a positive coefficient on $u_i$), and if state $i$ is underrepresented, then $u_i$ will be the underrepresentation. This problem is pretty compact; on my aging computer, Gurobi solves it in 0.14 seconds. Nice!
Here’s the comparison between the current and suggested allocation:
 State  Population  Current seats  Proposed seats 

 Alabama  4863300  9  8 
 Alaska  741894  3  3 
 Arizona  6931071  11  11 
 Arkansas  2988248  6  3 
 California  39250017  55  66 
 Colorado  5540545  9  9 
 Connecticut  3576452  7  3 
 Delaware  952065  3  3 
 District of Columbia  681170  3  3 
 Florida  20612439  29  35 
 Georgia  10310371  16  17 
 Hawaii  1428557  4  3 
 Idaho  1683140  4  3 
 Illinois  12801539  20  22 
 Indiana  6633053  11  11 
 Iowa  3134693  6  3 
 Kansas  2907289  6  3 
 Kentucky  4436974  8  7 
 Louisiana  4681666  8  7 
 Maine  1331479  4  3 
 Maryland  6016447  10  10 
 Massachusetts  6811779  11  11 
 Michigan  9928301  16  17 
 Minnesota  5519952  10  9 
 Mississippi  2988726  6  3 
 Missouri  6093000  10  10 
 Montana  1042520  3  3 
 Nebraska  1907116  5  3 
 Nevada  2940058  6  3 
 New Hampshire  1334795  4  3 
 New Jersey  8944469  14  15 
 New Mexico  2081015  5  3 
 New York  19745289  29  33 
 North Carolina  10146788  15  17 
 North Dakota  757952  3  3 
 Ohio  11614373  18  19 
 Oklahoma  3923561  7  6 
 Oregon  4093465  7  6 
 Pennsylvania  12802503  20  22 
 Rhode Island  1056426  4  3 
 South Carolina  4961119  9  8 
 South Dakota  865454  3  3 
 Tennessee  6651194  11  11 
 Texas  27862596  38  47 
 Utah  3051217  6  3 
 Vermont  624594  3  3 
 Virginia  8411808  13  14 
 Washington  7288000  12  12 
 West Virginia  1831102  5  3 
 Wisconsin  5778708  10  9 
 Wyoming  585501  3  3 
Or, in graphical form:
Roughly, it seems like big states get more seats and mediumsized states get fewer. Small states are already held up by the floor of 3 seats (two senators and a representative), so they don’t lose any seats.
Under this electoral assignment (and assuming no faithless electors), Clinton would have received 237 electoral votes  actually pretty close to the 232 (again, ignoring faithless electors) she actually received. Like in 2000, Florida would have been enough to swing the election, and instead of talking about ~50,000 people in Wisconsin, Michigan, and Pennsylvania, we’d be talking about ~40,000 people in just Michigan and Pennsylvania.
This result was actually a bit of a surprise to me: I thought with people generally moving towards cities in big states and cities generally voting blue, the difference would have been larger (though probably not large enough to change the overall election result). But in fact while this assignment of votes would have helped Clinton in places like California and New York, it would have hurt her in places like Connecticut and Texas.
I’m not sure there are meaningful lessons to take from all this. That electoral seats should be reallocated more frequently? Maybe, but the census is already underfunded. Perhaps this further emphasizes the terrible importance of voter suppression, which appears to have been quite successful in Florida in 2016. Mainly, integer programming is cool.

Counting D.C. as a state, which is why $i$ goes from 1 to 51. ↩