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It occurred to me that if one is a supporter of X party living in a far Y state, one might cast an indirect vote against Y in the house and presidential races by not taking part in the census. Due to apportionment, if there were enough boycotters, the state would lose electors and eventually representatives.
Besides the potential personal ramifications (e.g. a fine under US title 13), how might this backfire in other ways?
This is a bad idea for two reasons: it'll cause a decrease in state funding for where you live, and it may be tricky to actually cause your preferred party to get that representative/elector.
First, there are definitely other ways it would backfire, since census data isn't only used for apportioning representatives. It is also used for determining where federal and state funding should go:
How Our Data Are Used
...
To distribute more than $675 billion in federal funds to local, state and tribal governments each year.
Census data informs how states and communities allocate funding for:
Neighborhood improvements
Public health
Education
Transportation
Much more
If a large group of people of some political party in some state boycotted the census and it wasn't corrected, then that state would receive a smaller portion of federal funding, which would probably be spread fairly evenly over other states (i.e., both Party X and Y in other states benefit).
However, state funding for roads, hospitals, schools, etc. would only be diverted away from areas where the Party X boycotters are concentrated, to the benefit of Party Y supporters as they would receive a portion of the diverted funding.
Second, if you want a deep Y state to lose a representative, presumably you want that representative to go to a Party X state in order to represent the party you prefer. However, there is no guarantee that the representative wouldn't just go to another deep Y state.
After a census, representatives are reapportioned based on a formula intended to appropriate representation. The gist of it is that each state starts with 1 representative, and then the next 385 are given one at a time to states based on a ration between population and representatives it was already given. Here's the order of apportion for the last census.
So, not only would you need to get enough people from some party in some state to boycott the census and drop that state out position for what would have been its final representative, you wouldn't really help your party unless the new order of apportion would result in your party getting that extra representative.
Someone with a political science or mathematics degree (or just some spare time) might be able to calculate which states and how many boycotters are needed to give a new apportionment order that helps one party, but I can almost guarantee one thing: if tens of thousands of Party X voters suddenly disappeared from a deep Y state, somebody's going to notice and try to correct it.
Getting anywhere close to enough traction to even have an effect is a big hurdle. The average house seat represents 700,000 people, this is the upper bound for the number of people required to boycott a census. Though in the right state the required number can be drastically lower, 15,734 people could have moved a seat from Minnesota to North Carolina in 2010. If you manage to get enough people to all boycott and cause your state to lose a seat, the next problem is that it may end up being granted to a state you disagree with as much or more. The apportionment process grants seats to try and keep all states as close as possible for number of people per representative.
If your state has more than one representative the next problem could occur in the redistricting process. By boycotting the census you will cause whatever district you become a part of to grant less representation to everyone in it than intended, this disenfranchise you neighbors to some extent. There is also the possibility that the resulting districts will be drawn unfavorably for your goals or that a district though to lean in one direction actually lean more in the opposite direction. At the state level it would be a much larger impact as state representative districts could be drawn in ways that turn you in to an extreme minority further polarizing your state further away from your beliefs.
Interestingly enough, there's already a great deal of nonresponse in the census, they've calculated the Wave 1 nonresponse rate has grown to nearly 20% from 5%. This nonresponse bias is "corrected" by weighting those who did respond. The census already knows of your existence using administrative records (another way they correct for nonresponse) so apportionment likely wouldn't change but research that relies on your response (economic data, housing data, education data) that heavily influences government spending would be.
Fun fact: It's also a federal crime to not respond to the census, although it seems the government hasn't prosecuted anyone since the 1960 census.
Note that your boycott may be ineffective. The census bureau tries to correct for missing or inaccurate data in a process called imputation.
In addition to this being illegal, I also think it undermines the democratic system in much the same way as voter fraud and that harms everyone.
If you really are determined, you could actually move to a state in which you agree with the majority. You just need to establish usual residence there on census day (1 April 2020). That way your small contribution to apportionment will be transferred to that state in a legal fashion.
Remember also that if Party X won the House seat, there's a good chance that Party X controls the state legislature which controls drawing Congressional districts. So let's say you're successful in reducing the number of districts by 1. It's highly likely the redrawn district will favor Party X. Thus you would (in theory) help the party you want to harm, at least in the short term.
