The NRA successfully lobbied to stop gun control legislation. Several Democrats sided with Republicans to defeat it. But the NRA seems to have spent more than necessary to defeat the measures because they failed by more than a one-vote margin. It would have been enough to buy exactly the number of Senators necessary to prevent the bill from progressing through the Senate, no more than that.
But in fact the cost of defeating legislation is decreasing in the number of excess votes purchased. If the NRA has already secured enough votes to win, the next vote cannot be pivotal and so the Senator casting that vote takes less blame for the defeat. Indeed if enough Senators are bought so that the bill goes down by at least two votes, no Senator is pivotal.
Here’s a simple model. Suppose that the political cost of failing to pass gun control is c. If the NRA buys the minimum number of votes needed to halt the legislation it must pay c to each Senator it buys. That’s because each of those Senators could refuse to vote for the NRA and avoid the cost c. But if the NRA buys one extra vote, each Senator incurs the cost c whether or not he goes along with the NRA and his vote has just become cheaper by the amount c.
For the Vapor Mill: What is the voting rule that maximizes the cost of defeating popular legislation?
Cheap Talk 
5 comments
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April 30, 2013 at 12:48 am
Tayfun Sonmez
Hi Jeff,
Great model but in this particular case I wonder to what extent the cost to Senators with NRA-supporting vote is decreasing in the number of excess votes over that needed to fail the popular legislation. I look forward to seeing long term implications of the defeat.
April 30, 2013 at 3:29 am
Niall
I think this is the same intuition as Ernesto DalBo’s 2007 AJPS paper where a lobbyist conditions his bribe on pivotality.
April 30, 2013 at 8:02 am
Bryce
You can’t read too much into the margin, because the outcome becomes common knowledge among senators before the actual vote. Once they know they won’t be pivotal, some senators may take advantage of a low-cost opportunity to appease the NRA, even if they weren’t primary lobbying targets.
April 30, 2013 at 9:44 pm
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May 1, 2013 at 10:17 pm
mkovach
You can also get the same result with a competing lobbyist. If you buy a minimal winning coalition it’s easier for a competitor to “buy them back,” unless you pay a very large bribe. Thus the more powerful lobby buys a super-majority, which is ultimately cheaper.
http://www.jstor.org/discover/10.2307/2082886?uid=3739560&uid=2460338415&uid=2129&uid=2&uid=70&uid=4&uid=83&uid=63&uid=3739256&sid=21102134516571