Yesterday, the Senate—in line with expectations—rejected the most basic of gun control proposals.  In light of the Newtown massacre—an event that shook all of us—this might seem shocking.  For example, even leaving aside the emotional pull that perhaps we can as a nation call that horrible day back and make it right, the proposal arguably had/has 90% support among the public.

Does this demonstrate a problem with the Senate or, perhaps, democracy itself? Simply put, no.

Let me be clear: I have many family and friends who own and use guns for hunting and sport, and I do not believe that the debate about “taking away guns” is worth the breath or typing it takes to describe such a practically ludicrous concept. However, in the interest of full disclosure, I do not own a gun, and do not think that a gun is appropriate to keep in my house. Okay…that said…now I’m going to blow your mind. WITH SOCIAL SCIENCE.

First, the Senate ain’t done.  I’ll just note that and then spare you (for now) yet another installment of my “votes matter for signaling to constituents” argument (which would imply we might see the background checks come in through another, presumably bundled, amendment).

Second, and the “math of politics” part of this post, this vote demonstrates the not-so-gentle implications of the subtle interaction of psychology, indirect democracy, and multi-issue politics.

Before I continue, let me apologize for my shortcuts, I am about to unfairly but succinctly imply that gun rights advocates are all gun owners.  This supposition is demonstrably false in the general context, of course, because plenty of people cheer for teams representing colleges that they not only didn’t attend, but couldn’t have if they wanted to.  (You know who I’m talking about, right?)  That said, here we go…

A realist/game theoretic interpretation of democracy implies that it “works” (if it does) only because voters hold incumbents accountable for their decisions.  Taking this logic on its own and pairing it with the overwhelming empirical support for background checks, the intuitive conclusion is that democracy must be failing: clearly a few Senators at least must be ignoring the demands of their constituents.  Right? (Don’t worry, I’m not about to make an argument based on the (mal)apportionment of the Senate or sampling error.)

Umm, yes, perhaps…until you realize that Senate elections choose Senators, not positions. Once you realize this, you “think down the game tree” a bit as an incumbent and you think…

“Well, I vote on lots of different issues.  Each voter comes into the voting booth and does something like a weighted sum over the various positions I am seen as favorable/reliable on and then compares me with the challenger.”

In a nutshell, this means that every incumbent—when faced with a vote on any issue—considers the weight (or, in the terminology of political science, salience) of that issue with his or her electorate. (I’m abstracting from individual-voter-level differences for simplicity.)

Thus, the impact of an incumbent’s gun control vote on any given voter i‘s “approval/support” for the incumbent is basically something like

,

where is the importance of gun control (larger values imply gun control is more important to i) and is “+1” if i agrees with the incumbent’s vote on gun control and “-1” otherwise.

Note that this is just one issue among many.  The total approval/support of the voter for the incumbent is something like

.

Okay, that’s voting.  Now let’s think about psychology. In a nutshell, who cares most strongly about background checks? This is sort of a Olsonian–Tversky&Kahneman effect:

Those who encounter the policy most often care the most about it.

Gun owners (or people who think/fear they might want to buy a gun someday) will generally have (or be believed by incumbents to have) larger values of . And, to cut to the chase, many of them will not prefer to submit to (say) background checks.  (After all, most these voters are, indeed, good Americans.)

So, while 90% of the voters might prefer a vote for background checks (i.e., for a vote for yesterday’s amendment), few if any of them assign nearly as large a value of as the 10% of the voters who oppose background checks.

This matters because an incumbent—in the spirit of democratic responsiveness—is responsive to a voter on any given single issue only to the degree that the voter’s vote is responsive to the incumbent’s vote/stance on that issue. Congress deals with many issues. For better or worse, my argument here is that “gun control/gun rights” votes are generally more important/dispositive for voters who refer to the issue as “gun rights.”

Here’s a picture suggesting this, from Pew:

Revealed Differential Importance of Gun Rights/Control

In spite of that picture, note that, according to my argument, this is not about polarization in the classical sense (the two sides don’t necessarily have wildly different policy goals). While gun control is a fairly partisan issue, it is not actually a strongly partisan one. Rather, background checks represent an issue that (in line with the Olson shout-out above) present “concentrated costs” to an arguably much-less-than-majority group and “dispersed benefits” to a larger-than-majority group. If we had a referendum on the background checks amendment (and everybody had to turnout and vote), then I have no doubt that the Manchin-Toomey amendment would win in a landslide. But that’s not the way indirect democracy works. (And, for another day, thank goodness for that. AMIRITE, CALIFORNIANS WITH SCHOOL-AGE CHILDREN?)

So, I am sad that our nation tears and blows itself apart over both the issue and its instantiation. I cried (a lot) watching the coverage of Newtown and for a few days afterwards. That said, the institutional and psychological/preference realities of the issue mean that at least I sleep well at night confident in the notion that this strife does not imply anything untoward about our politicians or voters. To put it another way, as hard as it is to accept sometimes, democracy is about choices—the logic above is a convoluted (but more precise) way of saying “pro-gun rights voters will “throw the bum out” for a pro-gun-control vote…and a pro-gun-control voter probably won’t do the same to his or her incumbent for a pro-gun-rights vote.” As I disclosed above, I am more than happy to be proven wrong on this in 2014. And with that, I put my money where my mouth is and leave you with this.

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PS.  According to this article, Sen. Toomey (R-PA) “argued that `the Second Amendment does not apply equally to every single American…'”  I can’t resist the opportunity to suggest that this is wrong headed on arguably two counts.  First, it is distinctly poor taste and shortsighted to get into an Orwellian “some people have more rights than others” interpretation of the Constitution. Second, and more controversially, the Second Amendment is actually a guarantee of a collective (read: State) right, rather than an individual one.  I mean, one is the loneliest number even for militias?  (Also, are “poor taste” and “shortsightedness” synonymous in equilibrium?)