# Believe Me When I Say That I Want To Believe That I Can’t Believe In You.

A recurring apparent conundrum is the mismatch between Congressional approval (about 14% approval and 78% disapproval) and reelection rates (about 91% in 2012).  If Americans disapprove of their legislators at such a high rate, why do they reelect them at an even higher rate?  PEOPLE BE CRAZY…AMIRITE?

Maybe.  A traditional explanation is that people don’t like Congress but like THEIR representatives.  Again, maybe.  Indeed, probably.  But, in the contrarian spirit of mathofpolitics, I wish to forward another explanation.  This explanation is undoubtedly wrong, but raises an interesting welfare point that could still hold even if the microfoundations of the explanation are amiss.

The heart of the explanation is that a cynical belief along the lines of “all politicians are crooks” can help voters get better/more faithful representation from their representatives.  Thus, ironically, a deep suspicion among voters about the accountability of legislators can aid one in keeping those legislators behaving in accord with the voters’ wishes.  (PEOPLE BE CRAZY LIKE A FOX…AMIRITE?)

Now, let’s get to the argument/the model.

Theoretically, elections might help achieve accountability (i.e., make incumbents do what voters want) through their potential ability to solve two important problems: moral hazard and adverse selection. The essence of moral hazard is “hidden action”: I want my representatives to work smart AND hard, but I can’t actually observe them working.  Instead, I observe noisy real-world indicators of how hard they’re working: unemployment, budget deficits, health care costs, Olympic medals, and Eurovision.  To the degree that these are correlated with legislative effort and I condition my vote choice on these observables, I can provide an incentive for a reelection-motivated incumbent to work smart and hard.

The essence of adverse selection is hidden information: I want my legislator to take actions on my behalf when they present themselves.  The best way to “get this” is to have a legislator who shares my preferences.  (Think Fenno’s Home Style.) But, practically speaking, every aspiring representative will tell me that he or she shares my preferences.  So, regardless of how I discern whether my incumbent shares my preferences, the reality is that I will have a strict incentive to reelect an incumbent who expect shares my preferences…because, after all, I generally have little information about his or her challenger’s preferences.

So, in a nutshell, you’d like to solve both of these problems simultaneously: you want your incumbent to share your preferences and work both smart and hard.  In the immortal words of Bachman Turner Overdrive, you want them to be taking care of business and working overtime. Every day. Every way.

Unfortunately, solving both of these problems is frequently impossible.  The details of why can be summed up with the old saying that “a bird in the hand is better than two in the bush.” You see, once you believe that your legislator truly has your interests at heart, you will (and, in a certain way, should) be more likely to forgive/reelect him or her if he or she is a little lazy.

This fact ends up making it harder for a voter to discipline his or her representatives: in particular, think of the following simple world.  Assume there are faithful and faithless politicians (this is fixed for any given politician) and that every politician can either work hard or be lazy. And, for simplicity, suppose that the faithful type always works hard. (This is not important, it just simplifies the exposition.)

Furthermore, let $p \in (0,1)$ denote the probability that a random incumbent is faithful.

Here’s the rub: you don’t observe the politician’s type (faithful or faithless) or how hard he or she worked.  Rather, you observe one of three outcomes: Great, OK, or Bad.  Finally, suppose that the probabilities of observing these outcomes are

P[Great | work hard & faithful] = 0.35
P[OK | work hard & faithful] = 0.55
P[Bad | work hard & faithful] = 0.1

P[Great | work hard & faithless] = 0.3
P[OK | work hard & faithless] = 0.5
P[Bad | work hard & faithless] = 0.2

P[Great | be lazy & faithless] = 0.1
P[OK | be lazy & faithless] = 0.5
P[Bad | be lazy & faithless] = 0.4

(If you take a moment, you’ll realize that outcomes are more likely to be better for faithful types regardless of how they work and for those who work hard, regardless of their type (i.e., ceteris paribus). …JUST AS IN LIFE.)

To illustrate the problem at hand here:

Suppose that the voter observes an “OK” outcome.  what is the voter’s posterior probability that the politician is a faithful type?

$\Pr[\text{faithful}|\text{outcome=OK}] = \frac{0.55p}{0.55p + (1-p)(0.5 x + 0.5 (1-x))}=\frac{0.55p}{0.5+0.05p}>p,$

where x denotes the probability that a faithless politician works hard.  (Note that my judicious choice of probabilities obviates the need to worry about what this is.  YOU’RE WELCOME.)

The key point is that the probability that the politician is the faithful type, conditional on seeing only an “OK” outcome, is greater than p, the probability that a random challenger will be a faithful type.

This means that, upon seeing an “OK” outcome, you should reelect your incumbent.  He or she is a (probabilistic) bird in the hand.

So what?  Well, this all goes away if you believe that there are no faithful politicians.  That is, if you’re a hard-core cynic (as many of my FB friends purport to be), and you believe p=0, then upon observing an “OK” outcome you can credibly (and, concomitantly, “should”) throw the bum out.  If p=0, then (assuming that working hard is not too costly to the incumbent) the optimal reelection rule in this setting is to reelect if and only if the outcome is “Great.”  Furthermore, imposing such a rule in such cases will yield a higher expected outcome for you (the voter) than you can obtain in equilibrium when p>0.

In summary, it’s a lot easier to “throw the bums out” if you actually think they’re all bums. This, ironically, will both make you better off and lead to you throwing fewer out, because the “bums” will know what you think of them.

Tying this back to real-world politics, the mathofpolitics/logic of adverse selection and moral hazard suggest a somewhat subtle value of cynicism in politics.  (Which, perhaps seemingly oddly for a game theorist, is something I detest.)  What is also kind of neat about this logic is that it provides an interesting argument in favor of primaries: the whole logic of why adverse selection undermines the solution to the moral hazard problem is that the voter can not select a replacement who is “just/almost as likely as the incumbent” to have the same innate interests as the voter.  To the degree that we believe that this type of alignment between incumbents and voters is correlated with the incumbent’s partisanship, primaries offer the voters a (more) credible tool to discipline their incumbent’s moral hazard problem.

And with that, I am left thinking of Arlen Specter and accordingly want to leave you with this.