## Thursday, December 19, 2013

### A Game of Institutions Book 2: Beautiful Railway Bridge of the Silv’ry Tay!

I might be inclined to forgive the student who claims that Public Choice is the study of political rents. Even the very basic collective choice architecture work is undergirded by constituents' desire to organize for gain. The mere act of stumbling out of the Hobbesian jungle, twigs and brambles jutting from matted thickets of body hair counts as a political rent. But not all political rents are created equal. Last time, I asked you to consider the simple case where payoffs were symmetrical and alternatives to agreements were at worst, neutral. This time, I'd like to explore what happens when we play around with the symmetry condition. I'd also like to introduce a bit of heresy. Those of you with formal educations in economics may wish to consult a physician before reading further.

First, a quick review of some elementary ideas. If math gives you the heebie-jeebies, take heart—it's just shorthand for what goes on in folks' minds. Nothing fancy.

First up, something economists call the "discount rate". Have you ever wondered why you pay interest for a loan? Or why you earn money just by socking it away in a retirement account? It's not just because the value of a dollar is dwindling each year, though that's got a little to do with it. It's chiefly because on average, delaying the satisfaction of wants is uncomfortable. All else equal, I'd rather have a new pair of shoes right now than to wait another year. If we abstract from shoes, we can better say that all else equal, I'd rather have a claim on 100 bucks' worth of stuff right now than 100 bucks' worth of stuff a year hence. In keeping with tradition, let's use standard notation. A lower-case rho is our discount rate variable, and it's just a transformation of the interest rate r we see all the time:

$\bg_white&space;\rho&space;=&space;\frac{1}{1+r}$

So, if the interest rate someone would accept is 5%, her discount rate would be .9524 or thereabouts.

Again consistent with textbook conventions, a lower-case pi will be our payoff variable in each period, and for the accumulated payoff, we'll use an upper-case pi. Therefore, in generic terms, an infinitely repeated game will have payoffs for player i of:
$\bg_white&space;\Pi_i&space;=&space;\sum_{t=0}^{\infty&space;}\rho^{t}\pi_i$

You'll note, of course, that I'm assuming homogeneous discount rates, which as we all know right well from the behavioral literature, is poofy-headed of me. As we continue with this series, I'll relax this assumption to show how elites can turn time preference differences to their advantage. If you need to slap leather for an aphorism in the meantime, reach for the six-shooter labeled "patience is a virtue."

The next bit we need to do before we stop for the day (mostly for the sake of length than anything) is to think about the components of the payoff. In most elementary game theory lectures, this is just some parameter used to think about how we get to some equilibrium strategy or another. But in terms of economic behavior, the elements of the payoff are what we spend a lot of time thinking about here at EE. These are the moral intuitions, biases, jealousies, irrationalities, groping uncertainty, and beautiful human error that characterizes each and every one of us in our brief, blemished corporeal time on this tarnished earth. Ahead of our next installment, think about this little heresy:

$\bg_white&space;\pi_i_,_t_=_0(\lambda&space;\sum_{t=0}^{\infty&space;}\rho^{t}\left&space;(\pi_-_i_,_t-\pi_i_,_t&space;\right&space;),&space;\xi&space;\sum_{t=-n}^{-1}\left&space;(\pi_-_i_,_t-\pi_i_,_t&space;\right&space;),...)$

The lambda and xi parameters there are personal preferences for fairness in (a) prospect and (b) retrospect, respectively. When economists talk about this sort of stuff, some will acknowledge that (a) is of analytical concern (and often will consider it to be of normative import), but most (all?) will agree that (b) is an unforgivable logical fallacy. We call it the sunk cost fallacy, and if I had a nickel for every student who's fallen for it on a final, you can bet your bottom dollar I'd be posting this from my own private Caribbean island instead of my perch here on Fenrir's shaggy mane.

Why do I commit the sin of including a sunk cost in a payoff function? Because if it trips up college students by the truckload, imagine for a moment that it might be a pretty common fallacy among ordinary constituents. Strong enough even to be part of Zwolinski's argument in favor of BI (since this is still in the middle of being debated, I am electing to refrain from mining out the balance of the links. I do hope you understand) as a matter of practical observation. Remember that part of what we're doing here at EE is investigating the contours of commonplace moral intuitions, and that includes holding our nose and accepting the weird proposition that not everyone thinks like an economist.

Okay, so let's leave this to stew for a little while. Remember that one of the things I'm trying to model in this series is apostasy, so consider what things might look like when xi > 0. And consider this post a bridge as we move from symmetrical games of cooperation to asymmetrical games of cooperation. As we tiptoe from Locke to Rousseau.