##
Uncertainty Over Time (Lords)
*November 25, 2013*

*Posted by tomflesher in Micro.*

Tags: Doctor Who, intermediate microeconomics, Microeconomics, uncertainty

add a comment

Tags: Doctor Who, intermediate microeconomics, Microeconomics, uncertainty

add a comment

The previous post introduced a problem that arose in the Doctor Who 50th Anniversary special in which uncertainty was a core element. To explore that problem in depth, we’ll need some tools to work with uncertainty. Also, that problem was a little grim, so let’s take the edge off.

Let’s say that a friend of mine, Matt, is visiting and I need to order food. Matt’s favorite food is fish custard, so it makes sense that I should order him a dish of fish custard. Simple enough. Of course, if my other friend Peter were coming over, and I knew Peter didn’t like fish custard and much preferred haggis, it wouldn’t make sense for me to order him fish custard. Obviously I’d order haggis for him. To make this work mathematically, let’s say that each friend of mine likes his preferred dish about the same, so we could say that Matt gets utility of u_{M}(fish custard) = 1 from eating fish custard and u_{M}(haggis) = 0 from eating haggis, and Peter gets the reverse – u_{P}(fish custard) = 0 and u_{P}(haggis) = 1. If I want to maximize my friend’s utility, then I should do so by buying each friend his favorite dish. But what if I don’t know who’s coming over?

Intuitively, it makes sense – if Matt’s more likely to come over, I should order fish custard. If Peter’s probably on the way, haggis should go on the menu. If they’re equally likely, flip a coin. More mathematically, if the probability that Matt is coming is π_{M} and the probability that Peter is coming is π_{P} (and π_{M} + π_{P} = 1), the **expected utility** of my guest when I order fish custard is

E[u(fish custard)] = π_{M}u_{M}(fish custard) + π_{P}u_{P}(fish custard),

which reduces to

π_{M}u_{M}(fish custard) + π_{P}(0).

Since u_{M}(fish custard) = 1, then the expected utility of my guest from fish custard is just π_{M}. For haggis, of course, the same logic applies – Matt gets 0 utility, so we can ignore him, and the expected utility ends up being π_{P}. So, whichever friend of mine is more likely to show up should get his favorite dish ordered, and if I don’t know who’s coming, I might as well just draw names from a hat.

This gets a little bit more complicated if both of my friends like each dish. So, let’s consider what happens if u_{M}(fish custard) = 1 and u_{M}(haggis) = 0, but if u_{P}(fish custard) = 0.5 and u_{P}(haggis) = 1. Then, our calculation for the expected u(haggis) stays the same (it’s still π_{p}) but now our E[u(fish custard) ends up being

π_{M}u_{M}(fish custard) + π_{P}u_{P}(fish custard) = π_{M}*(1) + π_{P}*(.5)

So now, in order to figure out what dish to order, we need to know what the probabilities are! If it’s fifty-fifty, then E[u(haggis)] is .5, but E[u(fish custard)] is .5 + .5*.5 = .75! In that case, we should order fish custard, even though the guys are equally likely to show up, since Peter likes fish custard a little bit, too. We don’t get to flip a coin (mathematically, we don’t reach our **indifference point**) until

π_{M}u_{M}(fish custard) + π_{P}u_{P}(fish custard) = π_{M}u_{M}(haggis) + π_{P}u_{P}(haggis)

or

π_{M}(1) + π_{P}(.5) = π_{M}(0) + π_{P}(1) => π_{M }+ π_{P}(.5) = π_{P} => π_{M} = .5*π_{P}

That means we don’t get to the indifference point until Matt is twice as likely to arrive as Peter! Funny how these probabilities influence the choices we’ll make. Sometimes things get a bit more complicated, but that’s a post for another time.

… lord.