We have covered a lot of ground in Uncertainty Wednesday but we have yet to talk about measuring uncertainty. While most discussions introduce such concepts as mean and standard deviation early on, I have held off on them on purpose in order to develop a more comprehensive view of the sources of uncertainty and a hopefully better understanding of probability. Now is a good time to start thinking about how we might measure uncertainty.
Take our super simple model again of a world with only two states and two signal values. Now let’s think about the factors that go into uncertainty.
The first factor is the probability between the two states. If there are only two states and one state is extremely likely, then you face less uncertainty than if each state is equally likely. So that’s one aspect of uncertainty that we will want to make precise. It would also seem that if there are more than 2 possible states of the world that should increase uncertainty, e.g. a world with 100 states would seem to have more uncertainty in it than one with only 2. In upcoming posts we will spend a fair bit of time looking at different so-called probability distributions to study their characteristics and impacts on uncertainty. And as we have already seen when you receive a signal, you can use that to revise your estimate of probabilities in a way that reduces uncertainty.
The second factor are the outcomes for you that result from the different states of the world. A classic example is where in one state you gain money and in the other you lose money. For instance, you drill a hole in the ground and either oil comes out or it doesn’t. Because monetary outcomes are often of interest we may sometimes refer to these outcomes as “payouts.” More generally outcomes are the concept of a random variable, meaning a variable (such as the money paid or received) that takes on different values with different probabilities.
It is important to understand that these are two separate factors impacting uncertainty. Suppose you can either win $1 or lose $1. We can examine the impact on the uncertainty you face from changes in the probability between the two states of the world. Now conversely, let’s hold the probability fixed, say one state has 80% probability and the other has 20%. We can then examine the impact on the uncertainty you face from changes in the payouts between the two states.
But wait, there is more. Payouts are only the immediate outcomes. The value or impact of these payouts may be different for different people. What do I mean by this? Suppose that we look at a situation where you can either win $1 million with 60% probability or lose $10 thousand with 40% probability. This seems like a no brainer situation. But for some people losing $10 thousand would be a rounding error on their wealth, whereas for others it would mean becoming homeless and destitute. So even though the the probabilities (factor 1) and the payout (factor 2) are the same, the uncertainty that is faced can be quite different between people. So the third factor influencing uncertainty are the consequences or utility that results from different outcomes. This important difference goes by the prosaic name of “functions of a random variable.”
So from this intro alone it should be clear that it is unlikely that we can find a single all encompassing measure of uncertainty. Instead, as we will see over the coming posts there are measures and results that are associated with each of these three factors.
