COU 4: Modeling Multiple Uncertainties As Joint Uncertainty

Looking across nations and across provincial and municipal governments within nations, we see wide variation in both the total amount that governments invest in infrastructure and in the kinds of infrastructure (e.g. heavy transportation vs. commuter transit vs. energy generation vs. irrigation vs. flood control, etc. etc.) governments build. What explains this variation? Why do some governments build lots of stuff and others build little? Why do some governments build some kinds of infrastructure and not others?

In developing answers to these questions, political scientists typically start by looking at who pays for and who benefits from government infrastructure projects. Importantly, the groups benefiting from any given project are often small, relative to the population paying the cost. For instance, the benefits of the massive seawalls built in and around New Orleans starting in 1965 accrued primarily to the construction firms, landowners and residents of southern Louisiana, but were funded by the U.S. federal government out of taxes paid by the entire U.S. population.

Infrastructure projects, then, often appear to be instances of what political scientists call “targeted transfers” – i.e., government actions that transfer resources from broad swaths of the population to relatively small groups of beneficiaries. Understanding why some infrastructure projects get built while others do not, then, comes down to understanding the incentives politicians have to use targeted transfers to deliver benefits to some groups and not others.

In this and the next few COUs, you’ll apply techniques for modeling multiple uncertainties to depict uncertainties politicians face when they decide which of an array of groups to benefit through targeted transfers.

Imagine a politician who must choose just one of two alternative infrastructure projects – labeled Project A and Project B – to fund. Suppose that each potential project, if constructed, would benefit a group of persons completely distinct from those who would benefit from the other project. For instance, you might imagine that Project A is a flood control project in a coastal city that would only benefit the persons who live in that city and its surroundings, while Project B is a hydroelectric dam that would provide electricity to persons who live in an interior region of the country, hundreds of miles away from the coastal city. Call the group of persons who will benefit from Project A Group A, and call the group that will benefit from Project B Group B. Thought of in this way, the politician’s choice between building Project A and building Project B boils down to a comparison between the political advantage the politician can win by directing benefits to Group A vs. the advantages she can win by instead directing benefits to Group B.

Suppose that the political advantage the politician gains by directing benefits to either group depends on two things:

For instance, imagine that Group A consists of 100 persons. Suppose that 10\% of those are staunch opponents of the politician who will not support her regardless of whether she funds Project A, 30\% are staunch supporters who will support her regardless of whether she funds Project A, and the remaining 60\% are on the fence in that they will support the politician if she funds Project A and oppose her otherwise. Under these assumptions, the politician can increase her number of supporters by 60 by funding project A. This number, of course, varies both in the total size of the group and in the relative proportion of “on-the-fence” persons in the group.

Prompt 1

Draw a grid that depicts a probabilistic model of joint uncertainty on the part of the politician about two issues:

  1. The number of persons in Group A
  2. The proportion of persons in Group A who are “on the fence”

Do not assign specific numerical values for the joint probabilities in the model. Instead, add labels to the grid that use the notation for joint probabilities specified in the lesson to denote those probabilities.

Note that it is up to you to specify the exhaustive list of mutually exclusive possible resolutions of each of the two uncertainties you are modeling. As demonstrated in the lesson, make sure to include labels in the grid that make clear which axis of the grid depicts which uncertainty and which rows and columns of the grid correspond to each possible resolution of each uncertainty.

Finally, remember the academic honesty rules of this course: You may not look at, be told about nor use an answer to any COU developed by another person. Since it is up to you to specify the possible resolutions of the two uncertainties, there is no reason to expect that the resolutions you specify will be the same as those specified by any other person. Thus we will treat any answers that have the same possible resolution of either uncertainty as suspected incidents of academic dishonesty.

Prompt 2

Re-draw the grid you drew in response to Prompt 1, replacing the labels for the joint probabilities with specific numerical values. In selecting values to assign, remember that in any probabilistic model of uncertainty, the probabilities assigned to the exhaustive list of mutually exclusive resolutions of the uncertainty must sum to 1.

Also, recall the academic honesty rules of this course: You may not look at, be told about nor use an answer to any COU developed by another person. There is an infinite variety of specific numerical values that could be assigned in any probabilistic model of uncertainty. Thus we will treat any answers that have the same specific numerical values as suspected incidents of academic dishonesty.

Rubric

Prompt 1

A completely correct answer meets all the following criteria

  1. It is a grid with two axes.
  2. One axis is labelled in a way that clearly indicates that it represents uncertainty about the size of Group A.
  3. The other axis is labelled in a way that clearly indicates that it represents uncertainty about the proportion of persons in Group A who are “on the fence”.
  4. Each axis is clearly divided into two or more distinct rows (for a vertical axis) or columns (for a horizontal axis). The rows/columns on each axis are labelled in a way that makes absolutely clear what the exhaustive list of mutually exclusive possible resolutions of the uncertainty represented on that axis are.
  5. Each cell includes a label using the correct notation to denote the joint probability of the joint event depicted by that cell.

You can earn up to four points on this prompt:

  • 4 points if what you write fully satisfies all of criteria (a) through (e).
  • 3 points if what you write fully satisfies all of criteria (a) through (d) but does not fully satisify criterion (e).
  • 2 points if what you write fully satisfies all of criteria (a) through (c) but does not fully satisfy criterion (d).
  • 0 points otherwise.

Prompt 2

A completely correct answer meets criteria (a) through (d) from Prompt 1 and in addition has (instead of labels using the notation for joint probability) specific numerical values in each cell, with the values numbers between 0 and 1 that together sum to 1.

You can earn up to two points on this prompt: Two points if what you write fully meets all of the criteria required, zero points otherwise.