COU 6: Modeling Conditional Uncertainty

In a study of voter attitudes towards infrastructure and other long-term investments by government, Jacobs and Matthews (Jacobs and Matthews 2017) propose a model of how voters evaluate proposed government investments. From the point of view of voters, the costs of any proposed investment – whether in the form of new taxes, user fees, or foregone spending on other priorities – are certain. But the promised benefits – e.g. protection from floods, cheaper electricity, lower travel costs – are uncertain. For instance, Jacobs and Matthews ask a sample of the public to consider a proposed policy in which sales taxes on gasoline will be increased and the proceeds used to repair roads and bridges. Told of a proposed investment like this, Jacobs and Matthews claim, voters will feel certain that implementation of the investment will result in higher gas taxes, but will be uncertain whether the promised improvements to roads and bridges will in fact occur.

Voters, Jacobs and Matthews propose, understand that to build infrastructure, the government must give money and authority to some sort of implementing agency that will direct and oversee construction. For instance, many federally funded projects in the U.S. are implemented by the U.S. Army Corp of Engineers, which uses money and authority granted by Congress to build things by acquiring land, commissioning blueprints and architectural plans, letting out construction contracts, overseeing the work of contractors etc. etc. Voters, moreover, see these grants of money and authority as risky. They know that once an agency has a grant of money and authority to build something, deep-pocketed, well-organized special interests, such as real-estate investors, construction firms, and construction trade unions will try to push the agency to use its money and authority in their benefit, rather than to the benefit of the public. Thus voters expect such projects are likely to feature cost overruns, prolonged construction delays, shoddy construction, and even the diversion of funds to purposes entirely different from whatever was initially promised.

For these reasons, Jacobs and Matthews argue, voter support for proposed infrastructure investments will depend on the institutional arrangements through which those investments will be implemented. For instance, the power and authority needed to implement a project could be delegated to any number of entities, such as subnational governments, committees of elected politicians, the military, or civilian executive branch agencies. These entities differ vastly in their professional cultures, modes of governance, and political independence, and thus might (in voters’ eyes) differ in the extent to which they can be corrupted by special interests.

Imagine a voter asked in a poll whether she supports the funding by the government of a large construction project, such as a seawall, hydroelectric dam, or bridge repair. Suppose the poll names a widely-known public institutions (e.g., the military, a federal agency, a committee of the national legislature) and tells the voter that if the project is funded, it will be implemented by that institution. Suppose the voter is uncertain about two things:

The extent to which the public institution that will implement the proposed project can resist pressures from well-organized, deep pocketed special interests to alter the project to their benefit.
Whether the project, if funded, will be completed to the initially promised specifications, on time, and within its initially specified budget.

Prompt 1

Draw a grid that depicts a probabilistic model of joint uncertainty on the part of the voter about the two issues above.

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.

Prompt 2

Here again are the two uncertainties you modeled in response to Prompt 1:

The extent to which the public institution that will implement the project can resist pressures from well-organized, deep pocketed special interests to alter the project to their benefit.
Whether the project, if funded, will be completed to the initially promised specifications, on time, and within its initially specified budget.

For each possible resolution of (a) as depicted by the model you wrote in response to Prompt 1, draw one grid that depicts the voter’s uncertainty about (b) conditional on that resolution of (a).

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

Prompt 3

Re-draw each of the grids you drew in response to Prompt 2, this time writing specific numerical values for the conditional probabilities. Assign values that depict the idea that the likelihood that the project will be completed as initially specified, on-time and within-budget increases in the extent to which the agency implementing the project can resist pressures from special interests.

Rubric

Prompt 1

A completely correct answer meets all the following criteria

It is a grid with two axes.
One axis is labelled in a way that clearly indicates that it represents uncertainty about the extent to which the public institution can resist pressures from special interests.
The other axis is labelled in a way that clearly indicates that it represents uncertainty about the extent to which the project will be completed as specified, on-time and within-budget.
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.
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 satisfy 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 response to this prompt can only be evaluated in light of a response to Prompt 1 that earns 3 points or more (i.e. satisfies criteria (a) through (d) for Prompt 1). So, if your response to Prompt 1 earned less than 3 points, you earn 0 points on Prompt 2, regardless of your response.

That said, if your response to Prompt 1 satisfies criteria (a) through (d) for Prompt 1, a completely correct answer to Prompt 2 satisfies all of the following:

It consists of exactly one grid for each resolution of uncertainty about the extent to which the public institution can resist pressure from special interests, where the resolutions of that uncertainty are as depicted in the grid submitted in response to Prompt 1.
Each grid is exactly the same as the grid drawn in response to Prompt 1 with two exceptions:

The columns or rows representing the resolutions of the uncertainty about the extent to which the public institution can resist special interests other than the resolution represented in that grid are crossed out or shaded.
The only probability labels are in the un-shaded cells, and those labels use the correct notation for conditional probability, given the resolutions of uncertainty represented in each cell.

If your answer to Prompt 1 earned 3 or more points, you can earn up to 4 points on this prompt. Specifically:

4 points if your answer fully meets all of criteria (i), (ii)(A) and (ii)(B)
3 points if your answer fully meets criteria (i) and (ii)(A) but does not fully satisfy (ii)(B)
2 points if your answer does fully meets criterion (i) but does not fully satisfy (ii)(A)
1 point if your answer partially but does not fully satisfy criterion (i), by excluding some but not all of the realizations of uncertainty about the extent to which the public institution can resist pressure from special interests depicted in the answer to Prompt 1.
0 points otherwise.

Prompt 3

That said, if your response to Prompt 1 satisfies criteria (a) through (d) for Prompt 1, a completely correct answer to Prompt 3 satisfies all of the following:

It consists of exactly one grid for each resolution of uncertainty about the extent to which the public institution can resist pressure from special interests, where the resolutions of that uncertainty are as depicted in the grid submitted in response to Prompt 1.
Each grid is exactly the same as the grid drawn in response to Prompt 1 with two exceptions:

The columns or rows representing the resolutions of the uncertainty about the extent to which the public institution can resist special interests other than the resolution represented in that grid are crossed out or shaded.
Exact numerical values for probabilities are given in each unshaded cell.
The numerical values in the unshaded cells of each grid together form a valid probability distribution – i.e. they sum to 1.
The numerical values are ordered relative to one another across the grids in a way that depicts the idea that the likelihood that the project is completed on time, within budget and as promised increases in the extent to which the implementing agency can resist pressures from special interests.

If your answer to Prompt 1 earned 3 or more points, you can earn up to 4 points on this prompt. Specifically:

4 points if your answer fully meets all of criteria (i), (ii)(A), (ii)(B), (ii)(C) and (ii)(D)
3 points if your answer fully meets criteria (i) and (ii)(A), (ii)(B) and (ii)(C) but does not fully satisfy (ii)(D)
2 points if your answer fully meets critera (i) and (ii)(A) and (ii)(B) but does not fully satisfy (ii)(C)
0 points otherwise.

References

Jacobs, Alan M., and J. Scott Matthews. 2017. “Policy Attitudes and Institutional Context: Rules, Uncertainty, and the Mass Politics of Public Investment.” American Journal of Political Science 61 (1): 194–207.