COU 7: Using the Marginal-Conditional Method to Model Multiple Uncertainties

In some cases, investments in infrastructure trigger widespread and permanent changes in the technologies used by consumers and firms. While most persons benefit from those changes, there are always some groups who suffer severely. For instance, mitigating global warming by reducing CO2 emissions will require, among other things, “electrification” of a wide range of activities, such as transportation, steel milling and concrete production. And electrification on such a massive scale will require investments in new electricity generation and transmission infrastructure. Once this infrastructure is built, demand for fossil fuels and internal combustion engines will substantially contract. This will reduce the wealth of the owners of assets specific to those technologies and reduce the wages of workers with fossil-fuel-specific skills and knowledge.

The groups harmed by the technological changes that infrastructure investments trigger are typically well-organized, relatively wealthy, and thus politically powerful. Government investments in infrastructure that trigger technological change, then, are often politically impossible without the consent and cooperation of these groups. Thus, reformers who champion these investments typically succeed only when their proposals “bundle” the desired infrastructure investments with other policies that shower benefits on these powerful groups.

Acemoglu and Robinson (Acemoglu and Robinson 2001) study policies that attempt to compensate for technological changes by concentrating benefits on politically powerful groups harmed by those changes. They point out that politicians who wish to “buy” the consent or cooperation of such a group face a problem: If the proposed technological change is allowed to go forward, the wealth and income, and thus the political power of the group will be permanently reduced. To “buy” the consent of such a group to investments that will trigger that technological change, then, a politician must find a way to preserve the group’s political power into the future, even as changing technology reduces its wealth and income.

Imagine a lobbyist hired to represent the interests of owners of fossil-fuel-specific assets is bargaining with a politician over a legislative package that will fund investments in electricity infrastructure, and feature targeted subsidies, bailouts, regulatory privileges or other measures meant to buy the cooperation or quiescence of the lobbyist’s clients. Imagine the lobbyist is uncertain about how the proposed legislative package, if enacted, will affect her clients. Specifically, suppose she is uncertain about two things:

How politically powerful her clients will be in the coming decades if the legislative package is enacted.
Whether politicians will continue to use their power to protect and benefit her clients in the coming decades if the legislative package is enacted.

In the following prompts, you’ll use the marginal-conditional method to model this uncertainty.

Prompt 1

Start by drawing a grid that depicts a model of joint uncertainty about (a) and (b). 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

Now model the lobbyist’s marginal uncertainty about (a). Do so by re-drawing the grid you drew in response to Prompt 1 and then adding specific numerical values for the marginal probabilities to the ends of the rows or columns (depending on which axis you used to depict uncertainty about (a)).

Remember, a model of marginal uncertainty is just like any other probabilistic model of uncertainty in that it consists of an exhaustive list of mutually exclusive possible resolutions, along with probabilities assigned to those resolutions that sum to 1.

Prompt 3

Now build models of uncertainty about (b) conditional on each possible resolution of (a). Specifically, for each possible resolution of uncertainty about (a) specified in your answer to Prompt 1, construct one model of uncertainty about (b) conditional on that resolution of (a). For each of the models of conditional uncertainty you write, do not specify specific numerical values for the conditional probabilities. Instead, use the notation for conditional probabilities introduced in the lesson. Specify each model by re-drawing the grid you drew in response to Prompt 1, crossing out or shading the appropriate columns/rows, and then writing the appropriate conditional probabilities (using the notation from the lesson) in the un-shaded cells.

Prompt 4

Presumably, the likelihood that politicians will continue to use their powers to protect and benefit the lobbyist’s clients in the future is increasing in her clients’ future political power. Re-draw the grids you drew in response to Prompt 3, but this time assign specific numerical values for the probabilities. Assign values that depict the idea that the likelihood that politicians will use their powers to protect and benefit the lobbyist’s clients is increasing in those clients’ future political power.

As you assign values for the probabilities in each of these models, keep in mind that a model of conditional uncertainty is just like any other probabilistic model of uncertainty – I.e., it consists of an exhaustive list of mutually exclusive resolutions, with probabilities assigned to these resolutions that sum to 1.

Prompt 5

Re-draw the grid you drew in Prompt 1, this time writing specific numerical values for the joint probabilities. Compute these joint probabilities from the marginal probabilities you specified in Prompt 2 and the conditional probabilities you assigned in Prompt 4.

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 future political power of the lobbyist’s clients.
The other axis is labelled in a way that clearly indicates that it represents uncertainty about whether politicians will in the future continue to protect and serve the lobbyist’s clients’ interests.
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 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 future political power of the lobbyist’s clients.
The other axis is labelled in a way that clearly indicates that it represents uncertainty about whether politicians will in the future continue to protect and serve the lobbyist’s clients’ interests.
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.
Specific numerical values between 0 and 1 that together sum to 1 are at the ends of either the rows or columns (depending on which axis is labeled to depict uncertainty about the future political power of the lobbyist’s clients.)

You can earn up to four points on this prompt:

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

Prompt 3

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 3, 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 3 satisfies all of the following:

It consists of exactly one grid for each resolution of uncertainty about the future political power of the lobbyist’s clients, 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 future political power of the lobbyist’s clients other than the resolution represented in that grid are crossed out or shaded.
Each unshaded cell contains the correct notation for a conditional probability, given the events represented by that cell.

If your answer to Prompt 1 earned 3 or more points, you can earn up to 2 points on this prompt. Specifically:

2 points if your answer fully meets all of criteria (i), (ii)(A), (ii)(B)
0 points otherwise.

Prompt 4

That said, if your response to Prompt 1 satisfies criteria (a) through (d) for Prompt 1, a completely correct answer to Prompt 4 satisfies all of the following:

It consists of exactly one grid for each resolution of uncertainty about the future political power of the lobbyist’s clients, 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 future political power of the lobbyist’s clients 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 politicians will protect and serve the lobbyist’s clients in the future is increasing the lobbyist’s clients’ future political power.

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.

Prompt 5

An answer to this prompt can only be evaluated if your answer to Prompt 2 earned all 4 points available on that prompt and your answer to Prompt 4 earned 3 or more points (by fully satisfying criteria (i), (ii)(A), (ii)(B), and (ii)(C)). If either your answer to Prompt 2 earned fewer than 4 points or your answer to Prompt 4 earned fewer than 3 points, you earn 0 points on Prompt 5, regardless of what you submit.

That said, if you earned 4 points on Prompt 2 and 3 or more points on Prompt 4, you can earn up to 2 points on Prompt 5. Specifically:

2 points if all of the joint probabilities you write are correct, given the values of the marginal probabilities you wrote in answer to Prompt 2, the values of the conditional probabilities you wrote in response to Prompt 4.
1 point if more than half of the joint probabilities you write are correct, given the values of the marginal probabilities you wrote in answer to Prompt 2, the values of the conditional probabilities you wrote in response to Prompt 4.
0 points otherwise.

References

Acemoglu, Daron, and James A. Robinson. 2001. “Inefficient Redistribution.” American Political Science Review 95 (3): 649–61.