COU 4: Learning and Decisionmaking
Once again, think of the imaginary flood-prone city that we’ve used to demonstrate probability, learning and expected utility in Lessons 4, 5 and 6. Imagine a voter living in the city who is unsure about whether the mayor of the city is competent or incompetent. Further, the voter believes the mayor’s competence will partially determine whether the next hurricane that hits the city will cause the city to flood. Specifically, a competent mayor would have made decisions (for instance, sensible investments in infrastructure) that make the city relatively less likely to flood in the next hurricane. An incompetent mayor, on the other hand, would have failed to make adequate preparations, resulting in a higher likelihood that the city will flood.
Suppose that a mayoral election is coming up, in which the mayor is running for re-election against a single challenger. Then, just a few weeks before the election, a hurricane passes over the city. Given the voter’s beliefs about the effect of the mayor’s competence on the likelihood that the city is adequately prepared for a hurricane, the voter can learn about the mayor’s competence on the basis of whether or not the pre-election hurricane floods the city. Thus, whether or not the city floods might swing the voter’s vote between the challenger and the mayor in the election.
In this COU, you’ll analyze a model that combines the learning models from Lesson 5 with the expected utility models from Lesson 6 to depict this process. The first step in the analysis is to be clear about the sequence of events in the model:
- The voter has prior beliefs about the mayor’s competence and whether the next hurricane will flood the city.
- A hurricane passes over the city and either floods the city or does not flood the city.
- The voter observes whether or not the hurricane flooded the city. Based on that observation, she forms posterior beliefs about the mayor’s competence.
- Based on her posterior beliefs about the mayor’s competence, the voter decides who to vote for in the mayoral election.
Part 1: Learning from the Hurricane
In this first part of the analysis, you’ll work through the first three events in the model, calculating the voter’s posterior beliefs about the mayor’s competence in the event that the pre-election hurricane floods the city, on the one hand, and in the event that the pre-election hurricane does not flood the city, on the other.
To begin, suppose that the voter’s prior beliefs (i.e before the pre-election hurricane hits the city) about the mayor’s competence and whether the hurricane will flood the city are described by the conditional probability distribution depicted in the following diagram:
Prompt 1A
Fill in the following table to describe the voter’s posterior beliefs about the mayor’s competence in the event that the pre-election hurricane floods the city…
| Mayor’s Competence | Probability |
|---|---|
| competent | |
| incompetent |
…and mark up this diagram…
…to show how you computed the values for the blank cells in the table.
Prompt 1B
Fill in the following table to describe the voter’s posterior beliefs about the mayor’s competence in the event that the pre-election hurricane does not flood the city…
| Mayor’s Competence | Probability |
|---|---|
| competent | |
| incompetent |
…and mark up this diagram…
…to show how you computed the values for the blank cells in the table.
Rubric for Part 1
For each of Prompt 1A and 1B you get…
- 2 points if you write the correct values for both blank cells in the table and if you provide a diagram that correctly shows how to compute those values.
- 1 point if you write a diagram that shows that you used the correct procedure to compute the values in the table, but one or more of those values is missing or incorrect.
- 0 points otherwise.
Part 2
You’ve now depicted how the voter’s posterior beliefs about the mayor’s competence depend on whether the pre-election hurricane floods the city. We’ll now use your results to depict how those beliefs may or may not swing the voter’s vote in the election.
Suppose there are two candidates running in the election – the mayor, who is running for re-election, and one challenger, who is running to unseat and replace the current mayor. Thus, on election day the voter must choose between one of two possible actions: either vote for the mayor or vote for the challenger. Suppose that the voter’s choice will decide the election – i.e. the mayor will win if the voter votes for the mayor, and the challenger will win if the voter votes for the challenger.
Suppose that the voter doesn’t care about which candidate she votes for per-se. All she really cares about are the traits of whoever wins the election. Specifically, she cares about (a) whether the winner of the election is competent or incompetent, and (b) the extent to which the winner of the election shares her policy views.
We’ll start by specifying how the voter’s vote affects the competence of the election winner. Assume first that the challenger is new to politics and thus had no role in the city’s hurricane preparedness prior to the pre-election hurricane. Thus, whether or not the pre-election hurricane floods the city has no effect on the voter’s belief about the challenger’s competence. Specifically, at the moment the voter chooses between voting for the mayor and voting for the challenger, she has the following belief about the challenger’s competence:
| Challenger’s Competence | Probability |
|---|---|
| competent | \frac{1}{2} |
| incompetent | \frac{1}{2} |
As for the voter’s beliefs about the mayor’s competence at the moment she chooses who to vote for, you computed these beliefs in Part 1 above. You know that these beliefs differ, depending on whether or not the pre-election hurricane flooded the city. Thus, the voter’s beliefs on election day about the competence of the mayor and the challenger, depending on whether the hurricane flooded the city, look like this:
Now consider the effect of the voter’s vote on the extent to which the winner of the election share’s the voter’s policy views.
Assume the voter knows that the mayor’s and challenger’s ideal points are 0 and 10, respectively, and assume that the voter’s ideal point is 6, like this:
Given all of the above then, the outcomes that can result from the voter’s vote are:
- Winner is competent and has ideal point 0.
- Winner is incompetent and has ideal point 0.
- Winner is competent and has ideal point 10.
- Winner is incompetent and has ideal point 10.
To complete the model, we just need to specify the voter’s utility function. Assume that given the outcome, the voters utility level is given by:
\begin{array}{ccc} \underbrace{\begin{Bmatrix} 10 & \text{if the winner of the election is competent} \\ 0 & \text{if the winner of the election is incompetent} \end{Bmatrix}} & + & \underbrace{\begin{Bmatrix} 6-\text{winner's ideal point} & \text{if the winner's ideal point is greater than $6$} \\ \text{winner's ideal point}-6 & \text{if the winner's ideal point is less than or equal to $6$} \end{Bmatrix}} \\ \text{competence term} & & \text{policy term} \\ \end{array}
Notice that what we have here are, in effect, two different expected utility models. One model depicts the voter’s choice between voting for the mayor and voting for the challenger if the pre-election hurricane floods the city:
The other model depicts the voter’s choice between voting for the mayor and voting for the challenger if the pre-election hurricane does not flood the city:
The only difference between these two models are the voter’s beliefs about the mayor’s competence. Thus by comparing them, we can explore whether the pre-election hurricane changes the voters beliefs to an extent that causes her to change her vote.
Prompt 2A
Using the posterior beliefs you calculated in response to Prompt 1A and applying the voter’s utility function, fill out the following tables that state the voter’s possible utility levels from voting for the mayor and voting for the challenger and the probabilities with which those utility levels occur, if the pre-election hurricane floods the city…
| Utility Level | Probability |
|---|---|
| \, | |
| \, |
| Utility Level | Probability |
|---|---|
| \, | |
| \, |
…and explain, step-by-step, how you computed the cells in the tables.
Prompt 2B
Use your answers to Prompt 2A to compute the voter’s expected utility from each of her available actions if the pre-election hurricane floods the city by filling in a table like this…
| Action | Expected Utility |
|---|---|
| vote for the mayor | |
| vote for the challenger |
…and explain, step-by-step, how you computed your answers.
Prompt 2C
Using the posterior beliefs you calculated in response to Prompt 1B and applying the voter’s utility function, fill out the following tables that state the voter’s possible utility levels from voting for the mayor and voting for the challenger and the probabilities with which those utility levels occur, if the pre-election hurricane does not flood the city…
| Utility Level | Probability |
|---|---|
| \, | |
| \, |
| Utility Level | Probability |
|---|---|
| \, | |
| \, |
…and explain, step-by-step, how you computed the cells in the tables.
Prompt 2D
Use your answers to Prompt 2C to compute the voter’s expected utility from each of her available actions if the pre-election hurricane does not flood by filling in a table like this…
| Action | Expected Utility |
|---|---|
| vote for the mayor | |
| vote for the challenger |
…and explain, step-by-step, how you computed your answers.
Prompt 2E
Do the voter’s preferences over who to vote for change depending on whether the pre-election hurricane floods the city? For instance, does she get a higher expected utility from voting for the mayor than from voting for that challenger when the hurricane does not flood the city, but (on the other hand) a higher expected utility from voting for the challenger than from voting for the mayor when the hurricane does flood the city? State your answer, specifying how the expected utilities from the available actions are ordered relative to one another for each possible result of the pre-election hurricane (a flood or no flood) and explain your answer with specific reference to the results you calculated in response to Prompts 2B and 2D.
Rubric for Part 2
For each of Prompts 2A through 2E…
- You get two points if all the answers you give are correct and you give a correct explanation for how you computed the results.
- You get one point if some or all of your answers are missing or incorrect, but you give an explanation that shows that you correctly understand the process for computing the result.
- You get zero points otherwise.