COU 6: The Potential Candidate to the Right

Continue to consider the model in which the location of the potential candidate’s ideal point is specified as the parameter P:

A Model of the Choice to Run with the Potential Candidate’s Ideal Point Parameterized

A potential candidate chooses between two alternatives: to run or not run in an upcoming election
A current candidate is already running in the election, and is the only person other than the potential candidate who will run.
If the potential candidate does not run, the current candidate wins the election with certainty.
If the potential candidate runs, the potential candidate wins the election with certainty.
If the current candidate wins the election, he sets policy to the location of his ideal point, which is 10.
If the potential candidate wins the election, she sets policy to the location of her ideal point, which is a number P.

Given the location x of policy that ultimately results from the election and the potential candidate’s choice to run or not run, the potential candidate’s utility level is

\begin{array}{ccc} \underbrace{ -|x-P| } & - & \underbrace{ \begin{Bmatrix} 5 & \text{if the potential candidate runs} \\ 0 & \text{if the potential candidate does not run} \end{Bmatrix} } \\ \text{Policy Term} & & \text{Cost Term} \end{array}

In the lesson, we just showed that if P \leq 10, then the potential candidate prefers to run instead of not run if P < 5 and prefers to not run instead of run if 5 < P \leq 10. This leaves the case in which P > 10 unsolved. You’ll figure it out in this COU by responding to the following prompts:

Prompt 1

Assume that P > 10. Compute the expression for the potential candidate’s utility level if she runs. Show the steps you went through to compute it by first writing the utility function, with the two terms labeled as the “Policy Term” and “Cost Term”. Then show the expressions for the values of each of these terms that result when the potential candidate runs. Then show the expression for the value of the overall utility level.

Prompt 2

Assume that P > 10. Compute the expression for the potential candidate’s utility level if she does not run. Show the steps you went through to compute it by first writing the utility function, with the two terms labeled as the “Policy Term” and “Cost Term”. Then show the expressions for the values of each of these terms that result when the potential candidate does not run. Then show the expression for the value of the overall utility level.

Prompt 3

Using the expressions you derived in response to Prompts 1 and 2, write the inequality that defines the conditions under which the potential candidate’s utility level from running is greater than or equal to her utility level from not running. Then, re-arrange the terms of this inequality so that the potential candidate’s ideal point P is by itself of one side of the inequality. After doing that, describe the steps you used to re-arrange the inequality. (e.g. “First I added 4 to both sides, then I multiple both sides by \frac{3}{16}.”)

Prompt 4

Draw a diagram similar to the one that appears in the lesson just above the link to this COU. Unlike that diagram, this one should show the model’s implications for the potential candidate’s choice between running and not running in the case where P > 10. Like that diagram, it should show the range of possible locations of P from 10 and up along a line. It should also delimit and label the ranges of values of P where the potential candidate prefers to run instead of not run, and the ranges of values of P where the potential candidate prefers to not run instead of run. It should also use numerical labels to show the exact limits of these ranges.

Rubric

Prompts 1 and 2 are each worth 1 point. On each, you get 1 point if you give the correct expression and show the steps through which you derived it as specified in the prompts, and 0 points otherwise.

On Prompt 3 you get 1 point if you write the initial inequality asked for, rearrange it as required (showing any intermediate steps or not), and describe a steps that correctly produces the final inequality from the initial inequality. You get 0 points otherwise.

On Prompt 4 you get 1 point if you draw a diagram that correctly depicts the correct answer from Prompt 3, and includes all of the features required by the prompt. You get 0 points otherwise.