COU 1: Summarizing An Expected Utility Model
In this COU, you’ll make sure you understand the required elements of expected utility models by extracting the required elements from a narrative description of an expected utility model.
Below is description of an expected utility model that states all of the elements required of any expected utility model, but doesn’t explicitly identify which element is which. The prompts guide you through the process of extracting the required elements from the description and stating the elements of the model explicitly.
Imagine a voter choosing between two candidates in an election. One candidate called the incumbent is running for re-election. The other called the challenger is running to unseat and replace the incumbent. Assume the voter must choose between one of two available actions: Vote for the incumbent or vote for the challenger.
In real-world elections there are many voters, so that any one voter’s vote may or may not determine the election outcome. But this model will depict an extremely simplistic election in which the one voter in the model is the only voter who votes in the election. Thus her vote – whether for the incumbent or challenger – will decide which candidate wins the election and takes office.
Suppose that the ultimate outcome the voter cares about are the traits of whichever candidate wins the election and takes office. Specifically, each of the candidates has two traits that matter to the voter: his level of competence and his ideal point. Each candidate might have either of two possible levels of competence. Specifically, each candidate might be either competent or incompetent. Each candidate’s ideal point amounts to a number. Specifically, the ideal point of the incumbent is 0, and the ideal point of the challenger is 10, like this:
So, by casting her vote for either the incumbent or challenger, the voter will cause the winner of the election to be one of the following:
- competent with an ideal point of 0.
- incompetent with an ideal point of 0.
- competent with an ideal point of 10.
- incompetent with an ideal point of 10.
Suppose that the voter has direct and perfect knowledge of the candidates’ ideal points. Specifically, she knows that the incumbent’s ideal point is 0 and the challenger’s ideal point is 10. However, she is uncertain about each candidate’s competence. She believes that the incumbent is competent with probability \frac{2}{3} and incompetent with probability \frac{1}{3}. She believes that the challenger is competent with probability \frac{1}{2} and incompetent with probability \frac{1}{2}.
This means that the voter is uncertain about which outcomes will result from each of her available actions. If she votes for the incumbent, the winner of the election will be competent with an ideal point of 0 with probability \frac{2}{3} and will be incompetent with an ideal point of 0 with probability \frac{1}{3}. If she votes for the challenger, the winner of the election will be competent with an ideal point of 10 with probability \frac{1}{2} and will be incompetent with an ideal point of 10 with probability \frac{1}{2}.
The voter’s utility level from the outcome on the election is given by the sum of two terms, like this: \begin{array}{ccc} \text{competence term} & + & \text{policy term} \\ \end{array} The competence term depends on whether the candidate who wins the election is competent or incompetent. Specifically: \text{competence term} = \begin{Bmatrix} 10 & \text{if the winner of the election is competent} \\ 0 & \text{if the winner of the election is incompetent} \end{Bmatrix} The policy term depends on the distance between the voter’s ideal point and the ideal point of the candidate that wins the election. The voter’s ideal point is 8. So her ideal point is closer to the challenger’s ideal point (10) than to the incumbent’s ideal point (0). Specifically: \text{policy term} = \begin{Bmatrix} 8-\text{winner's ideal point} & \text{if the winner's ideal point is greater than $8$} \\ \text{winner's ideal point} - 8 & \text{if the winner's ideal point is less than or equal to $8$} \end{Bmatrix} Thus, the voter’s utility level from whichever outcome results from her choice 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} 8-\text{winner's ideal point} & \text{if the winner's ideal point is greater than $8$} \\ \text{winner's ideal point}-8 & \text{if the winner's ideal point is less than or equal to $8$} \end{Bmatrix}} \\ \text{competence term} & & \text{policy term} \\ \end{array}
Rubric
There is one correct answer to each prompt, although it may be possible to express that answer in a variety of ways. For each prompt, you get one point if your answer is wholly correct and zero points otherwise.