COU 3: Computing Expected Utilities

Part 1

Here is a slightly different version of the expected utility model you worked with in COU 1. A voter must choose which of two candidates to vote for in an election. One candidate is an incumbent who is running to be re-elected to an office she already occupies. The other is a challenger who is running to unseat the incumbent. In this simplistic election, the voter is the one an only voter. Thus her vote will determine which candidate wins the election and takes office.

The voter doesn’t care which candidate wins the election per-se. Instead, she cares about the traits of the whichever candidate wins. Specifically, she cares whether the winner is corrupt or honest and about the winner’s ideal point.

She is uncertain about whether each candidate is corrupt. Specifically, she think the incumbent is corrupt with probability \frac{5}{9} and honest with probability \frac{4}{9}. She thinks the challenger is corrupt with probability \frac{1}{2} and honest with probability \frac{1}{2}.

The voter has no uncertainty about the ideal points of the two candidates. She knows that the incumbent’s ideal point is 10 and the challenger’s ideal point is 0, like this:

The voters’ utility level from the result of the election depends on whether the winner is corrupt or honest, and the proximity of the winner’s ideal point to her own ideal point, which is 4. Specifically, her utility level from any given outcome is:

\begin{array}{ccc} \underbrace{\begin{Bmatrix} 0 & \text{if the winner of the election is honest} \\ -12 & \text{if the winner of the election is corrupt} \end{Bmatrix}} & + & \underbrace{\begin{Bmatrix} 4-\text{winner's ideal point} & \text{if the winner's ideal point is greater than $4$} \\ \text{winner's ideal point}-4 & \text{if the winner's ideal point is less than or equal to $4$} \end{Bmatrix}} \\ \text{competence term} & & \text{policy term} \\ \end{array}

Here’s a summary of the model:

Available Actions:

  • Vote for the incumbent
  • Vote for the challenger

Possible Outcomes

  • The winner is corrupt and has an ideal point of 0.
  • The winner is honest and has an ideal point of 0.
  • The winner is corrupt and has an ideal point of 10.
  • The winner is honest and has an ideal point of 10.

Conditional Probability Distributions

If voter votes for the incumbent
Outcome Probability
winner is honest and has an ideal point of 10 \frac{5}{9}
winner is corrupt and has an ideal point of 10 \frac{4}{9}
If voter votes for the challenger
Outcome Probability
winner is honest and has an ideal point of 0 \frac{1}{2}
winner is corrupt and has an ideal point of 0 \frac{1}{2}

Utility Function:

\begin{array}{ccc} \underbrace{\begin{Bmatrix} 0 & \text{if the winner of the election is honest} \\ -12 & \text{if the winner of the election is corrupt} \end{Bmatrix}} & + & \underbrace{\begin{Bmatrix} 4-\text{winner's ideal point} & \text{if the winner's ideal point is greater than $4$} \\ \text{winner's ideal point}-4 & \text{if the winner's ideal point is less than or equal to $4$} \end{Bmatrix}} \\ \text{competence term} & & \text{policy term} \\ \end{array}

Prompt 1A

Apply the voter’s utility function and the conditional probability distributions over outcomes in the model to derive the conditional probability distributions over the voter’s utility levels that can result from the voters action. Write your results in a table like this:

If voter votes for the incumbent
Utility Level Probability
If voter votes for the challenger
Utility Level Probability

Prompt 1B

Describe, step-by-step, the process you used to assign/compute the values in the table you provided in response to Prompt 1A.

Prompt 1C

Using the results of Prompt 1A, compute the voter’s expected utility from each of her available actions. Write your results in a table like this:

Action Expected Utility
vote for the incumbent
vote for the challenger

Prompt 1D

Describe, step-by-step, how you computed the results you wrote in Prompt 1C from the results you wrote in response to Prompt 1A.

Rubric for Part 1

Whether you earn any points on Prompts 1A and 1C depend on whether you earn points on 1B and 1D. So we will score 1B and 1D first.

For each of Prompts 1B and 1D, you get two points if you describe the correct procedure, with specific reference to the relevant components of this particular model, and zero points otherwise.

For Prompt 1A, you get one point if your answer is wholly correct and if you got full credit on Prompt 1B and zero points otherwise.

For Prompt 1C, you get one point if your answer is wholly correct and if you got full credit on Prompt 1D and zero points otherwise.

Part 2

Here is a slightly different version of the expected utility model you worked with in COU 2. A person is deciding whether or not to join an anti-government terrorist or guerrilla group. The person is unsure whether the group is competent or incompetent. She thinks they are competent with probability \frac{3}{16} and competent with probability \frac{13}{16}.

Her utility level depends on whether she is a member of the group and whether the group is competent. Specifically, for any given outcome her utility level is:

\begin{Bmatrix} 1 & \text{if the group is competent and she is a member} \\ 0 & \text{if she is not a member} \\ -10 & \text{if the group is incompetent and she is a member} \end{Bmatrix}

Here is a summary of the model:

Available Actions:

  • join the group
  • do not join the group

Possible Outcomes

  • the group is competent and she is a member
  • the group is incompetent and she is a member
  • the group is competent and she is not a member
  • the group is incompetent and she is not a member

Conditional Probability Distributions

If she joins the group
Outcome Probability
group is competent and she is a member \frac{3}{16}
group is incompetent and she is a member \frac{13}{16}
If she does not join the group
Outcome Probability
group is competent and she is not a member \frac{3}{16}
group is incompetent and she is not a member \frac{13}{16}

Utility Function

\begin{Bmatrix} 1 & \text{if the group is competent and she is a member} \\ 0 & \text{if she is not a member} \\ -10 & \text{if the group is incompetent and she is a member} \end{Bmatrix}

Prompt 2A

Apply the person’s utility function and the conditional probability distributions over outcomes in the model to derive the conditional probability distributions over the person’s utility levels that can result from the her action. Write your results in a table like this:

If she joins the group
Utility Level Probability
If she does not join the group
Utility Level Probability

Prompt 2B

Describe, step-by-step, the process you used to assign/compute the values in the table you provided in response to Prompt 2A.

Prompt 2C

Using the results of Prompt 2A, compute the person’s expected utility from each of her available actions. Write your results in a table like this:

Action Expected Utility
join the group
do not join the group

Prompt 2D

Describe, step-by-step, how you computed the results you wrote in Prompt 2C from the results you wrote in response to Prompt 2A.

Rubric for Part 2

Whether you earn any points on Prompts 2A and 2C depend on whether you earn points on 2B and 2D. So we will score 2B and 2D first.

For each of Prompts 2B and 2D, you get two points if you describe the correct procedure, with specific reference to the relevant components of this particular model, and zero points otherwise.

For Prompt 2A, you get one point if your answer is wholly correct and if you got full credit on Prompt 2B and zero points otherwise.

For Prompt 2C, you get one point if your answer is wholly correct and if you got full credit on Prompt 2D and zero points otherwise.