COU 1: Voters Learning about Prosecutors
In the United States, most District Attorneys – the local officials who prosecute criminal cases against defendants on behalf of the state – gain and retain their offices through elections. How do voters evaluate candidates for District Attorney in these elections? Do voters have the information required to vote responsibly and effectively?
In their paper “Citizen Oversight and the Electoral Incentives of Criminal Prosecutors”, Sanford Gordon and Greg Huber build and analyze a model of prosecutorial elections. In this COU, you’ll analyze a model of learning inspired by Gordon’s and Huber’s work.
Gordon and Huber point out that ordinary voters have access to just one piece of information bearing on the performance of incumbent district attorneys: The rate at which the cases they bring result in a conviction or guilty plea. This very limited information, of course, may not tell voters very much about whether an incumbent prosecutor has done a good job, or used her powers in ways consistent with voters’ values.
Imagine a voter who will vote in an upcoming election in which an incumbent district attorney is running for another term in office. Suppose that prior to casting her vote, the voter will observe the outcome of just one criminal case brought against a defendant by the incumbent. Specifically, the voter will observe whether the case results in a conviction or aquittal. Suppose further that the likelihood that the defendent will be convicted depends on how much effort the prosecutor gives to the case. Suppose the voter cannot observe the amount of effort the prosecutor puts in. But she can learn about it by observing whether the defendant is acquitted or convicted.
Specifically, suppose that the voter’s prior beliefs about the amount of effort the prosecutor puts into the trial and whether the defendent will be convicted or aquitted are described by the following Marginal-Conditional Diagram:
Imagine the voter observes the outcome of the case. Compute the probabilities in the voter’s posterior beliefs as follows:
- Compute the probability that the prosecutor put no effort into the case in the voter’s posterior beliefs when the case results in an acquittal. Write your answer as a fraction, not as a decimal number.
- Compute the probability that the prosecutor put a little effort into the case in the voter’s posterior beliefs when the case results in an acquittal. Write your answer as a fraction, not as a decimal number.
- Compute the probability that the prosecutor put no effort into the case in the voter’s posterior beliefs when the case results in an conviction. Write your answer as a fraction, not as a decimal number.
- Compute the probability that the prosecutor put a little effort into the case in the voter’s posterior beliefs when the case results in an conviction. Write your answer as a fraction, not as a decimal number.
For each answer your write, mark up a copy of the diagram above showing how you computed your answer.
You do not need to simplify any of the fractions you write, although you are welcome to do so if you’d like. For instance, a fraction written as \frac {\frac{1}{33}\frac{16}{342}} {\frac{1}{33}\frac{16}{342} + \frac{13}{33}\frac{87}{89} + \frac{19}{33}\frac{694}{879}} is perfectly acceptable, as long as it is correct.
Now imagine that the voter is not concerned per se with the prosecutor’s level of effort, but with the results the prosecutor produces. Specifically, suppose that in each criminal case the prosecutor manages, the defendant is in fact either innocent or guilty of the criminal act in question. Suppose the voter wants the prosecutor to act so that, to the extent possible, the accused in each case is convicted if she is in fact guilty, and is aquitted otherwise.
The problem the voter faces here is that any given district attorney manages thousands of cases per year, and each case is complex. The voter cannot look into the details of each and every case and try to judge whether its outcome seems to have matched the defendant’s actual guilt or innocence. All the voter can really know, when she decides how to cast her vote, are the rates at which defendants brought to trial were convicted. Just by observing a prosecutor’s conviction rate, what can a voter infer about the rate at which the actually guilty are convicted and the actually innocent are acquitted?
Think of a criminal case in which a single person is investigated as a suspect. Suppose that there are two features of the proceeding that drive its outcome, neither of which the voter can directly observe: First, whether suspect is in fact guilty or in fact innocent, and second whether the prosecutor does her job skillfully or unskillfully. Suppose the case culiminates in the suspect being either convicted or acquitted, and that the suspect’s actual guilt or innocence and the prosecutor’s skillful or unskillful management of the case drive the outcome as in the following Marginal-Conditional Diagram:
Suppose that the above Marginal-Conditional Diagram describes the voter’s prior beliefs, before observing the outcome (aquittal or conviction) of the case against the suspect. Imagine the voter then observes the outcome of the case. What can the voter learn from the outcome about only the in-fact-guilt or in-fact-innocence of the suspect? Using the identities connecting joint, marginal and conditional probabilities described in Lesson 4, one can show that \begin{array}{l} P \left( \text{suspect is in fact guilty} | \text{suspect is convicted} \right) = \\\\ P \left( \left. \begin{array}{c} \text{suspect is in fact guilty and} \\ \text{prosector acted skillfully} \end{array} \right| \text{suspect is convicted} \right) + \\\\ P \left( \left. \begin{array}{c} \text{suspect is in fact guilty and} \\ \text{prosector acted unskillfully} \end{array} \right| \text{suspect is convicted} \right) \end{array} and \begin{array}{l} P \left( \text{suspect is in fact guilty} | \text{suspect is acquitted} \right) = \\\\ P \left( \left. \begin{array}{c} \text{suspect is in fact guilty and} \\ \text{prosector acted skillfully} \end{array} \right| \text{suspect is acquitted} \right) + \\\\ P \left( \left. \begin{array}{c} \text{suspect is in fact guilty and} \\ \text{prosector acted unskillfully} \end{array} \right| \text{suspect is acquitted} \right) \end{array}
Using that result…
- Compute the probability that the suspect is in fact guilty according to the voter’s posterior beliefs when she observes that the suspect has been convicted. Write your answer as a fraction, not as a decimal number.
- Compute the probability that the suspect is in fact guilty according to the voter’s posterior beliefs when she observes that the suspect has been acquitted. Write your answer as a fraction, not as a decimal number.
For each answer your write, mark up a copy of the diagram above showing how you computed your answer.
You do not need to simplify any of the fractions you write, although you are welcome to do so if you’d like. For instance, a fraction written as \frac {\frac{1}{33}\frac{16}{342}} {\frac{1}{33}\frac{16}{342} + \frac{13}{33}\frac{87}{89} + \frac{19}{33}\frac{694}{879}} is perfectly acceptable, as long as it is correct.
Suppose that the above Marginal-Conditional Diagram describes the voter’s prior beliefs, before observing the outcome (aquittal or conviction) of the case against the suspect. Imagine the voter then observes the outcome of the case. What can the voter learn from the outcome about only whether the prosector acted skillfully or unskillfully? Using the identities connecting joint, marginal and conditional probabilities described in Lesson 4, one can show that \begin{array}{l} P \left( \text{prosector acted skillfully} | \text{suspect is convicted} \right) = \\\\ P \left( \left. \begin{array}{c} \text{suspect is in fact guilty and} \\ \text{prosector acted skillfully} \end{array} \right| \text{suspect is convicted} \right) + \\\\ P \left( \left. \begin{array}{c} \text{suspect is in fact innocent and} \\ \text{prosector acted skillfully} \end{array} \right| \text{suspect is convicted} \right) \end{array} and \begin{array}{l} P \left( \text{prosecutor acted skillfully} | \text{suspect is acquitted} \right) = \\\\ P \left( \left. \begin{array}{c} \text{suspect is in fact guilty and} \\ \text{prosector acted skillfully} \end{array} \right| \text{suspect is acquitted} \right) + \\\\ P \left( \left. \begin{array}{c} \text{suspect is in fact innocent and} \\ \text{prosector acted skillfully} \end{array} \right| \text{suspect is acquitted} \right) \end{array}
Using that result…
- Compute the probability that the prosecutor acted skillfully according to the voter’s posterior beliefs when she observes that the suspect has been convicted. Write your answer as a fraction, not as a decimal number.
- Compute the probability that the prosecutor acted skillfully according to the voter’s posterior beliefs when she observes that the suspect has been acquitted. Write your answer as a fraction, not as a decimal number.
For each answer your write, mark up a copy of the diagram above showing how you computed your answer.
You do not need to simplify any of the fractions you write, although you are welcome to do so if you’d like. For instance, a fraction written as \frac {\frac{1}{33}\frac{16}{342}} {\frac{1}{33}\frac{16}{342} + \frac{13}{33}\frac{87}{89} + \frac{19}{33}\frac{694}{879}} is perfectly acceptable, as long as it is correct.
Rubric
Prompt 1
For each of parts (a) through (d) you can earn up to three points. Specifically:
- Three points if you write the correct answer (regardless of whether or how it is simplified) and you provide a marked-up diagram that shows the correct method for computing the answer.
- One point if you write the correct answer (regardless of whether or how it is simplified) but you do not provide a marked-up diagram that shows the correct method for computing the answer.
- Zero points otherwise.
Prompt 2
For each of parts (a) and (b) you can earn up to three points. Specifically:
- Three points if you write the correct answer (regardless of whether or how it is simplified) and you provide a marked-up diagram that shows the correct method for computing the answer.
- One point if you write the correct answer (regardless of whether or how it is simplified) but you do not provide a marked-up diagram that shows the correct method for computing the answer.
- Zero points otherwise.
Prompt 3
For each of parts (a) and (b) you can earn up to three points. Specifically:
- Three points if you write the correct answer (regardless of whether or how it is simplified) and you provide a marked-up diagram that shows the correct method for computing the answer.
- One point if you write the correct answer (regardless of whether or how it is simplified) but you do not provide a marked-up diagram that shows the correct method for computing the answer.
- Zero points otherwise.