Statistical Profiling in Random Car Searches: Local law enforcement officials sometimes engage in “random” searches of cars to look for illegal substances. When one looks at the data of who is actually searched, however, the pattern of searches oftentimes does not look random.

A: In what follows, assume that random searches have a deterrent effect—i.e. the more likely someone believes he is going to be searched; the less likely he is to engage in transporting illegal substances.

(a) Suppose first that it has been documented that, all else being equal, illegal substances are more likely to be transported in pick-up trucks than in passenger cars. Put differently, if pick-up truck owners are searched with the same probability as passenger car owners, law enforcement officials will be more likely to find illegal substances when they randomly search a pick-up truck than when they randomly search a passenger vehicle. If the objective by police is to find the most illegal substances given that they have limited resources (and thus cannot search everyone), is it optimal for them to search randomly?

(b) Suppose the police decides to allocate its limited resources by searching pick-up trucks with probability δ and passenger cars with probability γ (where δ > γ). After a few months of this policy, the police discover that they find on average 2.9 grams of illegal substances per pickup- truck search and 1.5 grams of illegal substances per passenger vehicle. How would you advise the police — given their limited resources— to change their search policy in order to increase the amount of drugs found?

(c) Given your answer to (b), what has to be true about the probability of finding illegal substances in pick-up trucks and passenger cars if the search probabilities for the two types of vehicles are set optimally (relative to the police’s objective to find the most illegal substances)?

(d) If you simply observe that δ> γ, can you conclude that the police are inherently biased against pick-up trucks owners? Why or why not?

(e) What would have to be true about the average yield of illegal substances per search for the different types of vehicles for you to argue that the police was inherently biased against pickup trucks?

(f) Could it be the case that δ> γ and the police show behavior inherently biased against passenger cars?

(g) We have used the emotionally neutral categories of “pick-up trucks” and “passenger vehicles”. Now consider him more empirically relevant case of “minority neighborhoods” and “nonminority neighborhoods” — with law enforcement often searching cars in the former with significantly higher probability than in the latter. Can you argue that such behavior by law enforcement officials is not inherently racist in the sense of being motivated by animosity against one group, but that instead it could be explained simply as a matter of statistical discrimination that maximizes the effectiveness of car searches in deterring the trafficking in illegal substances? What evidence might you look for to make your case?

B: Suppose that the police have sufficient resources to conduct 100 car searches per day and that half of all vehicles are pick-up trucks and half are passenger cars. The probability of finding an illegal substance in a pick-up truck is p_t (n_t )= 9/(90+n_t )where n_t is the number of pick-up truck searches conducted. The probability of finding an illegal substance in a passenger car is pc (n_c ) = 1/(10+n_c ) (where n_c is the number of car searches conducted).

(a) Suppose that the objective of the police is to maximize the number of interdictions of illegal substances. Write down the optimization problem—with n_t and n_c as choice variables and the constrain_t that n_t +n_c = 100.

(b) According to the police’s objective function, how many trucks should be searched per day? How many passenger vehicles?

(c) If law enforcement conducts searches as calculated in (b), what is the probability of interdicting illegal substances in pick-up trucks? What is the probability of interdicting such substances in passenger cars?

(d) If law enforcement officials search trucks and cars at the rates you derived in (b), how many illegal substance interdictions would on average occur every day?

(e) How many of each type of car would on average be searched each day if the police instead searched vehicles randomly?

(f) If the police conducted random searches, what would be the probability of finding illegal substances in each of the two vehicle types? How does this compare to your answer to (c)?

(g) How many illegal substance interdictions per day would on average occur if the police conducted random searches instead of what you derived in (b)?

(h)Why is your answer to (c) different than your answer to (g)?

(i) Insurance companies charge higher insurance rates to young drivers than to middle aged drivers. How is their behavior similar to the behavior by law enforcement that searches pickup trucks more than passenger cars in (b)?