Question

Many cities regulate the number of taxi licenses, and there is a great deal of competition for both new and existing licenses. Suppose that a city has decided to sell 10 new licenses for \(\$ 25,000\) each. A lottery will be held to determine who gets the licenses, and no one may request more than three licenses. Twenty individuals and taxi companies have entered the lottery. Six of the 20 entries are requests for 3 licenses, 9 are requests for 2 licenses, and the rest are requests for a single license. The city will select requests at random, filling as many of the requests as possible. For example, the city might fill requests for \(2,3,1\), and 3 licenses and then select a request for \(3 .\) Because there is only one license left, the last request selected would receive a license, but only one. a. An individual has put in a request for a single license. Use simulation to approximate the probability that the request will be granted. Perform at least 20 simulated lotteries (more is better!). b. Do you think that this is a fair way of distributing licenses? Can you propose an alternative procedure for distribution?

Short answer

The solution requires the execution of simulation process to determine the probability of an individual obtaining a license. The fairness and alternative method of distribution is subjective and depends on personal interpretation of the situation.

Step-by-step solution

- Analyzing the request demands

First, you need to understand the demand for licenses. Among the 20 requests, 6 are for 3 licenses, 9 are for 2 licenses and the remaining (20-6-9=5) are for 1 license. This means there are a total of 6*3 + 9*2 + 5*1 = 35 requested licenses.

- Simulating the lottery

The city has only 10 licenses available. To simulate the lottery, you must randomly choose requests (without replacement) and give them licenses until all 10 are given. Note that a request can only receive the number of licenses they asked for or the remaining licenses, whichever is smaller.

- Counting successful requests

After each simulation, record if the individual with a request for a single license got his license or not. A 'success' indicates that the individual received a license.

- Estimating the probability

After completing at least 20 simulations, the approximate probability of the individual receiving a license can be estimated as the number of successful simulations divided by the total number of simulations.

- Discussing Fairness

The fairness of this method of distribution can be discussed by considering if this method is unbiased and if it equally considers the needs of all participants. It depends on personal understanding and interpretation.

- Suggesting an alternative

A possible alternative procedure for distribution can be proposed based on understanding of fairness and effectiveness of distribution. For instance, it might be fairer to distribute the licenses in such a way that it considers the number of licenses requested by each individual/company and the total number of requests.