Efficient Elicitation of Aggregated Rankings Under Kemeny’s Rule

Sampling with replacement involves selecting instances from a population multiple times, while sampling without replacement selects them only once. In this article, we compare the two methods through various scenarios using Mallows model. We observe that sampling with replacement leads to more even distributions of instances across the different steps, while without replacement tends to concentrate instances in certain steps.
One way to understand this difference is to think of a bag of candy. If you randomly pick pieces of candy from the bag and put them back in, you are using the sampling with replacement method. The distribution of candy pieces will be more even over time. On the other hand, if you take pieces of candy without putting them back, you are using the sampling without replacement method. Over time, the candy pieces will concentrate in certain areas of the bag.
Another analogy is a deck of cards. If you shuffle the deck and deal out cards one at a time, you are using sampling with replacement. The distribution of cards will be more even after each deal. However, if you deal out the cards without shuffling the deck, you are using sampling without replacement. Over time, certain cards will appear more frequently than others.
In conclusion, both sampling with replacement and without replacement have their advantages and disadvantages depending on the scenario. Sampling with replacement leads to a more even distribution of instances but may be computationally expensive, while sampling without replacement concentrates instances in certain steps but is faster to compute. By understanding these differences, we can make informed decisions about which method to use based on our specific needs and goals.