Question

Alice and March Hare are attending the Mad Hatter's tea party. The waiter refuses to offer different drinks per participant so they have to reach an agreement on what to request. The possibilities are: coffee, tea, or wine. The Mad Hatter proposes the following procedure: they take each pair of alternative drinks and let a majority vote determine their collective ordering. The drink that comes up top shall then be requested from the waiter. The individual rankings are as follows: - Alice: coffee, tea, wine - March Hare: wine, coffee, tea - Mad Hatter: tea, wine, coffee (a) March Hare suggests that they first consider coffee versus tea, and then the winner of that contest versus wine. Which drink will be ordered? (b) Instead, the Mad Hatter suggests that they consider coffee versus wine, and then the winner of that contest versus tea. Which drink will be ordered? (c) What order should Alice suggest, if she wants to get her favourite drink served? The tea party at the Mad Hatter's house was so wild that the house has become a mess. Since the Mad Hatter very much dislikes tidying up, he now contemplates leaving his house in Tulgey Woods. He finds beautiful mansions for sale in the following areas of Wonderland: Boro Grove, Mallow Marsh, and Whispering Woods. In terms of area the Madhatter prefers Mallow Marsh to Boro Grove, Boro Grove to Whispering Woods, and Whispering Woods to Mallow Marsh. Page 3 of 6 (d) Are the Mad Hatter's preferences about areas complete? ( 5 points) (e) Are the Mad Hatter's preferences about areas transitive?

Answer 1

(a) According to the proposed procedure, they first consider coffee versus tea. Alice prefers coffee and March Hare prefers tea. Therefore, tea wins the majority vote. Then, they compare tea with wine. Alice prefers tea and March Hare prefers wine. In this case, wine wins the majority vote. Therefore, the drink that will be ordered is wine.

(b) Following the Mad Hatter's suggestion, they first consider coffee versus wine. Alice prefers coffee and March Hare prefers wine. In this case, wine wins the majority vote. Then, they compare wine with tea. Alice prefers tea and March Hare prefers wine. Again, wine wins the majority vote. Therefore, the drink that will be ordered is wine.

(c) To ensure that her favorite drink, coffee, is served, Alice should suggest comparing coffee versus wine first. Since both March Hare and the Mad Hatter prefer wine to coffee, coffee will lose the majority vote. Then, they compare the winning drink, wine, with tea. Since Alice prefers tea and the Mad Hatter prefers tea as well, tea will win the majority vote. Therefore, by suggesting the order of coffee, wine, and tea, Alice can ensure that tea is ordered.

(d) The Mad Hatter's preferences about areas are not complete. While he has preferences for Mallow Marsh over Boro Grove and Boro Grove over Whispering Woods, there is no stated preference between Mallow Marsh and Whispering Woods. Therefore, the Mad Hatter's preferences about areas are incomplete.

(e) The Mad Hatter's preferences about areas are not transitive. Transitivity means that if A is preferred to B and B is preferred to C, then A should be preferred to C. However, in this case, the Mad Hatter prefers Mallow Marsh to Boro Grove and Boro Grove to Whispering Woods. But he also prefers Whispering Woods to Mallow Marsh. This violates transitivity, indicating that the Mad Hatter's preferences about areas are not transitive.