Question

Yakov, Ana, and Dina want to bake a cake. They are choosing among chocolate cake (C), vanilla cake (V), and red velvet cake (R). The following table shows each person's preferences for cake flavors. The three friends decide to vote on the kind of cake they want and agree to bake the kind that wins by majority vote.

Yakov Ana Dina
First Choice Chocolate Red Velvet Vanilla
Second Choice Red Velvet Vanilla Chocolate
Third Choice Vanilla Chocolate Red Velvet

1. Suppose Yakov, Ana, and Dina first vote on whether to bake a red velvet cake or a chocolate cake, and then they vote between vanilla cake and the winner from the previous vote.
Between red velvet cake and chocolate cake, the majority will vote for ________, and between this winner and vanilla cake, the majority will vote for . 2. Now, suppose Yakov, Ana, and Dina first vote on whether to bake a chocolate cake or a vanilla cake, and then they vote between red velvet cake and the winner from this vote.
Between chocolate cake and vanilla cake, the majority will vote for ________, and between this winner and red velvet cake, the majority will vote for ________.
3. Yakov, Ana, and Dina's democratic voting system produces transitive preferences.
True or False?

Answer 1

Answer:

The specified preferences of Charles, Diana and Juanita. Charles and Juniata prefer chocolate over red velvet cake. Dina favors red velvet over chocolate.

So, when there is voting among red velvet and chocolate cake, the majority will select the Chocolate cake and among winner and vanilla majority will vote for Vanilla cake.

This is because Dina and Juanita favor vanilla over chocolate.

Among Chocolate and vanilla cake, the majority will elect for vanilla and among this winner and red velvet cake, the majority will choose for Red velvet cake.

This is because Diana and juanita favor vanilla over chocolate. So, vanilla cakes get majority elects. When there is voting among vanilla and red velvet , red vanilla acquire majority votes as charles and dina favor red velvet over vanilla.

The statement is False. The preferences are not transitive.

In the first situation chocolate is chosen to vanilla and red velvet is preferred to vanilla. Although, in the second situation, chocolate is favored to red velvet and vanilla is preferred to chocolate. Thus, the preferences do not exhibit transitivity.

Answer 2

The specified preferences of Yakov, Ana, and Dina are based on the fact that Yakov and Ana prefer Chocolate over Red Velvet cake, while Dina favors Red Velvet over Chocolate cake.

1. When there is voting among red velvet and chocolate cake, the majority will select the Chocolate cake, and among winners and Vanilla, the majority will vote for Vanilla cake. This result is because Ana and Dina favor Vanilla over Chocolate.

2. Among Chocolate and Vanilla cake, the majority will elect for Vanilla, and among this winner and Red Velvet Cake, the majority will choose for Red velvet cake. This result is because Diana and Ana favor Vanilla over Chocolate. So, vanilla cakes get the majority of the votes. With a choice between Vanilla and red velvet, Red Velvet acquires the majority votes as Yakov and Dina favor Red Velvet over Vanilla.

3. The claim that Yakov, Ana, and Dina's democratic voting system produces transitive preferences is False.

The reason the preferences are not transitive is that, in the first situation, chocolate is chosen to vanilla and red velvet is preferred to vanilla. On the other hand, in the second situation, chocolate is favored to red velvet, and vanilla is preferred to chocolate.

Thus, the preferences do not exhibit transitivity, which describes a situation where Yakov, Ana, and Dina prefer Chocolate to Vanilla and  Chocolate Red Velvet cake. Therefore, they prefer Chocolate to Red Velvet cake.

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