This question is adapted from our textbook. Richard is an oldfashioned Englishma

Question

This question is adapted from our textbook. Richard is an oldfashioned
Englishman who likes his tea with milk and sugar. Though he prefers more
sugar to less, he cannot always distinguish between two cups unless the difference in
the amount of sugar is less than a third (< 1/3) or more than one (> 1) teaspoon. If
there are two cups of tea where the difference is between a third and one teaspoon of
sugar, he is indifferent between them.
(a) Is Richard’s strict preference relation complete and transitive? Explain or give a
counterexample.
(b) Is Richard’s indifference relation complete and transitive? Explain or give a counterexample.
(c) Is Richard’s weak preference relation complete and transitive? Explain or give a
counterexample.

Explanation / Answer

Solution:

We are given that Richard is indifferent if the difference in amount of sugar is between 1/3 or 0.33 tablespoon and 1 tablespoon, otherwise he can sense the difference.

a) Strict preference relation:

Completeness : For any two cups of tea, say 1 and 2, if Richard finds cup 2 more preferable, he should find cup 1 less preferable. For example, say if the amount of tablespoon in cup 1 is x, and in cup 2 is less than x + 1/3, or more than x + 1, then, cup 2 is preferred to cup 1, or cup1 is less preferred to cup 2. Thus, they can always be compared strictly, making the relation complete.

Transitivity : Take 3 cups of tea, 1, 2 and 3, then this says that if cup 3 is strictly preferred over cup 2, and cup2 is strictly preferred over cup 1, then cup 3 should be strictly preferred over cup 1. Let's take an example

Say cup 1 has 1 tablespoon of sugar; cup 2 has 1.25 (<1.333...) tablespoon of sugar, so cup2 is strictly preferred (you may use the sign of curvy >) over cup 1.

Also, say cup 3 has 1.5 (<1.25 + 0.333...= 1.5833...), so cup 3 is strictly preferred over cup 2. But if we compare the tablespoons of sugar for cup 1 and cup 3, the difference is 1.5 - 1 = 0.5 tablespoon, but 1/3 < 0.5 < 1, so Richard should be indifferent between cup 1 and cup 3. Thus, this relation isn't transitive.

b) Indifferent preference relation :

Completeness: Again this says if Richard finds cup 1 as good as cup 2 (cup 1 ~ cup 2), then he should find cup 2 as good as cup 1 (cup 2 ~ cup1). That is, the difference in the amount of tablespoons of sugar for the two cups shall lie between 1/3 and 1, so the two cups are again always comparable, and equally desirable (i.e, indifferent). So, the relation is complete.

Transitivity : Again take 3 cups of tea: 1, 2 and 3, then this says that if Richard is indifferent in cup1 and cup 2, and also indifferent in cup2 and cup 3, then he should also be indifferent in cup 1 and cup 3 or if cup1~cup2, and cup2~cup3, then cup1~cup3. Let's take an example

Say cup 1 has 1 tablespoon of sugar; cup 2 has 1.5 (which is >1.333...and less than 1) tablespoon of sugar, so there is indifference in cup 1 and cup 2.

Also, say cup 3 has 2.1 (so, 2.1 - 1.5 = 0.6 which lies between 1/3 and 1), so there is agin indifference between cup 2 and cup 3. But if we compare the tablespoons of sugar for cup 1 and cup 3, the difference is 2.1 - 1 = 1.1 tablespoon, which is greater than 1 (1.1 > 1), so Richard can sense the difference, and thus will prefer cup 3 strictly over cup 1 (cup 3 curly > cup 1, and not cup3~cup1). Thus, this relation isn't transitive.

c) Weak preference relation :

Completeness : If cup 1 is weakly preferred (you may use sign curly>~) to cup 2, or cup 2 is weakly preferred to cup 2 or both (both implies indifference), i.e, if Richard finds cup1 at least as good as cup 2, or cup 2 at least as good as cup 1 or both. So, the two cups are always comparable implying this relation is complete.

Transitivity : If Richard finds cup 2 atleast as good as cup 1, and cup 3 at least as good as cup 2, then he must find cup 3 atleast as good as cup 1. Using the examples for above 2 parts, we can easily see that this is always the case : cup 3 is either preferred to cup 1 or is indifferent to cup 1 (so in any case, cup 3 is at least as good as cup 1) Thus, yes this preference relation is transitive.

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