Question

Suppose that a decision-maker’s preferences over the set A={a, b, c} are represented by the payoff function u for which u(a) = 0, u(b) = 1, and u(c) = 4.

(a) Are they also represented by the function v for which v(a) =−1, v(b) = 0, and v(c) = 2?

(b) How about the function w for which w(a) =w(b) = 0 and w(c) = 8?

(c) Give another example of a function f:A→R that represents the decision-maker’s preferences.

(d) Is there a function that represents the decision-maker’s preferences and assigns negative numbers to all elements of A?

Answer 1

Suppose that a decision-maker’s preferences over the set A={a, b, c} are represented by the payoff function u for which

u(a) = 0, u(b) = 1, and u(c) = 4.

As 0 < 1 < 4 so the preference order is First c, Second b and Third a.

This means u prefers c between b and c, u prefers b between a and b.

(a) Are they also represented by the function v for which v(a) =−1, v(b) = 0, and v(c) = 2?

As -1< 0 < 2 so the preference order is First c, Second b and Third a.

This means v prefers c between b and c, v prefers b between a and b.

As both function u and v give the same preference order, so Yes, instead of u, function v can be used.

(b) How about the function w for which w(a) =w(b) = 0 and w(c) = 8?

As w(a) =w(b) = 0, so this means the function w is indifferent to a and b. There is no preference between a and b. So the function w is different from u and v.

(c) Give another example of a function f:A→R that represents the decision-maker’s preferences.

Consider f(a) =2, f(b)=3, f(c)=1

As 1<2<3, so the preference order for f is b is the most preferred and the a and finally c.

First b, Second a and Third c.

(d) Is there a function that represents the decision-maker’s preferences and assigns negative numbers to all elements of A?

Yes. Consider f(a) =-4, f(b)=-3, f(c)=-1

As -4 < -3 < -1 so the preference order is First c, Second b and Third a.

This is same as the preference function u.

Kindly like the answer.