Question

3. Consider the following set of alternatives W - la,b,c,d,e.f.g) and the following complete and transitive preference relation R W×W (recall that the interpretation is as follows: if (x,y)eR and (y.x)£R then x is preferred to y, and if (x, y)eR and (y,x)eR then x is just as good as y; the pairs of the form (x.x) have been omitted): (a) Represent the relation R using the symbols(for strict preference) and ~(for (b) Represent the relation R by listing the alternatives in a column, starting with the best at the (c) Represent the relation R by means of a utility function. As utility numbers choose indifference) top and proceeding down to the worst. negative integers.

Answer 1

W = {a,b,c,d,e,f,g}
R is WxW

Given, R = (a,b),(a,c),(a,f),(b,c),(b,f),(c,f),(d,a),(d,b),(d,c),(d,e),(d,f),(d,g),(e,a),(e,b),(e,c),(e,f),(f,c),(g,a),(g,b),(g,c),(g,e),(g,f)

a. Let us denote the relations using $\succ$ for preference and $\sim$ for indifference.

	a	b	c	d	e	f	g
a	$\sim$	$\succ$	$\succ$	$\sim$	$\sim$	$\succ$	$\sim$
b	$\sim$	$\sim$	$\succ$	$\sim$	$\sim$	$\succ$	$\sim$
c	$\sim$	$\sim$	$\sim$	$\sim$	$\sim$	$\succ$	$\sim$
d	$\succ$	$\succ$	$\succ$	$\sim$	$\succ$	$\succ$	$\succ$
e	$\succ$	$\succ$	$\succ$	$\sim$	$\sim$	$\succ$	$\sim$
f	$\sim$	$\sim$	$\succ$	$\sim$	$\sim$	$\sim$	$\sim$
g	$\succ$	$\succ$	$\succ$	$\sim$	$\succ$	$\succ$	$\sim$

In the table, the rows represent the first element of the relation, and the columns represent the second element.

b. Now, using the table we made above, we try to find the preference order, listing it in a column.

D is preferred over every other element, making it first. After D, G is preferred, so it occupies second place. After these two, E comes third. After E, A is the fourth alternative. Following A is B as the fifth alternative. C and F are equally preferred, at last, so we place them together.

D

G

E

A

B

C,F

c. Now we represent R by means of a utility function, where negative integers are chosen as utility numbers.

The utility function u : X -> R represents the binary relation $\succ$ on X if x $\succ$ y means that u(x) > u(y).
Since we've been told to use negative integers, let us create a formula to assign numbers. Let u(x) be such that it counts the number of elements preferred to x (with a negative sign).

So, for instance, with d, u(d) = 0, but since we want negative integers, we subtract 1. Thus, u(d) = -1.
u(g) = 1. We put a negative sign here and subtract 1. Thus, u(g) = -2.
Since d $\succ$ g, u(d) > u(g) -> this relation checks out.
Next, u(e) = -3
u(a) = -4
u(b) = -5
And since c and f are equally preferred, they will have the same utility number. This number will be the lowest since they are least preferred.
u(c) = u(f) = -6

Hope this helped!