Question

Chloe likes to read Latin American fiction: Fuentes, Vargas Llosa, Garcia Marquez, Carbajal, you name it. Though she prefers longer novels than shorter ones, she sometimes has trouble telling books apart. In particular, she is indifferent between any two novels that have the same number of pages. But she also is indifferent between two novels when the difference in the number of pages is greater than ten (> 10) and less than twenty (< 20).

(a) Is Chloe’s strict preference relation complete and transitive? Explain or give a counterexample.

(b) Is Chloe’s indifference relation complete and transitive? Explain or give a counterexample.

(c) Is Chloe’s weak preference relation complete and transitive? Explain or give a counterexample.

Answer 1

Consider an agent choosing between three boxes of Christmas ornaments (Schumm 1987). Each box contains three balls, coloured red, blue and green, respectively; they are represented by the vectors 〈R1,G1,B1〉, 〈R2,G2,B2〉, and 〈R3,G3,B3〉. The agent strictly prefers box 1 to box 2, since they contain (to her) equally attractive blue and green balls, but the red ball of box 1 is more attractive than that of box 2. She prefers box 2 to box 3, since they are equal but for the green ball of box 2, which is more attractive than that of box 3. And finally, she prefers box 3 to box 1, since they are equal but for the blue ball of box 3, which is more attractive than that of box 1. Thus,

R1≻R2∼R3∼R1,
G1∼G2≻G3∼G1,
B1∼B2∼B3≻B1; and
〈R1,G1,B1〉≻〈R2,G2,B2〉≻〈R3,G3,B3〉≻〈R1,G1,B1〉.

The described situation yields a preference cycle, which contradicts transitivity of strict preference
Consider 1000 cups of coffee, numbered C0, C1, C2, … up to C999. Cup C0 contains no sugar, cup C1 one grain of sugar, cup C2 two grains etc. Since one cannot taste the difference between C999 and C998, one might consider them to be equally good (of equal value), C999∼C998. For the same reason, we have C998∼C997, etc. all the way up to C1∼C0, but clearly C0 ≻ C999. This contradicts transitivity of indifference, and therefore also transitivity of weak preference.