Rating Movies on a Numerical Scale: My wife and I often go to movies and afterwards assign a rating ranging from0 to 10 to the movie we saw.

A: Suppose we go to see a double feature, first “Terminator 2”with the great actor Arnold Schwarzenegger and then the adaptation of Jane Austin’s boring novel “Emma”. Afterwards, you hear me say that I rated “Terminator 2” as an 8 and “Emma” as a 2, and you hear my wife comment that she rated “Terminator 2” a 5 and “Emma” a 4.

(a) Do my wife and I agree on which movie is better?

(b) How would your answer change if my wife’s ratings had been reversed?

(c) Can you tell for sure whether I liked “Terminator 2”more than my wife did?

(d) Often, my wife and I then argue about our rankings. True or False: It makes little sense for us to argue if we both rank one movie higher than the other even if we assign very different numbers.

B: Suppose that the only thing I really care about in evaluating movies is the fraction of “action” time (as opposed to thoughtful conversation) and let the fraction of screen time devoted to action be denoted x₁. Suppose that the only thing my wife cares about when evaluating movies is the fraction of time strong women appear on screen, and let that fraction be denoted x₂. “Terminator 2” has x₁ = 0.8 and x₂ = 0.5 while Emma has x₁ = 0.2 and x₂ = 0.4.

(a) Consider the functions u(x₁) = 10x₁ and v(x₂) = 10x₂ and suppose that I use the function u to determine my movie rating and my wife uses the function v. What ratings do we give to the two movies?

(b) One day I decide that I will assign ratings differently, using the function u(x₁) = 5.25x₁/6 1 . Will I rank any pair of movies differently using this function rather than my previous function u?

What approximate values do I now assign to “Terminator 2” and “Emma”?

(c) My wife also decides to change her way of assigning ratings to movies. She will now use the function v(x₂) = 590x₂^6.2. Will her rankings of any two movies change as a result? What approximate values does she now assign to the two movies?

(d) Suppose my wife had instead chosen the function v(x₂) = 10(1−x₂). Will she now-rank movies differently?

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