In the context of decision-making, I'm looking for a word that describes the relation of two values: "even if value a increases, value b decreases". For this relation, I have found ratio, proportion, rate and relation. Which is the most suitable, when I want to use it in the following sentence: "To weigh out possible advantages, ??? between the values have to be established"?
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The even if phrase at the beginning of “even if value a increases, value b decreases” suggests that value b decreases independently of what value a does. The phrase could be used in a sentence like “In previous times, b followed a; but in this time frame, even when a increases, b decreases.”
If instead you mean to say that increase in a implies decrease in b, and conversely when a decreases, you can say b changes oppositely to a, and might say b changes inversely to a. However, inverse typically refers to a multiplicative inverse. If a·b = k for some constant k, then a and b are inversely related. If instead a+b = k, you might speak of a zero-sum game.
In the sentence you ask about, none of ratio, proportion, rate, relation, inverse, opposite, zero-sum or variants thereof are appropriate. Instead write
A tradeoff is “An advantage or improvement that necessitates the corresponding loss or degradation of something else”.