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I have two vectors x and y ( shown here ).

I want to have a greater score for the vector which has a first greater value at any index/level starting from a to d. As it can be seen that vector x has higher value at index b hence its overall score should be higher than vector y.

Just a sum of these two vectors does not represent this behavior. Because score of a sum of these vectors gives a greater score to vector y than x. Is there any other mathematical way that I can use to score these vectors?

note:

(i) The highest value at any level is 1.0 and minimum is 0.0

(ii) To score a vector all its values have to be considered.

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    Is the precision of the numbers limited? Is it always three digits? – 2017-02-18
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    no it is not limited to 3 digits. but if anything works for 3 digits precision then it might work for me (as: "something is better than nothing") – 2017-02-18

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In case of limited precision of 3 digits, each of the variables $a$, $b$, $c$ and $d$ can take 1001 different values (everything between 0.000 and 1.000 in steps of 0.001). A place-value system of base 1001 will result in an appropriate score: $$ s = 1001^3 a + 1001^2 b + 1001 c + d $$ The problem is that the smallest possible difference at the $a$ must outperform the greatest possible difference at the $b$. Therefore I think it is impossible to get a formula in case of unlimited precision.

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    Well I think you are right it is kind of impossible for the unlimited precision case. Thank you for your answer for when the precision is limited :) – 2017-02-18