Someone will answer a series of questions and will mark each important (I), very important (V), or extremely important (E). I&#x27;ll then match their answers with answers given by everyone else, compute the percent of the answers in each bucket that are the same, then combine the percentages to get a final score.<p>For example, I answer 10 questions, marking 3 as extremely important, 5 as very important, and 2 as important. I then match my answers with someone else&#x27;s, and they answer the same to 2&#x2F;3 extremely important questions, 4&#x2F;5 very important questions, and 2&#x2F;2 important questions. This results in percentages of 66.66 (extremely important), 80.00 (very important), and 100.00 (important). I then combine these 3 percentages to get a final score, but I first weigh each percentage to reflect the importance of each bucket. So the result would be something like: $score = E * 66.66 + V * 80.00 + I * 100.00$. The values of E, V, and I (the weights) are what I&#x27;m trying to figure out how to calculate.<p>The following are the constraints present:<p>$1 + X + X^2 = X^3$<p>$E &gt;= X * V &gt;= X^2 * I &gt; 0$<p>$E + V + I = 1$<p>$E + 0.9 * V &gt;= 0.9$<p>$0.9 &gt; 0.9 * E + 0.75 * V &gt;= 0.75$<p>$E + I &lt; 0.75$<p>When combining the percentages, I could give important a weight of 0.0749, very important a weight of .2501, and extremely important a weight of 0.675, but this seems arbitrary, so I&#x27;m wondering how to go about calculating the optimal value for each weight. Also, how do I calculate the optimal weights if I ignore all constraints?<p>As far as what I mean by optimal: while adhering to the last 4 constraints, I want the weight of each bucket to be the maximum possible value, while having the weights be as far apart as possible (extremely important questions weighted maximally more than very important questions, and very important questions weighted maximally more than important questions).
Ask HN: How do I properly calculate category weights?