<p>Boo boo. Consider this counter example. Consider two colleges A and B, both of which only have room for 10 students in their incoming class. College A receives applications from 1000 students, all of whom have SAT scores at 1600, whereas Collge B only receives applications from 100 students, but these students all have SAT scores of 2300. Whereas College A is apparently more selective, College B perhaps has higher competition because the composition of applicants is more competitive. Indeed, this example is totally made up, but its simplicity sheds light on what is actually going on in more complex cases you consider. </p>
<p>What this example shows is that to make “selectivity” comparable, you need to hold (at the very least) composition of applicant pools (i.e., distribution of applicant characteristics over relevant dimensions) constant. But you don’t do that. More interesting is how selectivity of colleges with identical applicant pools varies, because that maybe says something about general perception of the prestige or desirability of the college. Without doing so, the selectivity you discuss doesn’t tell a consistent (or coherent) story. </p>