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I wonder what it would be like if the sororities merely accepted PNM's who wanted to join their sorority
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That's how it works at the "sign-in" eating clubs at Princeton. Some of the clubs use a traditional "bicker" system, similar to rush, to choose members. The other clubs are "sign-in" clubs...groups of sophomores sign into the club of their choice with their friends, and if there's too much interest, the club uses a lottery to determine which groups are admitted. The sign-in clubs are very popular, and they are doing very well financially and have lots of alumni support.
As tld pointed out, there's no prestige attached to membership in a sign-in club, but at Princeton, students don't really need additional prestige! Also, the bicker clubs are seen as undesirable by some students who view them as stuffy relics of another age.
The residential colleges at Harvard and Yale use random assignment, with 100% undergrad participation. These Houses/colleges serve many of the functions of Greek houses at other schools (smaller community, intramurals, study groups, parties and formals, community service, etc.). They have their own colors, coats of arms, traditions, etc. Again, lots of loyalty and alumni support for these institutions.
Now, I'm not saying that this would work everywhere. It's obvious that these three universities are ultra-selective, and any random sampling of the student body includes near-100% stellar academics and activities. But they're extremely diverse ethnically, geographically, economically, politically, and in terms of personality type. You might not think you could build strong social communities with random assignment or PNM choice, but at these schools, it does work.