If as soon as you read the phrase "I can prove mathematically" you thought, "Oh well no wonder he gets such a low response rate if he talks like that all the time", then I humbly submit that (1) while I like my Slashdot persona of a pedant banging people over the head with what I think is a brilliant idea, I do have other programmable settings, like conversation about movies, and (2) for once, it's not just me. Sites like eDateReview.com list hundreds of reviews for the most popular sites like Yahoo, Match.com and eHarmony, all of which got an average rating of about 2 out of 5. Another site hosting reviews of online personals services, DatingSitesReviews.com, posted a message urging people to take the predominantly negative reviews with a grain of salt, since users with a bad experience are much more likely to post a review than users with a positive one, but that only generated comments from the site's users reiterating, "No, they really do suck." Most of the complaints from men are not just about the number of obviously fake profiles (which led to lawsuits against Yahoo and Match.com), but about the low response rate even from women who are ostensibly real.
Not that I blame the women. Having watched over the shoulder of some female friends scrolling through their Yahoo Personals inboxes, some of them get far too many messages to reply to (and even if they had time to reply, they'd only be leading on most of the correspondents, since there would never be enough time to actually meet all of them). Yahoo Personals formats your inbox so that you see each person's picture along with the first few words of what they wrote, so if you have too many messages, all you can really do is scroll through the pictures (yes, women do care about that). In fact, Yahoo has a feature that lets you see the users who have viewed your profile -- which may have revealed more than Yahoo intended, since sometimes after writing to 20 or 30 people, I find out that none of them even clicked through to my profile anyway -- so if you're a guy, take Yahoo's advice about "polishing your profile" with a grain of salt. (In fact, many users with Yahoo Personals profiles are not paid members, which means they cannot reply to the messages you send them except with boilerplate like "I liked your message", and Yahoo blocks you from sending them your e-mail address. So Yahoo is listing them as people that you can contact through their service, even though Yahoo knows those people won't be able to write you back. If this strikes you as something between bad site design and actionable fraud, then you are not a Yahoo employee.)
So, yes, there is a problem worth solving. For the purpose of describing the response-rate system, I'm going to dispense with political correctness and refer to the people sending the messages as the "men" and to the people receiving messages as the "women". I hear from the pilot episode of Sex and the City that after the "mid-thirties power flip", the odds shift the other way (due to women getting older and the men accumulating more money, although the show doesn't put it quite so bluntly), so if you're in that age bracket, substitute the appropriate genders in the discussion below.
Note that I when I talk about listing women's response rates, I am talking about their response rates to men who meet their criteria. If you only want to meet men aged 28 to 29 who are interested in paddleboating, then your displayed response rate is only affected by the percentage of messages that you respond to from men in that group.
The mathematical argument commences: If you're a man writing to women on a site, for every women there are two essential variables: the probability P that she will reply to you, and how much value V you would place on getting a reply from her (say, on a scale of 1 to 10). For any woman that you write to, if P = (probability of getting a reply) and V = (value of getting a reply), then then probabilistic benefit of writing to her is P x V. (And I swear I didn't notice this until after I'd written the article, but I do not want to hear anything juvenile about framing a discussion of men meeting women in terms of the "P" and the "V".) If you have non-standard tastes, such as a preference for "Big Beautiful Women", that's great, since the women that you consider to be 9s and 10s may also be the ones that you have the highest chance of getting a reply from, since fewer other men are writing to them, and P x V for those women will be -- in a manner of speaking -- huge. Unfortunately, if your tastes are fairly typical, then the women you consider to be 9s and 10s are also getting lots of messages from other men, and have the lowest probability of replying to you. So as V = (value of getting a reply) goes up, P = (probability of getting a reply) goes down, and the product doesn't vary as much as you would like. There's nothing that the response-rate system can do about that.
In any case, if you're allocating your time rationally, you would first write to the members where you estimate P x V to be the highest, and then write to the members where P x V was the next highest, and then based on the cost-benefit principle, you'd continue writing until P x V of writing to the next person is exceeded by the value of the time it would take you to do it. (The incremental value of each additional minute of your time is not constant -- after a long time spent sending messages, you might get bored, and would require a larger incentive to spend an additional minute of your time doing it.) All of this is basically intuition and common sense, even if people don't think of it in terms of these equations (except me!).
But here's the advantage of the response-rating system: With a conventional personals site, you're only guessing the value of P x V -- to be precise, you know the value of V (at least as well as anyone can possibly know it from reading someone's profile), but you're only estimating P, based on how many messages you think she's probably getting from other people. Because of that randomness, that means some of the time you are sending messages without the best possible P x V value, and probably some of the time you are even sending messages where P x V would not even be worth the effort of sending the message, if you knew how low P was. Whereas with a system that shows a woman's response rate to people who meet her criteria, if you're someone who meets her criteria, you know roughly the probability P of getting a reply. (Actually, the probability P for you of getting a response, might not be the percentage-response-rate displayed by the site -- if you have an especially attractive or unattractive face, but there's no way for the woman to specify that in her criteria, then your chance of getting a reply might be higher or lower than the displayed percentage-response-rate. But, then you could just scale all of the displayed response rates upwards or downwards to gauge your probability P of getting a reply. It would still be better than making a total guess on a conventional personals site that didn't display percentage response rates at all.) So this is an unambiguous improvement from the point of view of the men. Another reason why men would be much better off is a specific case of this: they would avoid the time sink of writing to women who do not or cannot respond to most of the messages they're getting. With a response-rate system, those women's profiles would gradually display lower and lower response rates until the percentage was low enough to dissuade all but the most optimistic (or handsome) suitors. Without a display of the response rate, those users continue getting ridiculously large numbers of messages for as long as their profiles are active (as some of my female friends with profiles could attest).
Then consider from the women's point of view. Suppose you're a woman interested in meeting people who meet certain criteria, and you're sincerely interested in replying to at least a significant portion of people who fit those criteria. The problem is that of the men who meet those requirements, some of their attention is still going to be siphoned off by them writing to other women who only have a low chance of responding. Even if you have very specific criteria, so the men who meet your criteria have a great chance of getting a reply from you, on a conventional personals site they might not realize that. But if your response rate were displayed by your profile, then when men searched for women whose criteria they met (and who met their own criteria as well), you would be listed as one of the people with the highest chance of replying, and you'd have a greater chance of hearing from men who met your requirements. (This is not a huge benefit for women, because most women get enough messages that there will usually be some who meet their criteria anyway. The response-rate system would mainly be beneficial to men; I'm just saying it would not be worse for women and would in fact be a little better for some of them.)
Now there's one group of people who would not be better off: Women who create accounts mainly for the ego boost of getting huge numbers of messages and not replying to them. I talked with a few women who used the personals sites for this purpose; some I knew in person, some of them I talked to back when Yahoo Personals would display a woman's Yahoo Messenger screen name, and if you sent them an instant message they would sometimes reply out of sheer boredom and admit that that was what they were doing. These people would not be better off in a system that displayed response rates, since after their response rates dwindled low enough, so would the number of messages. So this would not be a true Pareto improvement, since the definition of a Pareto improvements insists on nobody being worse off, and doesn't make judgments as to people's reasons. Fine, but I submit that people who use the personals sites for the ego boost of ignoring messages are going against the site's purpose, and any improvement that pisses them off but improves things for everybody else, is still a good thing.
You might worry that the ego-trippers would continue to game the system by writing trivially short replies to all the messages they got, in order to keep their "response rate" high and keep the messages coming. I'm not sure, but I don't think that would be very common, because my impression from talking to the girls who do this is that the whole point of the ego boost is that the messages keep coming in without them having to do anything. If they had to exert themselves at all -- even long enough to reply to each message and say "yeah" -- then it wouldn't feel as much fun. Probably some would do this anyway. But of the men who kept getting responses like "yeah" and "I dunno", hopefully they would get the message quickly and stop wasting time. I could be wrong about some of these things, but my point is that it would not be any worse than the old system, where so many users already waste time writing to people who don't write back, and the new system would probably be better since it would eliminate some of the time-wasters.
There are some design decisions that I didn't specify here -- for example, do you display each user's response rate over their entire history on the site, or just over the last 24-72 hours, or both? A trickier question: Do you display the user's criteria that they have entered to specify what they're looking for, and which are used when the site calculates their response rate to all users who "meet their criteria"? Most sites do of course let users list what they are looking for. But suppose a woman is only interested in meeting men who make more than $75,000 per year, but she thinks it would be crass to list that on her profile. On the other hand, if she doesn't list it as a requirement, then her percentage response rate will be dragged down by all the men writing to her who make less than $75,000 but who she's not interested in replying to. One alternative would be that she could still have one set of public criteria displayed on her profile, and one set of "secret" criteria that included the $75,000 cutoff. Then men who made $75,000 or more would be steered toward her profile with the message, "You meet her criteria, and she responds to 50% of men who meet her requirements!" But you'd have the ticklish business of men who somehow find her profile, and meet all of her public criteria, but can't figure out why the system is telling them that they weren't a match for her -- and they contact the service to ask why, and tech support has to tactfully explain that sometimes you don't meet all of someone's secret requirements. In any case, a man would be able to reverse-engineer a woman's "secret" requirements by varying his own characteristics on his profile until the system said, "Ding! You're a match for her!" (But then what are you going to do, send her a message calling her a gold digger? Go ahead, it doesn't affect her percentage response rate anyway.)
In concluding that "everyone" would be better off under this system, I did make the type of assumptions that are common in mathematical/economic models, such as assuming that all participants are cold rationalists maximizing benefit to themselves. Such assumptions often do model human behavior pretty well, even in romantic pairings -- it explains why 10's usually end up with 10's, 9's usually end up with 9's, and so on. But these axioms may not take seemingly "irrational" preferences into account. For example, I've assumed that if it would be a waste of time for a user to write to 10 women who are not going to write back, then the new system is an improvement if it dissuades him from ever writing to those 10 women in the first place, because the end result is the same (nothing) and you've saved them the effort. But on a conventional personals site, after someone has written to 10 people and before they realize they're not getting any responses, they still have the hope that they might get answers, and that can be a good feeling. They'll be disappointed later on once they realize they're not getting any responses, but if they have a personality that is especially receptive to hope and especially resistant to disappointment, then it could average out to a better overall experience under the old system. Well, the old-style Match.com and Yahoo Personals would still be around for people who prefer to dream. I'm just saying the new-style system would better suit people looking for results.
The real problem with starting a competing personals site around this system (apart from pulling in enough users to reach "critical mass", but assume you had an ad campaign to do that), is that even if your system produced better matches for everybody in the long run, Yahoo Personals and Match.com would still be better at luring people in with the hope of getting a fabulous match-up. Even if Yahoo Personals got rid of all the fake profiles, and even if they gave anyone listed as a "member" a way to reply to people who send them messages (which among other things would bring them in line with laws against false advertising), their gallery would still be glittering with all the profiles of people who are getting too much mail to possibly reply to it all -- but as a new user, you wouldn't know that. On the other hand, with a personals site that listed criteria-specific response rates next to each profile, if you didn't have a good chance of getting a response from the most popular users, you'd know that from the beginning. You could then come back down to Earth and focus on the users whose criteria you met and who responded to most of their mail. But the site wouldn't be able to use its superstars to lure people in and string them along like Yahoo and Match.com can do.
So, to the business that launches a personals site around this system, this is what I'd propose to do: Since your system really does work better, contact a bunch of single reporters (I mean, higher up than me) and tell them to sign up for an account with Yahoo Personals, an account with Match.com, and an account with your new response-rating site, and spend twenty minutes on each site writing to users that they're interested in meeting. Or sponsor a controlled study where dozens of users try the same thing. Your site will be the only one where the participants can find and write to the members with the highest response rate for people meeting their criteria, and if the system really does result in more efficient matches, then the reporters and the study organizers ought to be able to verify that. Then you have your new ad campaign!
It's easy to list all the problems that would occur in this system: People could lie on their profiles, you can't always judge someone from a profile even if they're honest, people could waste your time starting a conversation and then bailing on you, just because people meet through the site doesn't mean they'll be a good match in person, etc. But these are problems with any personals site. This system only addresses the specific problem of efficiency; I haven't come up with an algorithmic solution to all of the problems of dating and love. It's only Tuesday.