I'll make it short: don't buy her a drink. Don't offer to buy her a drink neither agree to her demands ("I'm thirsty..", "Will you buy me a drink?" etc). Here's why - if you do, you supplicate. And women have nothing but scorn and disrespect for supplicating men. Let me give a few examples to illustrate this.
Her (thinking "Let's see if I can hook this sucker:)"): "Will you buy me a drink?"
You (thinking "Oh boy am I in luck, this woman must like me, she's asking me to buy her a drink:)"): "Sure!"
Her (thinking "Ha! Another supplicating male to buy me a drink just because I asked him. What a wimp. Do they really all think I'm gonna go in bed with them for a bottle of beer? Jeez… I'll take my drink and continue searching for a REAL MAN."): "Thanks! You're so sweet! Bye now!:)"
You (confused): "Hey… wait! Um…?"
So even if she stays for say five or ten minutes and has a nice little chit-chat with you - you started with a crash and burn, you've already been crossed out in her book of prospective partners because you SUPPLICATED!
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I find your question interesting because it comes off as if you are asking that if a man buys a woman a drink because she asked, does she instantly cross him out of her book of prospective partners because he supplicated? So, essentially by purchasing the drink when asked he is bowing down to her and begging her to be with him, thus she does not respect him and subsequently will never date him? I think it goes further then that actually.
Most women asking a man to buy her a drink don't even consider that man a prospective partner in the first place. My personal theory is it is the evolution of finding a sex partner. Many men would offer to purchase a drink for a woman in a bar hoping it will lead to other things...possibly sex. This is the "foot in the door" technique, often used by businesses. I give you something first, you feel bad and give me something in return. I bought you a drink, you feel you need to give me something in return, usually a conversation, but if the girl is really easy she will have sex with you. Over the years, realizing this is a technique, many heterosexual women use that to their advantage these days to go out and get loaded for free. It is the mating game that has been played since human beings started having sex.
So I would say, nowadays, MANY women do not go to bars to get picked up--they go to dance with their girlfriends and get some poor shmuck to pay for it all!
I wouldn't go as far to say buying a woman a drink is "supplication", but these days it would certainly be rare if it resulted in you getting laid or rarer still a relationship.
If you want a relationship, supplication is CERTAINLY a bad idea for either of the sexes. One of the partners bowing down and worshiping the other one is completely unhealthy and the other thinking he or she is "above" the other person can lead to abuse.
Have you seen the movie Superbad? The cops are right! You don't want to be with a woman you meet in a bar. You have to go to a farmer's market or a pumpkin patch, LOL.
There are women who DO go to bars to pick up and JUST HAVE SEX. If just sex is what you are looking for at the bar you won't get it from those asking for free drinks...look for the ones who are drunk already, NOT in a pack of other women and usually have their underwear on outside their skirts!
Good luck out there!