A client wants to meet up for drinks, and they've offered to pick up the first round. Do I offer to pick up the second round?
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First, be VERY cautious about drinking with clients and vendors. It may be well-intentioned, but they could be trying to manipulate you and make promises on your company's behalf while you are not in full control of your faculties. They may be also trying to get you to reveal confidential information.
That being said, I have found that an appropriate response is to order some "bar food" and pay for it. That slows down the drinking while not appearing to be a "freeloader" on the bar tab.
Finally, it is well worth your time to peruse your employee handbook about accepting meals / gifts from clients. And, as always, be sure your supervisor is well-informed about all your planned off-hours interactions with customers.
All good advice here, but in many companies/cultures having a few drinks with clients is expected. You need to do two very import things that haven't been mentioned.
1) Understand your companies policy regarding drinks (or other after hours meetings) with clients.
2) If you do have drinks, know your limits and don't pass them. A few drinks is one thing, but going over your limit is something else entirely.
This does not make sense. If somebody says "I'll get the first round" it is implied that there will be more rounds and that the speaker's offer does not extend to those future rounds.
If, however, you say "the drinks are on me" you are volunteering to buy a round of drinks with no obligation for people to buy a drink in return.
The best idea is to accept the drink and then, when the client has drunk about 3/4 of his drink, ask if he wants another drink. Do not leave it to the point where you are both looking at empty glasses! If he beats you to it and goes to order another round, then quickly stop him and say "No, it's my round!"