I had one player on here in a tournament offer a draw every move from move 1 because "The game looked drawish." I would say that is not appropriate etiquette.
So what is appropriate etiquette when it comes to offering draws?

Offering draws repeatedly, over and over is not considered infuriating at all and is quite customary apparently.
FIDE Laws (USCF have similar laws, and this site follows both):
12.6 |
It is forbidden to distract or annoy the opponent in any manner whatsoever. This includes unreasonable claims, unreasonable offers of a draw or the introduction of a source of noise into the playing area. |
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12.7 |
Infraction of any part of Articles 12.1 to 12.6 shall lead to penalties in accordance with Article 13.4.
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There is an etiquette in tournament play which is traditional: the player who is ahead in material or who has the only winning chances should be the one to offer the draw. There is another tradition which was once in the Laws of Chess but is no longer: that if a player offers a draw which is not accepted, he should not offer another draw until his opponent has offered one which was also not accepted.
That was probably dropped since it is a rare circumstance, and the rules already make it illegal to repeatedly offer draws to annoy the opponent.
There is no real etiquette concerning a higher rated player, but some do observe this mythical rule as well. My own practice has been not to offer draws unless the position was absolutely devoid of chances AND I am higher rated (just in case my opponent is refraining).
A little anecdote from the 1930s: in a tournament game, Alekhine and Vidmar had come down to a very balanced and equal ending. Vidmar offered a draw, which Alekhine refused. Two moves later, nothing had changed, but Alekhine now offered a draw which Vidmar accepted.
Asked why he didn't just accept the offer in the first place, Alekhine replied, "Against Vidmar, we make a draw when I want to, not when he wants to!"
ALekhine was pretty cool, thanks for sharing that knowledge

12.6
It is forbidden to distract or annoy the opponent in any manner whatsoever. This includes unreasonable claims, unreasonable offers of a draw or the introduction of a source of noise into the playing area.
I should have pointed that out when, in a memorable game against a friend that guy suddenly pulled out his guitar and started to play a menuett during my turn!
Anyway, my opponent accepted my draw and requested a rematch. But somehow I feel I violated etiquette and people should throw tomatoes at me.

Irina Krush says it is bad manners to offer a draw to anybody who signifgantly outranks you. After all, they have the better endgame, they should know if it is a draw or not!

in an OTB tournament, offer them a draw when your up a queen & when your about to shake hands tell them SIKEEEEEEEE, and laugh at their reaction.
Technicaly, you offer a draw, that draw offer stands until your opponent has made their next move. Be it 45 minutes later. (Even if you offered it during your own move.)

Offer a draw when:
1. You are in a drawn endgame! Offer a draw when you have an advantage such as being up a pawn, but can't see any way to advance. Don't offer a draw when you are at a disadvantage but can't see any way for your opponent to advance. If he has the advantage, it's up to him to decide when to give up, not up to you!
2. 3 fold repetition! 50 move rule!

Offer a draw once. If your opponent declines, then play it out, and let him make the next draw offer.
One draw offer is never offensive. It's when a player starts making multiple draw offers that a problem arises.

How do you guys do it in live chess, when you are not sure, whether it's a 3-fold-repetition? I tend to click on the draw button all the time then, because I don't want to miss the repetition draw. But if it isn't a repetition, it turns into a draw offer, although I don't want to offer a draw every move.
I hate pre arranged draws the worst. I was on the recieving end of 1 move pre arranged draws in my tournament career. Where 1st and 3rd agreed to a pre arranged draw in a grand prix event which was the final even and double points. 1st place gained nothing from the draw 3rd hopped over me from 1/2 point behind to 1/2 point in front. The second was in a championship which was all play all. The other 2 top players agreed to a draw after 1 move since this would secure their positions.
This was in days of being a junior player, if I was older and wiser like I am now I would have gone to the arbiter and asked for him to default the game for collusion.