Is it better to be overestimated or underestimated by your opponents?

Personally I am quite convinced, that in OTB tournament play not only your technical chess skill and knowledge and that of your opponents will decide your tournament result, but also how strong or weak your opponents think you are.

If they over- or underestimate you significantly enough, I'm sure it will lead to another result than that which you would normally get.

Let's assume we have a player who is really FIDE rated 1800 who participates in a 9 round tournament where nobody knows him.

Scenario 1: Our player is correctly listed as an 1800.
Scenario 2: Our player is incorrectly listed as 0 (no FIDE rating).
Scenario 3: Our player is incorrectly listed as a 1500.
Scenario 4: Our player is incorrectly listed as a 2100.

Assume that he gets to play the same opponents in all 4 scenarios and they are all rated between 1500 and 2200. He does always play his normal chess. Only in scenarios 2-4 he hides his true playing strength behind a false rating.

What do you think, in which of the scenarios he will score his best result and why?

If you are an 1800 you will not easily get a draw from a 2000 in an equal position. But if the 2000 thinks you are a 2100, he will usually happily give you a draw in an equal early middlegame.
On the other hand, if you're playing an opponent equal to your strength, and he thinks you're just a weak 1500, he might go into some risky, incorrect attack, because he wants to win more quickly and is overconfident you won't find the right defense.
If you're playing the weaker 1500 but he believes you to be a 1500 too, his equal, he might play with greater confidence than he normally would against you...

Is it better to be underestimated or overestimated?

That's an interesting thought experiment but I've gotta say that one quote "Play the board, not your opponent".

However humans don't really seem to always do that and let mystical rating numbers tell them how anxious they should be instead of whats happening in front of them. That being said, scenario 3 or 4.

It's far better to be under estimated than over estimated.

Look at it like this, and I'll give you the perfect example.

www.youtube.com/watch?v=84NwnSltHFo

The Bill Gates vs Magnus Carlsen game.

Carlsen accurately assessed Bill Gates skills, and his Carlen's terrible blunder was not punished.

Had Gates taken a lesson or 10 from some strong player before meeting with Carlsen on the down low, lets say Kasparov because he got the money, and he has the free time... and Kasparov gave him enough knowledge to be able to recognize and refute such an egregious blunder then Carlsen would have under estimated him.

Had Gates done that prep work he could have beaten Carlsen because Carlsen thought his skill level was low.

Now lets imagine for a minute Carlsen over estimated Gates skill level. Before the game Carlsen hears "Hey did you know that Gates has been taking lessons with Garry Kasparov, he might have a trick up his sleeve" This would get the ball rolling in Carlsen's head. Don't go easy on gates, treat the game like it's vs a Kasparov. Then by over estimating Gates ability he would be bringing a much less erroneous level of play and crush him.

Good example! At least for the situation when you face a player much higher rated than yourself.

As for lower rated opponents, I think I still prefer them to overestimate me. The more "technical respect" they have of my rating, the more cautious, that is to say passive, they will play. Once they think you are weak they'll attack with all they've got and that might get uncomfortable even for a stronger opponent. I think, that often the weaker players are not really so weak because of inferior chess knowledge or skill, but because they play too timidly. They just don't have enough self-confidence.

But as for higher rated opponents, yes, it is probably better to be underestimated by them.