The urge of sitting on your highest rating

Where does it come from? It's like online chess players want to keep their highest ratings for a while to show off, but this results in not playing when you were near your best and potentially going higher?

What is the psychology behind this? Do we just do it because others do it too?

I find it interesting, opinions are welcome

It is a paradox.
Because titled players will sometimes say not to concern oneself with ratings.
I stopped rated play with tactics and bullet and blitz.
I could play unrated - but too easy for opponents to be unmotivated ?
These days I'm content to do some unrated tactics puzzles.
That seems to be where the center of good play is.
'Detecting tactics' for both sides and then figuring out what to do about it.
There's also positional play.  Its better regarded as hand in hand with tactics than separate.  But understanding the tactics seems primary ...
and how not to make blunders and spot and exploit mistakes.

Very interesting  question! I think the motivation for this behavior is different for many people.

For some, it could be reflective of an avoidance behavior (avoiding a negative event, i.e. dropping from your peak). Avoidance is a pretty widespread defense mechanism and so to avoid feeling the negative emotions associated with dropping from your peak, in combination with the uncertainty of when you would reach it again, some people will avoid playing.
An avoidant behavior in this context can be further modulated by a lower level of confidence in one's own abilities and/or performance anxiety. I think to become really good at chess means that you need to have a high tolerance to failure (in chess at least).
 
/2cents

I don't think its avoidance.  Could be on some occasions.
I think its delegation of effort.
Rated play and ratings got some attention - so other more important things are attended to.
Everybody's different though.
In a related matter - there's the refusing of rematches.
Why should players be compelled to give rematches ?
Its not about avoidance unless one wants it to be.
Similiar - why allow takebacks ?
Finally - accepting draw offers when one should or shouldn't ...
Avoidance ?  One can neither supervise these choices - nor supervise anybody's perception of them - except by those who are very Persuasive.

Like I said, its different for different people and as you said it could be for some. But I was referring specifically to sitting on your highest rating and how for some, it could be a way to avoid losing their peak. The mechanisms behind accepting (or not) draws, takebacks, dont necessarily have to do with avoidance. 

Ratings are very volatile early on.
One could get an early rating that's 400 points over one's actual strength because a high rated early opponent or opponents had a bad day or were distracted or whatever ...
You know that some people keep making new accounts?
So there's a 'mechanism' right there to get a phony-high rating like a titled player might have.  After just a very few games of the 'Provisional' first 20 games.
Since people attach importance to rating - somebody might get 'status' that way - whether real - phony - imagined.

There's a joke about cheating at chess to get a high rating.
Its like you bought a gym membership - but then had somebody else do the exercises.

I do find the psychology of it very interesting. 

Consider a player that plays consistently at 1500 level. As they play many games their rating would nonetheless fluctuate due to good fortunate, tired opponents, maybe a couple of opponents in a row with bad connection etc etc.

It might sometimes fluctuate as high as 1600 rating. But from that point they are likely to get a number of losses in a row, and beat themselves up, feeling that they are "really" a 1600 player now, and they are somehow falling backwards.

After a couple of times of this happening they might "stick" at 1600 and not want to carry on playing. 

As I say, I find it very interesting, and I wonder if additional metrics might help for non-professional players to better understand where they are with the game. (FTR I definitely fell into this fallacy, as I have felt bad about failing to live up to a 1900 rating, and took a while to realize that that is just my high watermark). 

Aha welllllll

I stopped doing that and immediately dropped-

Usually it's because I get elated after reaching a peak the first time only to fall below right after...

Another dynamic is 'finding out what one can do'.
See what one is capable of.
And then - Mission Accomplished.
'Sitting on Rating' is an interpretation of many possible interpretations of something within a much bigger area of activity.
Idea: 'The psychology' would pertain to the entire big picture of chessplaying or recreation or competition or whatever.  
And 'psychology' can be distinguished from 'policy'.
'Policy' often dominates so many things.
Policy and psychology ... subsets of each other.  Overlapping each other.  Grey areas in between ...  Some way to try and digitalize them?
Could go on forever

Loss avoidance, placing a high importance on rating, showcasing your level.. lots of reasons. Lichess recently brought out a feature to hide all ratings on the site if you enable it. It still pairs you against players like usual but you don't see any ratings. Great idea

When I talk about mechanism, Im talking about a psychological pattern that belongs to an individual, not a strategy to keep a high rating. The mechanism you are talking about is a strategy/technique to maintain a high rating. Im thinking more in line with WHY would someone want to do such a thing. And for that we can talk for hours about development psychology because these patterns are rarely (if ever) found in isolation but permeate a person's behavior in all kind of activities. How a person keeps a phony high rating is one question, WHY a person feels the need to do so is another. Human beings have a natural aversion to negative stimuli and one method of dealing with aversion is to avoid it. Similarly, one way of dealing with ones own failures is to avoid them. This isnt anything groundbreaking and it definitely doesnt explain "why" EVERY single person does this pattern in chess but im sure quite a few do it for that reason.

You and userfriendly2 bring up very interesting points "Since people attach importance to rating - somebody might get 'status' that way" "placing a high importance on rating, showcasing your level"  but the question psychologists are interested (what is the psychology behind this urge, as the OP put it) is WHY someone attaches so much importance to rating to the extent that they need to make it phony. Saying "it's cuz they care a lot about their rating" doesnt answer the question.

For some people, it's because they don't want to tilt and lose their precious rating. Gaining rating is sometimes like gambling-- each game, you gamble on yourself on whether you win or lose, and you gain or lose rating as a result of the game. Once you fall from you peak rating, some (like me) feel the urge to climb back to where they were before. This can be a dangerous game-- the obsession can be bad and lead to hours (sometimes extended over several days) of playing.

Because of this, they sit on their rating and play rated games against other players at their level-- this happened to me once I reached 2000. I always wanted to reach 2000 bullet, and once I got there, I had a lot of fear that I'd lose it and have to grind it all back again. Due to this, I started playing basically exclusively unrated bullet games.

There are a couple of other players who do the same thing-- stay at their peak rating (or close there) and only play unrated games. The reason they do this (or at least the reason I do this) is so that I can get into the winning mindset and streak before playing rated (this helps avoiding the dreaded tilt).

I won't name their exact usernames, but you can see players like viking and sensei (2100+ bullet players) sitting in unrated open challenges from time to time.

We can always play unrated if we're on a losing streak. But once we play unrated we'll probably start winning. 

I simply like numbers and statistics, but unattached to any specific number.

Maybe the percentile is a better measure of how you compare - this time last year you would have needed a rating well over 1500 to be in the top 5%, now mid 1400s would make it

One can try to make all kinds of generalizations about this - but every person is different.
Regarding the business of protecting a high 'Provisional' rating -
I talked about how - somebody else says he's talking about why ...
I'm also talking about why.
We can all say 'we're talking about ...' and its still not going to change the endless arrays of situations this might happen in.  And not going to change the overall bigger pictures either.

One could try looking at one's own psychology and say -
"why would I want to question this and see it as some kind of oddball psychology because ''' I ''' don't do it ...  ?"
Introspection and objectivity about introspection can be uncomfortable and difficult.  Doing so thoroughly ...  impossible.
One is going to know about the introspection of the instropection of the (and so on) ... of the subconscious psychology of one's own mind ?

Some people get so upset when they don't get a rematch ...
feeling that its all about the 'psychology' of the person refusing them
(rather than their own psychology - very much the issue.  Why should anybody be obliged to give a rematch?)

A similiar thing happens with 'blocking' on this website ...
furious vehement reactions to being blocked - harassments and major operations against those using their blocking options ...
(such blocking maybe about as sinister or negative as locking one's bathroom door) ....

But then - a very odd thing happens when these behaviours are discussed publically ...
Instead of discussing the behaviours -
people Have and Manifest the behaviours ... 
It can be Astounding just how vehement and intense the reactions can be !
(related - compulsion to relieve cognitive dissonance)

Understandable if you set goals related to ratings. You hit a milestone, yet having done so you know a loss will un-achieve the goal you reached. It's bound to happen but sometimes postponing the inevitable can help you with acceptance in the end. It's why people quit smoking too. 

it's funny that people do this if you actually think about it. just because you reach a milestone, doesn't mean you aren't capable or deserving of it if you lose it sometime after. And say someone went to the extreme and never played again because they hit their goal rating... well what's the use in that? robbing yourself both the enjoyment of playing the game, and the possibility to improve even more? What if you are meant/capable of being 300 hundred more than your milestone? Now you'll never know cause you stopped playing/just play unrated.

I guess there is the fear that you won't ever get back to a rating if you lose it, but if you hit it once and never again, its still your peak rating and it doesn't mean you suddenly no longer possess the ability to play at that level (remember that rated people are pretty similar skill wise +- 200 of one another). I guess one could argue you were "lucky" or hit a string of opponents that made mistakes or were off their game, but no one is winning 10, 12 games in a row like that. You might hit 3, 4 people tops like that which is going to be something like 20 or 30 rating points at the most. 

I agree with the one poster in here. they need a mode like lichess where all ratings are just disabled from the view of the user (if they so choose to use that mode). Then those that are affected by this rating anxiety can still play and continue to grow and enjoy the game.

Perhaps you didn't know that both chess.com and lichess have had these modes for months?

It's called focus mode here and can be turned on/off in settings as well as with the icon below the button to flip the board.