Somehow I got the idea that our ratings might be inflated if our "average opponent" rating was significantly lower than our rating.  If this is true, I wonder by how much?

For instance, let's say P-1 has a 2000+ rating against "average opponents" rated <1400 and P-2 has an 1800 rating against average opponents rated 1800.  If P-1 played against P-2 would they be evenly matched or would the 1800 most likely lose?

What do you think?

Hmm, I think it's hard to say.

I don't play here much (played some blitz against whoever wants a game) and have an 1800 rating against avg 1400 opponent (if I remember right).  Slow time controls I'm a low strength class A (USCF) so I figure mines about right regardless of my low opp. avg.

I think it would tip me off to the possibility of an inflated rating, but would mostly judge by the strength of the moves made... also just check their history.  If the 2000 whoes avg opp is 1400 still regularly beats 2000 opposition, then it just happens that they don't mind easy games on this site.

There's a published paper for the USCF rating system that estimates that you will win 98% of your games against a player rated 700 points lower than yourself.  Based on a cursory glance at changes in my Chess.com rating, if you played 100 games with a player rated 700 points less than yourself, you would probably gain 50 points for winning 98 games, and lose 80 points for losing 2 games.  Not very productive.  (Note:  the paper is not applicable to Chess.com ratings, but I would expect a Chess.com evaluation to be similar) 

In USCF rating calculations, 400 points is the maximum rating difference considered.  If you are 700 points over an opponent, the game is rated as if he were only 400 points lower - because that's the minimum points you can win at your "k" level, 2 points for under 2300, 1 point over that, if I remember that.

The reason for this is your statistical probability of winning maxes out at 99% at 400 points, and you can never get to 100% under the rating theory. 

Therefore, if your average opponent is more than 400 points below you, you are getting more than you are statistically entitled to by winning.  Like Jude Acers or the late Claude Bloodgood, you can continue racking up huge numbers if you play constantly, even at 2 points a pop.  Both those players had their ratings frozen by USCF on more than one occasion for just this reason (with Bloodgood there was also substantial suspicion of manipulation of the system - he played in a closed pool in prison). 

So theoretically, the answer to the original question is that yes, this situation will tend to inflate the higher rated player's rating, all else being equal.  There is not enough information to say the player is 200 points (or any specific number) overrated, though, despite the theoretical possibility.  But given a similar and significant number of games played, one can definitely state the 1800 player in your example has a more statistically valid rating than the 2000 player.

I didn't realize USCF capped it at 400 like that -- that's interesting.

But on the 400 points being at 99% I understood it differently.  I don't remember the article but I was under the impression that:

100 = 64%
200 = 76%
300 = 85%
400 = 92%
500 = 96%
600 = 98%
700 = 99%

 The cap is because no player should get ZERO for a win while risking many rating points in a pairing he could not control.  My memory of the statistics is that the 99% level was reached around 350 points difference.  And this conforms with my experience and observations over many years - assuming established ratings, a player winning over someone rated over 350 points higher is a one-in-a-hundred occurence, or less.

You may have seen a statistical analysis of actual results which included all rated games.  Scholastic games really stink up the sample, because those players can easily either be 400 points underrated or through youthful inattention turn in results 400 points below their playing strength.  Only by including these games can I imagine that a player rated 600 points lower could possibly win 2 of every 100 games.  This is why USCF periodically considers ways to maintain scholastic ratings separately from the general pool, but no reasonable alternative has yet presented itself.