user2736: A games of chess with perfect play from both sides may be a draw. And while we don't have any perfect chess players it does seem that a match between two equally strong grandmasters is more likely to be drawn that a game between two equally strong amateurs (though I have no statistics on this).
I believe the closest thing we have to perfect play is world-class correspondence play. Are top quality correspondence games yet more likely to be drawn that OTB grandmaster games?
EDIT in response to @Rauan's question: I'm not sure whether world-class correspondence play allows for use of computer, but I would be interested in statistics either way.
EDIT: in response to @Spork's point engine vs. engine games, I looked around online. The CCRL list shows the strongest AIs drawing 45%-60% of the time at relatively brisk 40-moves-in-40-minutes time controls. I found a post on chess.org that says that human grandmasters "have a draw rate approaching 60% or more in their games with other GMs," but these would be at longer time controls than the 40/40. So I don't think there's a clear signal.
A site called KCEC has a regression analysis showing that stronger engines to tend to draw more than weaker engines even when correcting for the difference in rating between opponents.

user2429: 
Are top quality correspondence games yet more likely to be drawn that
  OTB grandmaster games?

Yes,  for the same reason that classical games between grandmasters are more likely to be drawn than blitz games. When you have more time to think, you can double check every move and avoid mistakes. In chess, if your opponent doesn't make a mistake, you cannot win. This hasn't been proved, but it is a general observation in chess. You will not find a game where a side has won and there is nothing the opposing side could do to prevent that victory. 

user5781: We can compile some "statistics" from the FICS games database.  It doesn't have correspondence games, but it has a range of time controls and a range of ratings, so we can extrapolate a little bit.  I'm putting "statistics" in quotes because this is low quality data (nothing against FICS, but online ratings aren't always the most reliable) and very sketchy analysis.
I've taken the rated blitz and standard games from January 2014 (just picked a month at random), restricted to "close" games (meaning that the ratings of the two players differ by less than 100 points), and grouped them into bands with ratings 900-1000, 1000-1100, etc.  (Each game belongs to the band for the average rating, e.g. if it's 1480 vs 1510, then the average is 1495 so it goes into the 1400-1500 band.)  Below you can see graphs of the drawing percentages for each band.
Overall, the draw rate is slightly higher for standard games (4.9%) than for blitz (4.1%).  Generally speaking, higher rated players seem more likely to draw, but there are some odd blips.  The very strongest players are mostly computers.


Draw rates for blitz games:
   rating band    draw rate         number of games
         650      0.14285714               7
         750      0.06250000             128
         850      0.02761001            1159
         950      0.02683616            6372
        1050      0.02766602           18145
        1150      0.02921338           42994
        1250      0.03549412           68941
        1350      0.03599521           93568
        1450      0.03968357          108080
        1550      0.04040593           96075
        1650      0.04451169           73037
        1750      0.04832920           50425
        1850      0.05731055           32542
        1950      0.06335272           17884
        2050      0.07692308            6058
        2150      0.07191781            1168
        2250      0.04819277              83
        2350      0.22137405             131
        2450      0.32692308             312
        2550      0.30275229             109
        2650      0.42307692              26
        2750      0.52356021             191
        2850      0.63428571             175

Draw rates for standard games:
   rating band    draw rate         number of games
         950      0.00000000               1
        1050      0.00000000              23
        1150      0.02554745             274
        1250      0.01761364            1760
        1350      0.02784332            6357
        1450      0.03215792           12563
        1550      0.03699803           15190
        1650      0.04336597           16649
        1750      0.05255485           14404
        1850      0.06633868            6301
        1950      0.09838646            2541
        2050      0.11631664            1238
        2150      0.16666667             210
        2250      0.30069930             143
        2350      0.26229508             183
        2450      0.39890710             183
        2550      0.47738693             199
        2650      0.56578947              76
        2750      0.44642857             112
        2850      0.25000000               4


user15223: Yes (actually, hell yes).
World class correspondence chess is very close to the infamous "draw death" of chess. If you thought 70-80% draw rate among the world elite at classical chess was bad, check out what it's like at world correspondence chess championship level: 9 decisive games out of 136 played, or about 95% draws.
World class correspondence chess has some pretty crazy statistics as a result:

Anyone can draw against anyone else (comparatively, a player rated ~2000 has no chance against a player rated 2700+ in OTB classical).
Because most games ends in draws, it's awfully hard to gain elo. You stay where you are!
Some openings are considered so drawish that a draw can be predicted right out of the opening, e.g.:


DN: What is your Centaur opinion over AlphaZero? Would you be able to beat it in a fair match?
WM: That whole spectacle seemed to be a spun tale with a stacked deck. If AlphaZero were to play in an actual engine tournament like the TCEC, I seriously doubt it would even make it to the semifinals. In terms of beating it though, I couldn't say if that would actually happen. It could play something like the Petroff and easily draw and that would look like some sort of small victory on its part. However, I will say it will NOT beat me in a Centaur CC game. That's just not going to happen.


Can you imagine an OTB tournament where there is no winner because every game is drawn? It's happened in correspondence chess.
I've seen one top correspondence chess player say he'd have some pretty good odds of drawing against a 32-piece tablebase (!) in some lines such as the Ruy Lopez Berlin, even as Black.

As engines keep getting better, this draw rate is going to keep increasing.
Edit: here's a graphic by former World Correspondence Chess Champion Leonardo Ljubičić with the draw rates in correspondence chess. Engines are clearly having an impact in the middle range and above.

2736: <p>A games of chess with perfect play from both sides may be a draw. And while we don't have any perfect chess players it does seem that a match between two equally strong grandmasters is more likely to be drawn that a game between two equally strong amateurs (though I have no statistics on this).</p>

<p>I believe the closest thing we have to perfect play is world-class correspondence play. Are top quality correspondence games yet <em>more</em> likely to be drawn that OTB grandmaster games?</p>

<p>EDIT in response to @Rauan's question: I'm not sure whether world-class correspondence play allows for use of computer, but I would be interested in statistics either way.</p>

<p>EDIT: in response to @Spork's point engine vs. engine games, I looked around online. The <a href="http://computerchess.org.uk/ccrl/4040/" rel="noreferrer">CCRL list</a> shows the strongest AIs drawing 45%-60% of the time at relatively brisk 40-moves-in-40-minutes time controls. I found <a href="http://www.chess.com/forum/view/general/draw-percentage2" rel="noreferrer">a post on chess.org</a> that says that human grandmasters "have a draw rate approaching 60% or more in their games with other GMs," but these would be at longer time controls than the 40/40. So I don't think there's a clear signal.</p>

<p><a href="http://kirill-kryukov.com/chess/kcec/draw_rate.html" rel="noreferrer">A site called KCEC</a> has a regression analysis showing that stronger engines to tend to draw more than weaker engines even when correcting for the difference in rating between opponents.</p>
 2429: <blockquote>
  <p>Are top quality correspondence games yet more likely to be drawn that
  OTB grandmaster games?</p>
</blockquote>

<p>Yes,  for the same reason that classical games between grandmasters are more likely to be drawn than blitz games. When you have more time to think, you can double check every move and avoid mistakes. In chess, if your opponent doesn't make a mistake, you cannot win. This hasn't been proved, but it is a general observation in chess. You will not find a game where a side has won and there is nothing the opposing side could do to prevent that victory. </p>
 5781: <p>We can compile some "statistics" from the <a href="http://www.ficsgames.org/" rel="nofollow noreferrer">FICS games database</a>.  It doesn't have correspondence games, but it has a range of time controls and a range of ratings, so we can extrapolate a little bit.  I'm putting "statistics" in quotes because this is low quality data (nothing against FICS, but online ratings aren't always the most reliable) and very sketchy analysis.</p>

<p>I've taken the rated blitz and standard games from January 2014 (just picked a month at random), restricted to "close" games (meaning that the ratings of the two players differ by less than 100 points), and grouped them into bands with ratings 900-1000, 1000-1100, etc.  (Each game belongs to the band for the average rating, e.g. if it's 1480 vs 1510, then the average is 1495 so it goes into the 1400-1500 band.)  Below you can see graphs of the drawing percentages for each band.</p>

<p>Overall, the draw rate is slightly higher for standard games (4.9%) than for blitz (4.1%).  Generally speaking, higher rated players seem more likely to draw, but there are some odd blips.  The very strongest players are mostly computers.</p>

<p><img src="https://i.stack.imgur.com/7A6SJ.gif" alt="Graph of drawing rates for FICS blitz games, January 2014"></p>

<p><img src="https://i.stack.imgur.com/Pxc5D.gif" alt="Graph of drawing rates for FICS standard games, January 2014"></p>

<p>Draw rates for blitz games:</p>

<pre><code>   rating band    draw rate         number of games
         650      0.14285714               7
         750      0.06250000             128
         850      0.02761001            1159
         950      0.02683616            6372
        1050      0.02766602           18145
        1150      0.02921338           42994
        1250      0.03549412           68941
        1350      0.03599521           93568
        1450      0.03968357          108080
        1550      0.04040593           96075
        1650      0.04451169           73037
        1750      0.04832920           50425
        1850      0.05731055           32542
        1950      0.06335272           17884
        2050      0.07692308            6058
        2150      0.07191781            1168
        2250      0.04819277              83
        2350      0.22137405             131
        2450      0.32692308             312
        2550      0.30275229             109
        2650      0.42307692              26
        2750      0.52356021             191
        2850      0.63428571             175
</code></pre>

<p>Draw rates for standard games:</p>

<pre><code>   rating band    draw rate         number of games
         950      0.00000000               1
        1050      0.00000000              23
        1150      0.02554745             274
        1250      0.01761364            1760
        1350      0.02784332            6357
        1450      0.03215792           12563
        1550      0.03699803           15190
        1650      0.04336597           16649
        1750      0.05255485           14404
        1850      0.06633868            6301
        1950      0.09838646            2541
        2050      0.11631664            1238
        2150      0.16666667             210
        2250      0.30069930             143
        2350      0.26229508             183
        2450      0.39890710             183
        2550      0.47738693             199
        2650      0.56578947              76
        2750      0.44642857             112
        2850      0.25000000               4
</code></pre>
 15223: <p><strong>Yes</strong> (actually, <strong>hell yes</strong>).</p>
<p>World class correspondence chess is very close to the infamous &quot;draw death&quot; of chess. If you thought 70-80% draw rate among the world elite at classical chess was bad, check out what it's like at <a href="https://www.iccf.com/event?id=66745" rel="noreferrer">world correspondence chess championship</a> level: 9 decisive games out of 136 played, or about 95% draws.</p>
<p>World class correspondence chess has some pretty crazy statistics as a result:</p>
<ul>
<li>Anyone can draw against anyone else (comparatively, a player rated ~2000 has no chance against a player rated 2700+ in OTB classical).</li>
<li>Because most games ends in draws, it's awfully hard to gain elo. You stay where you are!</li>
<li>Some openings are considered so drawish that a draw can be predicted right out of the opening, <a href="https://en.chessbase.com/post/correspondence-chess-and-correspondence-database-2018" rel="noreferrer">e.g.</a>:</li>
</ul>
<blockquote>
<p>DN: What is your Centaur opinion over AlphaZero? Would you be able to beat it in a fair match?</p>
<p>WM: That whole spectacle seemed to be a spun tale with a stacked deck. If AlphaZero were to play in an actual engine tournament like the TCEC, I seriously doubt it would even make it to the semifinals. In terms of beating it though, I couldn't say if that would actually happen. <strong>It could play something like the Petroff and easily draw</strong> and that would look like some sort of small victory on its part. However, I will say it will NOT beat me in a Centaur CC game. That's just not going to happen.</p>
</blockquote>
<ul>
<li>Can you imagine an OTB tournament where there is no winner because every game is drawn? <a href="https://www.iccf.com/event?id=68635" rel="noreferrer">It's happened in correspondence chess</a>.</li>
<li>I've seen one top correspondence chess player say he'd have some pretty good odds of drawing against a 32-piece tablebase (!) in some lines such as the Ruy Lopez Berlin, even as Black.</li>
</ul>
<p>As engines keep getting better, this draw rate is going to keep increasing.</p>
<p><strong>Edit:</strong> here's a <a href="https://twitter.com/LeoLjubicic66/status/1289682018851262465?s=20" rel="noreferrer">graphic</a> by former World Correspondence Chess Champion Leonardo Ljubičić with the draw rates in correspondence chess. Engines are clearly having an impact in the middle range and above.</p>
