user2429: Let's say a tournament decides to have special consolation prizes for players who finish in the 20th, 30th, 40th, and 50th positions. It does this to give incentive to players to keep playing all rounds even if they don't have a chance of finishing in the top 10, who would get the normal prizes. 
Now suppose player X playing the last round calculates that if he now resigns, he finishes exactly 20th, so he resigns. However, his opponent Y calculates that if he resigns now, he finishes 30th. Both players have incentive to resign. So as soon as X resigns, Y resigns as well. 
So what is the result of the game? 

user10182: From FIDE Handbook: 
Article 5: The completion of the game
5.1
b. The game is won by the player whose opponent declares he resigns. This immediately ends the game. 
So simultaneous resignation is not possible.

user7297: In such a hypothetical situation, I wouldn't be surprised if both players lose and will be eliminated from the tournament's results, thereby not qualifying for the consolation prize. It's entirely possible to classify their behaviour as unsportsmanlike and discrediting the game of chess, which is against the FIDE Code of Ethics.

2.2.10
  In addition, disciplinary action in accordance with this Code of Ethics will be taken in cases of occurrences which cause the game of chess, FIDE or its federations to appear in an unjustifiable unfavorable light and in this way damage its reputation.


user4418: 
So as soon as X resigns, Y resigns as well.

If the arbiter can establish this then player X's resignation, coming first, ends the game and player Y wins:

5.1.2 The game is won by the player whose opponent declares he resigns. This immediately ends the game.

But suppose the resignation really is simultaneous? 
Such a real world case was considered by Bozidar Kazic in his book The Chess Competitor's Handbook. Player A moved an unmoved pawn forward two squares, realised that he had allowed a devastating en passant capture, and resigned. Meanwhile player B overlooked the en passant possibility, thought that he was getting mated, and resigned.
The conclusion was that the resignation from player B took precedence because it was his turn to move.

2429: <p>Let's say a tournament decides to have special consolation prizes for players who finish in the 20th, 30th, 40th, and 50th positions. It does this to give incentive to players to keep playing all rounds even if they don't have a chance of finishing in the top 10, who would get the normal prizes. </p>

<p>Now suppose player X playing the last round calculates that if he now resigns, he finishes exactly 20th, so he resigns. However, his opponent Y calculates that if he resigns now, he finishes 30th. Both players have incentive to resign. So as soon as X resigns, Y resigns as well. </p>

<p>So what is the result of the game? </p>
 10182: <p>From FIDE Handbook: </p>

<p>Article 5: The completion of the game</p>

<p>5.1
b. The game is won by the player whose opponent declares he resigns. This immediately ends the game. </p>

<p>So simultaneous resignation is not possible.</p>
 7297: <p>In such a hypothetical situation, I wouldn't be surprised if both players lose <em>and</em> will be eliminated from the tournament's results, thereby not qualifying for the consolation prize. It's entirely possible to classify their behaviour as unsportsmanlike and discrediting the game of chess, which is against the <a href="https://www.fide.com/fide/handbook.html?id=9&amp;view=category" rel="noreferrer">FIDE Code of Ethics</a>.</p>

<blockquote>
  <p>2.2.10<br>
  In addition, disciplinary action in accordance with this Code of Ethics will be taken in cases of occurrences which cause the game of chess, FIDE or its federations to appear in an unjustifiable unfavorable light and in this way damage its reputation.</p>
</blockquote>
 4418: <blockquote>
  <p>So as soon as X resigns, Y resigns as well.</p>
</blockquote>

<p>If the arbiter can establish this then player X's resignation, coming first, ends the game and player Y wins:</p>

<blockquote>
  <p>5.1.2 The game is won by the player whose opponent declares he resigns. This immediately ends the game.</p>
</blockquote>

<p>But suppose the resignation really is simultaneous? </p>

<p>Such a real world case was considered by <em>Bozidar Kazic</em> in his book <em>The Chess Competitor's Handbook</em>. Player A moved an unmoved pawn forward two squares, realised that he had allowed a devastating en passant capture, and resigned. Meanwhile player B overlooked the en passant possibility, thought that he was getting mated, and resigned.</p>

<p>The conclusion was that the resignation from player B took precedence because it was his turn to move.</p>
