user24210: If a chess engine calculates in even depth, in the final position, it has to move again, whereas if it calculates at the odd depth, then the opponent has to move at the end of calculation (ignoring quiescence search here = no wild exchanges take place).
In these cases, does the expected strength of engine drastically differ? Meaning, would we get completely different "elo" formulas if we considered even and odd search depth separately?
It obviously makes no sense to compare engine of depth 5 with engine of depth 6, but I believe there would be a different gap from going even -> odd than odd -> even.

user9684: Assuming there's no horizon effect, there should be no significant difference between a position with only the who to move being changed, although the initiative does have some difference in the evaluation, it's hard to program in a static position.  (Since a move can both have a positive and negative effect, this can be ignored.)

user30545: In even depth search in the final position, the first player does not move again, but internally it has the side to move. At depth 0 it will normally call qsearch().
For odd depth search in the final position the second player has the side to move internally but it will not make a move.
All of these searches do need another condition that is on search depth extensions like check, pawn_push_to_7th_rank and others.
I took stockfish code and disable qsearch() and extensions. Instead of calling qsearch() I just return evaluate(). Here are the results for depth 4 vs depth 3 and depth 5 vs depth 4 in 1k games.
D4 vs D3
Score of Stockfish 070122 d4 vs Stockfish 070122 d3: 715 - 247 - 38  [0.734] 1000
...      Stockfish 070122 d4 playing White: 357 - 120 - 23  [0.737] 500
...      Stockfish 070122 d4 playing Black: 358 - 127 - 15  [0.731] 500
...      White vs Black: 484 - 478 - 38  [0.503] 1000
Elo difference: 176.3 +/- 23.8, LOS: 100.0 %, DrawRatio: 3.8 %

D5 vs D4
Score of Stockfish 070122 d5 vs Stockfish 070122 d4: 665 - 279 - 56  [0.693] 1000
...      Stockfish 070122 d5 playing White: 357 - 118 - 25  [0.739] 500
...      Stockfish 070122 d5 playing Black: 308 - 161 - 31  [0.647] 500
...      White vs Black: 518 - 426 - 56  [0.546] 1000
Elo difference: 141.4 +/- 22.6, LOS: 100.0 %, DrawRatio: 5.6 %


24210: <p>If a chess engine calculates in even depth, in the final position, it has to move again, whereas if it calculates at the odd depth, then the opponent has to move at the end of calculation (ignoring quiescence search here = no wild exchanges take place).</p>
<p>In these cases, does the expected strength of engine drastically differ? Meaning, would we get completely different &quot;elo&quot; formulas if we considered even and odd search depth separately?</p>
<p>It obviously makes no sense to compare engine of depth 5 with engine of depth 6, but I believe there would be a different gap from going even -&gt; odd than odd -&gt; even.</p>
 9684: <p>Assuming there's no horizon effect, there should be no significant difference between a position with only the who to move being changed, although the initiative does have some difference in the evaluation, it's hard to program in a static position.  (Since a move can both have a positive and negative effect, this can be ignored.)</p>
 30545: <p>In even depth search in the final position, the first player does not move again, but internally it has the side to move. At depth 0 it will normally call qsearch().</p>
<p>For odd depth search in the final position the second player has the side to move internally but it will not make a move.</p>
<p>All of these searches do need another condition that is on <code>search depth extensions</code> like check, pawn_push_to_7th_rank and others.</p>
<p>I took <a href="https://github.com/official-stockfish/Stockfish" rel="nofollow noreferrer">stockfish</a> code and disable qsearch() and extensions. Instead of calling qsearch() I just return evaluate(). Here are the results for depth 4 vs depth 3 and depth 5 vs depth 4 in 1k games.</p>
<h4>D4 vs D3</h4>
<pre><code>Score of Stockfish 070122 d4 vs Stockfish 070122 d3: 715 - 247 - 38  [0.734] 1000
...      Stockfish 070122 d4 playing White: 357 - 120 - 23  [0.737] 500
...      Stockfish 070122 d4 playing Black: 358 - 127 - 15  [0.731] 500
...      White vs Black: 484 - 478 - 38  [0.503] 1000
Elo difference: 176.3 +/- 23.8, LOS: 100.0 %, DrawRatio: 3.8 %
</code></pre>
<h4>D5 vs D4</h4>
<pre><code>Score of Stockfish 070122 d5 vs Stockfish 070122 d4: 665 - 279 - 56  [0.693] 1000
...      Stockfish 070122 d5 playing White: 357 - 118 - 25  [0.739] 500
...      Stockfish 070122 d5 playing Black: 308 - 161 - 31  [0.647] 500
...      White vs Black: 518 - 426 - 56  [0.546] 1000
Elo difference: 141.4 +/- 22.6, LOS: 100.0 %, DrawRatio: 5.6 %
</code></pre>
