Answers.Body: <ul>
  <li>Download <a href="https://www.pgnmentor.com/files.html#players" rel="noreferrer">all games</a> from top players. Around 341k games excluding the duplicates.</li>
  <li>Use <a href="https://www.cs.kent.ac.uk/people/staff/djb/pgn-extract/" rel="noreferrer">pgn-extract</a> to get BBvNB, BBvNN and BNvNN games.</li>
  <li>Run <a href="https://www.remi-coulom.fr/Bayesian-Elo/" rel="noreferrer">bayeselo</a> to get the stats.</li>
  </ul>
  <h2>Results</h2>
  <h5>Table 1</h5>
  <pre><code>Rank Name:   Elo    +    - games score oppo. draws   win  loss  draw
     1 BB  :    29    2    3 67811   55%     0   37% 24402 18049 25360
     2 NB  :     0    3    2 67811   45%    29   37% 18049 24402 25360
  </code></pre>
  <h5>Table 2</h5>
  <pre><code>Rank Name:   Elo    +    - games score oppo. draws   win  loss  draw
     1 BB  :    37    3    4  9896   56%     0   31%  3990  2873  3033
     2 NN  :     0    4    3  9896   44%    37   31%  2873  3990  3033
  </code></pre>
  <h5>Table 3</h5>
  <pre><code>Rank Name:   Elo    +    -  games  score oppo.  draws   win  loss  draw
     1 NN  :     3    4    4  31993  50.4%     0  35.7% 10421 10160 11412
     2 NB  :     0    4    4  31993  49.6%     3  35.7% 10160 10421 11412
  </code></pre>
  <p>The elo leads are significant for table 1 and 2. For table 3 the elo advantage is still within the error margins.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-01-22T13:17:48.877Z
Answers.Text: Please could you explain the numbers in the columns Elo, +, -?
Answers.UserDisplayName: Rosie F
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-01-22T15:28:50.557Z
Answers.Text: Margins of error at 95% confidence level. In table 3, NN has an interval of [3-4, 3+4] or [-1, 7], while NB has [0-4, 0+4] or [-4, 4]. Though NN leads by 3, it cannot fully claim superiority because it has a minimum of -1 while NB has a maximum of 4. In table 2, BB has [33, 40] while NN has [-3, 4], the minimum of BB which is 33 is more than the maximum of NN which is 4 so the lead of 37 is significant, BB is really superior.
Answers.UserDisplayName: ferdy
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-01-22T16:19:43.567Z
Answers.Text: Check the bayeselo link regarding Elo. The oppo. is opponent's Elo. For BB vs NN in table 2, the player with BB has an advantage of 37 Elo.
Answers.UserDisplayName: ferdy
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-01-22T17:40:49.503Z
Answers.Text: It would be interesting to split the data with regards to who has the material advantage.
Answers.UserDisplayName: Hauke Reddmann
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-01-23T02:06:02.010Z
Answers.Text: Number of pawns on the board may be a factor ([source](https://blogs.kent.ac.uk/djb/2018/10/14/data-mining-with-pgn-extract/)).
Answers.UserDisplayName: SecretAgentMan
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-01-24T06:08:51.540Z
Answers.Text: @SecretAgentMan [irrelevant](https://chess.stackexchange.com/questions/37436/on-average-how-many-pawns-are-there-at-the-start-of-the-endgame) ?
Answers.UserDisplayName: BCLC
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-01-24T13:46:17.093Z
Answers.Text: @BCLC I'm not sure what you mean after relooking at your linked post.  To be clear, I think the number of pawns on the board is relevant to KvB strength, though obviously there is the color complex interaction.
Answers.UserDisplayName: SecretAgentMan
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-01-22T03:24:17.817Z
Answers.LastActivityDate: 2022-01-22T09:53:42.487Z
Answers.LastEditDate: 2022-01-22T09:53:42.487Z
Answers.UserDisplayName: ferdy
Body: <p>I heard <a href="https://chess.stackexchange.com/questions/662/what-is-the-bishop-pair-worth">a lot</a> about the advantage of having a bishop pair and I am interested in the statistics about this. More specifically, if we collect a huge number of games (master games only, or games including non-titled players) for which at any point of the game, one side has exactly two bishops as his/her only minor pieces and his/her opponent has exactly two knights (or N+B) as his/her only minor pieces, what is the winning percentage?</p>
  <p>I will be interested in the winning percentage of:</p>
  <ol>
  <li>BB vs NB</li>
  <li>BB vs NN</li>
  <li>NB vs NN.</li>
  </ol>
  <p><strong>Edit</strong> I know this is extremely rare; but what will happen to the winning percentage for BB vs NB and BB vs NN if the two bishops are of the same colour? (<a href="https://chess.stackexchange.com/questions/23252/are-a-pair-of-bishops-of-the-same-colour-usually-worth-less-than-a-pair-of-bisho">Here</a> is a related question.)</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-01-22T21:58:13.323Z
Comments.Text: IM Watson did a data study and found that the winning advantage of BB, QN, and other "advantageous" piece combinations could be explained by the rating difference between the players.  He also posed the question if the higher rated player sought this advantage due to learned bias.
Comments.UserDisplayName: Mike Jones
Tags: <statistics><knights><bishops><bishop-pair>
Title: Can the "bishop pair advantage" be supported by statistics?
ThreadText: # Can the "bishop pair advantage" be supported by statistics?
  
  (Asked by: username_0 on 2022-01-21)
  
  I heard [a lot](https://chess.stackexchange.com/questions/662/what-is-the-bishop-pair-worth) about the advantage of having a bishop pair and I am interested in the statistics about this. More specifically, if we collect a huge number of games (master games only, or games including non-titled players) for which at any point of the game, one side has exactly two bishops as his/her only minor pieces and his/her opponent has exactly two knights (or N+B) as his/her only minor pieces, what is the winning percentage?
  
  I will be interested in the winning percentage of:
  
  1. BB vs NB
  2. BB vs NN
  3. NB vs NN.
  
  **Edit** I know this is extremely rare; but what will happen to the winning percentage for BB vs NB and BB vs NN if the two bishops are of the same colour? ([Here](https://chess.stackexchange.com/questions/23252/are-a-pair-of-bishops-of-the-same-colour-usually-worth-less-than-a-pair-of-bisho) is a related question.)
  
  **Tags:** statistics, knights, bishops, bishop-pair
  
  > **Comments:**
  > **username_1**: IM Watson did a data study and found that the winning advantage of BB, QN, and other "advantageous" piece combinations could be explained by the rating difference between the players.  He also posed the question if the higher rated player sought this advantage due to learned bias.  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_2 on 2022-01-22. Score: 13 | Accepted answer
  
  Download all games from top players. Around 341k games excluding the duplicates.
  Use pgn-extract to get BBvNB, BBvNN and BNvNN games.
  Run bayeselo to get the stats.
  
  Results
  Table 1
  Rank Name:   Elo    +    - games score oppo. draws   win  loss  draw
     1 BB  :    29    2    3 67811   55%     0   37% 24402 18049 25360
     2 NB  :     0    3    2 67811   45%    29   37% 18049 24402 25360
  
  Table 2
  Rank Name:   Elo    +    - games score oppo. draws   win  loss  draw
     1 BB  :    37    3    4  9896   56%     0   31%  3990  2873  3033
     2 NN  :     0    4    3  9896   44%    37   31%  2873  3990  3033
  
  Table 3
  Rank Name:   Elo    +    -  games  score oppo.  draws   win  loss  draw
     1 NN  :     3    4    4  31993  50.4%     0  35.7% 10421 10160 11412
     2 NB  :     0    4    4  31993  49.6%     3  35.7% 10160 10421 11412
  
  The elo leads are significant for table 1 and 2. For table 3 the elo advantage is still within the error margins.
  
  > **Comments:**
  > **username_3**: Please could you explain the numbers in the columns Elo, +, -?  
  > **username_2**: Margins of error at 95% confidence level. In table 3, NN has an interval of [3-4, 3+4] or [-1, 7], while NB has [0-4, 0+4] or [-4, 4]. Though NN leads by 3, it cannot fully claim superiority because it has a minimum of -1 while NB has a maximum of 4. In table 2, BB has [33, 40] while NN has [-3, 4], the minimum of BB which is 33 is more than the maximum of NN which is 4 so the lead of 37 is significant, BB is really superior.  
  > **username_2**: Check the bayeselo link regarding Elo. The oppo. is opponent's Elo. For BB vs NN in table 2, the player with BB has an advantage of 37 Elo.  
  > **username_4**: It would be interesting to split the data with regards to who has the material advantage.  
  > **username_5**: Number of pawns on the board may be a factor ([source](https://blogs.kent.ac.uk/djb/2018/10/14/data-mining-with-pgn-extract/)).  
  > **username_6**: @username_5 [irrelevant](https://chess.stackexchange.com/questions/37436/on-average-how-many-pawns-are-there-at-the-start-of-the-endgame) ?  
  > **username_5**: @username_6 I'm not sure what you mean after relooking at your linked post.  To be clear, I think the number of pawns on the board is relevant to KvB strength, though obviously there is the color complex interaction.  
  
  ---
  
  