Material Imbalances and Game Outcomes

This article looks at the effect of material imbalances on game outcomes

May 29, 2020 • by Patrick Coulombe, PhD

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We've All Heard It...

The bishop pair is desirable. Two rooks are stronger than a queen. Two minor pieces are stronger than a rook. We've all heard it.

But in practice, like really, what material imbalances should you strive to reach? Does common wisdom withstand the test of games?

Looking at Games With Material Imbalances

To investigate this question, I used the Caïssabase, a curated database of 4.02 million chess games updated regularly (with games going all the way to September 2019 at the time of this writing). As you would expect, the majority of games feature high-rated players, with most games featuring players rated 2100 or higher over-the-board.

I used the free chess database software Scid to search this database by material patterns. For each material imbalance below, I present the percentage of games won for each side (along with the percentage of draws for each imbalance):

        1. A queen vs two rooks or a rook and a minor piece (bishop or knight)

        2. A rook vs any combination of two minor pieces (two bishops, two knights, or one of each)

        3. Two bishops vs two knights or a knight and a bishop (the famous "bishop pair" test)

Queen vs Two Rooks or Rook and Minor Piece

Well well well... Not so wise after all. The graph above shows the percentage of games a queen won against two rooks (left bars) or against a rook and a minor piece (right bars). Contrary to popular wisdom, it looks like the queen rules them all—even the two rooks!

The left side of the graph shows that when a player has a queen vs two rooks (in a middlegame with other pieces or in an endgame), the side with the queen wins 49% of the time! On the other hand, the side with the two rooks wins only 31% of the time, and 20% of the games are drawn. I think this is the most surprising result: In practice, it looks like the player who trades two rooks for a queen has the upper hand, contrary to popular belief.

The right side of the graph is less surprising. It shows that when a player with a queen faces a rook and a minor piece in an endgame, the player with the queen wins 65% of the time (and loses only 9% of the time). A quarter of the games end in a draw.

Rook vs Two Minor Pieces

Popular wisdom says two minor pieces are better than a rook. By convention, a rook is worth 5 pawns and a minor piece is worth 3, making this a 5-to-6 disadvantage for the rook. But what do actual games say?

It does look like two minor pieces are superior to a rook by a large margin in the endgame. The graph above shows that when a rook faces two bishops (left bars), the player with the rook wins only 14% of the games, and loses 66% of them—two out of three! The rest of the games (one in five) end in a draw.

The middle and right bars of the graph show similar patterns when a player with a rook faces either a bishop and a knight (middle bars) or two knights (right bars). In those games, yet again the two minor pieces hold the advantage, and win almost half the time, while the rook wins only about a quarter of the time. About one game in four ends in a draw.

Two Bishops vs A Knight Duo (The "Bishop Pair")

And finally, here we are: the famous bishop pair. The Holy Grail, almost. But is it worth it?

The graph above shows similar outcomes when a player with two bishops faces an opponent with either a bishop and a knight (left bars) or two knights (right bars). Popular sayings didn't fail you—the bishop pair truly is an advantage in the middlegame and endgame. The player with the bishop pair wins roughly 41-46% of the time, and loses only 30-32% of the time. About one game in four ends in a draw.

What to Make of This?

To summarize, the following imbalances are favorable:

      • Facing two rooks (or a rook and minor piece) with a queen

      • Facing a rook with two minor pieces, especially two bishops

      • Facing a missing bishop with the complete pair

This means that, on average, players increase their chance of winning if they trade their two rooks for their opponent's queen, their rook for two minor pieces, or a knight for a bishop. Conditions may apply 😊

Another important takeaway is that sometimes, at least in actual play, common wisdom is just plain wrong! We've uncovered one such situation here (the queen seems to be better than two rooks combined), but what about other situations? Keep an eye out—there must be other knowledge in chess that we take for granted but that would fail the test when looking at tens of thousands of actual games.

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  • Chess Digits. Material imbalances and game outcomes. Retrieved on [today's date] from