Despite these endings being theoretical draws, I have heard that people play on because of significant winning chances. My question is which one has more chances? I would guess the bishop, but I do not know. Is it difficult to draw against the rook and knight?

  • 2
    Of course the side with an extra bishop...
    – SmallChess
    Jul 28 '17 at 3:07
  • @SmallChess : Reason?
    – Annatar
    Jul 28 '17 at 9:00
  • 1
    @Annatar King attacks three squares. Knight protecting king just duplicates attack on two squares. Bishop on the other hand prolong the line to 5 squares. Imagine Ke6 Be5, they cover from c7 to g7 and also mate can come from protecting rook on h8/b8. This setup reached, bishop already wins (Philidor position)
    – hoacin
    Jul 28 '17 at 10:26
  • I have once gotten into a R+B vs. R endgame (on the dominant side) in a classical game, and already after a few moves my opponent played inaccurately, allowing me to win. I also got into a R vs. R+N endgame once (but on the receiving end), and almost effortlessly kept the draw against an IM with the 50-move rule. I'd say there's a huge practical difference between those two endgames, with a R+B offering like 10x better winning chances than R+N.
    – TMM
    Jul 29 '17 at 0:01
  • There's a video on YouTube of Carlsen (R+B) playing Svidler (R) in blitz. Even though Svidler belongs to the world elite, he could not hold the draw.
    – TMM
    Jul 29 '17 at 0:04

TL;DR: Rook + Bishop offers significantly more winning chances against a lone Rook (38%) than Rook + Knight (only a few %).

Full answer:

'Secrets of Pawnless Endings' by John Nunn mentions the following about the ending with the Knight:

[the stronger side] has far fewer winning chances than in the ending of R+B v R and in practice the ending is often agreed drawn immediately. Top-level wins are very rare, but there was one famous example in J.Polgar-Kasparov, Dos Hermanas 1996 being one of the exceptions.

The chapter about R+B v R mentions 74 games of which 28 (38%) were won by the stronger side.

The book I already mentioned, and 'Fundamental Chess Endings' by Lamprecht and Müller, cover these endings. For the Bishop ending, they both explain the main defensive techniques, the so-called Cochrane defense and the Second Rank defense, and some key positions (discovered by Philidor in the 18th century already) which can be won by the stronger side, e.g. this one.

The endgame with the Knight is easier to hold in general. The defending side needs to prevent their King being forced to the edge; if that does happen, the chances of winning are actually greater than with the Bishop. An advantage for the defending side is that the Knight is worse in protecting its own King from checks by the defending Rook. If the defending King is confined to the edge, his rook should either pin the Knight or give annoying checks; this resembles the Cochrane defense in the Bishop ending. Some winning positions were published by Centurini in the 19th century.

In general, endgame databases are very helpful, at least for analyzing winning positions. For draws, they are trickier to use, because they don't tell you how difficult it is to hold a certain position. Especially the Bishop ending contains many 'marginal' positions which can only be held with secure play.

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