Bishop vs Rook: Why They Differ in Value

To many it might seem obvious why the gap between the piece value of bishop and rook exists, but this blog post might change your mind. It does not use philosophical arguments but data analysis to try to get to the bottom of why this difference exists. Using minor rule modifications and an automated engine based framework for analysis of piece values it explores what really explains the difference, and what doesn't.

Common explanations

In principle the bishop and rook both have four directions to move and an unlimited range, so conceptually their movements are similar, but the rook usually is significantly stronger. The by far most common explanation for that difference in strength is that the bishop is color-bound and can therefore only reach half of the squares. But that is not the whole story. A more complete list of mentioned reasons is:

1. Color-boundedness: Bishops are restricted to 32 squares and therefore more limited.
2. Board geometry: Rooks on average have more mobility than bishops on a rectangular board.
3. Checkmate potential: A rook can deliver checkmate with the help of its king, while a bishop can't.
4. Pawn alignment: Rooks can support own (passed) pawn pushes, as well as simultaneously block and attack opponent pawns.

But how relevant is each of these factors? And is there ever any way to actually decide that beyond a pure philosophical debate? It turns out there might be...

Decomposing the problem

As long as we keep the rules exactly as in standard chess, there is no way to separate these factors, since they are present at the same time. However, if we allow to make minor modifications to the pieces to allow for these factors to change while observing how this impacts the piece values, we can actually start to decompose the difference in value systematically.

Let's say we give the bishop the additional power to move one square orthogonally (without capturing). This gives the bishop the possibility to switch colors without significantly changing its general movements. Will this only slightly increase the bishop's value according to the few additional movement options? Or will it radically change its value by removing color-boundedness? Let's find out.

What if?

If we want to measure the impact of each factor, we will need to find rule modifications that mostly eliminate one factor while ideally retaining most others and without introducing many other new factors. The ideas I came up with are:

1. Color-boundedness: Give bishops additional (non-capture) movements to allow them to switch color.
2. Board geometry: Reduce the range of both rooks and bishops to reduce the impact of the limited length of diagonals restricting bishops.
3. Checkmate potential: Make the goal of the game to capture all pieces (except the king).
4. Pawn alignment: Make pawns move diagonally and capture straight, e.g. as in Berolina chess or Legan chess.

Since Fairy-Stockfish flexibly allows to make rule modifications, we can let it play games using each of these modifications. With the methodology I explained in a previous blog post we can then measure the piece values under each of these new rulesets and compare.

Results

As the main criteria for determining whether a rule change reduces the gap, and thus might explain the difference in the first place, I use the ratio of rook value to bishop value as obtained by the above mentioned methodology. Or alternatively to attempt to quantify how much of the gap in value is closed by a change also the ratio between the change in bishop value and the difference between rook and bishop value.

Color-boundedness

In order to remove color-boundedness for bishops, we can give them additional orthogonal moves. To reduce the impact, it is sufficient to allow only additional quiet moves, no additional captures. For completeness I also added the full Betza notation of each piece to unambiguously denote the movements. I used the three scenarios of adding the option to:

• move one orthogonal square in any direction (F0mW), like the wazir, which allows the bishop to switch colors
• leap two orthogonal squares in any direction (F0mD), like the dabbaba, which does not allow to switch color
• move one square backwards (F0mbW), which allows the bishop to switch colors, but in a very limited way

I then defined equivalent enhanced rooks as a control, just with the added movements being diagonal instead of orthogonal, but same in number and distance (W0mF, W0mA, W0mbrF). Now let's compare the ratios:

• rook vs bishop: 1.51
• rook with diagonal move vs bishop with orthogonal move (W0mF / F0mW): 1.316
• control experiment (W0mA / F0mD): 1.481
• minimal color switching (W0mbrF / F0mbW): 1.455

As can be seen from the data, allowing the bishop to switch colors significantly reduces the ratio of the piece values. And the same effect can not be seen when adding another similar movement that does not allow switching color. When only allowing a single way to switch colors, that seems to only contribute a small amount to reducing the ratio, perhaps because the piece is still "almost" colour-bound.

Instead of comparing the ratios, we can also compare the reduction in gap. Just adding the orthogonal move would reduce the gap by 51%. However, that would be an unfair advantage. After subtracting the effect of a similar non-color-switching move (two square orthogonal jump), the gap still reduces by 37%. If we want to further compensate for that these two movements are comparable, but not entirely the same, and subtract the same difference for the equivalent moves for the rook, we are left with a reduction of the gap of 23%. We could instead also directly subtract the improvements for the enhanced rook and bishop directly without involving the two square jump as a reference, and then we get a similar 21%.

So depending on how we define and calculate the color-boundedness effect, we can see that it seems to explain 20-30% of the gap between bishop and rook, or if you want to be extremely generous up to 50%. Regardless of how you spin it, it however does not seem to explain even half of the gap, despite it being the most frequently mentioned reason.

Board geometry

To investigate the effect of the board geometry, we compare the ratios for rooks and bishops with different ranges, i.e., when limiting them to distances of less than 7 squares. The below diagram shows the plot, which also includes the values of the "enhanced" rooks and bishops from the color-boundedness chapter.

It can be seen quite clearly that the limited range bishops start out even stronger than the equivalent limited distance rooks, but then very quickly stagnate, while rooks keep improving with range. This seems quite intuitive, since the diagonals often anyway aren't long enough for the bishops to benefit from a longer range, while rooks quite often do.

But why do bishops initially at short range start out not equal but even above rooks? That likely is because diagonal moves are faster in crossing the board than the equivalent limited distance rooks. Once the range gets larger, this effect diminishes, and the effect of the bishops running out of additional squares on the diagonals becomes more significant.

The seeming small decline in the bishop's value for larger ranges likely is just due to statistical fluctuations, but it still shows that there is not too much further growth in value as range increases, quite contrary to rooks.

Checkmate potential

To check the impact of checkmate potential, we modify the rules to either allow to win without checkmate, or even to discourage checkmate by ruling it as a draw. Comparing the ratios of the piece values between rook and bishop we get:

• Chess: 1.49
• Chess, but lone king is a loss: 1.49
• Chess, but lone king is a loss and checkmate a draw: 1.48

So regardless of the rule changes a rook remains worth roughly 1.5 bishops. This indicates that the ability to deliver checkmate likely is not a core reason for why rooks are stronger than bishops.

Pawn alignment

For the hypothesis stating that rook moves align well with pawn moves, and thus being able to well support or control (passed) pawns, we modify the rules in a way to make pawns move diagonally, which should take away this element. For the rook to bishop value ratio we now get:

• Chess: 1.49
• Berolina Chess: 1.59
• Legan Chess: 1.58

In this case the ratios suggest that the difference in rook vs bishop value might even increase. The data perhaps is not statistically significant enough to be incompatible with it remaining roughly the same, but definitely suggests that the rule changes do not help to close the gap between rook and bishop. I assume that there is a tension between pawns being able to move out of the way and the pieces being able to support them. For most of the game perhaps the possibility to increase the mobility of the pieces by moving the pawns out of the way of files or diagonals perhaps is more relevant, and only in endgames the support of passed pawns might become more relevant. Either way, the overall effect does not seem to be what the pawn alignment hypothesis suggests.

Conclusion

The analysis shows that the most frequently mentioned reason of color-boundedness likely explains only 20-30% of the difference in piece value between rook and bishop.

The most significant factor is the geometry of the board, which limits bishops much more in range, because they bump into edges more quickly due to their diagonal movement. This leads to the often mentioned lower average number of available moves, which makes the bishops less powerful. In contrast, when the maximum range of the pieces is relatively small to the board size, the diagonal move can be equally strong or even stronger than the orthogonal move, because diagonally traversing the board is faster than orthogonally.

Other often mentioned factors like the checkmating potential of the rook in KRvK or the aligment of rook moves with pawn moves do not seem to have any measurable contribution to the gap, or in the latter case even a potentially opposite contribution.

Bottom line: The gap between rook and bishop is mostly about geometry and less about color, checkmate, or pawns.

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Technical details

If you want to further dive into the details of the analysis, this part gives some more insights. The casual reader can feel free to skip it.

Limitations

While I tried to ensure to make this analysis robust and unbiased, there are still several factors that might have impacted the results. These limitations include:

• While the rule changes attempting to be as minimal as possible, it is unavoidable for them to have side effects
• Despite the large dataset the result still has non-negligible statistical uncertainty
• Fairy-Stockfish plays far from perfectly under the given conditions, so there likely are systematic uncertainties that are hard to quantify
• Possible other explanations for the difference were not investigated. Something like the possibility to cut of a king with a rook is such an inherent property of the movement of rooks and bishops that is hard to eliminate with a minor rule change. Many others like the possibility to line up pieces for a double attack overlap too much with other explanations, like the color-boundedness, so it does not make sense to consider them separately.

Appendix: Data

For the analysis with the above mentioned method I generated games with in total 20 million positions with a random mix of modified rooks and bishops, while keeping symmetry between white and black. In addition I generated 1 million positions each for the other rule modifications as well as for standard chess rules. In total that is 25 million positions.

The games were generated using a very short time control of 10ms per move and using classical/handcrafted evaluation in Fairy-Stockfish in order to ensure sufficient data and same conditions for all rulesets. Opening books were used in order to ensure enough diversity in the dataset.

The piece values for the big dataset regarding range and color-boundedness were as follows:

The normal bishop and rook were included as a double check. In theory they are identical to F7 and W7, respectively, but since built-in standard chess pieces and custom pieces work differently and are evaluated differently, it is a good check to make sure that there is no big deviation between them.