muskeetervalues

Muskeeter Chess Piece Values

"Muskeeter Chess" is the name of a modern Chess variant created by Zied Haddad. It is in fact, an evolution from several attempts to create a game to expand upon Orthodox Chess. During the 19th and especially 20th centuries, Chess Masters began to lament the overly academic constrained style of play that had become dominant in the professional chess world. Games had become mostly theoretical and "draw-ish". The solution, suggested by Grand Master Capablanca, was to expand the game, both in size and complexity, by the introduction of new pieces on a larger board. Taking the historical bishop+knight and rook+knight compounds, he rechristened them the Archbishop and Chancellor respectively, and created a 10x8 board to accommodate them. Some issues arose. There is a weak i-pawn which is not initially defending on the chancellors side, but more importantly the 10x8 board meant the construction of a custom board. This seems to have detracted from it, as well as the negative opinions of some grand masters who thought there was too much powerful material on the board.

Then another Grand Master, Yasser Seirawan, attempted to correct the failure of Capablanca Chess. Firstly the new game, Seirawan Chess, would not require a new board. The classic 8x8 chess board would be used. Secondly no change in the initial position of Chess would occur (addressing the weakness found in the initial position of capablanca chess). Instead new pieces would be introduced into the game of Chess by a new game mechanic called a "drop". The basic idea was that when a space was first made available on the first or eighth rank, it could be simultaneously occupied by a new piece waiting in the wings. Seirawan described this as a "double move", similar to castling in that two pieces move at the same time. If one failed to introduce a new piece on the vacated square, one could not use this square later and would have to wait for another square to be vacated. Seirawan used the same Bishop+knight and Rook+Knight compounds but called these the "Hawk" and "Elephant" respectively, and created new chess pieces to represent these. This lead to the popularity and wide availability of Hawk and Elephant chess pieces.

Haddad, taking inspiration from Seirawan Chess, wanted to expand Chess to allow a wide variety of fairy pieces to be introduced into the game. He created a new game called Muskeeter Chess. There are two major differences between Seirawan Chess and Muskeeter Chess. Firstly the drop mechanic was borrowed but slightly modified in Haddad's game. Now one didn't simply introduce a piece on any vacated square. Instead one chose (in advance) which square the new piece would "drop" on when vacated. Secondly, one now selected any combination of two "muskeeter pieces" from a wide selection of special pieces to introduce into the game. The choosing and placing of those pieces become "moves" themselves, before the "official" start of the game. In this way, the players actually chose the initial position for themselves. The procedure is as follows. First white chooses a "muskeeter piece", then black chooses a different muskeeter piece from the remaining available choices. This is move "-2". Next white chooses where to place the first selected piece, and black does the same. This is move "-1". Now white chooses where to place the second selected piece, and black does the same. This is move "0". Now the game begins. The only restriction on the placement of the new pieces is that the pair can not be placed in both a king file and a rook file. The reason for this is that castling would then vacate two squares at once allowing for the introduction of two pieces at the same time. Implicitly, only one piece is allowed to be introduced at a time, just as only one piece can be developed at a time in ortho-chess.

To assist in the playing of Muskeeter Chess, Haddad began to release a product series called "Modern Chess Variants". These are "kits" that include two fairy pieces in each color. They serve both as a "fairy Chess" construction kit, and also as the necessary equipment to play Muskeeter Chess, along with any staunton standard Chess board and chess pieces. Muskeeter Chess introduced 10 "muskeeter pieces", which include capablanca's Archbishop and chancellor design and rules, as well as another kit with the Hawk and Elephant, that can be used as Archbishop and Chancellor when playing Seirawan Chess, but also double as Muskeeter pieces with a different rule sets. Six further pieces are available: fortress, unicorn, spider, dragon, cannon, and leopard. The boxes come with cards that specify how the pieces move in Muskeeter Chess.

Although the pieces come in sets of two, that suggest these can be played in combination, the official rules stipulate that white's first choice can be any muskeeter piece, and black can be any of the remaining pieces. Therefore any combination of two is possible. Since the order they are chosen is not important we can compute the number of possible pairs using the triangular numbers. To make this easy to understand just assume there is an order that assigns 1-10 to each of the ten muskeeter pieces. If white chooses 1, black has 9 choices. If white chooses 2 however, we don't need to consider black choosing 1 as this was already covered. There are then only 8 choices to consider if white chooses 2, namely those greater than 2. It quickly follows that the number of choices is simply 9+8+7+6+5+4+3+2+1. The result is 45. This however does not account for the fact that both players can chose to place their pieces however they like, and they don't have to be symmetrical. The number of such arrangements for one player can also be easily computed. Here the order of the pieces matters as they can be distinguished. The first piece has 8 choices for which file to be placed in. 3 of those choices (king and two rooks) require special consideration. But if the other 5 or chosen, then it doesn't matter where the other one is placed. This gives 5x7=35 possibilities. If it is placed on the king however, the number of possible spaces is 5, since the remaining piece can not be placed on either rook. If one of the rooks is chosen first, then one can not chose the king next, leaving 6 possibilities. There are therefore 5x7+5+6+6 = 52 possibilities. But remember either player can have any of those 52 possibilities, so to consider each players set up we have exactly 52^2 = 2704 possibilities. If that wasn't enough, we now multiply 2704 by 45 to get all possible initial positions which is 121,680. By comparison Fisher Random has 960 initial positions, and at the time this was considered sufficient to render existing opening theory useless, or in fact, to making human memorization of all openings impractical. To make matters even worse, at least in Fisher Random, all the pieces are classic chess pieces, but in Muskeeter Chess we now have 10 new kinds of pieces, which means one can not solely rely on knowledge of classic pieces and formations. So Muskeeter Chess seems to be one of the better attempts to break up the monotony of the overly academic entranced chess literature.

Muskeeter Chess is just one of many attempts to get Chess beyond the reach of opening theory and computer analysis. The problem is that this can only really be a temporary solution as long as computing power grows exponentially (as long as moore's law holds). Even at 121,680 possible initial positions we are only talking about 5 orders of magnitude. This can be surmounted in approximately 17 doublings, or 25 years according to Moore's Law. Will Moore's Law hold for another 25 years? There is evidence that the doubling time is beginning to slow down, and computer engineers are starting to run into the fundamental problems of miniaturization as they approach the size of an atom. Transistors are already at the size of only 50 nanometers. It is not certain whether conventional computers will be able to surmount these last few orders of magnitude, but this doesn't take into account the possibility of quantum computers that could potentially leave conventional computation in the dust. One thing is certain ... for now this provides a buffer, if only a small one, between intuitive human players and massively calculating machines.

For our purposes this is however mostly beside the point. We're not trying to create a chess variant to baffle computers, we are simply trying to consider the merit of this particular variant under human play. The key thing for human play is that the game is not so complex as to be unintelligible, but not so simple as to be un-engaging. I believe Muskeeter Chess manages this, while at the same time opening up the possibilities of Chess to new kinds of interesting movements and formations.

Before we get into our primary analysis, for completeness, I will go over the 10 new pieces, and their movements. For those interested in purchasing such kits and getting into playing themselves, I have also provided some links. For those interested in playing online I have linked to the official Muskeeter Chess site and two places you can play it.

Without further ado, Here are the Muskeeter Pieces:

"The 10 Muskeeters"

I. Elephant

V. Cannon

VII. Fortress

VIII. Unicorn

IX. Spider

X. Dragon

Despite the fact that the white king has 10 pieces on his side (excluding himself), and the black king has only 5, black is clearly far ahead. There is of course where the pieces are placed, but the superior mobility of the queens themselves can not be discounted. Baring the immediate capture of a queen we could move the three queens to different locations and black would still have good winning chances. So here we clearly have an indication that not all chess pieces are equally valuable.

So not only does one have to consider how many pieces one has, but also what kinds of pieces. This too has a natural parallel in war. Cavalry would be counted better than footsoldiers, and elephantry better than calvary, and so on. Interestingly, the original game of chess, chaturanga, was exactly about this. Chaturanga simply means "four armies". The four armies were four different traditional divisions of the military: the infantry (pawns), cavalry (knights), elphantry (bishops), and chariots (rooks). So we see a clear idea that different armies move in different ways and some are more powerful than others.

This brings us to the simplest idea of piece valuation that is probably taught to every beginner. Namely that the pawn is the weakest piece, followed by the Knight, the Bishop, the Rook, the Queen, while the king is priceless since losing it is losing the game. In other words:

Pawn < Knight < Bishop < Rook < Queen < King

This is simply an ordering relation. No actual point values are assigned. For beginners this simply tells you when a trade of a piece for a piece is in your favor or not. Should you trade your queen for a knight? No. Should you trade your bishop for a rook? Yes. In fact this system can also answer other questions of material exchange. For example, it's a good trade to lose a knight and bishop for a rook and queen. But this system can only work if we can create a correspondence in which every member of one set can be shown to be equal to or weaker than a member in the other set. Some examples:

Pawn+Pawn < Knight+Knight

Bishop < Bishop+Rook

Pawn+Knight < Bishop+Bishop

etc.

It is however in capable of answering questions when such a correspondence does not exist, for example:

2 Pawns versus 1 Knight

2 Knights versus 1 Rook

Bishop and Rook versus 1 Queen

etc.

These situations can not be resolved because, although we know which pieces are more valuable individually, we don't know what their values are relative to each other. Just how much more valuable is a Knight than a Pawn, a Bishop than a Knight, a Rook than a Bishop, and a Queen than a Rook. This elementary method can not answer that question. This is fundamentally why scoring system is needed. It makes all of these trades computable.

But this leads naturally to the next question: just what values should they be assigned? Can we just give them any random values and use that? One might be tempted to use a progression such as pawn=1,knight=2,bishop=3,rook=4,and queen=5. Any set of values will of course give us a method of comparison, and allow us to figure out whether or not the "trade is good". By this heuristic we could say 2 pawns versus a Knight is an even trade, 2 knights versus 1 rook is an even trade, and a bishop and rook is worth more than a queen. We get answers, but what do they mean?

Attempting to assign reliable values to the pieces of chess is almost as old as the game itself. The practice rose naturally from the nature of the game. But figures given were just "rules of thumb", never official rules of the game. Furthermore they were usually given by masters simply based on their experience of what good trades were.

So what makes for good values? Is it all just subjective or a matter of opinion or taste? Well remember in the beginning when I emphasized not losing sight of the goal. Here is a perfect example. Even though values given were from experience, rather than some reliable method of calculation, the experience ultimately rested in the masters experience at winning and losing games. So the ultimate measure of the pieces value is still rooted in how it effects winning odds. Winning odds are about statistical averages that filter out other considerations, such as the specific position.

So with the masters experience of winning odds as our barometer, it seems the more experience a player has the better they are at estimating those values. The tradition of masters and grand masters offering their opinions of the value of pieces has continued into the modern era of chess. Chess engines themselves rely on precise measures of the piece values to make proper evaluations of the best moves to make. The problem is that all of these masters do not agree precisely on what these values are, or even precisely on what they measure! Chess engines each have their own set of values as well.

While we should certainly consider the opinion of a master to be higher than that of a novice, it would be better to have a mathematical argument in favor of a certain value, or at least a heuristic argument, to support a given value. This too has a tradition in chess. Various people did attempt to derive these "piece values" through some computational method or other. These too lead to a variety of results, and often times don't seem to incorporate all features that are relevant to a pieces value.

But despite all the variety of opinions, methods of calculation, or even debate about what all of these even are meant to represent, their does seem to be some approximate consensus. Enter the Reinfeld values. These are the values most commonly cited for the chess pieces. And like many of these values, they were popularized by a chess master, and kind of became standard. Many chess websites explicitly use the reinfeld values in calculating and scoring material advantage and disadvantage. Chess.com, one of the most popular chess websites uses these values. The Reinfeld values are:

Pawn=1

Knight=3

Bishop=3

Rook=5

Queen=9

In the Reinfeld values, like most of these point systems, a pawn is not simply "estimated" at 1, but rather it is the unit of measurement. Therefore it is more accurate to a say a pawn is defined as 1. The Reinfeld values tell us that a Knight is actually worth three times that of a pawn. One peculiarity of the reinfeld values, is that, according to them, the Bishop and Knight are equally valuable. However most beginners are taught that the bishop is more valuable than the knight. It is often said that the bishop is in fact worth slightly more than a knight, though no one can agree by exactly how much. The value can range anywhere from a half point to any smaller amount of a difference.

It is however acknowledged that these values are merely approximations of the actual values. If that is so, what are the real values? How are they meant to be computed, and what are they actually measuring? To make matters worse, material combinations are not always strictly additive. The most clear example of this is the Bishop pair, which is worth more than it's component parts. In other words, it's worth more than 6 points. The value of the bishop pair can vary as well, but awarding an extra pawn for a total of 7 is a good shorthand. By this reasoning alone the loss of the first bishop (4 point loss) is more valuable than the loss of the first knight ( 3 point loss), so this isn't really an equal exchange.

Revisiting our earlier combinations with the reinfeld values, we now get that 2 pawns is less than a knight, while two knights (6 pts) is worth more than a rook (5). Lastly a Bishop and Rook is less than a queen if it's the last bishop, but equal if it's the first bishop.

Without knowing where these values come from or what they represent, a beginner can still use them as a good heuristic for making good decisions about trades. If there is some doubt about the effectiveness of such a method, consider that this serves as an important cornerstone in chess engines in their so called "evaluation" functions, and computers can play chess quite well with these purely deterministic approaches (though admittedly it helps to also have an opening repertoire and endgame tables, which means that engines actually take a hybrid approach, but the use of values for various considerations is still indispensable given the huge number of possible games). As one progresses as a player, one begins to understand positional advantage, and learns that the rules of material advantage can be bent when the situation calls for it, though it's worth noting that a positional advantage ultimately converts to a checkmate or a material advantage anyway.

This is all well and good if we are merely interested in a rule of thumb for playing chess on a recreational level. The reinfeld values however can not tell us what the values of fairy pieces are, or what pieces are worth in chess variants, which brings us to the central issue of this article.

Muskeeter Values?

Unlike chess, that has become fairly standardized and regulated, the realm of chess variants and fairy chess is a wild an untamed area in which many inventors work independently of each other. As a result there is little agreement about what a "piece" is named, what it symbol is, or what kind of figure should represent it on the board. Muskeeter Chess side-steps this issue simply because it is an explicitly defined chess variant, where all the following has been established. For this article, when we use the name of a piece we are of course implicitly speaking of the "muskeeter" version.

There has been some research into the value of fairy pieces, but actual figures are hard to come by even in the age of the internet, and are far and few between. Explanations on general procedures for how to find these values is also scarce and far between. Initially, it might be hoped, that finding the values of the muskeeter pieces would be as simple as looking them up online, like the reinfeld values for the orthodox pieces. Unfortunately values, such as they are, are not to be found in one place and are the results of many opinions and methods. One can sometimes find guestimates for pieces by their inventors.

There is however at least one reliable an consistent set of figures for the most common fairy pieces: the chancellor and archbishop. Most agree that the Archbishop is worth about 7 points, while the chancellor is worth about 8. As usual, these are approximations and results may vary slightly from authority to authority.

In light of the absence of pre-established or generally accepted results on the relatively modern muskeeter chess (the earliest sets only occurring in 2010), we have no choice but to try and derive these values for ourselves. And that is exactly what we are going to do.

A Primitive

but Tentative Approach

Using Orthodox Chess as our clue, and using the reinfeld values as our "gauge", we can notice something important about the various pieces. Going with the standard accepted order: pawn, knight, bishop, rook, queen, we can see that the maximum number of squares these pieces can move to is in ascending order. The pawn can move to 4 spaces, the knight 8, the bishop 13, the rook 14, and the queen a whopping 27. The king can move to 8, but that's not really important as it is a royal piece and therefore it's evaluated differently. This is usually what is referred to as a piece's "mobility", or in this case the piece's "maximum mobility". Notice that this approach is not limited to the orthodox pieces. It can be applied to any piece. Just determine what the most squares a piece can move to in any given situation, and that can treated as synonymous with it's "value". Notice that a piece can not be infinitely valuable (except for the king) under this system. The maximum number of squares a piece can "move to" is 64 on an 8x8 board because that's all the squares there are. Most variants, and orthodox chess itself, stipulate that one piece may not simply move back to it's starting square as this would be the equivalent of passing a turn, which is implicitly forbidden. In this case 63 is the maximum.

Here we can give some initial estimates of the Muskeeter pieces by simply seeing what their maximum number of squares are. Here are the results:

Elephant = 16

Hawk = 16

Chancellor = 22

Archbishop = 21

Cannon = 16

Leopard = 16

Fortress = 20

Unicorn = 16

Spider = 20

Dragon = 35

This gives us the following order of pieces and muskeeter pieces:

1. Dragon (35)

2. Queen (27)

3. Chancellor (22)

4. Archbishop (21)

5. Spider, Fortress (20)

6. Elephant, Hawk, Cannon, Leopard, Unicorn (16)

7. Rook (14)

8. Bishop (13)

9. Knight (8)

10. Pawn (4)

So far so good. We have some way of comparing pieces, and it seems to align with our intuitive notion of strength. The Dragon is stronger than the queen, the archbishop is stronger than the leopard. Is the spider really as strong as the fortress? Are the elephant, hawk, cannon, leopard, and unicorn really equal?

What happens if we try to adjust this to match up with the standard practice of assuming pawn=1? Well in this case we just take maximum mobility for each piece and divide it by the maximum mobility of the pawn (4 squares). When we do this we get the following values for the orthodox pieces:

Pawn = 1.00 , Knight = 2.00 , Bishop = 3.25 , Rook = 3.50 , Queen = 6.75

As we can see this doesn't quite account for the reinfeld values. The knight seems too weak, the bishop and rook are much too close in value, and the queen appears a lot weaker than it should. What values do we get for the muskeeters? We get:

Elephant = 4.00

Hawk = 4.00

Chancellor = 5.50

Archbishop = 5.25

Cannon = 4.00

Leopard = 4.00

Fortress = 5.00

Unicorn = 4.00

Spider = 5.00

Dragon = 8.75

So we get "answers", but just how reliable are these? Well control of the board is dictated in part by how many and which squares your pieces can control, and they have to be able to "move" to them to control them. The main weakness here is that, pieces rarely can actually move to the maximum number of squares. How many squares a piece actually "controls" depends on where that piece is located and what conditions exist on the board. The fact that it still doesn't really answer where the reinfeld values come from also implies that this method is just too simplistic. The fact that so many pieces are seemingly "equal" despite very different capabilities should give us pause. Can we do better? Indeed.

A Better Method:

Average Mobility & the Leap-Square Formula

A better method for understanding the value of a piece is considering what it's average mobility is. Let's say we have an empty board. Imagine placing a given piece on every square on the board. Now take the average mobility across the board. There is some historical prescedence for this. There are 19th century diagrams that show the mobility scores for each square on the chess board. We can do exactly the same thing.

Knight Average Mobility Board

Let's consider the Knight for starters. There are 16 squares where the Knight can leap to 8 squares, 16 squares where it can leap 6 squares, 20 squares where it can leap 4, 8 where it can leap 3, and 4 where it can leap 2. This gives a total of 336. Dividing by 64 we get 5.25. This the Knights average mobility across the board. The Knight is not always going to be able to move it's maximum number of squares, and because of this the value of a knight actually fluctuates over the course of a game, but overall we can say it's value is about 5.25. We can do the same thing for the other pieces to figure out their average mobility. However checking each square individually for mobility is tedious.

A better way to compute these values then to figure out the number for each square individually is to develop a formula I call the "leap-square formula". From there we can generalize to all sorts of situations. The basic idea is to think of just a single square, situated away from the piece at position (x,y). Due to the symmetry of the board, the value of a leap square at (-x,y), (x,-y), (-x,-y), (y,x),(-y,x),(y,-x), and (-y,-x) will all be exactly the same.

Imagine a piece that can only leap to a single square defined by (x,y). It helps to imagine a specific example. Say it can only leap to (1,0). In this case as long as the piece is not in the h file, it can leap onto a space on the 8x8 board. Every other space it will be able to leap to. This means it can leap on 56 of the 64 squares leading to an average mobility of 0.875. If the leap is (1,1) then it can only leap on a 7x7 square with it's bottom left corner at a1. It isn't too difficult to see that this formula generalizes to:

Leap-Square Formula

(x,y) = (8-x)(8-y)/64

On an 8x8 board. I discovered this simple formula while thinking about an easier way to evaluate average mobility. Later I found that Ralph Betza, a Master who has done some work in the area of fairy piece evaluation also uses the leap-square formula in the form of:

(w-x)(h-y)/(wh)

Which is the generalized form to boards of dimension w x h. Now watch how easily we can compute the value of the Knight. The Knight is simply a (1,2) leaper. Let x=1, and y=2. Since it can leap to 8 squares of this type we find the value of the Knight as 8*(1,2). So we get:

Knight = 8*(1,2) = 8(8-1)(8-2)/64 = 5.25

We can now use this formula to quickly derive the values of other pieces. Next we might consider the bishop. This is a (1,1)-sweeper. On an empty board, we can treat every square a piece can reach as if it were a leap square. The value of a Bishop is therefore:

Bishop = 4*[(1,1)+(2,2)+(3,3)+(4,4)+(5,5)+(6,6)+(7,7)]

= 4*[7^2+6^2+5^2+4^2+3^2+2^2+1^2]/64

= 8.75

Here is the calculation for the Rook:

Rook = 4*[(1,0)+(2,0)+(3,0)+(4,0)+(5,0)+(6,0)+(7,0)]/64

= 4*8*[7+6+5+4+3+2+1]/64

= 14

The Queen's mobility is simply the sum of the Bishop and Rook since it can move like both and their abilities are disjoint. So

Queen = Bishop + Rook = 8.75 + 14 = 22.75

Before we move on to the Muskeeter pieces, let's consider some interesting things the leap-square formula reveals. The most important revelation is simply that leap-squares closer to the starting position of a piece are stronger than those further out. Thus those in the first perimeter are stronger than those on the second, and those on the second are stronger than those on the third, and so on. We can actually create a table of such values, up to the 4th perimeter, to gain some understanding of the distribution of points across the board.

Leap-Square Values

Remember earlier when we calculated the Elephant, Hawk, and Unicorn to be equal simply because they all had 16 leap-squares? Now we can begin to differentiate these pieces because not all leap-squares are of equal value. We also know closer squares are actually better. This suggests that the elephant is likely to be the strongest of the three. Here is the computation for the Elephant and Hawk.

Since the Elephant can move 1 or 2 spaces orthogonally or diagonally we can compute the elephant as:

Elephant = 4*[(1,0)+(2,0)+(1,1)+(2,2)]

= 4*(7*8+6*8+7*7+6*6)/64 = 11.8125

For the Hawk we have:

Hawk = 4*[(2,0)+(3,0)+(2,2)+(3,3)]

= 4*(6*8+5*8+6*6+5*5)/64 = 9.3125

So now the Hawk appears to be much weaker than the Elephant. For the Chancellor and the Archbishop we can use the fact that average mobility is the sum of the average mobility of their components:

Chancellor = Rook + Knight = 14+5.25 = 19.25

Archbishop = Bishop + Knight = 8.75+5.25 = 14.00

Next we have the Cannon and the Leopard. The cannon can be computed simply as the sum of it's leap squares:

Cannon = 4*[(1,0)+(1,1)+(2,0)+(2,1)]

= 4*(7*8+7*7+6*8+6*7)/64 = 12.1875

The Leopard can be computed by adding the value of it's diagonals plus a Knight:

Leopard = 4*(1,1)+4*(2,2)+Knight

= 4*7*7/64+4*6*6/64+5.25 = 10.5625

Next we have the Fortress and Unicorn. The Fortress is somewhat lengthy:

Fortress = 4*(1,1)+4*(2,2)+4*(3,3)+4*(2,0)+4*(1,2)

= 4*7*7/64+4*6*6/64+4*5*5/64+4*6*8/64+4*7*6/64

= 12.50

Unicorn = 8*(1,2)+8*(1,3)

= 8*7*6/64+8*7*5/64

= 9.625

The first reason I became interested in computing the muskeeter values was I wanted to know how the Fortress and Unicorn compared to the Chancellor and Archbishop. I thought the fortress in particular was a very strong piece and I expected it to show up as stronger than an archbishop because it had a max of 20 squares. Even though the archbishop had a max of 21 squares I figured, averaged across the board the Fortress would probably win. But when I made the calculations I found that not only were both pieces weaker than the archbishop, but also weaker than the rook. Although the fortress has a maximum of 20 and the rook a maximum of 14, when averaged across the board the Fortress quickly loses value dipping below the rook with 12.50. This means that on average the Fortress covers only 12.5 squares on an empty board.

Finishing up we have the Spider and Dragon:

Spider = 4*(1,1)+4*(2,2)+4*(2,0)+8(1,2)

= 4*7*7/64+4*6*6/64+4*6*8/64+8*7*6/64

= 13.5625

Dragon = Queen + Knight = 22.75 + 5.25 = 28.00

The Pawn, which serves as our unit of measure requires special consideration. It is unique from the other pieces in that it's the only one that attacks and moves differently. It also can never be in the first or eighth rank. For simplicity we can just assume that for squares it can not reach, it gets a 0. On each of the remaining squares we are interested in the maximum number of spaces it can move on that square. So here we are finding the average "maximum mobility" for all squares. Under this definition we don't have to change the values of any of the other pieces. Assume we are talking about a white Pawn. It can reach 4 squares from only 6 squares, 3 squares from 32 squares, and 2 squares from 10 squares. This gives:

Pawn = (6*4+32*3+10*2)/64 = 2.1875

Now to obtain the relative value of our pieces, we simply divide their average mobility by the average mobility of the pawn. This sets pawn=1.

Here is a table of values displaying the average mobility scores, the relative values (with pawn=1, knight=3, and rook=5), of the orthodox and muskeeter pieces:

The pieces highlighted in yellow are those who scored lower than a rook. Only three muskeeter pieces managed to score equal to or higher, namely, the Chancellor, archbishop, and dragon. I had expected the Fortress to do better than a rook, yet according to this it was only worth about 4+1/2 points or 4.46 relative to a rook's 5. I thought this was suspicious. You notice that all of the bounded pieces scored lower than a rook here. In practice I felt pretty certain that a fortress what a piece you would definitely want to have, maybe even more than a rook, because it could maneuver around in many situations in which the Rook would have great difficultly. Clearly leap-squares were more valuable than sweep-squares, yet average mobility does not take this into account. All reachable squares are weighted equally. So I tried to devise a weighting system. Just how much is a leap-square relative to a sweep-square?

Another reason to feel that this is only a partial solution to the question of the value of muskeeter pieces is that none of these values, regardless of which piece we take to be the standard, comes very close to the reinfeld values. The best fit seems to be to assume a pawn=1, in which we get a knight is 2.4 instead of 3, a bishop is 4 instead of 3, the rook is 6.4 instead of 5, and the queen is 10.4 instead of 9. Can we do better than this? Yes we can.

But before we go into the next section, let's see how the Muskeeter Ranks have changed...

Notice that nothing really got shuffled. Rather our new measurement is a refinement that allows us to differentiate pieces that formerly appeared to be equivalent. It now appears the Spider is stronger than the fortress, rather than them being equal. Out of the Cannon,elephant,leopard, unicorn, and Hawk it now appears the Cannon is the strongest of these and the Hawk is the weakest. So it does seem like we are making progress.

Before I discuss my next development I'd like to discuss where Ralph Betza went from here as it gives some insight into a more generalized approach.

Betza Coefficients

Here I'd like to introduce a concept I'm calling "Betza coefficients" in honor of Ralph Betza's work here on the evaluation of pieces values. As Betza rightly points out, the empty board, even taking average mobility into account, is still too idealized a situation. As he puts it "when is the chess board ever actually empty?". The correct answer is of course never (there must always be at least the two kings on the board at all times). I think it is worth pointing out however, that just because it's never empty doesn't mean we don't glean something useful from the average mobility on the empty board. We gain a pieces "ideal" worth, and that still counts for something. The real problem is that it oversimplifies things too much. As I pointed out, leap-squares are more valuable than sweep-squares, and yet average mobility doesn't take this into account. Squares are all equally valuable. If leap-squares are more valuable than sweep-squares however, we should expect the bounded muskeeter pieces to be much more valuable than they appeared under average mobility as they all heavily use leap-squares.

Betza's idea for taking piece calculation to the next level is simple. Rather than think of an empty board think of a full board. At the initial position of orthodox chess there are 64 squares, 16 white piece, and 16 black pieces, so a full 32 squares are occupied. Since it is impossible for the number of pieces to increase in orthodox chess this is the maximum board density. So assume a random distribution of 32 pieces on the board. Picking a random square there has to be a 1/2 chance of being occupied or unoccupied. Simple. Now comes the nice idea.

To evaluate a leap-square we simply notice that only half the time it will be unoccupied, and can therefore be moved to. So it retains 50% of it's ideal value under the worse case scenario of a "full board".

What about sweep squares? Well to reach a square that takes two spaces to move, ideally we need two unoccupied spaces, but this has odds of 1/4. So a "2-square" retains 25% of its ideal value. A "3-square" retains 12.5% of its ideal value, and so on, each new coefficient simply being half of the previous one. So here we get the "Betza-Coefficients":

Betza points out, of course, that these are merely simplifications. They are simply an easy rule of thumb that allows us to quickly calculate the value of pieces, sometimes even by hand. With the Betza Coefficients factored what happens to average mobility? Let's find out. First we consider the Knight. Let B_n be the nth Betza coefficient. Since all the Knights squares are leap squares we use B_1 to evaluate them. Combining this with the leap square formula we obtain:

Knight = 8*B_1*(1,2) = 8*0.5*7*6/64 = 2.625

The Knight is now worth half of what it's ideally worth. But let's now see what happens to the Rook:

Rook = 4*[B_1*(1,0)+B_2*(2,0)+B_3*(3,0)+B_4*(4,0)+B_5*(5,0)+B_6*(6,0)+B_7*(7,0)]

= 4*[7*8/2+6*8/4+5*8/8+4*8/16+3*8/32+2*8/64+1*8/128]/64

= 32*[7/2+6/4+5/8+4/16+3/32+2/64+1/128]/64

= (7/2+6/4+5/8+4/16+3/32+2/64+1/128)/2

= 3.00390625

It's interesting to note that more than 50% of the rook value, according to this calculation, comes from only it's leap squares! In fact the leap-squares contribute to 58.3% of the value. The values of squares further away rapidly decreases.

Now let's consider the relative values. Before when Knight was defined as 3 rook was 8.00 which was clearly too high. If we set Knight:=3 here we get the relative value:

Rook/Knight*3 = 3.00390625/2.625*3 = 3.43

Now the Rook looks absolutely pathetic and barely stronger than a Knight.

Before when we let Rook:=5 we had the knight was a measly 1.88. But now we have:

Knight/Rook*5 = 2.625/3.00390625*5 = 4.37

Which now makes the Knight seem quite powerful. What is going on here?

Skipping to the chase we can easily rework the values of the Orthodox and Muskeeter pieces factoring in the Betza coefficients. Only the pawn deserves special consideration here. Previously we considered it's optimal mobility for every position on the board. Betza's method works fine for the squares directly ahead of the pawn, but for the attack spaces, a vacancy actually means it can not attack. So the pawn is weird in that, what would normally be a worse case scenario for most pieces is actually quite beneficial for the pawn which has more opportunities to snag pieces. However it can only attack if the piece is an enemy. There are a maximum of 16 enemy pieces to 64 squares, so we use a special Betza coefficient of 1/4 just for the attack squares of the pawn. With that clarified the rest should be clear enough if you care to check and verify my calculations.

The table below shows the results of these new calculations:

The pieces highlighted in red are those that are worth more than a queen. Suddenly we have 5 pieces past the queen when before we had only 1. Furthermore, some of the same pieces that were weaker than a rook are now stronger than a queen. What's going on here? Weren't these supposed to be more refined measurements than average mobility? But we can see that the orthodox values are not really any closer to the Reinfeld values. The queen went from 10.40 to 7.322, alternatively larger and then smaller than the reinfeld values. With the rook we see the same trend going from a high of 6.40 to a low of 4.090. On the flip side we see leapers going from lower than expected to being too high. The knight goes from a value of 2.40 on an empty board to 3.574 on a full board.

Betza himself acknowledges this and notes that the pieces are now alternatively too high or too low. But the answer is actually quite simple: these conditions are overly "pessimistic" in the same way that the empty board is too "optimistic". We've simply traded out a best case with a worse case, but the average case would of course be somewhere in the middle. We need an average of the best and worse case to get a truly representative measurement of the pieces value over the course of the game. Betza doesn't appear to go on to do this, but it would be a simple matter to simply take the average of the two mobility scores we get for the empty and full board. Below is a table that does just this, and then takes the average best-worse case for the pawn to be equal to 1. Let's see how close this gets us to the reinfeld values:

Now it looks like we are starting to get somewhere. We can see the beginnings of the Reinfeld values in the last column. Furthermore the Chancellor is around 8 points and the Archbishop is just a half point below 7. The Rook is finally 5 points and change, the queen is 9 points and change, and the knight is closer to 3 points than 2. The Bishop appears to be too strong being closer to 4 points than 3. None the less it's the closest fit we've seen yet! But there is still some issues. The Hawk, Leopard, and Unicorn appear to be weaker than a Rook. Have the muskeeter pieces ranks been shuffled? Let's see...

Almost all pieces have actually improved in their relative value, with the only exceptions being the Dragon and Chancellor which were both slightly overvalued apparently. The Archbishop has fallen a rank, the Spider has climbed, the Fortress has fallen and been overtaken by the Cannon and the Elephant. This despite that fact that the fortress has 20 squares where as the elephant and cannon have 16, however the fortress really only has 12 leap-squares as 8 are sweep-squares which accounts for it slowly losing out to pure leapers.

These values still don't seem quite correct or precise enough. It seems unlikely they are even correct to the first decimal place. What might be the source of the error? Well the Betza coefficients are oversimplifications for one. Firstly, shouldn't mobility take into account captures and not just empty spaces? Secondly the Betza coefficients are based on the naive assumption that we can compute a sweep simply by considering the probabilities for each square in isolation. This would only be true if we truly made a random choice for each square, leaving it empty 50% of the time, filling it with an ally 25% of the time and with an enemy 25% of time. If we actually did this for each square however the board could actually end up being more than 50% occupied sometimes, or less than 50% occupied. Or it might in fact have more than 16 allied or enemy pieces, which should be impossible. The key thing is that if we actually arranged the 32 chess pieces randomly on the board our choices are restricted by the choices already made. Pieces can not occupy the same square, and once we place a piece it can't be somewhere else.

So while a leap square should hypothetically have a 50% chance of being occupied, it we are doing a sweep and traveled at least one square, and are moving to the second then that square must necessarily be vacant. We therefore have the probability 31/63 for the next space being vacant not, 32/64. Taking this into account the probability of two consecutive spaces being both vacant is actually 24.6% rather than exactly 25%. Furthermore, the Betza coefficients only consider a full board, what about the full spectrum of possibilities between a full board and an empty board?

We could of course work very hard to calibrate the betza coefficients to get more accurate results, but why do this when we could simply investigate all of these probabilities experimentally instead. We can take an empirical approach and simply run a computer program to randomly generate a large small of board conditions and take the average over all of these. This is exactly what we will do next...

An Empirical Approach

What if we could hypothetically consider every possible board state, consider the mobility of a piece on every possible square in every possible board state and then take the average. Hypothetically this would give us a very strong indication of the pieces average value over the course of the game. If we want to pick a standard by which to measure the values of pieces, this measurement would be a fairly accurate approach for a large range of possible fairy pieces. The problem is, even if we assumed that this method would give us the "actual" value of a piece it is completely infeasible. The number of board states in regular chess, let alone muskeeter chess, are so astronomical that even with computers we really have no hope of brute forcing such a calculation. To give some perspective the number of positions in standard chess is something on the order of 64!/32! which works out to about 10⁵³. Even assuming we could crunch a quadrillion boards every second this would still take 10³⁰ years! Simply put, no amount of existing computing power is even close to such a feat.

But no need to worry because we can do the next best thing and take a randomly generated representative (hopefully) sample of the whole.

I wrote a simple program to generate random "positions" and then compute the average mobility of pieces across a specified number of such randomly generated positions. A simple function takes as input the number of allied and enemy pieces to be placed on the board. Pieces are indistinguishable from edge other except for their affiliation. An 8x8 matrix of integers is used to represent the board at all times. A "0" indicates an empty space, a "1" indicates an enemy piece, and a "2" an allied piece. The number of enemy pieces ranges from 0 to 18, and the number of allied pieces from 0 to 17. The reason for this is that in Muskeeter Chess the maximum number of pieces of one type that can be on the board is actually 18, since there are the 16 orthodox pieces plus 2 muskeeter pieces. So there can actually be up to 36 pieces on the board at maximum. Why is there only a max of 17 allied pieces then? Simple. The 18th piece is the piece whose average mobility we are trying to compute.

Every combination of enemy and allied pieces is covered with equal probability. Since there are 19 possible states for the number of enemies and 18 possible states for the number of allies, this means there are 19x18 = 342 combinations. A single "trial" consists of running through a randomly configured board with each combination of material. So a single trial is actually average mobility over 342 boards or 21,888 squares. Theoretically, one can get a better representative sample simply by increasing the number of trials. Tests indicate that the values of the pieces tend to converge and become increasingly stable over multiple test runs with larger number of trials. This is good because it suggests these samples are representative of the whole spectrum of possibilities.

With a single piece on a single square, how is mobility actually measured? Simple. We just consider every available move that piece can make. Unlike Betza mobility, the pieces are allowed to capture enemy pieces. So for example, if the knight has 5 vacant spots, 2 enemies in sight, and 1 friendly piece a knight's move away, then it has a mobility of exactly 7 squares. The pawn follows the conventions previously proposed. Although it is averaged over the 64 squares, it actually scores 0 for the first and eighth ranks since it can not legally occupy them. It is only allowed two forward moves on the second rank. It is only allowed to reach it's attack squares if there happens to be an enemy piece there. One extra consideration is made for the pawn however. I read somewhere that a square a piece can only move to is considered worth half of a square that one can move and attack on. Therefore the forward squares are worth 0.5 points instead of 1 point as usual.

The program is only around 1000 to 2000 lines of code in C++ and works relatively quickly.

Here are three tests with just a single trial:

Even with just a single trial we can already have some confidence that these values are far more exact and reliable than previous measurements. We see the first reliable occurrence of the reinfeld values. Just look at the first digit of the pawn, knight, bishop, rook, and queen. The queen finally is valued at 9 points. The king's mobility score is also provided in this program and it's works out to about 4 points, which is often cited as it's "attacking value". The chancellor is around 8 points, and the archbishop is around 7.

Even more incredible, even with just a single trial, this values are incredibly stable. We can already reliably reproduce the first decimal place over and over again.

Now that we finally have a method that seems worth putting some trust in we can learn some interesting things the reinfeld values don't tell us. For one, it appears both the Knight and Rook may actually be more valuable than the Reinfeld values suggest. The Knight works out consistently to a value of 3.15 and the Rook often comes out as 5.25. So the Rook appears to be worth about a 1/4 pawn more than usually stated. The Knight is about 3/20ths more than it's nominal value. The Bishop appears to be a bit of an outlier. Average mobility suggests a much higher value than even the most generous estimates of about 3+1/2 pawns. It comes out consistently as about 3.75. The Archbishop appears to be just slightly weaker than 7 points, and the Chancellor is actually closer to 8+1/2 points than 8, as some sources state. The value of the queen, for some reason, seems to fall almost exactly at 9 points.

If there is still some doubt about the precision of these values we can simply up the number of trials. Here are some values obtained for running 1000 trials. That's 342,000 unique randomly generated boards, and 21,888,000 squares. 1000 trials takes about 3 minutes of computational time, which makes it possible to do multiple tests still with relative ease.

Here are the results:

It can be clearly seen that the values are much more stable with 1000 trials. More digits have stabilized and the variance has gone down across the board by about two orders of magnitude. The Queen is so close to 9 points that even with a 1000 trials we are not able to definitely say whether it is more than or less than 9 points. The Rook clearly is very close to 5+1/4 pawns. A Bishop is about 3+3/4 or 3.74 to be more precise.

There is no reason not to try to push this approach to it's limit to get as close to the exact value of the pieces as reasonably possible. Theoretically these values are converging to a precisely defined value that we get when we consider every single possible position that a piece can be in, in a game of muskeeter chess. As stated earlier this is impractical to compute directly because the number of possibilities is too vast. But what we can do is keep cranking up the number of trials to get more and more digits to stabilize. In this way we can approach this theoretical value. Of course even if we could truly obtain these theoretical values, they still wouldn't really be the "values" of the pieces. This is because there are many considerations that go into the value of the piece besides those that are addressed in this method. However it is generally agreed that mobility is a very large part of what makes a piece valuable and so we are actually looking at something fundamental and useful about the piece. It's just worth remembering that mobility and value are not exactly synonymous. We will look more into this later, but for now let's just finish what we are doing now.

Here are the results for 10,000 trials:

ADSF

100,000 Trials

1,000,000 Trials

At last we have the results of mobility averaged over 1,026,000,000 randomized boards! The variance is no more than 1 or 2 parts per 100,000 at this point which is about as stable as we can expect at this level of precision. Now let's order these pieces from weakest to strongest and then I will offer my recommendations for point values in the style of the reinfeld values. For this I assume that anything equal to or less than 0.3 of a pawn in difference is too small for the average player to really notice, and too small to make a difference in the endgame. For this reason it should be sufficient to round the pieces to the nearest half a pawn. This makes certain pieces appear equal, such as the Archbishop and Elephant. Quarter pawns are also simple enough not to get to bogged down by the mental arithmetic. In any case here is the table:

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