A Statistical Approach to the Point Value of Pieces in FFA

Lots of interesting comments, thanks a lot! And thanks for the kind words.  

@pjfoster13 I also suspect that 1-point queens is the wrong way to go, and that we should aim more for always having pieces on the board that are worth their point value. 

@JKCheeseChess I wouldn't mind sharing the spreadsheet, the thing is just that I don't know how to share xlsx files here on the forum. Any ideas?

@Sigma_1984 it doesn't matter who owns which piece, we're only looking at which type of piece captures what. If a rook captures a knight, the rook as a piece wins 3 points and the knight as a piece gives away 3 points. 

@BabYagun I am subtracting how many points the piece gives away by getting captured, which is affected by two things: the value of the piece and how easy it is to capture. What kind of pieces would be harder to capture than others? Depends on the mobility of the piece, I'd think. Unless it has a completely different capture mechanic like the king, which is easier to capture because you only need to threaten it. You could create a board where one piece is completely isolated from the others, so it will never be captured. But then it won't ever capture anything either. 

@grable many good points here, first of all I'm impressed you've already recounted the game.  In your example with the knight fork from the game, the rook moved away so it couldn't be captured, but it's hardly relevant for your point. Maybe people would take the bishop over the rook, might make sense in this case because red is more easily attacked by green's bishop than green's rook. There are many factors that play in on every move, sometimes you might decide to capture a lower value piece. But the goal is always to, in the end, have more points than your opponents. 

It's important to question the reliability of the method. As you say it only looks at captures and ignores any other aspect of the game. If you make a positional sacrifice, it's because you expect it to make you win back points later, or give away less points. Which type of piece is included in a sacrifice and which one in the cleaning up afterwards will give unfair results with such a crude looking method, but once again the idea is that this will even out with a large enough sample size. 

Time controls and other rule adjustments will affect these numbers. Should piece values be adjusted for every case? Maybe not, but some factors like board size seem obvious. Actually I've been thinking the correct way to go about it might be to make only random moves, with randobots for example. You should try doing the same analysis yourself, it gives insight in which variables have consistent patterns and which don't. I found surprisingly consistent results even in a hyperbullet format, bishops almost always underperformed for example. I have calculated the deviation of each EPG as well.

Wow just wow

Nice Job master Hest

Algorithm to find balanced piece values for any FFA format:

1. Let 4 randobots play a certain amount of games, say 20. 

2. Make a script that counts all captures (doing this manually takes a while). 

3. Make a script that finds the combination of piece values that gives the lowest total EPG deviation from 0. 

> 1. Let 4 randobots play a certain amount of games, say 20. 

100 or more

> 2. Make a script that counts all captures (doing this manually takes a while). 

This is what I though while reading your article. Some parser is needed to process PGN4.

> 3. Make a script that finds the combination of piece values that gives the lowest total EPG deviation from 0. 

I think that MS Excel or Google Spreadsheet can easily do that. I used to solve such tasks using MS Excel Solver: https://support.microsoft.com/en-us/office/using-solver-to-determine-the-optimal-product-mix-c057e214-962f-4339-8207-e593e340491f

There is another approach (more simple comparing to creating a PGN4 parser from scratch). We just need to donate a good bottle of wine to @Spacebar to motivate him add EPG calculation during all FFA games, enable it on the server for a day and log the results. This way we'll get data from real (not randobots) games, and it is 100 times simpler to parse a log with pre-calculated values than parse PGN4 and calculate values based on moves.

Still I think that 1pt Queens should be a separate piece. Mixing it with pawns isn't right.

@Babyagun Great, you make it sound very doable. I wasn't sure how to make those programs.  

The reason why I'm thinking about using randobots is that with "real" games, the piece values actually affects how the games are played. Like with my spreadsheet I can change the king value to 10 and see if the numbers improve, but the more you tinker with the values, the less legitimate the sampled games become. You don't have this problem if all the moves are random, it feels more pure and objective. 

Why isn't mixing 1pt queens with pawns right? Much of the value of a pawn lies in its ability to transform into a stronger piece. I wouldn't sacrifice my knight for a pawn that's about to promote if it was never going to promote in the first place. 

I understand and I would also like to draw your attention to another point. When a player is in check and has to wait 2 more turns before he can respond to that check other players can benefit from that checked player by capturing his pieces, in FFA at the very least. But then if you look at it from a strategical perspective only one player of the remaining 2 would attack the checked player.

So from here on we have 3 possibilities : Either both players attack the checked player and capture each a piece or only one of them attacks and captures or none of them. Continuing from this logic that would mean that there will be a wasted chance to capture a piece by default, considering that it would be beneficial for the opposite player to not capture pieces of the checked player, since it would be semi-teaming in the first phase. You have already stated that a piece should be able to be captured to be accountable data. In statistics this is called, as you know, 'conditional probability'. So to make your calculation more accurate you might consider to add another condition besides the one you already have. 

The value of a piece is the expected amount of points that the piece will capture, granted that it becomes captured.

To this I would also add the probability of a player benefiting from a check, because it would obviously affect the EPG. So this is maybe a factor to give some thought.

Next step: write a bit of code to do this automatically from PGN4 format, instead of counting manually; would provide more statistically significant data, and I'm sure @spacebar would be happy to run it on a database query of games from a particular rating range : )

Not a coder, but I've started working on something with copy/paste functionality into a spreadsheet. We'll see how it goes.

Very cool analysis! Would love to see the output from a larger sample size using a script, and possibly even see the points adjusted as a result.

I agree that pawns have increased value because of the promotion part and 1 pt queens are just like that. 

Only real man proves his opinion using math. I really liked your approach to solving this problem. It is ideologically simple (everything ingenious is simple) and its results can be logically explained.
But there is one point that I do not like. You combine the cost of the mate and the cost of the king. The king has a double function, +3 is its cost price, +17 is the price of defeating the player (mathematically this can be estimated as the price of dead pieces left after the mate). In your example (# 3541595), -40 points for two not captured kings should not go minus to kings (in EPGA). This mathematically simplifies the task, since the total balance of points will be zero, otherwise I don’t know how you will calculate the combination of piece values ​​that gives the lowest total EPG deviation from 0. Therefore, it is better not to consider double checks - these are points from the void (by the way, I have already forgotten the reason for introducing points for double checks, where was this discussed?) and it is better to use in the sample only games with the 1pt queen, which is considered as a pawn.
Points for mates are a controversial topic.
In your example, blue did mate the yellow one not only with one queen, there was a whole complex of pieces (even the green queen) that participated in the mate (there was also an alternative: this mate could be implemented with a blue bishop, this is +25 to the bishop (in EPW), this is as much as all the bishops earned during the whole game), maybe all these pieces (two bishops and two queens) should have received some part of the points from the mate (for example, it can be considered with a coefficient dependent on the number of fields taken away from the king, not counting the field occupied by the yellow pawn, then gain +13.3 (and +5 for the yellow bishop) for queens, +6.7 for bishops)
Or you can do it differently, just subtract -20 to kings per mate (in EPGA), but add only +3 (in EPW) per mate to the balance of the matting pieces and +17 to the balance of kings (and all +20 to the balance of kings for not captured king). I like this option more, but the explanation is not mathematical: Chess was created as a model of a real war, and who gets the lion's share of trophies for victory? Of course the king, because the army does everything for the glory and prosperity of their king! Without a king, there is no army. By the same thinking, the king in 2 pс is priceless.
It would also be interesting to compare the FFA and Solo, perhaps the debuts would affect the experiment.

I like your opinion

Maybe comments should have like buttons to see how much a comment is appreciated. Here I look at the admins for a further approach. 

@sigma_1984 how would you implement benefiting from a check? It sounds complicated and I'm not sure it would even change the numbers. The activity of the pieces ignores the strategy of the game, or rather the strategy of the game is caused by the activity of the pieces.

@Arseny_Vasily the king is indeed the tricky piece. I am not sure how much of the king discussion belongs in this thread.

The principle of this method is to measure the causality between moves and point generation. Everything that creates points, including double checks, needs to be taken into account in order to find the true point generating ability of each piece.

Whether or not double checks should exist is a discussion that I do not think I have ever seen.

You are right that the total sum being different from 0 makes the average EPG different from 0, so it is more correct to say that we are try to find the EPGs that deviate the least from their average. There is no mathematical problem with that, so keeping count of points appearing or disappearing into nowhere can very well be done.

It is cool to keep in mind that chess is supposed to be a representation of real war, we chess players can get stuck in our "perfect" theoretical universe. But giving the points for mate to the mater's king due to medieval traditions would go beyond the action->point causality, so I would not do it for this system.

One could consider the king as the player's representative on the chess board, just like the medieval king was considered to be god's representant on earth. For 2pc and Teams this is quite fair because losing the king means losing the game, but not in FFA because a player can lose his king and still win the game. 

I think there are two ways to handle the king with this system: 

1) Treat the king like any other piece and try to minimize its EPG. 

2) Keep the same ways of calculating all the EPGs, but ignore the king EPG in the minimizing process.

In the first method we try to get as close to 0 for every EPG, which would strip the king of all its bonus value. The king would even end up being undervalued due to its inability to put itself in danger (check) and the rule where leftover kings give away points at the end of the game without getting captured.

The second method allows setting the king to a certain value, like 20, and then try to balance out the other pieces compared to each other. The other pieces should still have EPGs as close to each other as possible, but not necessarily close to 0. Then the bonus is evenly distributed among all other pieces and the pieces can still be considered balanced. 

We could try out either of these models, I would recommend starting with method 1) to verify the base value of the king and then try to add bonus points and use method 2).

I think you may have confused this topic with 2 player chess 101? A lot of this doesn't even apply in four player chess. There is no center to speak of like in 2pc, because there are too many squares to effectively hold control over. The center and space in general function very differently in 4PC because there are often a myriad of squares pieces can choose from which they can be highly mobile. This also means the queen is free to come out early because it can often find safe squares, and it can be a strong strategy to develop the queen early in both FFA and teams. Lastly, there are many cases where castling is contraindicated. So these 2 pc generalities don't really apply here and can be downright terrible advice.

One of the major pitfalls of new players is to try to play 4PC like 2PC, but you really have to start to think about the game from the ground up. If you ask yourself why certain ideas in 2PC make good general rules to play by, you will find the same doesn't also apply in 4PC. There are of course exceptions (speedy development is also good in 4PC for example).

Love this analysis, thanks for the legwork @hest1805 ! After all the discussion, what would be your proposal? It seems most of us would agree that:

1. Bishops should be worth more than knights as they can cover way more ground on the much larger board
2. There's been much debate on the value of bishops vs rooks:
   - https://www.chess.com/clubs/forum/view/how-many-points-should-rooks-and-bishops-be- https://www.chess.com/clubs/forum/view/bishops-are-worth-more-than-rooks-in-4-player- https://www.chess.com/clubs/forum/view/value-of-bishops-vs-rooks
3. Keeping the value of the bishop and rook the same seems sensible
4. The queen probably should be worth more than 9. 10 seems reasonable.
5. Pawns probably should be worth more than 1 point. 2 seems reasonable.

I would also suggest considering increasing the value of pawn queens. Why? Two reasons:

1. If a queen is the most powerful piece, and a pawn queen has the same powers as a queen, then a pawn queen should arguable be worth the same as a normal queen.
2. It balances the game a bit more. If your enemy gets a pawn queen, that's great for them, but if you manage to kill it, you should get more than just 1 measly point, which is quite an imbalance.

So maybe meet in the middle. If we increase a normal queen to 10, then a pawn queen is worth at least half of that, so say 5.

Additionally, if we increase a pawn to 2, then we should increase the knight to 4.

So in sum, here is a proposal:

1. Pawns: 2
2. Knights: 4
3. Bishops: 5
4. Rooks: 5
5. Queen: 10
6. Pawn Queens: 5

It would truly be great if we could tweak value pieces in a custom game setup, and experiment that way @chess.com