Using maths to ruin chaturaji... or mathematical game theory

As promised my own post here. I was thinking whether to bother this club with it by my own initiative; but since I got an invitation I just had to. Feel free to post your opinion. I am talking here from a purely mathematical perspective on the game.
My hypothesis: Player #4 is doomed and there is a strategy ('soft teaming') for player 1 and 3 that benefits them egoistically to cooperate. I am not even sure that player 2 can help against that even if he intends to disrupt; when in fact he benefits by doing nothing. In this theory all players are of equal strenght and (as in mathematical models) are smart players focussed solely for their own gain, aka top 2.

First I will explain the strategy, then I will point out why I think it is mathematically correct/right. It starts with player #1 (red) moving the kings pawn and player #3 the knight pawn.

Case 1: player #4 (green) moves 1. ... g2-f2
It does not matter what blue plays, as red can check blue to protect yellow. And he should do it to make sure his strategy works (= even if blue wants to disrupt, he cannot). Next move looks like this
The key point is yellows check. Greens King is in a mating net already, which will be finished by yellows knight move. Reds next move should be Nc3 (does not matter which moves blue played) to solidify his soft teaming with yellow by a)not fucking him up and b)protecting him from blue just in case. Green can choose which player he gives his king to, but RED & YELLOW took out blue without any risk (no backstabbing)

Case 2: Green plays f3-f2. Position after first moves is
Here I have to "force" blue to play the bishops pawn. At the same time, from a rational point of view, as red threatens Bd2 it kinda does not really makes sense to play otherwise, right? 
Here is my point. Green is to move. Again yellows pawn push is the key. Green is to move, but he is fucked anyway. Also, there is no worries for red or yellow to be backstabbed anyway, since the next move the position will look like this
Even IF blue intends to disrupt the strategy, green is still mated as red can save blue, and has 0 reasons not to.

Case 3: Green plays g4-f4
I could not find a way to mate Green here, though in my game it seemed counter-intuitive to play that way for Green. Certainly he will have a rough game, whereas red and yellow did not diminish their chances... 

But to reinstate my point, there is also this possible setup, which could be really fool-proof Green-killer
with the point beeing the following position
If King moves, then red can still give check to make sure blue does not screw with yellow. In the end Green is mated, or has to play 1. ... g4-f4 again which is a huge disadvantage...

Even if one of the players will need to part with a bishop there (though I did not even saw a way how Green could even make a dilemma out of it, like moving the king to a square where both yellow and red could take but should consider not taking at all) - I think it is still the best way to proceed. For sure you will be top 3 players, and in a perfect game (meaning smart equal players) it is a 3-way so beeing a piece down won't give you less chances. Like in some scenarios the best path is to soft team between the weak players against the strongest, therefore is must be (in theory) up to chance; but practically around equal chances in the resulting 3-way fight. Aka every single living player benefits by mating green at the start. Which in terms, if it should be that way, results in a game of luck (not to be #4)
As a methematician, just the fact of "they are to dumb to think that way", and simply the unlikeliness of my opponents to play that way does not solve the problem for me. It goes down to basically: in theory you are dead as green, but you can roll the dice whether your opponents cannot play the game properly. When playing properly means kick out GREEN and then share the points (on average 2nd placement after all)









 

Could this be a possible continuation for green instead of pushing their knight pawn to f2? By playing Nf3, green stops red from giving a check to blue, and also stops yellows mating threat. Even if red decides to check blue anyways, blue can ignore that and check yellow because they know green will take the bishop.

this is just for the first diagram by the way

What if red doesn't play d3?

Good point actually. Other than red moves 1.d2-d3 and blue (without any information about yellow and green) moves b5-c5, then yes; green is saved from the g7-g6 idea.
But 1 practical point is: are you ready to rely on green moving the knight so save you? (also in theory, are you really sure green will actually take the bishop instead of letting you die : backstabbing dillema? )
The theoretical issue (mathematical perspective) is, should red start by 1.d2-d3 and blue go for b5-c5 (that is, for whatever reason it is rational for blue to save green to get best chances), then red, yellow AND green can 'team up' against blue; since for red and yellow the goal was to kick one player, does not matter whom

Well green is kind of forced to capture the bishop. If they don't then yellow and red will again be able to go through with their plan of trapping green.

My point is, if I understand correctly RED HAS TO play d3 to get the best chances by brute calculations. At the same time the other have to kinda follow the strategy to maximize their own selfish chances.
In short (if these calculations are correct): Red has to play 1.d3 if he is not idiot. Others have to follow up if they are no idiots. Obviously this is all just a theoretical discussion, since there will be at least 1 'idiot' who ruins this perfect strategy.
But in a higher sense from a mathematical view point the question of "what if red does not play d3?" is the same as "what if red does not take the free piece??"- why should he not? The only reason is that red is too dumb too understand it; since quite possibly it is a mate in 3 right from the beginning. Why should red play anything else other than d3?

What I meant by this (one certain subcase) is:
If you as a player have the choice
1)trade a knight for a bishop (+2 points)
2)grant another player +3 points but eliminating a player completely
I am not sure which one is better. I certainly will consider as green to allow Red to kill blue. Which in terms mean blue cannot be sure green will take the bishop 100%. Besides, this is not the main point obviously; but rather details of a certain opening-constelation. 
There are 2 aspects in this thoughts I gave you
1)there has to exist a strategy, which screws 1 player, probalby player 4 really hard; which means a)you should use it and b)game is not fair at all. 
2)how to implement such a strategy correctly if you agree with me that it should exist. 

We can gladly argue about 2), but to be frank, as a player, I am more worried about 1) - as if the game is not fair, that means it is not fun when played correctly/competitivly. Like it loses all my interest and I have to regard it as a game of luck. e.g. we could make tournaments about it, and player #4 will be always last, and the whole enjoyment part is which ranks will player 1-3 share in the end...

I think your analysis is quite accurate. The only part I think you need to look deeper into is 1.g f4, which is the only move for green in some cases, but it is not as bad as it might look at first glance, even if green has to step out of checks on the next two moves. Same counts for blue, after 1.r d3, 1.b c5 is not only ok, but to be recommended in my opinion. 

Here, for example, blue and green have to spend one or two moves getting out of check, but their counterplay will come quickly afterwards. Worst case that I have found from my example with best play for blue and/or green, they have to give bishop for knight in order to stabilize. 

I looked at that too, but it seems very bad for green. Red can move their king to e2, cutting off the escape of the green king, and yellow can attack green all they want.

what is green supposed to do after this? Greens king will get hunted down, and if red really wants to eliminate green, they can keep checking blue.

The right move for yellow was probably Nh6, but this just doesn't look good for green

In your picture, 3.g Kh4 is fine. The discovered checks aren't that scary, and the yellow bishop will be chased away by blue. 3.y Nh6+ is scarier, but green hangs on after 3.g Kh5.

Anyways I think after 3.r Ke2, which I don't think is good, blue should go 3.b Nc8, keeping c6 for the king and trying to trade the knight for the yellow bishop.

When I revisited 3. r Ke2, it didn't seem as impressive as I had thought, although after 3. y Nh6+ 3. g Kh5, green's king still isn't very well placed in my opinion, and red may be able to try something like 4. r Bc3 to try to shift their king over to d2 or c1

also maybe this is stupid but I was just showing an idea

lol this is all too big brain for me

One short conclusion that nobody spoke against is:
-In a strict teamgame chaturaji, red&yellow forcefully win in a few moves (ref: Sauce)
-By the laws of Nash/Game Theory (to be precise, looking for a local optimization aka not analyzing who has the best chances in the resulting 3-way fight; but rather assuming it is about equal), Red and Yellow can team up to eliminate, or at least hurt Green a lot without losing anything (e.g. 1 pawn and grating another player +3). Blue cannot do much against it, though he does not gain anything by doing that. In fact, best he can do is to get as a perfect starting position for the 3-way fight, but the more ambitious he plays, the more likely is he to sabotage this very situation that he benefits himself (elemination of Green)
-Finally the most imporant piece: This club is for the chaturaji masters. Meaning maybe we will have tournaments about it, or encounter each other. I think it is a usefull information to know, that by perfect play Green is doomed. As you intend to play and understand this game mode properly, it results to be essential to know this information and to (ab)-use it. If you are in a game, you can soft team red & yellow and elimitate green, does not matter how strong he is.
-On the other side, if you are green, and you are playing against "professional" chaturaji players, like members of this club, you can essentially leave. Well, you can decide whom you grant your 3 points or maybe even 5 points if you have a preference to a certain player. But sadly, that is all you can do

I know that computers have found a relatively quick forced win in Teams for ry in the BNRK setup, but I don't think it's the case yet for RNBK. If Sauce knows this, I'd be curious to hear more about it. Anyhow, that doesn't necessarily mean that green is doomed in the FFA version, because in Teams ry would win by getting an initiative against both blue and green, and there would be many lines where blue is the one mated in the end. 

Assuming that the resulting 3-way fight is about equal in all cases is at best a very rough assumption. I think in many cases where green dies first, blue will most likely finish 3rd afterwards. But that's only my speculation, none of this has been proved as far as I know. 

Also claiming that the best thing blue can do is to sit and watch green get eliminated, without yet showing any concrete lines against 1.g f4 where green is dead, is rather premature imo.

There is a big difference between hurting a player and eliminating a player. If all ry can do is hurt green, there is no doubt that blue should be active and try to exploit the situation. You claim that green is doomed, and certainly there are lines where it is true. You can find ways to make any player doomed even with best play, that's the nature of 4pc FFA. But that green is doomed in all scenarios where the other 3 players play well, is still far from certain. 

I do not disagree about the case that I did not prove the case for every single deviation yet, and this is a valid point; though I hope we can at least agree that 'suffers a lot' may suffice enough to show RED & YELLOW should just play it and they both benefit without risking anything; when Green is either objectivly dead, or has really low chances just by beeing green... (e.g. 25% -> 10% is still nowhere close to beeing fair). Furthermore I am sure that we did prove that green does suffer a lot to fulfill this

As for the other points, there is also another viewpoint.
-In a team game Red & Yellow vs Blue & Green the first party wins, but sometimes not by mating green, but mating blue first, yes. But if everyone fights for himself, why should blue try to safe green like in a teamgame and risking his own game? In fact, he might be supporting Red & Yellow if it benefits him. At the very least, he may not activly sabotage the attempt.
Besides, for yellow and red it is important to mate one of them, they don't care who it will be in the end. And saying then "green is not screwed" but rather "green or blue are screwed" does not diminish the issue

-Lets assume a 3 way fight, as we consider Green simply dead right at the start; possibly with Yellow or Red having to lose 1 pawn or something. IF you can prove that blue is the underdog then (having less than 1/4 chances as he did at the start), then yes, he should be trying to support green. But I am not convinced that blue is as screwed in a 3-way game.
I do agree that he is trapped between 2 armies, so he is in a lot of troubles. BUT, exactly because the payoffs are '3' , '1', '-1' it will be his saving grace. E.g. Red may help yellow to kill blue with 0 problems. But if red would do this, when yellow would win all the points for himself, is this a rational behaviour? Or if yellow is starting to go for promoting his pawns to rooks, red has an incentive to stop this by leaving his king there (fighting yellow). Blue is fighting yellow too. Red might even stop harassing blue so yellow does not got ahead. 
ALSO: imagine Green is dead and Red starts trading with blue everything, including the kings (so that yellow gets no points) In the end Red and Blue have some points, while yellow still having all his army, lots of rooks stays alive in the end but has no points
... and so on. By my understanding (in theory) the 3 players are equal as long as 1 cannot outfight 2 together. This is not the case so the equilibrium will be something equal as I said. This is the case for AI's not for short- and simple minded humans

Generally speaking, I don't think we can say that we have proved anything by showing a couple of example lines and discussing vague terms such as "suffers a lot". It is clear that it is better to be red than it is to be green, but the magnitude of the difference is still highly unclear imo. Game theory needs to develop a lot more before we can't talk confidently about certain % chances for each color, except obviously the average 25% chance for each player.

So let's look at some more lines instead. I came across the following interesting case based on the example from @Ihavethesauce: 

Red to move.

From here there is the line 4. Bxf4 d6 g6+ Kxh6 5. Bxh6# 6xe7 leading to the following position: 

Yellow to move.

4.r is only correct if 4.y is correct, which I don't think it is, because yellow is massively unfavored in the resulting 3-way fight. In Teams this line would just win for red+yellow. So some aggressive opportunities that are strong/winning in the Teams format are not viable in FFA.

I agree that blue shouldn't try to save green on a general basis like in a team game, but I think blue should play actively and try to take advantage of what goes on on the board, which coincidentally due to the way the players are situated often results in improving green's chances. Just like how coincidentally it is good for red and yellow cooperate early on, even though it is not a team game. Remember that the goal of this game is to get the most points, not to survive the longest.

Thinking about the starting position with green dead is interesting. As you said, red can force many trades with blue so they get a big point lead on yellow. Then the game may end with yellow in 3rd place if either red or blue accepts 2nd place. But they can also for some time both play on for 1st without much risk. Evaluating when it gets too risky and yellow's long term advantage of being able to promote all the pawns easily becomes significant, is the tricky part, and too much greed from red and/or blue might easily give yellow chances anyways. Also, red and yellow can trade bishops with each other to gain 5 points on blue, which is hard for blue to catch up with, considering his own bishop has no natural trading counterpart.