f4 seems to be the only way green can stay in the game.
Using maths to ruin chaturaji... or mathematical game theory
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Same question here: why should yellow go for Ng8-h6+? Ng8-f6+ is better imo. Yellow and green won't forcefully mate green, but the whole premise of the operation was the gametheoretical approach
-and I will be more specific this time- it is not about percentages, but specifically about expected outcome. If we assume that by normal play (no cooperation) each player has the same chance for each position (maybe red a bit more than green, but lets neglect this single move advantage for later stages). The expected value for each player, assuming all have same rating ist then with no strategies E(Red)=1/4* first place (+3) + 1/4*2nd place (+1) + 1/4*3rd place (-1)+ 1/4*4th place (-3) = 0 = E(Yellow)=E(Green)=E(Blue).
Now assuming (which is not true, just for the example) that green will be dead for sure, then for each player the chance might be E(Red) = 1/3* 1st pace + 1/3*2nd place + 1/3* 3rd place = 1.33 on average. It is quite possible that someone, like blue has actually the worst chances in the end, e.g. E(Blue | 3-way fight after green is dead ) = 20% * 1st place + 20% * 2nd place + 60% * 3rd place = 0.6 + 0.2 - 0.6 = 0.2 . So even with a terrible position blue is incentivised to follow it, since 0.2 is better outcome than complete randomness as with no strategy
Now back to the question #20 : so far I do believe that for red, the move 1.d2-d3 increases (or better to say maximizes his chances compared to other moves) his expected outcome. That is why he should play it. Yellow should follow the strategy by the same logic. Coming now back to your post: the move Ng8-h6 does not maximize his chances. True, it may kill of green, but this is not the purpose of yellow egoistical play. Instead Ng8-f6 is a risk-free approach that slowly kills green, where he is not under threat to lose too much, or is in danger to be backstabbed (as in beeing left behind). Same for red, Bxf4 I do not see as the best move, but instead any move that develops his pieces, as green is fucked long-term and cannot do anything while beeing dominated.
You do not need to point out this is not a concrete strategy how green dies. I hope to show you that if you consider yourself beeing green you cannot do anything in this situation and all you can hope for is that blue attacks yellow soon enough (which red can sabotage by attacking blue himself) before green can do anything usefull for his win. Yellow on his end might even consider (this is for the AIs to prove) diminish his expected outcome by doing that in the first place...
I started with d3 opening style as green in this game and finished 4th and lost 55.7pts, while I only needed 8 more points to reach 2201 in rapid https://www.chess.com/variants/chaturaji/game/19413248/102/4
Gellart believes that I can't play safely when starting with d3 as green
Would b3 or c3 be the better opening style to solve this chaturaji problem?
it depends on what red yellow and blu do but if u can u should prob play b3, and if that isnt possible c3, d3 would be a last resort for when red and yellow do the thing
@terkenal
I think best move in this situation is 2. ... Nh2-f1! The issue is, they may mate you, yes; but this would be a bad rational choice (besides that they are humans and not cooparating anyway), as red needs to protect b2. So he will be forced to play
3 r: Kd1-f1 b: xxx y: Bg7-f6+ and you as green can move Kh4-g4 and then run to f3 to save yourself. If red plays Kd1-e2 yellow should be rather taking b2 instead of mating you. So you have double safety (a salfish and a rational) not to be mated
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.
Yeah, so I guess f4 is the best option for green if red and yellow go for the same opening. Also the line that you showed is interesting, but like you said, I don't think yellow should go into this. Yellow succeeds in killing off green, but in doing so will basically be destroyed by red and blue. Yellow can't really capture blue's pawn(In your second position), as that would allow blue to check yellow. This would give red a chance to freely eat up yellow's rook. Considering this, I do not think red should capture green's pawn. Hest's line goes 4. r Bxf4, b d6 , y g6+, but after looking at what happens after green is checkmated, I think yellows best move is just to move the bishop out of danger. So instead of playing 4. y g6+, yellow could just play Bf6. Because of this, green is free to capture red's bishop, and red doesn't gain anything from this. After this, I think green is fine here, and red is just down a bishop.