In my opinion, if you can't make a forcing ffa puzzle, then it isn't a puzzle. It should always be assumed that players will play the optimal move in a sequence otherwise, as shown by your elaborate rhetoric, things get very messy and puzzles will be less fun to solve. The point of ffa (whatever point system is in place) is to get first and we must assume opponents will play optimally for the integrity of the game as computers do in 2pc tactics. Also, in the case with people like me, I'm going to be way too lazy to keep up with any rhetoric in an attempt to address cases like "throwing", "kingmaking", etc. and it may give people the idea that such low plays are to be commonplace and implement them themselves in fits of erratic emotion if they see such "themes" demonstrated in 4pc tactics.
About the topic of FFA Puzzles
In my opinion, if you can't make a forcing ffa puzzle, then it isn't a puzzle. It should always be assumed that players will play the optimal move in a sequence otherwise, as shown by your elaborate rhetoric, things get very messy and puzzles will be less fun to solve. The point of ffa (whatever point system is in place) is to get first and we must assume opponents will play optimally for the integrity of the game as computers do in 2pc tactics. Also, in the case with people like me, I'm going to be way too lazy to keep up with any rhetoric in an attempt to address cases like "throwing", "kingmaking", etc. and it may give people the idea that such low plays are to be commonplace and implement them themselves in fits of erratic emotion if they see such "themes" demonstrated in 4pc tactics.
The other issue being, is that if we do not disallow kingmaking, most of FFA puzzles will be highly trivial, and they will almost never be complicated move sequences that last over 10 moves, unlike the teams puzzles. Two players may just decide to team up to such an extreme extent so the other player stands absolutely no chances, i.e. most FFA positions are impossible to solve. Hence, providing that at least one player will not try to throw the game is in fact already enough so that one player can force a win, that is already enough to win some FFA puzzles.
Now, let's assume the 4P puzzle: checkmate a flank. Everyone may refuse to cooperate in fact, you may get 3v1ed, anything can happen. So this means FFA will never have interesting puzzles, and at least 50% of them will be "find out how does the key player not win".
In the above puzzle, given that one player can be in any rating range, the other non-key player must do not throw the game and play optimally. So, both players CAN play maximally optimally, but then blue can still win by points in a certain line. However, if both players will team up just to screw the game up for blue and let the 1st place get decided by blue which is an extreme example of teaming and throwing, well in most FFA puzzles no one would be exempt from this, but most importantly this will close-to-never happen in the real play.
Blue in fact has a forced mate on red, so it may either end in a 2-way tie if yellow and red do play perfectly, but if RY want to at least somehow fight for a win they should opt for a 2-way tie assuming "everyone plays like a computer".
So what do we assume? That everyone will play like a computer or at least two players will? Then it is more than enough to make puzzles like an above one, where it is in any case a forced win or a 2-way tie for blue for 1st.
Do we assume that at least two players can play very emotionally? Then that does not get interesting, most of the puzzles would be incredibly simple for an expert player. So far I haven't seen a single such puzzle that would be uneasy to solve: they all are trivial.
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What is crux for FFA puzzles is that any player can throw the game in the favor of an another player, as such making most of the positions practically unsolvable, hence FFA puzzles are usually too easy to solve, and are never complicated. As such, I wanted to amend this issue by introducing the following FFA puzzle classification format:
a. Constancy. This defines how well should players play in order for the key player to win.
1. Fixed Constancy. This means no matter what the other sides play, the key player will always win, and there is absolutely nothing the others can do about this.
2. Irresolute Constancy. This means that the key player can always win given that at least one player will not throw the game but instead try to maximize the winning their winning chances.
3. Practical Constancy. This means that players will only throw the game when they cannot maximize their winning chances, but otherwise will play optimally as to halt an immediate win / advantage.
4. Deterministic Constancy. This means that every player will play optimally and will not throw the game as long as there is a very slim chance of winning, i.e. imagine a 2700+ game.
b. Goal.
1. Victory, win by force, quite obvious I assume.
2. Practical Victory, a victory that can be prevented only by extreme teaming play or by extreme throwing. Extreme here counts as an incredibly accurate move sequence that players would cease to even think of in practical games, to see this idea one must see all the attacking idea aforehand.
3. Practical Edge, obtain a non-winning advantage that in a real, live game the key player would be definitely able to convert against players of the same level as the key player.
4. Practical Advantage, obtain a certain advantage, i.e. eliminating a flank as to secure one's own comfortable gameplay. No one guarantees the key player will win against same-strength opponents, but they must be equal at least.
c. Any FFA puzzle that has a line that ends up in a forced king-making or encourages one to throw as one cannot change the game outcome is considered as "practically invalid".
d. It is assumed that all the players are of the same level for practical constancy and deterministic constancy.
Let me take the following example puzzle:
[StartFen4 "Y-0,0,0,1-0,0,0,0-0,0,0,0-32,41,39,0-0-{'royal'
'h2','a7','h14',''),'pawnBaseRank':3,'noCorners':true}-x,4,X,1,yW,X,3,yU,x/1,x,x,2,X,yU,yK,X,2,x,x,dU/1,x,x,3,X,X,3,x,x,1/2,dγ,1,dα,5,yR,dγ,1,dΔ/4,bK,1,bδ,2,yN,2,dC,1/X,X,bU,1,bY,bδ,2,yδ,yM,2,X,X/2,X,bR,3,dI,3,X,dY,dW/bW,bN,X,2,rδ,2,yG,1,yδ,X,dK,dK/X,X,1,bδ,2,rδ,rN,dF,1,dδ,1,X,X/2,bγ,1,bδ,2,rδ,1,rγ,rγ,dδ,dC,dN/bC,5,rK,rY,1,rB,2,dγ,dR/1,x,x,3,X,X,3,x,x,1/1,x,x,rR,1,X,1,rW,X,2,x,x,dU/x,4,X,2,X,4,x"]
[Variant "FFA"]
[RuleVariants "AllowPassing Anonymous BarePieceLoses PromoteTo=M Prom=10 Sideways"]
[CurrentMove "0"]
Yellow to move, Irresolute Constancy, Practical Victory, Blue to win.
Difficulty: pretty difficult.