x^4 can only have 2 bumps, x^6 can have 3. I had to plot the graph myself to make sure it has 3 bumps because the x axis obfuscates the picture.
Local optimum is a very intuitive concept — it is a maximum in a local area, but not necessarily the highest peak total. People understand what a hill’s summit is even if they can see a mountain in the distance.
No you can’t.
We live in the real world. AoE2 might be a virtual game but is played by physical humans.
Also there is no limit to changing those parameter.
Let’s say AoE4 would be more balanced then AoE2DE then theoretically that would be a different local maximum in the same parameter space.
But even at the core the question is impossible to answer.
AoE2 includes randomness. Not just are the maps random or the chance of arrows missing.
But also are player choices random.
If opening A counters opening B which counters opening C which counters opening A.
Then if you pick a random one of those the game can be decided by the random choice the other player makes.
You can’t necessarily scout that either.
This game is not chess.
This is also not really a good question to ask.
It will always be able to improve it indefinitely.
The more interesting question would be if it should change a lot at once to reach a better state.
Like a big expansion usually does.
The Conquerors and The Forgotten Empires added a lot of changes to the game at once but they are the oldest expansions.
Do people want a big change? For the sake of balance and more importantly fun.
For those people that get tired of playing the same strategies again and again, not because they are unbalanced but because the variety is too low.