Rare sight of an uncontested knocked out area. Third round grand conquest is always a wild one.

It has been while since I wrote anything about FE Heroes as I have no interest in covering metas and crying about how many times I ended up with 20798 lifts in Aether Raids. The most recent one was about voting gauntlet written in 2021 but just like the last AHR we had, people just don't care as much anymore. So today let's look at another mode where theoretical behavior differs so much from reality.

Grand conquest.

In the past infernal difficulty is almost impossible with the warps. As time goes by, the difficulty goes up by including more inflated units into the pool. That however, does not take account into the facts that (1) old units are still there meaning that the average opponent strength is not inflating as quick and (2) recent units also enables more disgusting combos with such potential inflating faster than pure unit strength. As a result, those 'top difficulties' became much easier. Similar phenomenon happens in most PVE modes. Remember 10th Stratum in 2016, or Veronica as final boss in the first Tempest trial?

For inferal difficulty with warps, my solution is to use units with multiple actions so that I can overwhelm a certain front constantly in a single turn so that opponent can't warp. Summer Edelgard, Lynja, Ninjorrin and many more disgusting units would do the job well as long as terrain allows early aggression.

But I am not going into that in detail. My question is, how would grand conquest looks like in an ideal play? Here are the assumptions and simplifications:

Assume that players would score 2000 per sword consumed. Assume that teams consist of equal number of players who will play at all time and waste zero resources. WLOG assume that scores do not flow to neighboring area (because you can adjust number of times you participate in those areas to achieve the same score regardless) and ignore the 'help out' function because that help would be of negligible effect when everyone scores so much (that will be an interesting random perturbation under a perfect game from both though). Ignore also the mechanism that scores multiply with players participated in the area because I believe it's worthy to invest at least an epsilon in every area or otherwise opponent is always happy to take the free area.

This is now the game where each player starts with K units of resources in the reserve (which do not recover, resembles the swords). In each turn the teams take turns to assign resources onto contested area (say, teams take turns to assign 1 unit every time). Each turn players can distribute at most N (= 4 x head count) plus arbitrary amount left in the reserve into those contested areas. At the end of each turn, area with opponent spending more onto it would have the ownership flipped. The score calculation would be the same as in FEH: summation of [area hold/5] plus area count at the end.

How would the game look like then?

...the more I think about the possibilities the more complicated it could be like, that at a point I simply give up.

We simplify by thinking of a game of two players (maybe with 20 areas). At first it's of course the concept of choke points and defending the choke points, so you want to reduce the number of points that you are defending to increase the average resource that can be put in one area. You may also think of going into an equilibrium where it's not worthy to take any more lands except for the last turn, and the examination can go much longer.

However all these become useless with the extra resources K assigned to each team at the beginning because they can then breakthrough the defense with much higher firepower. This is now a game of distributing proper resources in each turn to balance between offence and defense.

Similar problems occur in real life, quite commonly actually. One practical example is to participate an auction with limited amount of money. Assuming that each participant has equal amount of money and each auctioned item has a fixed and transparent intrinsic value, how do you maximize profit in the auction?

Going into these 'games' would be extremely complicated and far beyond what we want to discuss here, so I decided to cheat a bit.

What if teams are allowed to react against opponent's resource distribution bit by bit within the same turn? Say, when a team is to assign 1 unit of resource onto an area then the other team can assign 1 unit after the first assignment.

Well then this will be a draw as long as the team with the second move replicates everything. Zermelo then tells us that since player has a forced draw strategy, there will be no winning strategy.

Everything is going well on a two player game as this is the classical setup. What about a three player game then?

All classical tools broke down here and I have absolutely zero intention to head into an active research field. But my guess is that it depends on the maps. In a two players game, the contested area is always symmetrical to both, hence the draw. In a three players game this is sometimes not the case, especially when two teams combined always overwhelm the third one.

We can generate perfectly symmetrical map (like the star graph if you like) where by symmetry will end up in a forced draw. There are also one-sided maps like three teams based in the three vertices of a V-shaped map where the two teams will simply squeeze the middle team to death.

There are configurations/states where forced draws will happen from there, which we call that an equilibrium. Equilibrium may occur with unequal scores like 9:10:11 or 5:9:16, as it depends on the contested areas only.

There will be maps with asymmetry that a team will start with negative maximum possible expectation, meaning that they are going to lose somewhere they had at the beginning. Since trading (between two teams) is a waste of resources, the team would cleanly lose that area and fight for an equilibrium afterwards. Similar strategy will be adopted by the team who are supposed to gain that piece of land who will quickly take that and go for the equilibrium from there.

The last paragraph is surely the most shady argument/guess. One immediate question is: is it possible for an oscillation to happen? The answer is yes and obvious if you set K=0 and resources available to 3 (instead of 4) then by parity there will be exchange each turn. If we take cyclic states as an equilibrium as well, then I ultimate guess is that given enough rounds, all maps will end up in an equilibrium (a 2:14:14 in the V-shape map is also an equilibrium!).

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I think I have my time wasted in writing all these because grand conquest will never be played in such an ideal/competitive way. It is still more fun to play the mode as it is, rather than giving the team a chatroom probably temporary discord server as well with players monitoring and reacting 44 hours nonstop.

FEH is really declining with less groundbreaking updates, lower reaction against new units due to saturated barracks, less votes in popularity contests like CYL and AHR, PVP modes completely scaring away most players and so on. I would still support the game till the end, but please give me a premium unit that is +10-worthy both in terms of art and competitiveness. Not now, but when I have enough orbs...

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