Ok - I simulated (and calculated) orb distribution probabilities for an early game scenario.
Conditions -
- 8x10 arena, both players with 20 pieces each controlling 15 empty tiles on their half of the board. 10 neutral tiles exist in contested territory between squads.
- 10 orbs spawn (to simulate the effects of 2-3 orb spawns)
Scenarios -
#1 - Orbs spawn randomly (probabilities calculated - see graph on the right)
#2 - If an orb spawns on a tile adjacent to an orb, it is rehashed once (simulated - see top graph)
#3 - Orbs are not permitted to spawn on tiles adjacent to an orb (simulated - see bottom graph)
Graphs - (Correction: Forbing defined as R ≤ 2B orbs)
Discussion -
Getting forbed is defined by your opponent getting twice as many (or more) orbs as you in their territory. They can get 8 and you get 1, or they get 2 and you get one (with the rest landing in neutral territory). Under the random orb drop rules, this happens 18.7% of the time in this slightly contrived scenario. In each of the graphs, the chance that the red squad gets X orbs to land in its territory is shown in the graph. The portion of the bars that are pink represents instances when the red squad got forbed - i.e. the blue squad got twice as many or more orbs in its territory. The sum of the pink bars is 18.7% in the graph on the right.
When you add Shmoo's rehash an orb if it lands next to an existing orb rule, almost nothing changes (scenario #2, top graph). The chance of getting forbed goes down to about 17%. You can see that it is slightly harder to get 5 or more orbs to land on the red side. It also a bit more likely that you'll get 3 or 4 orbs out of the hash. But you chance of getting forbed doesn't move much. Your chance of getting really screwed (4 to 1 orb ratio) goes down too, but it wasn't all that high to begin with at 3.9%!
If you get strict and say no orb can spawn on a tile adjacent to an orb, you still get little change! This is more dependent on the configuration of the squad. If arrayed in a checkerboard pattern, all empty tiles are on diagonals and therefore, wont inhibit each other. If all the tori are in rows, the the chance of mutual inhibition is higher. Using the squad arrangement in the figure, you see that the orb distributions change more. It now becomes harder to get a large number of orbs on your side. The odds of forbage don't change as much as you might think - only down to ~13.4%. You chance of getting really forbed (4-1) are down to 1.5%. As I mentioned, the configurations matter. The blue squad has a slight advantage by having two fewer tiles with two neighbors. This translates into a 3% better chance of getting more orbs that the red squad.
Conclusion -
Rules restricting random orb distribution have little effect unless the severely limit the chance of an orb spawning on a given tile. Even then, the chance of forbage is not affected much.
In my opinion, the benefits of improving "fairness" by implementing such a strategy do not outweigh the added complexity involved.
Furthermore, most orb spawns are small and orbs are collected quickly - meaning that neighboring orb inhibition does little to stem uneven orb distribution over several spawns.
An alternative might be to "turn off" a tile for two or three orb spawns after it has received an orb. Another alternative might be to rehash any uncollected orbs on the third spawn after they were created. These ideas would need to be explored further to assess their impact.
I'm happy to vary the simulation (i.e. different board or squad configurations, smaller number of orbs per spawn, different rules governing orb spawns). Just let me know what you want to see - assuming your dorkometer has not already exploded.
