Threshold in wargames

Well, I need to be more specific.

The one that I am talking about is when players use a number of pieces. And the game results changes drastically.

"The 1 tank beats 7 soldiers, but 3x7 soldiers will beat 3x1 tank." Kind of change. The threshold in that regard would be at 3x the army composition.

And with occurence, I mean that the threshold is a concequence, not a test.

Edit:
anyone interested in how this effect occurs on mathematically basis?

The effect is very noticable in games like Starcraft(2).

Let me put this in mathematical terms.
Obviously, I like to compare simple soldiers with a tank.

And if you want to follow what I am saying here. I suggest that you play things out on paper as well. Just to get a feel of what I am doing.

Let's say the statistics are:

Soldier;
6 Costs
6 Health
1 Damage

Tank;
36 Costs
36 Health
6 Damage

A Tank can one shot a soldier.
For every tank, you have 6 soldiers.

You can have these battle's yourself as well. Even if it is only on paper.

The health of the tank will go down. But the overal damage for the soldiers will go down as well.
So, we can follow the health of the tank versus 6 soldiers:
36, 30, 25, 21, 18, 16, 15
What happens if we add one more soldier?:
36, 29, 23, 18, 14, 11, 9, 8
It balances out when you have 8 soldiers:
36, 28, 21, 15, 10, 6, 3, 1, 0
The soldiers win with 9 or more:
36, 27, 19, 12, 6, 1, 0, whereas 3 soldiers survive.

The tank wins on equal grounds, while costing the same.
The threshold is when we have more than 8 soldiers.
Here, the threshold is not feasable when you give both players the same ammount of resources. This is a fake threshold to look at.

What happens at higher scales?
If we increase the ammount, like for example 10 tanks against 60 soldiers. Both sides will suffer from damage depletion.
But the tanks will always have the upper hand.
Now, I am good with excel in this regard. So I will post some results here:

Number of tanks and 6x soldiers; % of remaining tank health
1x ; 42%
10x ; 18%
100x ; 7%
1000x ; 2% (2% of 1000 tanks is 20 tanks remaining!)

It is more interesting to see the lower scales though. But before I post that. What can we do about the tanks having the upper hand? Most designers will adjust the statistics. We can choose to change the health, the damage or the costs. Let's say, I change the costs. One tank doesn't cost 36, but 42 instead. Now we can compare the results to the original design.

Once again, we put in a number of tanks and soldiers. The first percentage will show the remaining health on the winning side, concerning the original design. The second percentage will show the ramining health on the winning side when concerning the adjustment.
A negative number will indicate the soldiers health.

1x; 42%; 22%
2x; 36%; 0%
3x; 35%; -24%

The first percentage goes down, but never will reach 0.
The second percentage shows that it starts with the tanks winning. Then a balance is found. And eventually the soldiers win.
The threshold is at 2x the army compositions. Or a choice between 14 soldiers or 2 tanks.

This is very suitable to use for games that allow up to 21 soldiers or 3 tanks. Because players will have windows of opportunities to attack or not.
Also, giving the same resources will result in interesting battle's.
If fog of war is present. A player needs to scout the enemy. If the enemy is training soldiers, the other player can decide to build a tank and attack asap.
If the enemy is building tanks instead, but shows no signs of attacking. Then the player can decide to swarm soldiers instead.

A threshold like this one can be considered a RPS effect in time and decision making.

Of course, if you want the threshold to be at a higher army composition. You can fine tune the statistics.
This is what happens when I adjust the health of a tank instead. Whle keeping the costs at 1 tank for 6 soldiers.
Threshold; Health each tank
1x; 21
2x; 25-26
3x; 28
4x; 30-31
5x; 30-31
6x; 32-33
7x; 32-33
8x; 32-33
9x; 33-34

One last thing to consider is. The reaction time of players to an event. Where they have to make a choice. While a threshold is at the exact middle for a balanced game. When time is an issue. And armies also need to cover distance. It is often recommended that the threshold is placed at 2/3th. So that 1/3th can be used for the time of movement.

Of course, having reinforcements right away is a must. This is why proxies are being build. Suffice to say, if I want to be really nitty picky, a well balanced game will have the threshold at roughly 7/12th.
So if you want the army composition to balance out at 7, you need to have a maximum of 12. And in this example, the tanks health has to be between 32 and 33.

I will try to answer questions as good as possible on this.
Cheers, X3M

When I say; "the threshold of defeating 1 battlecruiser is around 10 marines."
And: "the threshold of defeating 2 battlecruisers is around 18 marines."
Would that be understandable?

Some people don't understand the change in the numbers though. So I try to explain it. How it happens. And then how it can be used to balance your own game in a sense.

Once understood. It can be used to the players advantage over other players.

PS. I stopped making games. I am only analysing the ones that I play.
Because this whole thing started when people didn't understand that.

There exists an (overkilling) advanced version to that one.
"The units health is not proportional to the current combat events."

The best example is the infantry in C&C td that crawls when they are under fire.
Their remaining health "doubles" from the moment they took a shell. Every other projectile afterwards will only do half the damage if you will.

I studied that one. And I would love to chat with the guy who thought of that one. Because it is brilliant!!
More than 20 years ago, this was thought of. Before RTS where even a thing.

The damage done by an army of 6 soldiers could be considered like the following:
6+5+4+3+2+1=21
Now, if they double in health once they are under fire:
6+2x(5+4+3+2+1)=36
That is exactly the same as one tank of equal costs.
And it doesn't stop there.

So, to balance, we have several things to choose from.
But most RTS games go with the threshold balance.
Yet almost no one understands that one :(