The fine line between a game and a simulation

The question though, where does the game end, and where does the simulation start?

Would it be possible to set some levels to it?
In a sense, for a wargame, the SD compared to the average result is a good way for me to calculate how strong the randomness even is.

For MtG....well, in a sense, this is more or less impossible. But yeah, simulation, totally.
Just like chess. The randomness is mostly, what is your opponent going to do.
The difference with MtG and chess is. Will the opponent have 20 pawns, or 4 queens? You don't know that.

Still, the gaming part is the unknown. Chess is a puzzle battle as how i see it. And the best brain wins.
Stratego has a FOW, and thus hidden information. Which is also a puzzle battle, with a gamble portion. Thus more of a game now. It doesn't look like a simulation to me at all.

In a sense, Stratego has several different pieces. And there is a chance on meeting certain pieces when moving. We can assign a chance and thus also an "average" and "standard deviation" to a situation.
Thus a game.

MtG has these battle's like a 1/1 vs 1/1 creature.
The result if one attacks and one blocks is always the same. Both perish.

Ogame has battle's with a random factor in it. If a certain ship hits another certain ship, it may attack again.
With only a few ships. If a cruiser attacks, it has a rapid fire against light fighters. The RPS here is that heavy fighters are thus slighlty better than cruisers in the long run. The light fighters are really just fodder and can nullify the biggest attacks. But at the first 3 ships, the cruiser attacks. Let's say a defender has 3 light fighters and 3 heavy fighters. The cruiser has a 50% chance it can attack again. On average, 2 attacks can happen by the cruiser.
The game rolls this, so there are situations that the cruiser hits only once. But 3, 4 or the maximum of 5 are also possible. So the true average is 1.94. And the funny part is, if sufficient cruisers attack. The light fighters go down faster than the heavy fighters in the 3 sub battle rounds. (i think there were 3)
Thus the average attack of the cruisers goes down with more cruisers.
But the game doesn't stop at less than 10 ships in a battle. There are battle's where a player has over 1000 ships. Some even have this in the millions, depending on how long the game runs.
So, the randomness.....simply dissappears. And it is more of a simulation than a game now.

***

Here is some math I wanted to redo. And others can take note of the method and use it to their own. Although I did not had the confirmation yet that this is correct.

Now that I have more "info" on how to calculate the SD for more rolls, based on 1 roll. I could take a look again at Warcraft2.

This game "rolls". Between 50% and 100% of an attack. And it only yields whole numbers.
If an attack is 6 basic, 3 piercing. And the target has 2 armor. The damage is 7 in total. Rounded to the next even number, 8. The roll is between 4 and 8. But the die here, due to rounding: has 1x 4, 2x 5, 2x 6, 2x7, 1x8 as faces. The target has 60 health.

First you determine the average and SD of 1 roll.
Average = 6
SD = 1.225

For the formula later on, we use:
u1 = 6
o1 = 1.225

The most difficult step is to determine how many rolls on average you need. And you cannot round this yet. The number of rolls is most certainly higher than just dividing the health by the average. This is specially true for having low yield dice, including zero's. And low health value's. I still remember that my 3 health soldier actually had 3.662 hit points

Either way, there are 2 methods for determining this.

1. Excel worksheet.
This requires a lot of knowledge and pretty big table's depending on how much health you target. I once went to 20 health for a d6. But now we need 60 health and a d8. And much more copies of a table per round. Since the average seems to be 10 rolls. Here I need to go for option 2.

2. AnyDice. With tracking in Excel.
What you do is you roll x number of dice. You start at the lowest expected possibility of your target being destroyed.
In this case, it is 60/8=7.5 or 8 dice.
You roll them in anydice, take note of [at least] reaching 60 and the chance for it. Lucky you, you can also divide the roll by 60 and get the same result here.
Then you work your way up until the chance is 100% on at least 60 damage. In this case, we can expect to stop at 60/4=15 dice.

The table will be big depending on the number of rolls. Especially if you have 0's in the attack damage. But the bigger the table, the bigger the SD. This is the first sign that the battle is a game and not just some computing for the outcome. 8 to 15 rolls is just a sad little table. Still a lot of work for getting an individual result for every roll. And we aren't there yet.

In order to get the average rolls. We need to first see how much the chance increases per extra die. We take the chance and subtract the previous one. For the roll of 8 dice here, we subtract 0. For the 9 dice, we subtract the [at least] chance of 8 dice. Etc.

We get a list of chances that we need x dice for defeating the enemy. If done correctly, we see that 10 and 11 are both pretty big chances. Multiply the number of dice needed with the chance for those dice. Then sum up all the results. I got to an average number of rolls of 10.438. (This is 11 attacks on average, not 10)

For the formula we use:
a = 10.438

In order to calculate the average TOTAL damage for all possible number of dice. We use the next formula:

ux = u1 x a
ux = 6 x 10.438 = 62.628

62.628 is a bit more than 60. I don't know why...but it seemed important. It means that if you roll the average number of dice. You actually deal more damage. Not that this is true, but you need this result for later.

Now to calculate the TOTAL standard deviation that belongs to this average damage.

ox = o1 x sqrt(a)
ox = 1.225 x sqrt(10.438) = 3.958

And that.... is ridiculous low.

What we want is to know what 2/3rd of the time is happening here. What is the total damage happening from low to high...that is needed to destroy the target. Note, this is lower and higher than the health of the target. But....it is still destroyed. It is math...vague math. But it seems to work.

The damage needed goes from:
(62.628-3.958) to (62.628+3.958)
or
58.670 to 66.586

Now, we divide this by the average damage per die. In order to see how many attacks we need in 2/3rd of the time. And round this upwards.

I get 10 to 12 rolls, or attacks. Although that 12 is a big rounding. It still happens in the 2/3rd of the time.

What I didn't do in the BGG probabilities questions topic is. Dividing the SD by the average damage as well.
Here we have only 0.66 rd as a roll. This on top of 10.44 rolls. Which leads us to that same 10 to 12 rolls. But also shows us that the 12 is a very small chance. Either way, dividing the total SD by the total average rolls gives me a percentage of:
0.66/10.44 = 6%
I would like to call this the Individual Gamble Chance Level or IGCL.

This 6% is too low for my taste... In a sense, there is only a 2x 6%^2 = 0.7% chance that the outcome of this 1 on 1 battle is in favor of 1 of the 2 fighters in either direction. This is a simulation from my point of view. And the whole randomness is rather useless.
I think I like to call this number the Duel Gamble Chance Level or DGCL.

In that BGG topic. The 3 targets have:

Attacking a Goblin with a d6-3 weapon.
3 health,
3.556 average health,
1 average damage,
1.155 SD,
3.556 total average damage
2.178 total SD
61% IGCL
75% DGCL (interesting, this is higher than the IGCL)

Attacking an Orc with a d6-3 weapon.
5 health,
5.654 average health,
1 average damage,
1.155 SD,
5.654 total average damage
2.746 total SD
49% IGCL
47% DGCL

Attacking a Warlord with a d6-1 weapon.
20 health,
8.534 average health,
2.5 average damage,
1.708 SD,
21.335 total average damage
4.989 total SD
23% IGCL
11% DGCL

While the targets in the BGG are not in a real duel. And the player has much more life. The IGCL are in place here. As for that last one, the warlord. I need the DGCL for my ratio 8 wargame. This means that it is only 11% of a gamble when having a 1 on 1 fight.

One thing I forgot to mention is "the last attack".
If a game is more of a simulation than a gamble.
You run the simulation. But you look at what decision you make at the end.
Now, if the 2 fighters are at their last health. The outcome is more of a gamble if you decide to continue the fight.

If you retreat, you can heal up/repair. No matter what game, if it is a dungeon crawler, wargame or any other combat related game.

Of course, if the targets have less health than the minimum damage, it is a simulation where the outcome is a certain defeat. If the targets have the health equal to the average damage roll, it is as close as possible to a gamble.

In case of Warcraft 2. If the fighters have 6 health remaining. There are only (mostly) 1 to (rarely) 2 rolls left.

6 health,
1.375 average health, (i just realized that this is named wrong, "average hits needed")
6 average damage,
1.225 SD,
8.250 total average damage
1.436 total SD
17% IGCL
6% DGCL

6% DGCL for the last exchange... not sure how I feel about this.
Edit: The AI reacts a bit slow in some games. So, that 17% is still valid. Especially for when the game is turn based like a Catapult against a melee target. If it fails in killing in 1 shot, it is "over".
So, let's say, it is 17% on the moment the player makes a decision for a final exchange. And 6% for when the battle is already taking place.

Same with RNG....
But I described IGCL and DGCL in this topic :D

Individual Gamble Chance Level:
A score in percentage of how big the Stardard Deviation (SD) is compared to the average outcome. When all actions of randomness are summed up for a certain set or situation. Of 1 player.

Duel Gamble Chance Level:
Same score, but then there are 2 players involved. Calculating this is different for the game mechanics. The score can be calculated from the game, or from the IGCL.

I did explain... Somewhere in this topic at their first mention...

I created a score for myself to see how much randomness there is in relative terms. For specific situations.

If your average roll is a 1. This is also your goal. And your standard deviation (SD) of that roll is 1. Your IGCL is 100%.
You either roll 0, 1 or 2 at 2/3rd of the times. A gamble in my eyes.

If your goal is 4 hits. Yet your total SD is 2 at this point. Your IGCL is now 50%. This is a game in my eyes.

If your goal is 400 hits. Your total SD will probably be 20 here. Your IGCL will become 5%. Which is very low. And thus more of a simulation to me.

The relative part would be based on the fact that soldiers and tanks are actually the same type of units in my game. A soldier works with lower numbers, a tank with higher numbers. 20 hits with a SD of 1 compared to 400 hits with a SD of 20 is the same. Both yield a IGCL of 5%. Simulation.
4 hits with a SD of 2 and 400 hits with a SD of 200 are also the same. Both yield a IGCL of 50%. Game.
1 hit with a SD of 1 and 400 hits with a SD of 400 are the same once more. Both yield a IGCL of 100%. Gamble.