Probability

But - what does that really mean?

Let's start first investigating what 'chance' is supposed to express and discuss 'winning' later.

What is probability?

However, there is another concept known as propensity which is expressed in a probabilistic framework - that holds that a given situation is not fully knowlable, but if repeated often enough will lead to a given distribution of outcomes. A prime example is the collapse of the wave function in quantum mechanics - the theory does not tell into which result a measurement will resolve the state, indeed it claims that such knowledge is impossible - yet it will give probabilities for finding outcomes.

Why is this important for a battle simulation?

Consider for instance the battle Alexander the Great fought at Granicus: Its main thrust was a cavalry attack across the river, then up onto higher terrain into the Persian cavalry. General understanding is that this isn't sound tactics as it puts Alexanders troops at a disadvantage in difficult terrain - it should not have worked to win a battle against a numerically superior adversary who had the river as defensive line.

Analyzed in a lack of knowledge framework, the conclusion is however that we might have a naive understanding of the situation - perhaps Alexander realized something that we today can not, a weekness in the Persians perhaps, and so the victory was inevitable. In this case, a simulation of the battle should result in a Macedonian victory every time if set up correctly.

This is different when understood in a propensity interpretation - here it is possible that the actual odds were against Alexander (say 90% for the Persians) - yet he got lucky. So even if set up correctly, the simulation would produce a Persian victory most of the time.

Generally I adhere (for reasons too complex to explain here) to a propensity interpretation as far as things like battle results are concerned, and so the actual simulation will do the latter - if repeated often, it will offer a range of possible outcomes with numbers according to how likely they are.

How often do we roll the dice?

Consider a soccer simulation - do we assign 70% for the better team to win and roll for the whole game? We'd lose the information how many goals were scored though. Or do we assign the 70% for every action - first when a player acquires the ball, then three times when he moves through the defeners, again when he passes to another player and finally when that player tries to score a goal (which takes perhaps a minute or so).

The probability of course needs to be multiplied, so the better team has now a 11% chance to score a goal - but as we probe this every few minutes, there might be a number of goals in a match. Of course the other team now has only 0.073% chance to score... which is 150 times less chance rather than the 2.3 times it was when we only threw a dice once per match.

So what is happening here? The law of large numbers asserts itself - the averaged result of random events is very predictable if there are sufficiently many events. So by introducing randomness at every corner in the simulation, we'd actually make sure that it by and large departs and we end up with the (even slightly) better unit winning practically always.

Which in essence means we're trying to find the right amount of randomness.

But what is that? It seems that often battles are shaped by key events - a unit that manages to break through the enemy lines early and then starts to attack them from the rear may in doing so already decide the encounter, regardless of what the performance of all others is. So randomness needs to be at least on the level to allow such events to happen (or not) - but this also means that the law of large numbers doesn't necessarily take over quickly, because what happens at key events matters more than what happens on the rest of the battlefield. So the right amount of randomness is the one which allows the key events to happen at the chances intended by the user.

In practice the simulation will roll dice for combat once per minute - which seems a reasonable compromise between trying to fine-grain the combat timescale to a degree that it does not freeze units 'forever' in place while a fine-time-scale movement of other units happens around, yet trying to keep enough randomness in an encounter of two units to allow key events (which a combat simulation at the finer movement timescale of a few seconds would not do). However, the actual numbers are under user control and can be changed if the perceived randomness is too low (or too high).

Coming back to Alexander at Granicus - the framework needs to be able to account for the key event that the uphill cavalry charge is a quick success (even while it has a low success probability) that turns the tide of the whole encounter.

Continue with Winning a tactical advantage.

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