Computational Combat Statistics

Well, if there ever was a post title that would cause readers' eyes to gloss over and tempt them to skip a post, today's post might fit the bill. Had I mentioned that this would be an exercise in mathematical simulation, I may have lost everyone.

What am I doing and why?

Motivation

In the first game of Madonna dell'Olmo (see Madonna dell'Olmo), King Emmanuele III automatically holds the initiative on Turn 1.  With Spanish grenadiers defending the redoubt positioned in front of the cathedral, the prospect of storming the position looked daunting.  Given that taking this position is one of the criteria for eventual victory, the cathedral and redoubt represent a key piece of real estate.  With Spanish support nearby, attacking on Turn 1 may offer the best chance at securing the position before the Spanish grenadiers are reinforced.  What the Piedmontese must accomplish is to either destroy the grenadiers or throw them out of Madonna dell'Olmo on Turn 1.  While success may be short-lived, the loss of the Spanish grenadiers and occupation of the cathedral will set back Spanish plans on this wing for at least two turns.  This is valuable time the King could put to good use. 

Turn 1 starting positions.

In last week's game, the King opted to use his first turn initiative to activate other formations and stand pat with Corbeau's Far Left Wing in front of Madonna dell'Olmo.  Is this the only viable strategy?  Consider a few choices of action. 

Options

1. Do nothing.
2. Use first fire volleys with Grenz and attack with the grenadiers.
3. Attack with grenadiers and grenz.  Corbeau does not lead the charge.  
4. Attack with grenadiers and grenz.  Corbeau leads the charge.   

Only one of the grenz regiments and Sardinian grenadiers can reach the redoubt on Turn 1.  Can one of these four scenario options lead to success using the rules in play?  Can players assess intuitively beforehand if any of these options provides a path to success?  Intuitively, it may seem that (4) may offer the best chance of success but will this be enough to dislodge grenadiers in the heavy cover of the redoubt?  Perhaps combining an assault with preparatory volleys is useful?  

Discrete Event Modeling

Now, if these events required only a single computation, getting an answer may be straightforward.  However, there are many events in play both sequentially and simultaneously.  To get a grip on possible solutions, conditional probabilities are involved.  To accomplish this task, discrete event simulations need to be built and executed.  Once built, these combat models will be executed 500,000 times per scenario to create a large sample size from which to draw inferences.

Let's looks at each one.

1. Do nothing.  No computations needed here.

2. Grenz volley followed by grenadier assault.  This scenario requires first a simulation of volley fire and defender morale tests followed by paired simultaneous close combat attacks and morale tests by both attacker and defender. 

The grenz' first fire bonus offsets the heavy cover penalty of the redoubt.  The results demonstrate that grenz first volley fails to score a hit and resulting enemy loss through morale failure about 42% of the time.

Assuming the grenz fail to make a mark against the defenders, then the subsequent assault yields the following result.  That is, the defender is only forced out about 30% of the time.

What if the grenz manage one hit?  Well, then the results change to,

with the chance of defender retreating jumping from 30% to 38%.  Still, prospects for success are not great.

Is this worth the risk especially given the risk to the attacker where "Attacker Retreat" and "Fight On!" results carry losses to the attacker?  A “Retreat” result signifies an actual retreat or unit destruction.  In either result, the unit vacates the hex.

3. Attack with grenadiers and grenz.  Corbeau does not lead the charge.

This scenario sees the Sardinian grenadiers assault the redoubt without any preparatory volley from the grenz but the grenz join in to support the assault.  Corbeau does not join in to lead the attack.  This scenario requires paired simultaneous close combat attacks and morale tests from both attacker and defender.   

These results show, that on average, the defenders will be ejected from the redoubt about 57% of the time.  Again, the attacker suffers losses through failed morale tests about 40% of the time.

4. Attack with grenadiers and grenz.  Corbeau leads the charge.

This scenario sees the Sardinian grenadiers assault the redoubt without any preparatory volley from the grenz.  The grenz join in to support the assault while Corbeau leads the attack.  Like scenario (3), paired simultaneous close combat attacks and morale tests from both attacker and defender are required.

With Corbeau leading a coordinated attack at the head of both grenadiers and grenz yields success in throwing the Spanish grenadiers out nearly 75% of the time.  For me, that is a risk worth taking.  Of course, the attackers will suffer casualties but the price for gaining the redoubt seems not too high. 

For this situation, a coordinated assault with an attached general provides the greatest payoff.  Of course, if Corbeau is lost or the Sardinian grenadiers are badly damaged, holding the position may be difficult unless reinforced quickly.  Still, aggressive action by the King on Turn 1 could set the direction for the entire battle.  

Without tackling the complicated and intertwined computations underlying the game engine, itself, it can be difficult to "guess" which approach is best.  Even with a correct guess, confirmation is valuable. 

What would I do?  Attack straight away with full force.  Your mileage may vary.