Thursday, 11 August 2011

Crusade of Light convention

On 10 September, Gaming Club Fallen Angels in Leuven together with De Witte Ridder from Leopoldsburg are organizing their first joint convention, titled Crusade of Light:


Crusade of Light


We will be there with our Holowczyn game.

The entry is free, so there's no reason not to go and support this new convention :)

Tuesday, 9 August 2011

Bavarian Generals

Some freshly painted Bavarian leadership. First up, Kronprinz Ludwig, commander of the 1st Division in 1809:

GL Kronprinz Ludwig

Next up, GM Von Zandt, commander of the cavalry brigade of 1st division:

GM Von Zandt

And finally, GM Von Rechberg, commander of the 1st Infantry Brigade of said division:

GM Von Rechberg

The figures are Front Rank (Ludwig & Von Zandt) and Foundry (Ludwig's escort from 1st Chevauxlegers and Von Rechberg). The size difference between these two makes is obvious when one compares the figure of Ludwig with his midget escort :)

Saturday, 23 July 2011

The Lanchester Model for Combat explained with an application to unit activation mechanisms

First of all, this is a long post, so I apologize for those who are used to short snippets.
If you don't want to read the whole thing, skip to the conclusions in section 4. Or if you don't even want to do that, here's the conclusion verbatim:

If you want to write a ruleset that rewards manoeuvring, be sure the rules allow to inflict as many casualties proportional to the number of units. If you want a ruleset that puts less emphasis on manoeuvre, and more on clever use of terrain, use a fixed number of units that can activate every turn.

Anyway, the long story:

When I was reading through some old issues of Wargames Illustrated, I came upon the model of battle proposed by Frederick Lanchester during WW1 ('Aircraft in Warfare: The Dawn of the 4th Arm'). The fundamental question he was faced with was how to judge the relative strengths of 2 opposing forces, and more specifically two groups of aircraft facing each other.

This led to the well-known 'law of squares': when 2 forces are meeting each other, and when trying to predict the outcome of battle, one should take the square of their strengths, subtract them from each other, and take the square root of the result. The outcome is the number of soldiers/units/... left to the numerically biggest side.
E.g. suppose Red with 10 soldiers (or units, tanks, aircraft ...) is facing Blue with 6 soldiers. Let's also assume the quality of soldiers on both sides is equal. Red, due to its numerical superiority, will be victorious. But how many soldiers will Red have left after the fight? Lanchester's law provides the answer: 10.10 - 6.6 = 100 - 36 = 64; sqrt(64) = 8; hence Red will have 8 soldiers left after all Blue's soldiers have been eliminated. This is a simple example, but Lanchester's model also allows for the difference in troop quality to be factored in.

The classic application of Lancester's model is to show how a large force can be defeated by a smaller force, by splitting the large force in 2 smaller forces and defeat them one by one. Real-world examples cited in this context usually are Trafalgar or a number of Napoleonic battles. Suppose Red has 8 soldiers/unit/ ... and Blue has 10. Through clever manoeuvring Red has managed to split Blue's force in 2 halves, one of 4 strong and another of 6. Red takes on the first half first, emerging victorious with sqrt(8.8-4.4) = sqrt(48) = 7 soldiers. Red then continuous to battle the remaining Blue force of 6 soldiers, again victorious with sqrt(7.7-6.6) = 3.6 units left.

This model is simple enough, and relatively well-known, but as I was reading through the article, I was wondering what the underlying mathematical model was; exactly what assumptions were being made; and how it could be applied to some issues of games design. Often, the assumptions stated are along the lines of: 'What's important is that all units involved have an equal opportunity to kill or to be killed'. This seemed a bit vague to me, so I thought I could quickly reconstruct the mathematics myself by applying some discrete probablistic Monte Carlo model, but never arrived at the square law. So, I was even more intrigued about the underlying mathematics.

Theory

A quick search on the net provided me with the following article (2006), which explains the mathematical model underlying Lanchester's model (download pdf from http://arxiv.org/abs/math.HO/0606300 - Warning: this article uses differential equations, so if you're not familiar with those, stay away :-)). The fundamental assumption is as follows: During a unit of time, the number of casualties lost on side A, is proportional to the number of troops on side B. Or, in other words, if your opponent would have twice the number of troops as he has now, he would inflict twice as many casualties on you. The number of casualties inflicted scales linearly with your troop number. Is this is a valid model? The idea is that ALL of your troops add to the casualties inflicted: every tank/soldier/plane shoots at the enemy all the time during the battle. No troops are held in reserve, no troops are kept away from the fight.

Now, over time, both sides will lose troops, linearly proportional to the number of troops on the other side. By integrating the underlying equations, the square law as stated above is achieved. More specifically, the square law states: R(t).R(t) - B(t).B(t) = constant, with R(t) and B(t) being the number of troops at any point in time t. Since the constant value will never change, the sign of the difference never changes, and the strongest side always wins. Since the equation holds at time 0 (when the battle starts), we can say that for any point in time:

R(t).R(t) - B(t).B(t) = R(0).R(0) - B(0).B(0)

When the battle ends at time tfinal (and without loss of generality, we assume R is the biggest force, and hence Red wins), we have:

R(tfinal).R(tfinal) = R(0)*R(0) - B(0).B(0)
or
R(tfinal) = sqrt(R(0).R(0) - B(0).B(0))

The interesting thing is, that is you changes the assumptions, a different model comes out. Suppose that the number of casualties inflicted over a unit of time is not proportional to your strength, but is a fixed number. E.g. you have 100 soldiers, but only 5 are ever engaged, and when these are eliminated, others step in. Your actual number of 100 soldiers is not relevant for the casualties inflicted, but only the 5 you throw in the fight during a given time unit.
Using similar principles, the following equation is arrived at:
R(t) - B(t) = constant
or when the battle ends:
R(tfinal) = R(t) - B(t).
In other words, Red is still victorious, but only with the difference of actual numbers in troops left. Hence the linear comparison of strengths, not the quadratic comparison.

So, in order to keep as many troops alive at the end of the battle, you should throw in ALL of your force (square law), and not enter the battle piecewise (linear law).

A quick example:
Suppose Red attacks a force of Blue of 6 with 10 soldiers.
  • According to the quadratic law (ALL units are engaged in battle all the time and inflict casualties), Red wins with sqrt(100-36) = 8 soldiers left. Tactically, this means all 10 soldiers of Red are thrown at all 6 soldiers of Blue in one big attack.
  • According to the linear law (fixed number of casualties per time unit, e.g. units pick each other off 1 by 1), Red still wins, but only with 4 soldiers left. Tactically, this means that e.g. both sides send in 3 soldiers first (they eliminate each other). Then, both sides send in the next 3 (kill each other as well). Red remains with 4 soldiers. Note that the fixed number (3 in this example), is rather irrelevant (the example works with any number small enough).

Tactical Consequences

As stated above, the main advantage of the square law is that you can defeat a bigger force by trying to split the force in half. Would this also work in case of the linear law? Working out a few examples show that it doesn't. There is no advantage in cleverly outmanoeuvring your opponent - at the end of the battle the linear difference is always the result. But it does pay off to do that when the square law is in effect.

Now, let's go to the wargaming table. Our unit of time is the turn. So, during a single turn, can you inflict a number of casualties equal to your strength, or only a fixed number of casualties irrespective of your strength? And will it affect your tactics?
  • Suppose the number is fixed (linear law): clever manouvring doesn't help (see above). The only thing you can do to gain an advantage on top of the linear difference is to try to prevent your opponent from killing his quotum in any given turn. How can you do that? By making it more difficult for your opponent to hit you (e.g make use of terrain and go into cover).
  • Suppose the number is proportional to your strength (quadratic law). Now, if you are the weaker force, you have every interest to try to outmanoeuvre your opponent and hitting with everything you can on small chunks of his force. So taking the initiative is much more important than when the linear law would be in effect.
Whether we are working in one model or the other is governed by the rules system. So, the rules used might affect your tactics on the wargaming table.

Application to unit activation

Let's consider 3 broad categories of unit activation in wargaming rules:
  1. All units are activated during your turn all the time (classig IGO UGO).
  2. Only a FIXED number of units are activated during your turn. Examples are card activation such as Memoir44/Battlecry: typically on average, you can expect to activate 2 units, irrespective of the total number of units in your force. Or e.g. in rules like Black Powder or Blitzkrieg Commander (roll against command value to activate a unit), you can activate an (expected) fixed number of units per turn per commander. It doesn't matter whether the actual number activated is by itself a stochastic variable - what does it matter it that it does not depend on the total number of units in your army.
  3. A number of units PROPORTIONAL to your total force is activated. E.g. in a card-driven system, you can vary the # cards played, or change the content the single card you can play during your turn. When using rules that use general activation as the central mechanism, command quality of generals or the number of generals should scale with the number of units. However, it should remain proportional over the course of the battle. E.g. if you lose units, the number of units you can activate in one turn should scale down proportionally (less cards can be palyed, generals should also be lost, etc...). Otherwise, if this last thing doesn't happen, you're really in case b.
In case 1, the quadratic law applies: all units can attack in every turn, and you can expect to inflict casualties equal to the number of troops. So, manoeuvring and splitting the opponent's force is a valid tactic.

Case 2 results in the linear law. It doesn't matter that much how many units you have, the stronger force usually will win (everything else being equal, of course). In a game such as Memoir44 this is very obvious: since you only can activate a fixed number of units each turn, your excess units are actually a reserve - attacking in force and using your numerical advantage is often not possible in the sense of the quadratic law.

Case 3 is again the quadratic law, providing the amount of units that can attack each turn scales down linearly as you lose units. Keeping the same #cards (or quality), or keeping the same #generals (or quality), actually moves you to case b.

The analysis above makes of course abstraction from other factors: initial setup, the amount of units that can be thrown at the enemy given their dispositions at the start of the turn, etc. But it's always possible to make the analysis on subparts of the battlefields.

Conclusion

If you want to write a ruleset that rewards manoeuvring, be sure the rules allow to inflict as many casualties proportional to the number of units. If you want a ruleset that puts less emphasis on manoeuvre, and more on clever use of terrain, use a fixed number of units to activate every turn.

I know this is a long piece of text, but I hope that some game designers did find this useful. I was mostly intuitively familiar with most of the concepts outlined here, but having written them up concisely actually clarified my thinking a bit.

Friday, 8 July 2011

Ingermanlandski Regiment

It's been a while but here's some stuff I painted again. This is the Russian Ingermanlandski Regiment of the Great Northern War.

Ingermanlandski Regiment

The Ingermanlandski was a regiment raised in Ingria, the bit of Russia that Peter nicked from the Swedes and build St. Petersburg in. They were a high profile regiment, being paid to the same level as the Guard Regiments.

The figures are Musketeer Miniatures, the flag is scanned from a uniform book I own (for the nitpickers, yes, it's the 1712 pattern flag, so after the 'highlight' of the Great Northern War) and the flag tassel is from Front Rank Figurines.

Even though my focus is now shifting to Napoleonics, I do plan to occasionally paint some things for the Great Northern War, and this was one of them. One cannot have a Russian GNW army without the Ingermanlandski, and that oversight has now been corrected.

Monday, 13 June 2011

The battle of Holowczyn: scenario

Update February 2, 2016: Apparantly the link was broken to the document describing the scenario. This link is now restored.

We've fought this one twice now, and there's at least one more time to follow, so I think the scenario is as finished as it will get. Here's the battle of Holowczyn.

Wednesday, 25 May 2011

Big Joe and friends

And away West we go:

Big Joe and friends

These will feature in the next 'big scenario game' at my place. Incidentally, I've come up with a name for these, with thanks to Eddy to suggest part of it. As of now, they will be called Command Post games. Playing with toy soldiers tends to involve commanding them, and I live above a post office -- hence the name :).

Sunday, 15 May 2011

Shootist in 54mm

I recently tried my painting skills at some old 54mm Britains Deetails Old West cowboys. These figures have been in my collection for almost 40 years - I got them when I was a kid aged 5 or 6 as a ‘Sinterklaas’ gift (Belgians and Dutch wargamers will know what this means, and for all the others: it’s the original Santa Claus in the low countries). The original Britains figures are very crudely painted; they are toy soldiers after all. I completely repainted the figures, maintaining the original color schemes, but applied some layering to give the figure more depth. Since I also wanted to maintain the toy soldier look, I didn’t go overboard with painting too much detail. The sculpting of the figures don't really allow it either.

I couldn’t resist setting up some scenes using my old wooden fort (same Sinterklaas gift), adding some scenery items from my wargaming collection.

scene1.jpg
Shootout at the Oklahoma - Minneapolis crossroads. The roadsign is also over 40 years old and is part of my wooden fort set.
scene2.jpg
Two gunmen defending their hideout. The 'Dead Man's Gulch' sign is also from the Britains Deetail range.
scene3.jpg
Shootout in front of Fort Delaware.

Just for comparison, below you see a picture (ripped from the internet somewhere) showing original Britains Deetail figures - but in different color schemes. Apparently, there was a lot of variation to be found in the painting style when they were sold. Britains Deetail figures were apparantly sold mostly during the seventies.

A complete overview of the Britains Deetail range (Wild West figures) can be found here: http://www.angelfire.com/biz/toysoldierhq/Britwestd.html

I still have some Indians and US Infantry figures lying around as well, but I think I'll return to painting 28mm figures for now.

Update: In the mean time I discovered that the oval-shaped bases date from a later period, and that the square-based bases are the original early-seventies ones. So somehow, some 'newer' figures have entered the box in which everything was kept in the attic during all those years. Perhaps my younger brother acquired some figures during later years, and they were all tossed together. I also don't exclude the possibility I bought some additional figures myself when I already started wargaming, but I have no recollection of that. Fact remains, I did receive a wooden fort with some cowboys and indian figures when I was a 6-year old kid :-)

Friday, 6 May 2011

1st Chevauxlegers

Finished and based up, two squadrons of 1st Chevauxlegers, _König_ of the Bavarian army of 1809:

1st Chevauxlegers

They have another two squadrons coming (they had four in the field in 1809) but I'll paint those as a separate unit for Black Powder.

Tuesday, 3 May 2011

The Battle of Saint-Clair radar station

On Easter Monday, Alan and Koen came over to Zemst to play my April 'big game' (I try to organise one big scenario game each month - so far March's game was Holowczyn and now April's was this) which is the Battle of Saint Clair radar station, a semi-historical scenario covering a British para drop during D-Day, June 6 1944. The British paras must, as part of the larger airborne operations on the day, capture and disable the German long range Wassermann radar at Saint Clair and then retreat westwards, preferably blowing up the bridge across the Dives river behind them. For those knowledgable about D-Day, the semi-historical part of the scenario is to be found in the fact that I moved the radar station over to the other side of the Orne river and adjusted the opposing forces to match my collection of figures.

Alan and Koen took on the role of British para commanders while I played a dual role as umpire and German commander. Alan took 1st Battalion, the Parachute Regiment (1PARA) under command, while Koen played 2PARA and 1st Parachute Brigade Regimental HQ and AT attachments (two troops of 6 Pdr guns). The following photo shows the British commanders working out their plan before the game (as unlikely as it may seem, this photo was not staged :) ):

IMG 2213
Koen and Alan making plans ...


1PARA, landing between Saint-Clair and the radar station north of it, decided to profit from the surprise of the drop to rush the radar station. An action in true paratrooper fashion, to be sure, but perhaps not the wisest one, given that the station was defended by, among others, a deadly quad 20mm AA gun. But perhaps the daring paras might be able to storm the station while the gun crew was still waking up?

IMG 2217

While 1PARA was dashing to glory, 2PARA rushed the town of Saint-Clair. The paras quickly defeated the company sized German garrison of the town, taking only light casualties in doing so. The AT guns which had landed in the same area as 2PARA were hitched to their jeeps and started off across the fields in the direction of the main road and bridge west of town. So far, D-Day was going well for the Red Devils.

IMG 2220

The German defenders meanwhile, except for the unfortunates in Saint-Clair itself, were fairly slow in manning their guns (they had after all only just settled in, having spent most of the night on alert) so 1PARA succeeded in dashing a company straight through the gates and up to the entrance to the bunker complex under the radar station before the first defenders were seen in the gun pits of the complex.

Late to the party the Germans might have been, but they were just in time. Before long, the battalion of paras attacking the now fully alerted defenses of the radar station was smashed to pieces. Between the fire of a 75mm AT gun, a company in the trenches outside the bunker complex and especially the deadly quad 20mm AA gun, 1PARA was quickly reduced to a single company in cover behind some hedges, with just a few scattered survivors of the rest remaining. 1PARA had effectively seized to exist as a fighting unit. The high tide of the advance was the charge of an unnamed sergeant, the last remaining survivor of the company that had made the intrepid dash to the bunker complex, who managed to reach the main radar control bunker. A VC was definitely in the making there, but fate intervened when the Germans inside the bunker came out on top in the ensueing struggle.

Clearly, 2PARA would have to fulfill the mission now. And that it did. Undaunted by the fate of its sister battalion, 2PARA mounted a company in some German trucks captured in Saint-Clair and rushed them down to the radar station. There, supported by fire from their battalion mortars and a troop of pack howitzers from the divisional artillery, they were able to suppress the crew of the AA gun with accurate small arms fire. Profiting from the lack of heavy return fire resulting from this, they then made short work of the German company in the trenches. This disheartened the defenders of the radar station to such an extent that they decided discretion was the better part of valour and surrendered themselves and the radar station. The primary task of the mission was completed!

There now remained the secondary task. The paratrooper force needed to relocate to Bricqueville, west of the Dives river and in closer contact with the rest of their division. The bridge over the Dives they would cross on the way there also needed to be blown up to deny its use to the Germans. At this point, they had 2PARA virtually completely unscathed, with two companies passing through Saint-Clair on foot and a third mounted on German trucks in the radar station. 1PARA, with about one and a half to two companies (out of five) of effectives remaining was regrouping between the radar station and Saint-Clair and the AT guns were deploying on a small rise overlooking the road to Bricqueville. German resistance in the area had stopped after the surrender of the station garrison.

Just then however, a column of German trucks was spotted arriving from the direction of Bricqueville which wasted no time in dismounting a company of what seemed to be Fallschirmjäger in some woods by the road. It seemed that the small staff car the paras had spotted racing west earlier in the day had alerted German reinforcements. Clearly, the fighting was not over yet.

While the AT guns, which were now the closest to the enemy the British had, started exchanging fire with the newly arrived German troops, the paratroopers on foot started redeploying to counterattack the German reinforcements. The Germans, who over time were reinforced until a small battalion of Fallschirmjäger blocked their British counterparts' progress towards Bricqueville, scored a major victory when they managed to eliminate the crew of the British AT guns. This would prove to have serious consequences soon.

Some time later the British were getting the upper hand in the fighting against the German reinforcements when an ominous clanking and creaking was heard from the west. Two platoons of PzIV tanks from 21st Panzer division had arrived, and the British were out of AT guns. Things did not look well.

IMG 2223

The Panzers went to work immediately, one platoon advancing along the road and the other swinging out on the flank and wrecking some French farmer's fields to outflank the British paras along the road and in Saint-Clair.  With the AT guns out of action, it fell to the PIATs in the parachute battalions to deal with the tanks. Another opportunity for a VC presented itself. However, when the first PIAT team dashed into range of the Panzer IV on the road, missed it and was mercilessly machine gunned afterwards, the British quickly decided to change tack.

Given that they were mostly powerless to do anything about the roving panzers, the British quickly redeployed and set about getting as many of their men as possible across the river towards Bricqueville. Again showing great resourcefulness the paras set up a relay of jeeps, Bren carriers and captured German trucks to organise a veritable taxi service through the fields westwards. With the exception of a company in Saint-Clair besieged by panzers and a second company dispersed after running into one of the panzers in the fields, most of the remaining British managed to make it to safety. When the final paratrooper set the fuse on the demolition charges on the bridge, the secondary task of the mission was also accomplished. The final photo shows the final few paratroopers marching across the bridge escorting a number of German prisoners:

IMG 2225

It was a major British victory, though at great cost. Out of an original 10 companies of paratroopers that dropped, only an estimated three companies' worth plus brigade assets made it across the Dives river to Bricqueville.