December 01, 2013

One part history and one part math

Upplagd av Minondas

This post is something I’ve been both wanting to get of my chest for quite some time. At the same time, I’ve been hesitating to write it, because its content will quite probably be received as “heretical” by some Lardies. Also, I’m afraid that the end result of my efforts will be a mix of a rant and seemingly pointless ramblings. Rest easy though, I will try to get to the point I’m trying to make… eventually.

The history part
By history, I mean my personal wargaming history. If you take a closer look at this blog, it will rather quickly become clear to you that my wargaming revolves to a large degree around rulesets from Too Fat Lardies. ‘I Ain’t Been Shot Mom’, ‘Dux Britanniarum’ and ‘They Couldn’t Hit an Elephant’ are the rulesets I use most frequently and enjoy tremendously. The reason for me liking products from Too Fat Lardies is simple – basics of these rulesets are intuitive and easy to learn while the games they provide are usually quite eventful and unpredictable.

At the same time, I must say that I can’t help but feel that the controversy surrounding the game mechanics routinely used by Two Fat Lardies that ensures the unpredictably I’ve just expressed me liking so much isn’t completely unfounded. Those who are ‘in the know zone’ know of course that I am talking about the ‘staple’ components of most TFL-s rulesets – the card driven game engine and the dreaded ‘Tea Break’ card.

For those unfamiliar with TFL-s rulesets a quick explanation may be in order. Most of TFL-s rulesets share same central game design component  - sequence of the game turn is controlled by a deck of cards. This deck is usually prepared before the game and its content can vary between the games depending on opposing sides or special conditions. It is used to decide the sequence of a game turn – each card, when turned, usually activates a unit or a leader. A card can also trigger an event, give bonus actions, grant temporary advantages and so on. Since the deck is being shuffled at the start of each turn, the sequence of activations is always random and provides a rather ‘chaotic’ game.

The ‘Tea Break’ card (it has different names in different rulesets, but works same way in all of them) adds additional twist to the inherent chaos of card-driven game turn engine described above. When it’s drawn, it signals end of on-going game turn. Oh, your card for that unit posed to charge across the field wasn’t drawn yet? Well, sorry about that, but you’re shit out of luck, let’s hope Lady Fortuna will treat you better in next turn! OK, things aren’t that bad most of the time, units that hadn’t been activated can usually still perform limited set of actions (for example shoot at close range), but the ‘Tea Break’ card usually puts an end to any movement during the turn.

I’ve actually written about these two game mechanisms and their effects on the gameplay before, both in my reviews of TFL-s rulesets and in after action reports for games in which TFL rulesets were used. It is actually hard not to at least mention them whenever TFL-s rulesets are discussed, simply because they are such central part of the whole package. Also, it is hard not to mention them at least once or twice , because they tend to evoke emotions among players… at times those emotions are very strong indeed!

When is “fog of war” fog of war and when is it just plain chaos?
The discussion about pros and cons of turn sequence engine consisting of a card deck and randomized end of turn had been at its height in my small wargaming group few years ago, at the time when we used second edition of ‘I Ain’t Been Shot, Mom’ as our primary WWII ruleset. As my review once indicated, I was quite fond of that ruleset and was willing to accept fair amount of ‘glitches’ as long as we got to use it, but my buddies grew steadily fed up with its unpredictably. The thing is that even I had to agree with the main argument against the game mechanism – it did feel far to random and made any sort of planning nearly impossible.

Now, I am fully familiar with the argument of the Lardies. Lad, dust off your Clausewitz and von Moltke and see for yourself -  the ‘elders’ clearly state that friction of war will make even easiest tasks very hard to accomplish and that no plan survives the contact with the enemy. QED, case closed, who are you to object this ancient wisdom?!

Well… let me start by saying that I completely agree with the above statements. At the same time, I can’t help but point out that a surprising amount of plans in “real” wars did actually work out as planned, at least to a certain degree! Or, if seen from a different perspective, the number of occasions where real world plans turned into a complete FUBAR-s, while always memorable, is surprisingly small in military history when all things are considered.

Over the time, my opinion of the card deck driven turn/random turn end-system crystalized into this - it does the the right thing, but is far too random in its original form. Once I arrived to that conclusion, I couldn’t help but ask the follow-up questions – how random is this system and can anything be done to tweak it into something more manageable?

One part math
Let’s quickly recapitulate how the game turn mechanism works in majority of the rules from TFL. Each side has a set of cards activating a unit or triggering an event. For sake of simplicity let’s call the opposing sides A and B. In addition, the card deck contains at least one ‘neutral’ card (end of turn card), but usually there are other cards that can influence either of the sides (for example ‘Cautions or political leader’ which in ‘They Couldn’t Hit An Elephant’ hamstrings next leader with either of these characteristics that is activated). Let’s call these cards N. So the deck consists of A+B+N. Let’s call that total for T.

Now that we have terms in place, let’s take a look at how the infamous ‘Tea Break’ card works. It’s a
single card in a set of T cards. What are the chances for it being in a specific position once the deck is shuffled? Simple – it’s 1/T.

Simple… and quite devastating. This simple realization shows clearly the volatility of this game mechanism – since the chance of ‘Tea break’ card landing in a specific place in the deck is exactly equal for all locations, there is no ‘average’ for where it will land. Thus, the end of turn is totally random for every single turn in a game.

All right, so why is it so important and ‘grave’ news? To figure that out, we need to move to the subset of mathematics called combinatorics. Among other things, it can answer the question that is quite interesting for us – in how many ways can our deck be ordered. The quick answer to that is yet again quite simple – it’s T!, where ‘!’ stands for factorial. Factorial is a ‘short’ for multiplication of a series of numbers, each of them one less than preceding. So a factorial of 5 (written 5!) is 5 * 4 * 3 * 2 * 1 = 120.

T! gives us a total number of so called permutations for our card deck – that’s number of unique ‘sorts’ where actual order of cards does matter. The idea is this – if we have T number of cards then there are T possible cards for first position in a permutation. Next, we only have T – 1 cards, so only T –1 possibilities remains. Next T –2, T – 3, and so on. We multiply these values with each other to see in how many possible ways they can ‘mix’.

In this particular scenario a simple permutation gives us something slightly different than what we’re really after. Permutations give a number of all possible orderings of the cards, with consideration taken to the order of a specific card. But that’s not really all that interesting to us – what is of real interest to us is in how many different ways card subsets A, B and N can be mixed.

Clarification – look at this like that: if we have x of blue cards, y of yellow cards and z of red cards, how many combinations can we order them in. Since one blue card activates same side as another blue card, we don’t care about where specific cards are placed, only in number of possible colour arrangements.

I will skip a number of ‘maths’ steps required to arrive to the formula below and simply say that as long as we’re considering the whole of a deck, then the number of such selections (as opposed to permutations) is:

T! / (A! * B! * N!)
What’ we’re doing above is really quite simple and logical. First we calculate the total number of possible ‘sorts’ (that’s T!). Next we calculate a total number of possible ‘sorts’ for A, B and N, multiply them with each other to get a total number of combinations between those sorts. Last, we divide the two totals with each other and get a total number of combinations between A, B and N, regardless of how individual cards are placed.

I know that it’s a weird and very un-mathematical explanation, but you’ll just have to take my word for it – this is the formula we’re after. Or, better yet, look into basics of combinatorics. It’s actually quite fun… and somewhat mind-boggling.

Right, so we now have this formula, what’s so interesting about it. Let’s put in some real numbers into it and see what comes out. For example, in the game that I’ve run yesterday (after action report coming soon) I had ten cards for side A, ten cards for side B and finally four cards that could influence either side. This gives us following total number of selections or in wargame terms, number of ways in which opposing sides could be activated and events triggered:

T = 10 + 10 + 4
A = 10
B = 10
N = 4
24! / (10! * 10! * 4!) = 1963217256

A bit larger number than expected, isn’t it? Now, keep this number in mind and recall that the chance for ‘Tea Break’ card being in a specific ‘slot’ is 1/T or in this specific case 1/24. This means that my card deck provides 81800719 possible combinations for every single position for ‘Tea Break’ card.

Finally… yes, we’re getting to the end now… Consider how many turns are usually played in a single game. 30, maybe 40? It doesn’t take much afterthought to realise that these facts, when put together, clearly illustrate that the card deck driven game turn sequence mechanism is completely unpredictable in the scope of a single game.

2 kommentarer:

Thomas Nissvik said...

Interesting stuff! My feeling, however, is that you are over-analysing the issue by calculating the position and spread of the Tea Break card. What you need to focus on is the same thing the commanders on the ground focus on:the plan. Each turn there is a 50% chance that your unit card comes up before the Tea Break and you can act in accordance with your plan. If you put a Big Man next to your important unit, there is a 50% chance that his card will come before the Tea Break each turn. This will give you a better than even chance of getting where you want to go with what you want to get there...at least before the enemy starts to affect your movement.

If your group feels that the regular system is too random, I strongly suggest the method of having two Tea Break cards and only ending the turn after both have been drawn. It makes things a lot more predictable. How much, I leave up to you to calculate.

Minondas said...

Yes, I know what you mean and that's probably also what Mr. Clarke means when he says that you'll have 50 percents chance to activate a unit. But the problem is that this statement is a bit of mixing apples and oranges. True, as long as the relative position of every single card versus the tea break card is regarded as a single event, then you're reducing the issue to same level of complexity as the flip of the coin.

Unfortunately it's not that simple in our model - the 50/50 chance you're describing occurs in it only if the 'Tea Break' card is located exactly in the middle of the deck and number of available 'slots' before and behind it is exactly equal. As soon as the 'Tea Break' card is in any other position, the chance of any other card being in front/behind also changes. The variable that both you and Mr.Clarke seem to forget about is the limited number of 'slots' in a card deck and how their availability is affected by position of the 'Tea Break' card. This is best illustrated by looking at the extremes of this model - with 'Tea Break' card in first 'slot' of the deck, all other cards have zero percent chance to be in front of it, while with 'Tea Break' card in last position, every other card in the deck has one hundred percent chance to be in front of it.

I've been using two Tea Break cards (or 'Coffee Break' cards) in my TCHAE-games for quite a while now and this little tweak does a lot to 'even out the rough edges' of the card deck. But this is only part one of a longer series of posts regarding the card deck driven turn sequence and this adjustment is part of the discussion, so I won't disclose too much at this time. :)

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