Just a quick heads up regarding ongoing sale of e-books at Pen&Sword – they run a ‘buy one, get one for free’-campaign. Considering the fact that their e-books cost £5 per volume, personally I think it’s a pretty sweet deal. Wide range of topics, from classical warfare to current conflicts; take a look, maybe you’ll find something interesting.
December 30, 2013
December 26, 2013
This little thing called probability
This is the second part of posts dedicated to somewhat different analysis of card deck driven game engine used in many games from Too Fat Lardies. On this occasion I will look into another ‘heresy’ in world of Lardies – game balance and fairness or, as it turns out to be, probabilities.
Before I start however, I feel that a couple of words need to be said about the comment that Thomas was kind enough to write in response to my previous post. In it he makes a very interesting observation, which finally explained to me why Mr. Clarke often says that a player has about 50 percent chance to activate half his units when the card driven turn sequence game mechanism is in use. Every time I heard or read that statement, I always asked myself: “How does he arrive to that conclusion?”. Thomas has finally clarified the issue for me – every card can come either before or after “Tea break” card and with only two possible outcomes for each card, chances for each card to be in front or behind “Tea break” card are fifty percent.
There is but one problem with this assumption and unfortunately it is a rather serious one. It is correct only the under condition that the ‘'’Tea Break’ card is located in the middle of the card deck. As already shown in previous post, that is highly improbable, as chances for ‘Tea Break’ card to land in a specific position is always 1/n where n is total number of cards in the deck.
Thomas’ suggestion to use two ‘Tea Break’ cards is also worth closer examination. What difference does the second ‘Tea Break’ card really make? Well, let’s start with examining the function of a single ‘Tea Break’ card – when a deck is shuffled it will land in one of n positions, where n is total number of cards. So if we have total of 10 cards, it has 10 possible ‘slots’. If we now add another ‘Tea Break’ card, we increase number of cards in the deck to 11. So the first ‘Tea Break’ card now has 11 possible ‘slots and after it’s been ‘placed’, the second ‘Tea Break’ card can be located in one of the remaining ‘slots’, which in our example have now been reduced to 10. Combination of these two gives us 11 * 10 possible ‘permutations’. Since repetitions don’t interest us, we need to divide 110 by 2 to get total number of ‘combinations’ (after all ‘Tea Break 1’ in second position and ‘Tea Break 2’ in seventh is exactly equivalent to the ‘’Tea Break 2’ in second and ‘Tea Break 1’ in seventh position).
So by adding a second ‘Tea Break’ card, we are actually increasing the ‘unpredictability’ of end of the turn by increasing the number of possible combinations of ‘Tea Break’ within the deck from 12 to 55.
Balance? We don’t need no stinkin’ balance!
In the days when I frequented TMP, two arguments used to flare up on that site’s forums with surprising regularity. The first one was ‘Is our hobby a game or a simulation?’. The second was ‘Balance – do we need it or not?’. My personal conclusion was that many Lardies are of opinion that games played with TFL’s rulesets can be regarded as a historical simulation and not expecting balance in a game is a crucial issue if one is to achieve the ‘simulation’ goal. Thus the unpredictability and at times tangible ‘unfairness’ of the card deck driven turn engine should not only be tolerated, but actually appreciated as a model reflecting reality closer than for example the venerable IGOUGO.
I tend to agree with that opinion; after all, if one plays a scenario set on Eastern Front in 1941, one expects for the generic German company to be more efficient than its Soviet counterpart (I know, I know, it was far from certain, but in general Germans did kick some ass in 1941). So how does the card deck driven turn sequence engine manage to re-create such situations? Well… quite frequently by giving the presumably superior side more cards than the other. The reasoning behind this apparent ‘imbalance’ is simple – the side with units that are judged to be more efficient, better led, with superior training/morale or simply dressed in camouflaged uniforms should be allowed to do more. The choice to give the superior side more cards seems at first glance simple, clean and logical game mechanism. And yet, it is a mechanism with, in my opinion at least, an embedded fatal flaw. You see, it is one thing to say that one side should have greater chance to activate units for this or that reason. It is a completely different thing to say that that side is to have more opportunities to actually participate in the game.
Right… by now you’re thinking: “The poor lad has lost his mind, what is he rambling about?!”. I assure you though that I am completely sane and I do have a point. Let me use a quick practical example to visualize my point: two players play above mentioned game on Eastern front using IABSM ruleset. The Soviet player has three platoons and two leaders – one card for each of them means five cards in the deck. German player has three platoons infantry, an attached MG platoon, three leaders, an additional card for bonus actions for machine guns and an artillery spotter – that’s nine cards.
The player on the Soviet side has obviously a hard task in front of him, but the disparity in forces could be regarded as ‘historically correct’. It needs however to be observed that in a deck consisting of five cards for one side and nine cards for the other, there is also a mathematical disparity which creates a ‘double penalty’ for Soviet player – not only does he have inferior forces at his disposal, he will also have much lower chance to actually do anything with them.
Back to maths
I will not bore you with mathematical formulas this time around. Instead let’s look at another practical situation. Consider a deck consisting of 14 cards of two types – blue and red. For the sake of convenience we place ‘Tea Break’ card in the middle of the deck and will always draw seven cards before the end of the turn. Since we know the position of ‘Tea Break’ cards, it can be disregarded it in the calculations. The table below shows chances for drawing certain number of blue cards, depending on total number of blue cards in the deck.
In my opinion, there are several things worth interest in this table, but one issue is especially interesting – it doesn’t take much ‘imbalance’ in the deck to create a scenario where the side with inferior number of cards in the deck is pretty much guaranteed to lose; not because of the inferior number of cards (or ‘units’), but simply based on mathematical probabilities.
In our game with nine German and five Soviet cards we find the probabilities for the Soviet player in the middle column. He has just above 50 percent chance to activate either two or three units, which means four or five German cards being activated in same turn. If he loses one unit and reduces number of his cards to four, the probability to activate three units in a turn falls down to about one in four, while chances to draw four cards become very miniscule indeed.
Fair or not fair?
Based on a 10+ years of usage of different rulesets having card driven turn sequence at their core, I’ve always been regarding them as ‘unpredictable’. After spending some time on proper mathematical analysis of that mechanism, I can’t help but regard it as definitely unbalanced. Drawing this conclusion doesn’t however have to automatically mean that it’s also unfair or unplayable. Or it doesn’t mean that until two final questions are answered.
The first question is this – has the game designer accounted for this phenomenon in his game design? If the answer is yes, then there isn’t much room for further discussion – any disadvantages that the player with lower number of cards will suffer have been accounted for (or at least should have been) and such player needs to be regarded as accepting the challenges that follow out of inherent imbalance. If the answer to the question is no, then the ruleset is in my personal opinion seriously flawed.
The second question is perhaps of much more importance; are the players aware of the imbalance embedded in the card driven game turn sequence and if so, do they accept it as part of the game? I dare to say that answer to that question isn’t as clear-cut as one would like to think. Based on observations of my rather limited wargaming community, I think that there is a peculiar unwillingness to look ‘under the hood’ of rulesets and an almost child-like belief that ‘if it’s published, it must be right’.
In the end, of course, it’s ‘to each his own’. Personally, I am freely admitting that I don’t like what I found ‘behind the curtain’. I have therefore done some significant changes to the TCHAE before our latest game. What those changes are and how they afflicted the outcome of that game will be the topic of the final part of this trilogy about card driven turn sequence game engine.
December 01, 2013
One part history and one part math
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:
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
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.
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.
Call of the Kickstarter sirens
Normally I am very sceptical to Kickstarter ventures, but I must admit that every once and again there is an announcement of a new venture that manages to bypass even my jadedness. And it must be said, West Wind’s Productions offering to produce a whole shedload of 15/18mm armies for “ancients” if financial backing is provided beforehand is very hard resis; after all, “ancients” has always been my “first love” and those test minis look soooo sweet!
http://www.kickstarter.com/projects/832150598/war-and-empire-the-miniatures-game-of-ancient-warf
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