December 26, 2013

This little thing called probability

Upplagd av Minondas

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.

Probs

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.

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