## 02 January 2011

### More Dice, More Information, but not as much as you think

I may have created some confusion in my previous posts on the information in dice (1,2). That's understandable, because the concept of information is complex and has multiple interpretations, most of which I would not claim to really understand either. Let's see if I can sort this out without making an even bigger mess.

Shannon Information measures the information content of a random distribution of discrete events. A coin flip (heads/tail), a To-Hit roll (hit/miss), and a Hit-Location roll (arm, leg, torso, etc.) are all examples of this. What really matter here is not the number of dice rolled or coins flipped, but the number of possible outcomes and the probability of each. So you might use 4d6 to determine the result of a to-hit roll, but there are still only two outcomes - hit or miss. In game terms, you can think this as the information you don't know yet, just before you roll the dice, or the variability of outcomes of that roll.
Shannon information is measured on a logarithmic scale (base 2), so each additional bit represents a doubling of information. In absolute terms, the difference between 10 and 11 bits of information is MUCH more than the difference between 2 and 3 bits. Be careful with this sort of comparison though, because it's a bit like comparing apples and oranges (or comparing to-hit and hit location rolls). Such comparisons may not be meaningful.

There is an extension of Shannon Information to continuous outcomes, but this also requires changing the definition somewhat. I won't go too far into this, but there is one key point I'd like to make. When dealing with the sum of multiple dice, the distribution of the sum tend to become more like a normal distribution as the number of dice increases. Calculating the entropy of the sum of 10D6 is a bit of work, but the entropy of a normal distribution is easy to calculate. Long story short, I can get a good guess at the entropy for the sum of a large number of dice by using the normal distribution as an approximation.

I had speculated about calculating the information in an entire game. This was rather silly of me, because this would mean framing the outcome of an entire game as a single probability distribution. I can't do that, but now I know how to make an educated guess. If I only considered the win/lose aspect of a game this would be easy, because that is just a complicated sort of discrete "coin-flip" outcome. The more interest way to look at this is to consider ALL the ways a game might play out, and to treat this as a sort of continuous outcome. The law of averages comes into play, some ways the game will play out are more likely than others. For instance, if at some point in the game you make multiple attacks to achieve an objective, perhaps to destroy an enemy tank, then in the final outcome of the battle it might not matter which attack was successful, so long as one of them was - they all lead to the same outcome. This might be stretching the concept too far, but I should be able to use the entropy of the normal distribution to approximate the amount of information of a very complicated random distribution - like that of an entire game.

Now I can make an educated guess about the information in a game. I'll use a Battletech example, but there is surprisingly little dependence on the game. The most common random event in Battletech is weapons fire, which includes the to-hit and hit-location rolls, which each have about about 3 bits each (as calculated here). In a two player game where each player has 4 battlemechs, and each mech makes about 5 weapons attacks per turn, there will be about 10 random events per turn for the first 5 turns or so, about 200 random events, then a decreasing number of attack for the next 5 turns, call it 150 random events. That's 350 random events in one game, but I left out anything else that might require a die roll, so I'll round it up to 400. The basic random event in Battletech is about 3 bits, and 400 repeat random events adds about log2(400) or 8.6 bits, for a total of 11.6 bits.

Sooooo ... now that I've gone through all that, it seems that the information in a game is just log2 of the number of random events, plus a few bits of overhead. Does this mean anything all? I need to think on that.

Here is a partial answer: A game doesn't need any randomness at all, it could be completely strategic, like Chess. Add just a little bit of randomness, the whole game may depend on just a few rolls of the dice. As randomness increases the law of averages will come into play - Games like Risk and Battletech have many dice rolls - so many that the average effect of many rolls is almost always more important than single roll. Too much randomness, and players lose the ability to affect the outcome. The trick is getting is the right balance of randomness, and I don't think there is a formula for that.

PS: My friend Ashely also recently noted that it might be a good idea to eliminate any rolling of the dice that doesn't significantly add to the game. Wisdom!

More:
Dice and Information
Dice and Information, So What?

DevianID said...

Howdy again. The bit about the coin toss win/loss seems familiar heh!

As for your bits of randomness in a game being about 11.6, might I posit some more food for thought? Long winded rant coming on... In an earlier comment of mine I talked about the bits of information in a movement phase. Suffice it to say, the movement phase has a huge amount of unique outcomes that can play out similar to a dice toss. While we may try and discount many of these unique board locations as being inferior to others, (IE moving 5 hexes on a walk is better than 4 hexes on a walk if all your weapons will be in the same range band since 5 hexes gives you a better target modifier) this is still potential information.

As a quick refresher, a mech with 3 JJ using the jump movement, has 6 unique facings, 3 torso twist settings, and 37 potential hexes. Thus, 666 combos, more if you can also flip your arms. That is a max of 9.38 bits of info right there. Add in walking and running options, the options when standing still, ect, and movement in one phase for 2 units with 3 jump jets each has more raw info than the entire game's firing of 5 weapons/turn x 10 turns x 2 units. And in the 5 weapon/turn example you used, you assumed that all the weapons hit and thus generated the hit location extra roll.

The reason I mention all this? You brought up the idea of trying to get the right balance of randomness. While there is no best answer, you can get a formula for amount of info the player controls (movement, picking which weapons to fire and when to fire them)versus the amount of info the dice controls (to hit and location rolls). This ratio would allow you to group games into categories, like spaghetti sauce types, and players could pick their favorites like extra chunky or smooth and balanced. (Yeah, I just watched the link Pink provided, and it seemed to fit too well not to state)

As a sample, chess is 100% player driven. With some crunching, Battletech may formulate out to 90% player driven (just a guess right now). Choosing heads or tails is 50% player driven (player calls head or tails, randomness determines heads or tails).

Some games may be less than 50% player driven, for example in a squad based system where multiple random events are resolved based on 1 decision, there is more randomness in the outcome than the decision. As an example, in Warhammer if you have a squad of 10 standard marines with bolters, and you are in rapid fire range of 2 units, the players decision is 1 bit of data (unit A or B). However, that is 20 d6 dice rolls to hit, up to 20 more to wound, and a final 20 possible to save. Thus, with a A or B decision you generate 60 potential d6 rolls for a total of >1% player input for that shooting attack. If such randomness versus player input is repeated elsewhere in a game, it may very well have less player input than guessing a number at random.

Now, perhaps some people like games with less player input (or more surprise) and some people like games with more player input (more chess-like). In designing the perfect random game balance, you would really be designing the perfect balanceS of game randomness, where you have several random generators that are picked based on what level of player input the gamers are interested in. For battletech, for example, one variant would remove all dice factors all together--all damage and hit locations would be purely statistical and the winner would be the one who maneuvered better. Then there would be battletech as it currently is, and then a variant where your total randomness is restricted ala quickstrike, thus increasing player choice (though quickstrike also removes some player options so who knows). Finally, on the farthest end of the spectrum, we would have battletech with no player interaction, where the entire game is random. This would be the kind of result where you just roll some dice to see an outcome.

Anyway, longwindedness aside, its all just food for thought.

Dan Eastwood said...

Bradley, your comment got caught in the SPAM filter. Sorry for the delay.

Very interesting thoughts! I should note that movement choices are not random, so the concept of information doesn't really apply. BUT this may still be a useful way to measure the number of options a player has to choose from, AND it might be interesting to what happens IF players move randomly. This is a difficult topic I've been trying to work up to writing about for 2 years now, maybe it is time.

I should reply at length to your comment, but I don't have time at the moment. I'll try later, or maybe work my reply into a new post.

Or here.