Shannon Appelcline: Dice Games, Part Two: Just the Stats, Ma’am

It’s a Dice Fest!

That’s an oft-seen complaint on certain internet bulletin boards whose readers think that all games should be entirely strategic, with no chance for random elements to intrude upon carefully made plans.  If that’s really the sort of game that you like, then no problem. But, don’t buy blindly into the concept. I think dice games can provide a lot of benefits that you don’t find in a “less” random game, the greatest of which is the visceral and encompassing joy that can fill you when you receive an unlikely, but badly needed roll. Besides that, if you’re wanting to simulate reality in any form, then you need to accept that randomness happens. Just ask Hillary Clinton or Constable Charles d’Albret (of Agincourt).

This isn’t to say that a good dice game is totally random. Instead, it uses additional mechanics to turn that luck into another game element that can be controlled by a good player--which is the topic of this week’s article.

Dice Skills: Mechanics & Skills

You roll the dice and something good happens. Or something bad. In a game like Craps there’s no control over this. However as dice games have evolved, they’ve given players the opportunity to choose their risks, to reroll, and to make decisions based on how the dice ended up. By taking advantage of these mechanics, a well-skilled dice player can do better than his peers, pointing out the core strategic basis of dice games.

Probability Knowledge is the most important skill in any dice-based game. You have to know what the chances are before you make any decision. Are you going to roll dice for an attack in Struggle of Empires? Then you should know what the odds are beforehand.  Similarly it’s helpful to know how likely you are to fit your dice into a particular category in To Court the King or Yahtzee and what your odds are of getting a particular airship in Airships.

Designers can build this mechanic into their game by making their random rolls relatively easy to parse. You can look at Settlers of Catan and know that your “6” region comes up 5/36 of the time--thanks to the handy printing on the production chits. Similarly if you’ve got four “5"s in Yahtzee, it should be obvious that you have a 1/6 chance of getting the fifth one in a final roll.  The more dice you roll, the more you roll dice in opposition to each other, and the more generally esoteric the results, the harder it will be for players to calculate probabilities. They can make for better games, but it’s also a danger.

(And I’m going to return to the topic on calculating simple probabilities toward the end of this article.)

Risk/Reward Assessment is the close cousin of probability knowledge, and perhaps the most important skill in any dice game. Once you know the odds, you then compare them to the returns, and decide if you want to take in action.

In order for a game to incorporate this mechanic, it needs to allow for a wide variety of rewards. The recent Airships offers a good example of this: you can choose between perhaps a dozen or more items to roll for at any time. Some may be somewhat unlikely, while some may be entirely automatic. You have to balance that range of probabilities against what each object offers in return. There are, of course, some dice games which don’t offer a risk/reward mechanic at all. Take Kingsburgh: you roll once and you can’t change your results, so there’s no risk/reward implicit to the die roll (although how you use your dice can be risky, based on chaotic interactions with other players). But, we generally shouldn’t expect to see any of these mechanics in all dice games.

Randomness Avoidance is the art of making the randomness matter as little as possible by increasing a player’s set of choices. Sometimes a player will be penalized for risk avoidance (under the typical risk vs. reward rubric), but at other times minimizing risk can just be a good idea.

Klaus Teuber has done a good job of showing how to introduce a randomness avoidance mechanics in several of his games. In his classic The Settlers of Catan if you spread out among hexes with many production numbers, you’ll always be moving forward (though I’m not certain this is by any means the best strategy). Similarly in his Catan Dice Game you can build your empire in such a way as you can always build either a city or a settlement--and thus no matter what dice roll you start off with, you can try and build on it successfully (and here I’m more certain this is a good strategy).

Randomness Responsiveness is the final core randomness mechanic that I think dice games should consider. It means that there are mechanics and/or player skills which can help offset a bad roll.

Kingsburgh shows off both mechanics and skills in regard to this mechanic. First, the system helps you out. If you get a bad roll, you get to place your dice first. Second, a good player can try and get better use out of his low dice than a bad one. In some ways the game is entirely about resource gathering, and thus a 13+ roll is great, because it guarantees you 3+ resources. However, you can actually get three resources with a roll as low as a 9, which if split into a 2, a 3, and 4 can give you two gold and a wood (or vice-versa). Afterward there’s yet another level of player-driven randomness responsiveness because a player burdened with a bad roll can decide to switch up what he’s building, taking advantage of the dice results he actually got.

Without any of these four mechanics, I’d agree that a dice game can be just a dicefest, but as they’re added to a game it can quickly become more strategic and more thoughtful, to the point where the strategic puzzles embedded in a dice game are just as notable as those embedded in any “non-lucky” game.

The Probability Appendix

I’m sure that eyes are already glazing over at the “p” word, but let me generally state that I’m not the type of gamer who likes to do complex calculations when playing a game. Instead, I prefer my games to be light enough that I can have fun. Nonetheless, I’m happy to calculate a bit of probability when I’m playing a dice game if it’s easy. I usually limit my calculations to expected value and simple odds. With those I have a good enough thumbnail to know generally what’s going on without knowing the precise values.

An Expected Value is just the average result you’re likely to get on a roll. On a single normal six-sided die it’s 3.5, and thus on two dice it’s 7, etc. There’s no simpler probability calculation that you can do, and yet it’s very useful. If you know you need a result of 6+ on a 2d6 roll, then you can figure the odds are with you. Since the expected value is a 7, that means on average (e.g., more than half the time), you’ll get a 7. Thus your odds of getting a 6+ are even better.

Sometimes when you have weird dice you have to figure out the expected value by hand. As a rule of thumb, just add up the values of all the sides, and then divide by the total number of sides. Take Airships as an example. The white dice have sides 1,1,2,2,3,3 (expected value: 2), the red dice have sides 2,3,3,4,4,5 (expected value: 3.5, but with less variance than a normal die, meaning that you can’t see the same highs or lows), and the black dice have sides 4,4,6,6,8,8 (expected value: 6). When you put those numbers together, it suddenly becomes a lot easier to see which tiles you should go for and which you shouldn’t.

Simple Probability is the other thing that I think is worth considering in any dicing game. The basic precept is that the odds of an event happening are the number of opportunities for that event to occur divided by the number of total possibilities. So, to take things simply, the chance of rolling a 3+ on a six-sider is 4/6 because there are four opportunities (3,4,5,6) out of six possibilities (1,2,3,4,5,6). It’s similarly pretty easy to figure out the odds for any pair of dice. There is 1 way to roll a “2” (1+1), 2 to roll a “3” (1+2,2+1), 3 to roll a “4” (1+3,2+2,3+1), 4 to roll a “5”, 5 to roll a “6”, 6 to roll a “7”, 5 to roll a “8”, 4 to roll a “9”, 3 to roll a “10”, 2 to roll a “11”, and 1 to roll a “12”. Yes, those are the pips shown on Settlers of Catan pieces. To figure out the chances of rolling a number of higher, you just add up the cumulative opportunities and divide by 36 (the total number of ways to roll 2 dice: 6x6). Thus, for example, our aforementioned chance to roll a 6+ is 5+6+5+4+3+2+1/36 or 26/36, which is indeed pretty good odds.

Besides understanding this basic concept of probability, it’s also important to be comfortable guessing. For a more complex probability if you can reduce it to a simple probability that’s not far off, then you’ve got as much as you need to know for a game.

And that’s about as much as you need to know to be a superior player of dice games.

Around the Corner

My recent reviews have included Ticket to Ride: The Card Game, Martin Wallace’s Toledo and the comic book Hercules: The Thracian Wars #1.

I’ll be back in two weeks to finish up my look at dice games with mini-reviews of everything that I’ve played.

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