Saturday, 17 November 2007

mechanisms for randomness

One fine day, I spent some time thinking about what are the mechanisms used in boardgames to create randomness. It's one of those things that a boardgame hobbyist (a.k.a. geek) does. I tried to distill all the mechanisms that I know of down to the basics. I tried to categorise them. I came up with a list. But first, let's define "randomness". In the context of this blog entry, I define randomness in games as anything that players (1) mostly cannot predict, and (2) mostly cannot control. E.g. what cards you draw from the deck, the outcome of a die roll. This is as opposed to something that a player can predict or can manipulate, something that is deterministic. E.g. in Puerto Rico if you take the role of craftsman and produce lots of corn, the player next to you who also produced lots of corn will probably then take the captain role so that he can ship all his corn for victory points before you can do the same. In Pirate's Cove, you can discourage others from fighting you by upgrading your ship to be the most powerful ship in the Caribbean.

Randomness is important in a game, because without it, there is no game. If you can predict and control everything in a game, then it is not a game. It is a machine, with a known input and a known output. Chess is a perfect information game, but there is still randomness (well, at least according to this blog post) because you cannot always predict what your opponent will do. If you think about it, starting a game without knowing who will win is randomness. Randomness is important in making one game different from the next. It is the spice. It creates variety. It creates surprises. Games with more randomness can level the playing field between experienced players and newer players, and thus making it more fun for both because everyone has a more or less equal chance of winning.

After thinking about this, I find that there are only four basic mechanisms for randomness in games. Every mechanism in every game (well, I guess I can only say so for the games that I have played) that creates randomness can be categorised as one of these four. Here they are:


One of the most basic mechanisms for creating randomness. With one normal (i.e. not unbalanced) six-sided die, each of the six numbers are equally likely to appear. When you roll this die, you expect an average of 3.5, because assuming you roll many many times, the results should average out to be 3.5. There are dice with more than six sides. There are dice with less than that. Dice in Samurai Swords and Axis & Allies: Battle of the Bulge have 12 sides. There are games that use more than one die. The Settlers of Catan and Monopoly use two dice. Can't Stop uses four. Risk Express, Pickomino, Yspahan use a lot of dice. When more than one die is thrown, you can calculate probabilities. With two dice, 7 should appear much more frequently than 2 or 12. The possible outcomes are not equally likely anymore, and you can use probabilities to guide your decisions. Just remember probabilities are just that - probabilities. In the long run the results will reflect the probabilities, but your game may not be long enough for that to happen. So don't whine when you build your settlements next to 6 or 8 spaces in The Settlers of Catan, but noone throws 6 or 8 throughout most of the game.

Wargames often use dice as a mechanism to manage probabilities. Often you throw dice to fight. E.g. in Axis & Allies, an attacking bomber scores a hit if you roll four or less. So the probability of a bomber being successful in an attack is 67%. In Hammer of the Scots, a full strength unit (i.e. your soldiers have not been killed or injured yet) rolls more dice than a depleted unit, and thus can score multiple hits with one attack.

Some games use special dice. Lord of the Rings has a six sided die, showing different numbers of different icons, and even has a blank side. Pickomino's dice have numbers 1 to 5, and 6 is replaced by a worm. Die Macher's dice are numbered 0, 1, 1, 2, 2, 3. By having special dice, these games customise the probabilities and possible outcomes to suit the game.

Very colourful dice in Risk Express.

Even with standard dice, there can be interesting ways of using them. In Struggle of Empires, when fighting a battle, you roll two dice, and your strength is the difference between the two dice. So your possible outcomes range from 0 to 5. 1 is most likely (10/36 chance), followed by 2 (8/36), 0 & 3 (both 6/36), 4 (4/36) and 5 (2/36). In Yspahan, you throw 9 dice (you can pay to add more), and then group them by numbers, and then these groups of dice are assigned to up to six possible actions that all players can choose for this round. Sometimes when the numbers are evenly distributed, all six actions will be available. Sometimes when the results are lopsided, then some actions may not be available. This is quite innovative.

An important aspect of dice, and dice-like mechanisms, is that there is no memory. Every die roll is independent of the previous roll, and no history is kept. If you roll a 6 for ten consecutive rolls, it does not make rolling a 6 more or less likely in your next roll. The probability is still 1 out of 6. You cannot take known history into account when determining the probability of the next die roll. This is easy to misunderstand, because naturally people will think: What are the chances of rolling a 6 for eleven consecutive rolls?! So, those people at roulette tables at casinos who record past results are not behaving rationally, at least based on my theory here.

There are other mechanism which are not dice, but behave the same way, and I categorise them together with dice. E.g. the spinner in Twister.


The other commonly used mechanism for creating randomness is cards. The most important difference between cards and dice is the memory element. Consider a standard card deck. If you already have three aces in your hand, then the chances of drawing the fourth one from the deck is quite small. So the memory element exists because once you draw a card from the deck, then there is a smaller probability of that type of card appearing again, or in the case that every card in a deck is unique, then the card will never appear again. The "memory" is "reset" only when the deck is reshuffled. This memory element is important for many games.

One of the mini expansions for The Settlers of Catan replaces the two dice with a deck of cards. The cards are mostly just numbers from 2 to 12, representing the possible outcomes of throwing two dice, and the 7s appear more than the 2s and 12s, reflecting the probabilities. I think there are some special event cards too. By replacing the dice with cards, this expansion tries to make the "die rolls" more "fair", i.e. you are less likely to get a situation where a lot of 4s get "rolled" and no 8s get "rolled" at all. Different people may have different opinions about whether this is good or bad. I am just using this example to illustrate how the memory element is being used. This mini expansion is using the memory element to try to make the "die rolls" more "normal".

Using a deck of cards allows much flexibility for the game designer to control probabilities and the distribution of possible outcomes. The number of cards in the deck, the distribution of cards within the deck, the hand size of players, how many cards are drawn every turn, are all factors that determine randomness. In Princes of Florence, if you draw cards from the deck you draw five and choose one. This makes it less likely that you will be stuck with a card that is completely useless to you. So the game is basically trying to minimise bad luck to you due to randomness, and is trying to give you more choices. In Bohnanza, the more valuable cards occur less frequently (e.g. red beans) than the "cheap" cards (e.g. wax beans).

When playing a game with cards, you should make use of the memory nature of cards. In Ticket To Ride, if you remember that many green trains cards have been picked up by your opponents or have already been played by your opponents, then you probably should not be trying to claim a new long green route.

There are many mechanisms in games that are of the same nature as cards, although physically they are not cards. Mahjong is one example. I have actually seen a card version of Mahjong. That really saves space, and is much more quiet. But of course you won't get the enjoyment of shuffling mahjong tiles the traditional way, the "swimming" way. The tiles in Carcassonne is actually also the card mechanism. If you are like me and remember that there are exactly only two cloisters with roads (in the base game), then if both have appeared, you don't bother hoping to draw another one. The tiles in Tigris & Euphrates is a card mechanism, but since there are only 4 types of tiles in the bag, you can't really keep track of how many are drawn and it's quite difficult to predict what will come next. The tiles in Scrabble is also a card mechanism. Similarly, tiles in Ingenious, event tiles in Lord of the Rings (if you have drawn a lot of good ones, you'd better not dilly-dally anymore and quickly finish the scenario before you start drawing one event tile, i.e. usually bad, after another), building tiles in Attika (as the game progresses, you gradually narrow down what are remaining in your stacks), treasure tiles as well as forest tiles in Tikal.

In addition to being a mechanism for creating randomness, cards are also a very convenient mechanism for hiding information from your opponents.

Tiles in Taluva is basically a card mechanism, but in this game the luck element in the tiles is not very high, and also it is quite difficult to remember tiles (i.e. card counting). Well, at least I never bothered to do so.

The event tiles in Lord of the Rings (the square tiles in the middle of the photo with dark green backs) are a card mechanism. This photo shows Lord of the Rings with the Battlefields expansion.

Players' Intentions

I hesitated a little before deciding to add this to the list. It is a fair assumption that players play to win, and will make decisions that improve their chances of winning. That is something predictable. What is not so predictable is what your opponent(s) considers as the best move at a specific point in the game. He or she may feel that a certain move is the best, which you do not agree with. He or she may have some grand scheme behind that one move that you are not aware of. There may be some ulterior motive. Maybe he likes you and wants to let you win. Or maybe he hates you for beating him so soundly in the previous game that he is now working harder to make sure you lose than to make sure he wins (jerk).

Taking a step back, I realise that players' intentions are a random factor in every game! Well, at least in non-solitaire games. You won't be able to fully predict or control your opponents' moves. If you could, there is no point in the game.

Randomness caused by players' intentions is commonly called "player chaos" in the boardgame hobby. Some games allow you to better guess your opponents' decisions. Some games make it very difficult to predict your opponents' moves. Games with mostly or all open information are usually the former, e.g. in San Juan, if you see your opponent hurriedly building 4 or 5 cheap indigo plants, he probably has a guild hall in his hand. In many games you can observe what your opponent is doing and more or less guess what he is trying to do, what is his strategy, or what kind of information he is hiding (e.g. cards in his hands, how much hidden money he has left). In some games it is harder, e.g. in Category 5 (a.k.a. 6 Nimmt) it's hard to predict what cards your opponents have and what they will play. In some games guessing your opponent depends on how good he is. Good opponents may sometimes be easier to guess because he knows the game well. A new player may be quite clueless. In some games guessing your opponent well requires knowing him well, e.g. some people just enjoy being the assassin or the thief in Citadels, or some people just want to be them to protect themselves.

Hidden information in a common mechanism that can be used to hide player intentions. Your cards in Jambo, the number of cubes (i.e. your score) behind your screen in Tigris & Euphrates, your ticket cards in Ticket To Ride. Maybe this is another topic I can explore in the future.


The last thing that I can think of, which probably will not apply to most boardgames, and will only apply to dexterity games, is physics. Most boardgames are about information, and the physical components just represent information. Boardgames can easily be implemented as a computer program. Then you just have pixels on the screen instead of physical components representing and presenting information. Physics does not come into play in these cases. But when it comes to games like Carrom and Loopin' Louie, physics do come into play. The strength of your finger flicking the disc, the friction of the table, the slight tilt of the table etc.

Now let's see how my hypothesis can be applied to my collection of games. I'll mostly skip the 3rd and 4th, player intentions and physics, since the former is applicable to all games, and the latter is applicable to very few games. Let's see how card and dice apply to some of my games.

  • Acquire - The tiles are a card mechanism.
  • Age of steam - Has dice.
  • A Game of Thrones - Although you have cards, everyone has a fixed hand, and the key element of randomness is actually player intentions, because you don't know which card your opponent will play. Also you don't know when your ally will backstab you.
  • Amun-Re - Province card = card mechanism. Worshipping (or stealing from) Amun-Re = player intentions. I guess blind bidding = player intentions.
  • Ark - It's a card game, so obviously...
  • Blokus - No card, no dice. Only player intentions. This is a perfect information game, like Chess, and yet it is a very newbie-friendly game. I guess it depends on how seriously you want to take it.
  • Ca$h n Gun$ - Definitely player intentions, plus a bit of card mechanism in the 5 money cards revealed each turn.
  • Caylus - I can't think of any randomness other than player intention. No, wait, the first six pink public buildings are set up in a random order. And I think that's the only randomness.
  • Chicken Cha Cha Cha - Card.
  • En Garde - Card, and memory element is important because the card deck is so small.
  • Gulo Gulo - Card.
  • Hammer of the Scots - Cards you draw at the start of every year, and dice during battles and some events.
  • Hansa - Card mechanism, for the distribution of goods at each city.
  • LOTR: Confrontation - Mostly player intentions, but you can say there's a bit of card mechanism, in terms of how your opponent's characters gradually get revealed. The cards in this game doesn't really have the card mechanism that I'm talking about, because you know exactly what your opponent has in his hand. It is his intention that you have to guess.
  • Modern Art - Definitely card.
  • Monopoly - Mostly dice, and that's probably why many Eurogamers dislike it. Card too for the Chance and Community Chest card decks.
  • Mykerinos - Card, in terms of what "lands" get drawn and how they are laid out at the start of every round.
  • Power Grid - Card, for the power plants.
  • Puerto Rico - Card, for the plantations. And that's the only random element, and the impact is not very big.
  • Ra - Card.
  • Risk - Lots of dice. Cards in terms of your card rewarded after a successful attack.
  • Samurai - Card, for your hand of tiles.
  • Through the Desert - There is some randomness in the setup, in where the oases are placed, and where the water holes of values 1 to 3 are placed. I guess you can say the setup has a card type randomness, in that if you've placed a 3-value water hole, then there are fewer 3-value water holes for you to place. The game itself doesn't have any dice or card type randomness.
  • Traumfabrik / Hollywood Blockbuster - Card.
  • Villa Paletti - Here's one with physics.
  • Wallenstein - Another one with physics. When you throw cubes into the cube tower, you will not know exactly how many will fall through and come out, and how many will be stuck inside.
  • Yinsh - Perfect information game, i.e. only player intention.

Hey, my generalisation seems to work!

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