Are you a classical or quantum gamer? Classical gamers like control, knowing that the outcome of the game will largely be determined by their decisions and the mistakes of their opponents: their preferred gaming system does not invite fortune to sit at the table. Quantum gamers accept that decisions made during game play can only affect the probability of the outcome but don’t determine it: their preferred gaming system is more open to the vagaries of chance.
I am a quantum gamer, myself, with an affinity for that peculiar love of statistician and gambler alike: dice. In this I am not alone. Dice are as old as civilization itself. Our ancestors diced with the gods and believed fate could be foretold in a single throw. Even the course of history has turned on the roll of a die. Julius Caesar's legions crossed the Rubicon only after the die was cast, plunging the Roman Empire into civil war. Such is the mystique and power of dice.
This should come as no surprise, given that the games we develop are dice-based, although always using custom dice. They can be enjoyed by virtually anyone, including young children, but gamers may like the challenge of finding tactics to move the odds in their favour. This article explains how the game, which recently won best game in the ToyTalk awards, was developed using computer technology.
Jurassic Wars uses 20 Dinocards featuring well-known dinosaurs, and seven D6 custom dice (‘combat dice’). We use D6 dice not only because they are familiar to the mainstream market but also they are economical when we are committed to manufacturing only in the UK. The game has two types of combat dice--red for carnivores and green for herbivores. As you may expect, the red dice are more likely to score a hit against an opposing dinosaur than the green. In fact, the red die has a hit symbol on three of its faces, the green die on only two. This is not an arbitrary configuration but the result of statistical analysis--we will look at this later. First, let’s examine the basic combat mechanism and how it evolved.
Initially, Jurassic Wars was to have a set of several different types of combat dice with the better combat dice typically being used by carnivore Dinocards, reflecting their greater attacking prowess. The herbivores would typically have a greater number of hit points, reflecting their greater strength. When a dinosaur sustained hits these were to be recorded on the cards themselves (which would be laminated so they could be wiped clean after use). But not only was this tedious and cumbersome–wiping down cards is not a good use of anyone’s time–but it also raised production costs by adding extra components. Other mechanisms were considered but were rejected because the additional components increased costs or reduced the portability of the game.
Eventually it was decided the most elegant mechanism was to actually make the number of combat dice the store of hit points. Each time a hit is suffered during a combat a die is set aside. This is realistic because, as in nature, a positive feedback loop is set up. The more hits are sustained, the greater the injuries suffered; the greater the injuries suffered, the less likely a dinosaur will score a hit. Clearly, the larger herbivores need to use a greater number of dice to reflect their greater strength but the dice also need to reflect their weaker offensive capability. This necessarily implied two types of dice--red for carnivores and green for herbivores, with the red dice more capable of producing a hit.
A key design point was to define hits scored during a round of combat as the difference between the hits rolled by each player. If two players roll the same number of hits then the dice are re-rolled until a difference is obtained when the hits are scored against the player with the lower throw. Dice are set aside accordingly and combat resumes until one player has no dice left.
As soon as the basic mechanism was established one question loomed large: what is the optimum difference in hits between the two types of dice? For example, why shouldn’t the red die be configured for five hits and the green for four? For this answer we turned to computer technology.
It’s probably no exaggeration to claim that Jurassic Wars is one of the most tested modern games in history: over 10 million combats were simulated by computer even before the first prototype was developed. The kernel of a dice-based game mechanism is easy to model and can be programmed on any computer using a compiled language. The simulation is invaluable because it provides the game designers with a full sensitivity analysis on game parameters.
One of the parameters tested was the configuration of the combat dice. For example, take the combat between a dinosaur with a dice rating of 3-Red (ie a player starting combat with three red dice) and 4-Green. These are the highest dice ratings in Jurassic Wars and are used by the most fearsome carnivores and best-defended herbivores respectively. It was important for game integrity that this combat should be close and the edge, if any, rest with the carnivore. It would be a little disappointing, to say the least, if the mighty Tyrannosaurus was often defeated by a brainless leaf-muncher. If we limit the disparity between the two types of dice to one hit, there are only four possible combinations. These are shown below.
It can be clearly seen that the 3 and 2 combination provides the closest outcome for a 3-Red and 4-Green combat, with the edge to red. If the hits per die are increased then green starts to win more than red. Why? Because the dice are also the store of hit points and as combat effectiveness converges the advantage lies with the player with the greater number of hit points, ie 4-Green.
If the red die has greater than a one hit edge over the green then the red becomes far too strong, winning an overwhelming number of combats. Given this data and the design objectives, it is obvious that the optimal dice arrangement is a red die with a hit on three faces and a green die with a hit on two. Using these two types of dice, Jurassic Wars has seven different dice ratings: carnivores have a dice rating of 1-Red to 3-Red, while the herbivores have ratings of 1-Green to 4-Green.
Understanding the game mechanism means appreciating the differential effectiveness between the two types of die. The probability the red die will make a hit is ½. We will write this as Pr (Red Hit) = 0.5. It should be obvious that Pr (Red No-Hit) = 0.5. A green die, on the other hand, has an asymmetric probability distribution: Pr (Green Hit) = 0.33 and Pr (Green No-Hit) = 0.67.
What is the basic probability that a 1-Red beats a 1-Green? We can calculate this in two ways. First we can calculate it theoretically. The table below shows all four outcomes of a throw each between a red die and a green die.
We can see that there are three outcomes: red scores a hit, green scores a hit and a draw. We can ignore draws because in the event of a draw both dice are re-thrown until a difference is obtained. From the table we calculate that Pr (Green scores a hit) = 0.5 x 0.33 / 0.5 = 0.33. A similar calculation confirms that Pr (Red scores a hit) = 0.67. As we are dealing with only one die for each combatant if follows that whoever scores the hit wins the combat. Therefore, we have theoretically arrived at the following two basic probabilities: Pr (1-Red beats 1-Green) = 0.67 and Pr (1-Green beats 1-Red) = 0.33.
Checking the probabilities by computer simulation revealed that 1-Red beats 1-Green on 6,670 times out of 10,000 combats. The full probability distribution is included in the Probability Tables document. This equates to a probability of 0.667 - and hence the computer model is extremely accurate. The advantage of using the computer model is that some of the theoretical calculations with larger numbers of dice become fiendishly involved and complex.
The computer model revealed some surprising results. What is the probability that 2-Red beats 1-Red? A little reflection probably suggests that it would be 0.67 or something close to it. In fact, the computer model showed that 2-Red won on 9,005 times out of 10,000 or a probability of 0.9. This can be verified theoretically.
There were two other design points to be included in the standard game rules. One was the introduction of the Game Period, which acts like trumps in a card game. The Game Period is determined by randomly selecting one of the three Timeline cards: Jurassic, Early Cretaceous or Late Cretaceous. Any dinosaur matching the Game Period increases its dice rating by one. Not only does this add variety to the standard game but by varying the Game Period mechanism increased tactical opportunities can be introduced into different versions of the game. The other – and more tactical – design point was the introduction of Combat cards. In the standard game there are three types (expansion packs will introduce more). These are Surprise Attack, First Blood and Extreme Aggression.
Each player is entitled to play a Combat card at the beginning of combat, but they are a scarce resource and should be played carefully. Surprise Attack reduces by one the dice rating of the opposing dinosaur. This has a significant effect. For example, if a Troödon (1-Red) makes a surprise attack against Utahraptor (2-Red) then the probability of a Troödon victory increases from 10% to 50%. First Blood is also an effective card. It increases by one the hits rolled on the first throw, which can profoundly affect the outcome of combat. In the above example, it increases the probability of a Troödon victory from 10% to 33%. Extreme Aggression entitles a player to re-throw any dice that are not showing a hit– for one throw only. Its probabilistic effects are hard to calculate because the effect of a re-throw is unpredictable but the card is probably best used with carnivores as they are more likely to obtain a hit on a re-roll.
This article has tried to show that a simple game mechanism is not necessarily one that is simple to create nor is it one without tactical interest for gamers. Central to our philosophy, above all else, is that a game should be enjoyable for many - and since the dawn of civilization the simple die in some form has been part of our leisure time.
Classical gamers will still shake their heads and may paraphrase Einstein: ‘God doesn’t play dice, nor should we’ but quantum gamers will be confident that the fundamentals of the universe do not support such a supposition. ‘It’s the probability, stupid,’ they declare as they roll the dice. And then there are a few gasps as the mighty Tyrannosaurus suffers a hit from an angry-looking Iguanodon…