Double It

In the back of one of the manuals for the Civilization computer games – I forget which one – there’s a section that talks about how the game was designed. This contains a nugget of wisdom for game designers of any stripe: “If something isn’t working, double it or halve it.”

I think that this principle is a very sound one. Suppose you have a situation like “The penalty for juggling while skateboarding isn’t high enough – nobody feels its that difficult in game, though everyone agrees it would actually make it quite a lot harder in the real world. Plus the guy who can force people to skateboard feels way underpowered because putting people on skateboards is barely impacting their juggling at all.”

It might be tempting to increase the penalty by one and see if it makes a difference. The problem is that playtesting tends to produce a fairly limited set of data points, because of random variance (Even if the game has no randomness the skill of the players you have is a factor and even if you are testing with the same players over and over they will have good days and bad days). It’s fully possible to have an experience where something is weak, so you make it more powerful and then to play a half dozen games in which it does even worse. Just by chance.

So to actually see the difference there’s a sense to making big changes. We’ll not increase the penalty by one – we’ll double it! It may well be that doubling makes it waaay too hard to juggle while skateboarding, but if that’s the case you know the undoubled variable was a little too small and doubled was far too high and that gives you information to pick the right point in between. Compared to the very real chance of learning nothing with a one point bump, there’s a good argument for doubling.

The thing about doubling is that you really have to understand where your zero point is.

Imagine two systems for a game.
In system A I roll D6 + 3 and need to get a 7+
In system B I roll D12 + 3 and need to get a 9+
The chance of success in both are the same. But if (for whatever reason) we double that +3 then they change a lot. I don’t know about you, but I’d be super confident of rolling 7 or higher on D6+6.

The trick is to understand what doubling something actually *is*, specifically it’s making it twice as big relative to zero. However in our first game zero is clearly not the default state, presumably when we set the target as “7+” we anticipated that under normal conditions the player would have some sort of bonus to the roll. Perhaps in the average situation a player has +1 to this roll – in that case we should treat that average case as zero and double relative to that. So when we want to double our +3 modifier rather than doubling the number itself we double the difference between it and the zero point, making it a +5.

If you’re feeling fancy you could look at the ratio of success to failure and aim to double that rather than doing anything with the raw numbers. D6+3 will have a ratio of 1:1 between successes and failures, so we might want to double that to 2:1, which means wanting a 66% chance of success. That means we want 3/6 sides of the dice to result in success and so can make the modifier D6+4 – it turns out that the random factor and target numbers we’re working with are sufficiently small that in this case a one point bump does make a big enough difference!

I’ve been doing a lot of this with Genesis recently, having built up datasets to the point that they show me which things it’s great to be a god of (Fire, Showmanship) and which ones tend to lead to failure (Madness, Undead) I wanted to make a new version to rebalance things. However it takes a *lot* of plays to get a decent amount of data for Genesis since a lot of the gods aren’t used in any given game so keeping the number of iterations down is important. That means adopting a policy of making significant changes that are going to show enough of a difference to meaningfully drive decisions, but doubling the level of a card in this game isn’t something that makes sense. A card is level 1-10 so fully half of them can’t be doubled without extending the range. It’s been better to look at “How many other cards in the game will this beat” and to aim to change that number significantly, though of course changing any card slightly changes that number for every other card ðŸ˜‰

The point is that while doubling a thing feels dramatic making big changes during testing is healthy. You learn more than small changes and sometimes it turns out that a big change is what you needed and you can keep it as is. The rule of “double” isn’t necessarily important so much as the overall philosophy, but if you do want to keep it the trick is to be mindful of what you’re doubling and relative to what zero point: Raw numbers often don’t tell the whole story.

Thematic Dice Rolling Systems

A while back I wrote an article comparing different ways to roll some dice and get a result that produced the same average result but had different probability profiles. Today I wanted to talk about a way to make use of some of those features: Making the method of dice rolling tie into the theme of a game.

Consider Zombicide. If we grab ourselves a precision rifle and fire on a zombie we roll 1D6 and will kill it on a 3+. One the other hand if we go for the submachinegun we roll 3D6 and will kill it if any of them work out as a 5+. The rules never explicitly state “Accurate weapons roll fewer dice but need lower numbers to hit, inaccurate ones do the opposite” but the pattern is established through repetition on most of the games weapons.

This is a good feature for a game to have. Ideally a game should have a lot of little things that are needed for the game to progress and serve other gameplay purposes building on each other to establish links between the theme of the game and what the players are actually doing. Even if the probability is the same if you set off some kind of huge explosion throwing a large pile of dice is more satisfyingly “explosiony” than throwing a small pile.

However as well as the visceral feel we should also remember what different systems do to the probability curve. Thematically you might want an explosion to be and feel chaotic compared to – say – a sniper. Throwing a boatload of dice has that visceral feel, but in terms of what it actually does the “many dice” thing tends to be more reliable because when you roll several dice together you get a bell curve. That is to say that the average result occurs more often and the extreme result less often, compared to rolling one dice where rolling a middle number and rolling the highest or lowest number all have the same chance to happen.

So are we stuck in a situation where the visceral feel of a thing is directly opposed to its actual effect on the game? Of course not!

We can think about how we apply modifiers. If someone is rolling some D6s and needs X+ on each one to get a hit we’ve got three levers: We can change how many dice are rolled, we can change what result is needed for a hit and we can change how many hits are needed.

On a side note, it’s generally desirable to modify as few things as possible so that when someone sees “-1” they know what it applies to because the game is consistent – but it’s still worth considering which one to modify.

If a modifier applies to the number of dice it has a more profound effect on “1D6 3+” than “3D6 5+”. Adding an extra dice increases the chance of success by 23% for the first pool, only by 9% for the second. Taking one away is even more profound since with zero dice the first pool can’t succeed at all.

By contrast if it applies to the number that has to be rolled it will affect the “Big dice, high number needed” pool more dramatically since that target number applies to every dice and it has more dice to be affected.

So if we’re looking to make it so that we can use a big pile of dice for explosions or shotguns or risky behaviour in general but also have the probability profile make those things riskier rather than more certain – where does this mean we put our modifiers?

The answer is “It depends”. Specifically it depends on why you might be modifying a dice roll. If the most common sort of modifier is a positive modifier because the character doing it is really skilled it makes the most sense for it to be a dice modifier. That way a skilled character gets the most out of doing a precision activity – the game will “feel right” when a champion marksman adds more to a sniper rifle than a scattergun.

On the other hand if the most common modifiers are negative situational modifiers because the weapon’s old or you moved or whatever then applying them to the target number may have the desired effect. The chaotic weapon will break down when the circumstances are against it but the reliable one will – well – be reliable.

Modifiers aren’t our only option either. A lot of games use rerolls as a means to provide reliability. If you’re happy that your game won’t be slowed down too much by the extra decision and roll step involved in having one then providing them can be a way to distinguish risky but dynamic actions from reliable less dynamic ones. That way we can still give our grenade a huge dice pool but make our sniper rifle’s smaller dice pool stick more closely to the bell curve – if the pool is smaller but less than half the size then providing one reroll makes it effectively larger in terms of reliability – so it can be small for the visceral action of picking dice up and doing things with them but large in the abstract mathematical sense of how it actually behaves.

Finally – the most important thing to consider is how a game breaks free of a fail/success binary. The examples above talked about “Roll one hit to succeed” but even in the Zombicide example, there’s a reason to roll more than one hit. If the player scores several hits they kill several zombies – good times.

This feeds into the theme as well. It means that “3D6 5+” has an advantage over “1D6 3+” not only in having a better average and being able to more consistently score a success – but also in that it can potentially kill up to 3 zombies where the other version will only ever kill 1.

It didn’t have to be this way – the rule could have been “Look at only the highest dice, if it’s below the target number nothing happens, otherwise it kills a zombie plus an additional zombie for each point it exceeds the target by”. Then a 3+ could kill up to 4 zombies and 5+ could only get 2 even if it had lots of dice.

The rule makes sense for that game – it’s intuitive that a spray of automatic fire could kill several zombies were one very well placed shot usually wouldn’t – but perhaps your game isn’t about shooting zombies. Perhaps your player is rolling for their ad campaign to convince the nation that keeping the planet inhabitable to humans is more important than saving a few quid on groceries. If your critical success is “Convinced some persuasive celebrity” then maybe a “Brute force the campaign with lots of money” approach should roll lots of dice and be consistently effective, but you want the best chance of a critical effect to lie with a “Carefully targetted ads” approach.

There’s not only one way to do these things – the point of this article is to remind us to be conscious of them. There are a hundred ways to resolve “I’d like the player to throw some randomisers to see if this works” into a rule. It’s worth taking the time to pick the one that not only makes your game work best as an abstract mathematical model, but also the one that’ll help players to be invested in the theme and for the things that happen to seem like they intuitively make sense from the actions the mechanics are describing.

Consequences of Precise Probabilities

Pathfinder

I’ve been playing the digital adaptation of the Pathfinder Adventure Card Game lately. Besides the bugs (and dear me there are a lot) it’s a pretty faithful recreation of the physical game – yet I’m finding one aspect of it is making me play very differently: Every time you do something to modify a die roll the game shows you the probability of the roll succeeding and permits you to undo that modification.

Probabilities in Rise of the Runelords

Many of the bonuses in the game are represented by extra dice. At any given point you usually have access to a whole load of these bonuses, but use of them has a very noticeable opportunity cost leading to you seeking to use as few as possible. On a typical turn you might hit into a situation like this:

You’re righting something that needs a total of 22 to beat.
You subtract one from each die rolled.

You can punch at D6 or attack with a sword at D6+D8 or cast a spell at D10+D6+D6+2
You have four blessings which will give you a D6 if you’re attacking physically or a D10 if it’s a spell.
One of those blessings gives two dice instead of one if it’s used for a D6.
You could discard the sword, losing the ability to use it again and get an extra D6
Your mate you shuffle part of his hand into the deck to provide an extra D4+2
You have an item you can discard for a bonus D4 but that applies to all fights this round (Which is at its best if you save your blessings since each blessing could also be used to flip another card and maybe get into another fight if it’s not used here)
One of the other players has a spell that gives +3 to strength rolls so works with the sword but not if you’ve cast a spell.

Computing the probability of success using each combination of bonuses would be a massive headache. Heck even the base combinations would take some doing. But the game does it instantly and displays the number for you.

How does it change the game?

The obvious first degree effects are that it means I make optimal decisions. If I can get a bonus one of two ways and the cost to both is the same then I’ll always pick the bonus that’s best in the situation. That’s not such a big deal, I’d probably have done it anyway most of the time.

The major impact it’s had has been in how I’ve learned the game over repeat plays.

In tabletop I doubt I’d notice the difference between a 91% and 97% chance of success. It’d always be at the point of “I’ve got a big pile of dice and the average expected result is at least twice what I need to roll – it’s gonna be fine.” Yet providing the probabilities makes me sensitive to it. It’s a game that can be won or lost on a single roll – tripling the chance of failure for that roll (from 3% to 9%) is actually a really important difference in those critical situations.

It’s harder to pin down how these things are altering the emotional experience of the game. I’m making better moves, but am I enjoying the game more or less for that? The reaction to a roll seems different – on the one hand I get “Well I decided to stick at 95% and not throw another card in, 1 in 20 chances happen all the time” where I’d have got “That roll is so absurdly below average, I hate you dice.” but on the other hand failing a >99% roll is a worse kick in the teeth than it’d have been if you weren’t aware of just how good the odds were.

There’s also a conflating factor in the game being single player. “Shall I throw in my other characters extra card to squeak an extra 2% chance to win out of this roll” is a fundamentally different question to “Shall I ask Jane to give up her extra go so I can have a 2% extra chance to succeed?”

Overall I think being aware of the exact probabilities sharpens the game. Dice are fickle, but over time, over many rolls, fair. It makes me more aware of smaller changes which in turn means I think about choices that I might otherwise have discarded out of hand. Ultimately it frees me from arithmetic to enjoy what the designers intended the game to be.

What’s the take away for designers?

Games can be more or less explicit about the probabilities of success involved in doing things. A magic computer box that does the numbers is not a necessity: Settlers of Catan dots its pieces to show dice probabilities. It would not be possible to do something for more complex mechanics like the examples shown here – but the mechanics themselves can be streamlined.

It’s also not an all or nothing approach. For instance Race for the Galaxy includes a card that allows you to guess the cost of the top card of the deck and draw it if you were correct. That card contains a little table showing how many of each cost of card are included in the deck. This doesn’t mean you can calculate the probability (I mean you could if you wanted to sit and count the number of cards of each cost currently in play and that you’ve personally discarded) but it makes it less obscure – you have a better idea of what it might be than if you didn’t have the contextual information.

Most games are likely to benefit from giving the players more information and more powerful tools to make decision – if the rest of the game supports that. There are some games that being able to work out the odds of each move *is* the gameplay and these calculations are intentionally on the cusp of human ability – but for more games working out the best move is the gameplay and understanding the odds of various outcomes is a tool in reaching the more interesting factors that define the best move in that particular game.

Card Probabilities

I’ve been playing a lot of Magic: Duels lately and consequently have been spending a certain amount of time swearing at the frequency with which I get hands that either consist entirely of or are entirely devoid of lands. Thinking about how likely that is opened the floodgates to a ream of ideas about card games in general, so let’s dig in:

Dice: Rolls, totals and pools

It’s going to be a mathsy one this week, I’m interested in thinking about different ways to interpret dice rolls and what they mean for game design. It’s not going to be as off the wall as I was in 6, this isn’t about considering stuff like the position of the dice relative to the other dice, but even within those constraints there are still a lot of ways to interpret the number showing on the dice.

Don’t tell me the odds

So there’s this cool psychological effect whereby if you ask people “What are your odds of drawingÂ a full house?” you get higher numbers on average than when you ask “What are the odds of drawing a full house?”. People are almost universally aware that the cards drawn off a deck does not change depending on who is doing the drawing, but when asked – on average – will assign themselves higher odds. How do we use that?

Randomisation Timing

I’ve been playing the new Hearthstone expansion recently and I’ve not had a particularly good time with it. I’ve noted before that I’d enjoy Hearthstone more if they deleted every card with the word “random” printed anywhere on it and this expansion seems to be pushing towards more cards of that nature (A brief aside: There is a difference between “I don’t like this” and “This is bad” – I think this is actually going to be a good direction for the game as a whole). The thing is that there are other games which involve more randomisers than the average game of Hearthstone, because what bothers me isn’t the presence of the random factor,Â but the timing surrounding it.

Cards and randomness

Cards are a great way to add randomness to a game, they’re incredibly flexible in terms of what you can do with them. You can fix a deck, or part of a deck. You can add things to it without consulting what they are. You can view a few cards and get rid of a few of them selectively. You can put new (or previously used) cards into a deck and they can go on the top, bottom or be shuffled somewhere in between. Cards are magic.