# math and boardgames

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supercool designer
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In games involving mathematical values is solid math used to determine the values or do most game designers estimate the values and then work it out in playtesting?

Kirk

X3M
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Joined: 10/28/2013
It depends

It depends on the game.
The game type (board/card/video).
And the game designer.

Most games have estimates (is what I can tell from this forum). And are adjusted during immense play testing.

There are games however that require a starting point with math. Or you can forget getting the game done within a life time.

A perfect balance is impossible. Even with math.
But there are math tricks to get a jumping start.

questccg
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Joined: 04/16/2011
Playtesting

The outcome of playtesting - is what "breaks" or "makes" a game. For example in my WIP, originally we would trade 2x card to buy cards. Well it just took too long. And so I modified it to 1x. So you trade 5 points for cards = 5 points.

Another idea that I scrapped because "in practice" it sucked was "charging weapons". In one of our scenarios, you battle an alien vessel. Well originally the aliens would charge their weapons. It turned out "not enough charging" made it virtually impossible to attack your homeworld (Planet). And so we had to alter the mechanic such that it's an alternate way to lose the game ("Critical mass"). Now we roll 1d6 to determine the firepower and damage that alien starship does...

So I would conclude IDEAS that are NOT playtested are worth 0%. You need to see if they work in the context of the game. The ideas may sound great but unless you actually TRY them... I wouldn't set them in stone...

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Joined: 09/02/2014
I have a pretty good head for

I have a pretty good head for estimating how changing numbers will impact on the balance of my games.

But if in doubt or seeking better information I work out a spreadsheet that crunches everything for me.

(also, I don't think there are any games that don't involve "mathematical values" in some small way, math is huge)

gilamonster
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Joined: 08/21/2015
For what it's worth (I've not

For what it's worth (I've not published anything yet - this is just a hobby for me) I often do initial calculations/test runs using a python script or occasionally a spreadsheet (for instance, in a card or tile game involving drawing, I use these tools to calculate odds of certain desirable/undesirable occurrences/combinations) then I check my results with as much playtesting as possible.(Because I cant write code to predict human fun-levels). Of course I only do this when something is non-obvious/non-deterministic and obviously important enough to balancing the game.

Jarec
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Joined: 12/27/2013
When dice are involved, I

When dice are involved, I always try to break the mechanics down to the bare numbers, and use the average values from the dice rolls. That would be my base line to fiddle around in play testing.

X3M
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Joined: 10/28/2013
Jarec, that doesn't always

Jarec, that doesn't always work.
A couple of months ago, I added something "new" to one of my games. And the average showed something totally different than what the real outcome would be.

Probabilities is something where you can use averages. But you need to keep your eye on all other possible numbers. And their true effects.

Math in games contain simple calculations, probabilities AND cause and effect situations.
If an effect situation is not wanted, you need to adjust the cause.

Soulfinger
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Reiner Knizia holds a

Reiner Knizia holds a doctorate in mathematics if that says anything about the importance of solid math in successful game design.

McTeddy
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Both. I come from a video

Both.

I come from a video game background, so it's slightly skewed, but I've met many designers who are pure math. They believe that numbers and statistics are supreme and they design based on it.

But they tend make those games games where you grind for 20 hours to get an item your friend got in 1 try. (Yay for "Replay Value") or casino games/mobile equivalents. This IS a valid way to look at things, and it is profitable when the house always wins.

But honestly, in the gaming realm, you also need to consider psychology. Two statistically equal events can feel VASTLY different from one another, and 99 of 100 players don't know much math and will assume their initial feelings are true.

Mathematics can give you a starting point, but you need proper playtesting to feel out the human element. Skew some numbers based on player response, remove elements that people feel are vastly different (Overpowered/Weak) than the math suggests, etc.

I've also chatted with many of the big name board game designers about design methods and math rarely came up. Playtesting ALWAYS did.

This isn't to say math isn't useful or that they don't use it... but that simply to stress the importance of live human testing.

X3M
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Interesting points

McTeddy wrote:

But honestly, in the gaming realm, you also need to consider psychology. Two statistically equal events can feel VASTLY different from one another, and 99 of 100 players don't know much math and will assume their initial feelings are true.

This is true.
In most war games you have a fast paced combat and slow paced combat.
If both combats would get you the exact same result. Players still choose the fast paced combat (even if there is a bit more risk to it or the results are a bit less). I guess, I am that last one of the 100 who actually chooses the slow paced combat.

McTeddy wrote:

remove elements that people feel are vastly different (Overpowered/Weak) than the math suggests, etc.

Do you mean that if the element is good balanced by math. Players still might feel like it is overpowered or weak?
I can imagine that this can happen. But I can't imagine the situation where it happens. Can you give an example?

McTeddy
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Off the top of my head, I

Off the top of my head, I can't think of any specific examples either. But I know I've thrown out cards and full mechanics due to strong player response.

Certain effects can have added psychological impact like "Skip a turn", "Cancel opponents move", or even "Win the game if..."

Even if the odds of that instant win were one in a billion, the player who loses to it will often feel cheated. A player may have only lost 1 or 2 turns or actions... but this can often be exaggerated by emotions. It doesn't take much for a player to feel cheated.

For actual examples of Math Vs. Perception, two Sid Meier examples.

When developing one of the Civilizations he was using pure math. He had testers coming because it was "Broken", they had a %50 chance to win, but they lost 3 times in a row!

It was even worse in larger number battles... yeah 1 vs 2 should be %50, but now it's 10 on 20!

Eventually, he skewed the math to make it more intuitive. It'll cheat regularly to be a little closer to the "Perceived" Odds.

- - -

The same goes for the new X-Com. Alot of people complain about cheating dice rolls on higher difficulties, but that's not quite right.

Turns out, they added a bonus to your rolls on the easy difficulties so you can't miss 99% shots, and 50% is likely to hit regularly.

It's only when they start playing fair that things seem off.

- - -

And some gambling games (Including mobile F2P games) do the exact opposite. They skew the numbers to ensure they hit the minimum legal payout rates.

Keep in mind, board games ARE different because you are rolling the dice by hand. People are quick to blame AI for cheating, but slower to jump to the loaded dice conclusion.

Just keep in mind that player perception matters as much as reality. This can both be a valuable tool and an awful curse.

Masacroso
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Copy the nature... The easy

Copy the nature... The easy way to balance something is feedback mechanics, cybernetics. This is how nature is stable and very flexible.

The question is that many times is hard to do that mechanics doesnt come exponential/snowball functions. We must try to do linear functions but you cant do this many times.

To solve this we have feedback-mechanic, in the same way that nature works to balance and stabilizing life.

X3M
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You make absolutely sense.

You make absolutely sense. But I didn't understand the last part.

I agree on copying nature. It worked for me a lot, even if it feels counter intuitive to some.

I also agree that a snowball effect needs to be stopped. They might slow down the game exponential. That is if they increase the number of steps in your game. Or they create infinity.

Exponential effects are indeed dangerous. Unless you have a linear effect cancelling them into becoming a linear effect themselves.

But what do you mean with a feedback mechanic? The cancelling that I just spoke of? Or something different?

Masacroso
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X3M wrote:You make absolutely

X3M wrote:
You make absolutely sense. But I didn't understand the last part.

I agree on copying nature. It worked for me a lot, even if it feels counter intuitive to some.

I also agree that a snowball effect needs to be stopped. They might slow down the game exponential. That is if they increase the number of steps in your game. Or they create infinity.

Exponential effects are indeed dangerous. Unless you have a linear effect cancelling them into becoming a linear effect themselves.

But what do you mean with a feedback mechanic? The cancelling that I just spoke of? Or something different?

There are a lot of feedback mechanics, the more classical and simple example are logistic curves.

I dont have the topic now ready to explain in a easy way but essentially a feedback mechanic is when any action creates an automatic reaction that diminishes or increases the effect of the action, so your power or effect is limited.

I dont have, by now, a clear example applied in a game but it can be done.

Many times balance mechanic very similar to feedback exist: in Risk-like games generally when you have more territories is a lot harder to maintain it or defend all your frontiers, so the expansion of your army is a perfect example of automatic balance of it power (some kind of abstract "implicit-feedback"... but generally feedback is considered a explicit thing, that is, that not belongs to the first action, something external), indeed this was used a lot in real wars to weaken a strong army, by example Russian army do it vs Napoleon.

Sorry I dont have now a real example. I must think one.

pelle
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Mathematics can be important

Mathematics can be important and using something like python to script details is very useful at times.

But any part of your game that can be balanced using maths alone without playtesting has been solved and is not something very deep or interesting to play. So if your excel-sheets or python-scripts or whatever can tell you what the outcomes will be, with certainty, then you need to add something to the game to make the outcome more interesting and difficult to solve.

Of course small subsystems are OK if they can be solved. You do not need every little detail to be as deep as chess or go. But if for instance tactical battles are an important part of your game and you want player decisions related to battles to matter it is a really bad sign if you can balance the battles using spreadsheets without playtesting.

Masacroso
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pelle wrote:Of course small

pelle wrote:
Of course small subsystems are OK if they can be solved. You do not need every little detail to be as deep as chess or go. But if for instance tactical battles are an important part of your game and you want player decisions related to battles to matter it is a really bad sign if you can balance the battles using spreadsheets without playtesting.

Why this is bad? I think is the opposite: if you can predict easily the outcomes you can focus in deeper things as strategic, in meta-decisions.

When more obscure is a game less deeper becomes... simply because you must spent your energy and time focused in different things than in only few.

Chess and go are perfect examples: they are games that you cant focus on strategy very much if you dont have a lot of years learning before. Just masters and very experienced players (that dont wast energy trying to figure things) can focus mainly in the strategic level of the play.

But it is very important differentiate complete information games than incomplete information ones. If you make very clean and predictable outcomes for the first then the game is completely mechanic and very AI friendly/human unfriendly. But this is not the case for incomplete information games.

pelle
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If a system (entire game or

If a system (entire game or part of a game) has any decision in it, you will have to make sure that your calculations can account for the best possible ones, or they will not say anything at all about game balance, because a human player will sooner or later find the strategies you did not cover.

If the decisions are so trivial that you can easily script something (even in a spreadsheet!) that covers all possible decisions by both players, some player is going to solve that as well. It just can't be very meaningful decisions because if the correct/best answer is known there isn't really a decision at all, you just know what to do and do that.

Scripts are great for optimizing tiny subsystems for mechanics that run automatically, like a single battle in a larger strategic game, or when there are only a few player choices that you can enumerate. But if you can automatically for instance balance your unit types without serious playtesting then that means how players use those units, what decisions they make, are no longer important.

If I can calculate with maths that unit type A is exactly as powerful as unit type B, that means nothing either player does to tactically use those units matter unless you have managed to account for all possible tactical moves they can make ("wait, what if unit A moves into a forest hex, then takes special action X, retreats two hexes, takes special action Y, THEN it can easily beat a unit of type B at range 7... no wait what if that other unit is in an area with a unit of type C and then makes its special move Z to ..."). If your game is not full of interesting tactical decisions to make, why bother playing? So your script is better able to simulate all the possible combinations of actions that the units can take (and in combinations with other units) before it can say anything about game balance.

The most important use of maths is probably to ensure that the game can not be solved, which is the opposite of being able to simulate it for auto-balance.

gilamonster
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pelle wrote:...But any part

pelle wrote:

...
But any part of your game that can be balanced using maths alone without playtesting has been solved and is not something very deep or interesting to play. So if your excel-sheets or python-scripts or whatever can tell you what the outcomes will be, with certainty, then you need to add something to the game to make the outcome more interesting and difficult to solve.
...

Actually, I usually use computer calculations (python scripts, etc) more to ensure that certain elements of the game design are not going to seem broken to the players (usually probability-related choices, like the chances of drawing randomly drawing a certain hand or multiple hands simultaneously from a deck). Sometimes a few back-of-envelope calculations suffice, but occasionally when there are multiple potentially workable solutions, a small script makes it easier and quicker to compare them and choose the best one. Of course any game which can be completely solved in a few minutes with a few lines of script is not going to have much replay value. Anything more than simple design blunders and I really think you have to rely on playtesting.Although running a few simulated rounds/hands/tactical combats (particularly when there are random elements or it is not easily solvable) can be a good way of getting an idea what play may look like before playtesting (again, as a sanity check - not a method of completely solving the subgame).

If I understood correctly, I agree on the feedback principle - in control-theory terminology, player-scoring should be stable (I think - control-theory is a bit of a black art to me) - so ideally it should be gradually harder for the leading player to advance their lead (but not so much that it becomes impossible to maintain it, or the game will never end, or end randomly). I haven't yet created a game that is sufficiently complex to require significant numerical modelling to find this sweetspot (but I would like to at some point as a learning exercise)

pelle
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Yes, calculations as input to

Yes, calculations as input to playtesting is a great idea. As you say for simple things like calculating probabilities of drawing cards etc that do not rely on player choices they can provide very useful data. For slightly more complex interactions they can be used to give a good idea what values to use, but I would not put too much effort into making the scripts "perfect" because playtesting is too likely to reveal that the initial values were way off anyway.

Masacroso
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Incomplete information cannot

pelle wrote:
The most important use of maths is probably to ensure that the game can not be solved, which is the opposite of being able to simulate it for auto-balance.

Incomplete information cannot be solved. This is why I difference between complete information games and incomplete ones.

I understand what you says now pelle, I misunderstood about what you consider "solvable on an excel sheet" or not.

From incomplete information point of view is a lot better simple games than complex ones, as I said before. See Poker, by example, or GOPS. You can easily predict all the outcomes but this doesnt mean you have a winner strategy, you must play with uncertainity but you dont need too much uncertainity, just a little, to make the game unsolvable.

Rock-paper-scissors is balanced and easy to predict all the possible outcomes... but is unsolvable.

P.S.: sorry, I had a mistake... it is not incomplete information for GOPS. It is imperfect information. Anyway imperfect information games are not solvable in a deterministic way, i.e., they are not really solvable.

X3M
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I only use it to get games

I only use it to get games balanced. The "solvable" part. Sure the game has an unsolvable part. But where would you put the following?

Unit A and B are balanced. Now to see what kind of rock paper scissors can be made with both A and B in an army. By using spreadsheets you know that there is not one super team. But 2A2B beats 3A1B, which beats 1A3B, which beats in turn 2A2B.

Would you say that I solved it myself? Would you say that I made sure there is a RPS present at a bit more complexer level? What if other players solve this, would it matter?

I think it is important to check balances by math. Of course this means that you check the best playstyles as if players try hard.

McTeddy
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One thing to keep in mind

One thing to keep in mind about using RPS as a base or argument for balance, is that it's usually not talked about in a positive manner. I don't know if I've ever met a designer or player who said it was a good game.

It's generally looked down on because there is no meaningful player choice. You have 3 options, all equal, and no way of knowing what your opponent will choose... it's just a coin flip.

(At the pro-level it's actually a little deeper with predicting your opponents strategy or reading their body... but this is a little out of the scope of most discussions)

The pattern is often used in video games because they are full of micro-decisions. Your opponent builds a rock army so you add paper troops which he counters with scissors and so on.

It's a simple pattern to understand, but it's also not the point of the game. The game is actually about constantly changing tools to overcome whatever your opponent is doing.

Masacroso
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McTeddy wrote:One thing to

McTeddy wrote:
One thing to keep in mind about using RPS as a base or argument for balance, is that it's usually not talked about in a positive manner. I don't know if I've ever met a designer or player who said it was a good game.

It's generally looked down on because there is no meaningful player choice. You have 3 options, all equal, and no way of knowing what your opponent will choose... it's just a coin flip.

(At the pro-level it's actually a little deeper with predicting your opponents strategy or reading their body... but this is a little out of the scope of most discussions)

The pattern is often used in video games because they are full of micro-decisions. Your opponent builds a rock army so you add paper troops which he counters with scissors and so on.

It's a simple pattern to understand, but it's also not the point of the game. The game is actually about constantly changing tools to overcome whatever your opponent is doing.

Im agree. It was only an example. RPS is just purely random game... but it was an extreme example about incomplete information game.

Not all incomplete information game are purely random, in the 1/3 is a pure luck game, or very close (maybe you can use, after all, some complex analysis and psychology as in poker). When probability move from the mode to the extreme it is when the game make sense.

The point for any strategy game is making some plans and try to play it. But to make plans you need a really simple way to do it, other way you are playing a non-strategy game. As I said: chess and go are perfect examples, only masters and very experienced players can play a real strategy game, other people it is just trying random things because to see ahead you need a lot of experience and practice.

Poker and many other games are just the opposite: they are more closer to a pure strategy games... you only play strategic actions, not mechanic-algebraic-computer ones. The same for all kind of incomplete or imperfect information games as Stratego, Bridge, Truco, GOPS, maybe Hive and so on.

pelle
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X3M wrote: Unit A and B are

X3M wrote:

Unit A and B are balanced. Now to see what kind of rock paper scissors can be made with both A and B in an army. By using spreadsheets you know that there is not one super team. But 2A2B beats 3A1B, which beats 1A3B, which beats in turn 2A2B.

Question is how you know that 2A2B beats 3A1B for all possible tactics that both players can use, all different parts of the boards, for all different goals that players want to achieve. Maybe I am happy just blocking the enemy units, preventing them from breaking out of some corner of the board. Maybe the next time I must eliminate them. Maybe the next I need to just bypass them to get to something more valuable behind them. If there is a simple "2A2B beats 3A1B" for all possible combinations of situations it just sounds like a very simplistic game.

This is also why I do not believe in RPS-mechanics. There should be enough going on in the game that a unit A is sometimes able to beat B, sometimes not, and their relative values are different depending on the current situation in the game. That makes interesting units to play with, not units that are A beats B beats C beats A.

X3M
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I never use only RPS. I only

I never use only RPS. I only showed that as example. Just an example is not a game.

There are plenty of stuff to add to games.
Perhaps we should see RPS as an helping ingredient to a game. Not the game itself.

Just purely RPS is madness. We need a story around it. Other factors that are the game.

It might sound strange, but a lot of FPS these days also use RPS. Not as the core of the game, but as an extra. Somehow people love these games. Why? Perhaps they have now better ways in planning ahead. Buying the right weapons. Having enough ammo. etc. You have an economy/planning part. Which is strategy.

Playing a game with RPS, that is not just a random pull of luck. I agree there. But there are plenty of other mechanics to help with this. If it is board or video.

- Scouting, what is the other player building.
- Trying to trick the other player into building something by showing something that you build yourself. Here you use the RPS mechanic against your opponent in an entirely different way.
- If there is a possible battle, fight or run/retreat to better grounds? Perhaps you have created an ambush. You need to look a bit weaker for lurking in your enemy into the trap.
- If a direct fight occurs, which target should be destroyed first. Does the player have a helping tool for that ready? (ex. Event Card)

RPS might steer players into a certain direction. But it is up to the designer to make sure they keep plenty of choices in other area's.

And you can always make a game with several different rocks, papers and scissors. Or a scissor that is also paper.

How strong is the difference between the RPS? That is also important. Especially when you have randomness added.
A>B could be 10>1 or 5>4 or 5434>5433
If B is the weaker one, a player still might use B against A. And if B wins, B can have it's true use further ahead. Like blowing up a building or removing an important card.

However, 10>1 is very extreme. This is a hard counter.
5>4 can be a hard counter, or a soft one. Soft counters are the best to make a game, yet keeping some sort of RPS.
5434>5433 is probably just a result of trying to balance a game where A=B. But we all know that perfect balancing is almost impossible.

Unless you get infinite counters. Like a rock that beats scissors that beats paper,.. no exceptions.
And it is for that reason that people don't like a game purely based on RPS. Because then it is indeed just a random pull of luck.

McTeddy
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X3M wrote:I never use only

X3M wrote:
I never use only RPS. I only showed that as example. Just an example is not a game.

There are plenty of stuff to add to games.
Perhaps we should see RPS as an helping ingredient to a game. Not the game itself.

I agree entirely.

amilsan
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Joined: 10/31/2015
playtesting is unavoidable

first, i understand mathematic to be more related to a model, and then you have the probability

for example:
a game for 2 players, with a deck of 2 colours numbered from 1 to 15, each player chooses his colour

the mechanic is simple
the players draw 3 cards each
actions:
1 - deploy a card on the table
2 - draw a card or cards, according to the sum of values of deployed cards the player has, the total sum of both its the max cards possible that can be in hand
3 - put cards in a reserve deck
4 - attack, this can be done if you have deployed the same number of card that your opponent has, if u win the attack u take that card (from your opponent)
5 - add a card to attack or defence from your hand or the reserve

victory condition: to leave opponent without cards

so if u put less valuable cards you will have less card in your hand, but if u put more valuable cards you have less valuable cards to fight.

the mechanic works, but its incredible boring, even more boring than in the papers. Its long, few decisions, its almost a sum 0 and all the bad things you can say. But until you dont test you cant have the very real criteria.