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Board game design seems like mathematical modeling

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larienna
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TL;DR: Maybe mathematical modeling techniques could be used in board game design when searching and creating mechanisms or when stuck in a design.

In spring I had a very annoying and frustrating math class about derivative/integral calculus. Normally when I do math, I learn a technique and apply the process, but in this class, for what ever reason, the feeling was very different. You had to be creative about solving limits for example which felt almost like if you had to invent the wheel on your own.

Now from my experience it seems easier to design video games than board games which seems to have more trial/error and some struggling to find the right mechanism for the game. And I have been wondering for year about why I would have an harsh time designing board games compared to their digital cousin, and I think I have found the reason.

It would be that board game design would be more closer to mathematical modeling than video game design and that I am not used to use mathematics creatively. Here is a concrete example, Let say you want to solve this simple problem:

"What are the odds to roll 5 or 6 on a D6"

Before the invention of computers, there was 2 ways to solve this: A) Create a mathematical model to find the answer, B) make many die rolls and sum the stats. The problem with method B is that not only it takes a lot of time, but it would also give an imprecise answer. If you use method A, you would have found that the odds can be found by using a simple formula as: Desired number of faces OVER total number of faces.

There is 2 things to point out here:
- first: The face value is not used in the math formula, which means that the math model allows to solve a reality problem, without using reality exactly as it seems.
- Second: When you know the formula, you can easily apply it. But when you don't know the formula, it's a bit more complicated to solve the problem.

Now how can it be transposed to board game mechanics:
- The mechanics of board games does not always reflect reality accurately. For example, in the android board game, you try to incriminate or innocent people rather than trying to find the murderer. In the ends, it gives a similar feeling to what reality could be like without being a perfect replication.
- Then once you know a mechanism, it's easy to apply it to other game ideas but when you don't know, it's harder to design your own mechanism. So for example, if you know deck building games, it's easy to apply the mechanics to another game, but much harder if deck building have not been invented yet.

But if you asked me, how to find the odds of the math problem above, my reflex would be to roll the die multiple times and count the stats. And I know that ... because I actually did it at least once during my youth. Why would I prefer this method, well possibly because I am a practical person, and therefore have an hard time representing reality abstractly. Which could in the end explain why I can barely design new mechanism in board games.

I do have a fascination for board games which could be comparable to my fascination for math. Like I said to a math teacher (or almost), in computer programming, any problem can be solved, but math allow solving the same problem more efficiently and elegantly. So it's hard in this context not to love math, but it does not mean I am capable of solving problems by creating math models by myself.

So I think this is where my limitation lies, and it would explain why many game designers are very good in math. It's not because they necessarily need to know how they compute odds, it's because the process to design board game mechanism is very close to solving mathematical problems.

So for me, the tips I could give myself is focus on video game design and create board games using only mechanisms that already exists. But still, I am wondering, even if I do not have the flair for abstraction, is there some way I could help myself when I am stuck with a board game idea.

This is where the calculus story comes back into play. The books I used in the class was written by James Stewart who seemed pretty lunatic as he said that derivative calculus was ... beautiful. Now one part of the book explained various methods he used to solve mathematical problems creatively. I thought that maybe those techniques could be transposable to board game design if math and BG mechanics are so close to each other. I googled a bit and found this page:

http://stewartmath.com/media/5_inside_focus.php#

which shows some example where he use his problem solving techniques. I'll list what's on the page bellow, but I think there was more techniques originally found in his book. Still, it would be interesting to see how those techniques could be used to design board game mechanics.

- Recognizing something familiar/patterns
- Taking Cases
- Analogy
- Introduce something extra
- Indirect reasoning
- Drawing a diagram
- Working Backwards
- Establishing subgoals
- Mathematical induction

I can see a few application in board games, for example: recognition of patterns could try to map a reality behavior with a game behavior. Introducing something extra consist in adding component to the game to allow expanding the possible mechanics or allowing new interactions. Drawing a diagram can sometimes make reality be seen in a different way (Reminds me of rummy-kub which was displayed in a video game as a 4 by 13 grid for tiles played).

What do you think?

Have you used similar techniques when designing board games?

Do you think the board game design is related to math problem solving?

jfeast
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Great Link

You have included a really cool link http://stewartmath.com/media/5_inside_focus.php#
Thanks!

X3M
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Great post

I often search for math tricks. So that I can simplify AND improve the game. Yet the whole complex math stays hidden as much as possible.

The main goal for all that math is to keep balance. Yet for allowing a dynamic game play.

I could give examples. But that would be a TLDR post. You can find some of them and the search for it, in BGDF.

Edit:
I like to add, that the phsycological aspect of board games. Are very important. And that while mathematical the game looks perfect. The human mind might perceive it different.
A nice math curve that steadily progresses from 0 to 100%, might be a yes/no situation in the game.

gxnpt
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process

- Recognizing something familiar/patterns
- Taking Cases
- Analogy
- Introduce something extra
- Indirect reasoning
- Drawing a diagram
- Working Backwards
- Establishing subgoals
- Mathematical induction

Sounds about right to me. The sequence is not rigid, but that is a good description of general problem solving, not just game design.

FrankM
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Mathiness

There is a branch of math called Game Theory that takes these concepts and makes them precise. Precise as in complete with mathematical proofs that could make your eyes bleed.

The basic notion is that players ought to take opponents' reactions into account when choosing a move. Random elements are represented by a player called Nature who always chooses moves according to a probability distribution (for example, rolling one d6 shows up as Nature choosing a number 1-6 each with 16.67% probability). Without going off the deep end with extensive-form notation and Nash equilibria, you want to avoid what are called "dominant strategies" and "dominated strategies."

A dominated strategy is a potential move that no rational player would ever make, while a dominant strategy is one that every rational player would make. If there is never a time to use a particular move, it shouldn't be in the game. If there is always a "right answer" every turn, you've actually removed decisionmaking from the player (which could be fine for a kids' game).

Don't rely too heavily on Game Theory, though, because it describes rational behavior and we know that players are never purely rational. There are theoretical games (in particular the Ultimatum Game) where human players systematically deviate from what the math tells them they should do.

larienna
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I know math is used a lot in

I know math is used a lot in game design, but here what I am trying to state is that game design process is math. If I have problems using math creatively, those difficulties would transpose to board game design.

For a comparison, you could say that board game use art, but the design of a game itself could also be a piece of art even if no art is used in the game.

X3M
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Perhaps you are trying to say

Perhaps you are trying to say that you only feel secure about your designs. When there is logic to them?
And that you don't see an easy way, doing without this logic?

I guess we share a common ground in that.

You are absolutely right if it comes to math being used as a tool. Yet there are so many designers that do without.

Most of them do trial and error, which cringes me. Yet they get the job done. And they do this without actively using math.
They guess. And I think this is because they work from the human nature as focus point.

Dare I call it them using their instinct?

So while some will agree, there are others who disagree. But I feel that it is a bit of both.

***

I like your comment about a board game being an piece of art itself.

***

At the beginning of your first post, you talked about derivatives and integrals. I feel I need to warn you about the trap of these. They are not useful tools.

larienna
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Quote:Most of them do trial

Quote:
Most of them do trial and error, which cringes me.
Yet they get the job done. And they do this without actively using math.

This is what I need to rely on myself and I hate it. It's very unpredictable and unproductive. It multiplies development time generously. This is why I am thinking more in doing video game design.

Quote:
At the beginning of your first post, you talked about derivatives and integrals.

It's the first time I have been confronted in using creativity in math, but I am not sure if it's because the teacher was bad, or if it was because the subject required it. I agree that I see little use to it even in video game programming unless doing physics simulation. Most useless math class I ever had.

X3M
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Not just math tools, but which one?

For probabilities, I have made a step backwards from simulations (super fast trial and error). Which is 1 trial and error in the big picture. Instead, I am learning, how to use Excel and Anydice.com more effectively.
I only have examples for war games in this matter.

***

Regarding the integrals for video games. That is a different matter then board games.

When some one falls down, to their surface/injury/death. That has to be calculated before the drop finishes. I think that is what they want to teach you. And if it is a landing in water, how deep?

In an old Minecraft version, you found yourself inside the ground. Then placed back on the surface with some damage. It was funny if you fell to hard, then you where placed in the wrong spot too. :)
It is nostalgic for me, since I tried to make a "Artillery" clone, back when I was a teenager. The cannonball ended up inside the ground, all the time. And I didn't even know how to correct that, back then.

There is, for all I know, no example for board games in this matter.
I find myself using Summation equations more often then other calculations.

https://en.wikipedia.org/wiki/Summation
https://en.wikipedia.org/wiki/Mathematical_induction

I know, I need to get better at those. But I am feeling an idiot at the moment when applying.
I want to learn how to put summation in actual formula's. Without having to create the table, then put a program to use to give me the formula.

FrankM
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Derivatives and Integrals

For the specific examples of derivatives and integrals:

A derivative is a description of how something is changing. An example is a speedometer, which describes how quickly your position is changing (assuming your tires aren't spinning uselessly in the mud, but that's an implementation detail that's not so important here). People handle simple derivatives instinctively (an object that appears to be enlarging is moving toward me, and I can usually guesstimate when it will get here). There are plenty of situations when a derivative would be helpful for the "board AI" to craft an appropriate response to player actions (e.g., attack the unit that moved the furthest this turn), but the complexity of the rules involved mean this is typically the province of videogames.

There are higher-order derivatives like acceleration (rate that speed is changing), jerk (rate that acceleration is changing) and snap (rate that jerk is changing), but human senses don't process these well which means they don't belong in a board game unless there is a pressing need to simulate specific physics. And even then, you would probably do it in terms of the more familiar first derivative "speed" (for example, tracking the current speed of a rocket in space, then thrust points can change the speed).

An integral is trying to back out a thing's state when all you have are the changes. I can't think of any situation where this would be appropriate in a board game. If you need to recall the state later, just keep a record of the state. Players might want to do some simple integration in their heads to make projections, but that's on them. By the way, summation is a special case of integration, so I'm curious how this came up in a game design.

For mathematical modeling in general:
A big part of designing a game is getting the simulation aspect as simple as possible while maintaining the essential elements. You can hide a lot of complexity in a physical component, with dexterity elements being the ultimate shortcut. You could work out the sci-fi physics of a tractor beam in exquisite detail and present it as a dice-vs-dice challenge with a laundry list of modifiers... but it's a lot more fun to try to toss a ring around the enemy pawn from your side of the board.

larienna
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Quote:A big part of designing

Quote:
A big part of designing a game is getting the simulation aspect as simple as possible while maintaining the essential elements.

Which is similar to the concept I used to call: "Fitting an elephant into a shoe box".

This is why I want to focus more on video game, because I only have to cut and abstract the things I want. I am not forced to do it, I do it because I find the design more elegant.

Which would give me more flexibility on how abstract or detailed I want the game to be. For board game, I have bare minimums to meet for the game to be simply playable.

larienna
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It feels weird to say, but it

It feels weird to say, but it seems that in order to design a board game, I would need to design a video game first, play that game intensively and then abstract the game as a board game.

It's a lot of work to do a simple board game. Else I could see some financial benefits of publishing the video and the board game of a same concept. People who know the video game might also play the board game.

Jerry
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Math is hard

Your comment about game designers usually being good at math stuck out to me.

For me, I've always struggled with math, all through my life. It's made it that much harder to balance games because of this. One of the bigger problems I run into is, (For example) How many cards do I put into the game? When designing a deck of cards, how many of each type of card? How many types of cards? What effects should they have? Figuring out odds has never been my forte.

Most of my games are balanced by testing numerous times. I will put a mechanic into the game, and if it doesn't work, I will tweak it slightly. This is rather than try to "math" the problem away.

X3M
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Why even go for the board

Why even go for the board game, if you have a playable video game? It almost sounds like that you are stuck between 2 "worlds".

Perhaps you need to conferm you design for yourself through video games.
A positive side to all this is that you can test your game. Then adjust, until it feels right. And until it feels like a board game that is turned into a video game.

larienna
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Quote:Why even go for the

Quote:
Why even go for the board game, if you have a playable video game? It almost sounds like that you are stuck between 2 "worlds".

There could be an interest in playing a board game of a video game.

- First to get a similar but different experience. For example Xcom.
- Second, be able to play the game in multiplayer face to face with more social interaction.
- third, get to play a similar game in a much shorter time frame.

I don't say it would apply to all type of games, but it could be interesting.

I want to focus more on video game design, but I am currently on a forced break because of my wrist wounds.

Still most video game turn based strategy could be designed around a Relational Database. This is what makes it much more easier for the design aspect.

It's like if I said that all board game ideas can be implemented as deck building game. In that case, there is no need to search for mechanisms, you know you are going to use deck building.

lewpuls
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Three kinds of games and game

Three kinds of games and game fans: math, people, story
https://youtu.be/8680nw61VWM

I confess, the more mathematical a game becomes, the more it's like a puzzle, not a game.

The Professor
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Don't confuse two ideas

Jerry,

While math can certainly undergird your design, I would take caution if you think math will necessarily get you to a balanced game.

For instance, in my game, TAU CETI, each asymmetric race possesses benefits, but I will not portray that they are balanced, but provide the perception of balance. To Dr P's point, this is why certain games are simply solveable, which is not want I want to design, much less play.

Cheers,
Joe

questccg
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I know FUN is relative

lewpuls wrote:
Three kinds of games and game fans: math, people, story...

I really don't care about Math, People or Story. In my mind it kind of goes like this: is the game FUN to play??? Usually this means cool and innovative mechanics... Some "power struggles" between players... And a tiny bit of randomness "to make it just a little less predictable".

I don't like purely deterministic combat. Maybe Magic has the monopoly on this... And the mechanics work real well with the cards... But in most cases I don't want the cards to 100% dictate the outcome of a battle. I want some dice action to take into account "other" variables that come into play.

But Magic may be deterministic, there are a BUNCH of "take-that" cards you can play to offset the determinism and make it much LESS predictable. All those +1/+1 and Flying, etc. cards make it more variable who will win the battle. And so I really LIKE "take-that" cards (Magic Instant cards). And if you have those cards to affect the outcome of a battle.

So I don't care about "Math", "people" (somewhat: is everyone enjoying the game? I would hope to play with people who are open-minded... Not like "I don't like games with dice...") and "story" (is something that I would like to get more of - be more involved in creating my OWN stories for people to enjoy!)

I played "Metro" at a game night. And we were all laying tracks to earn points. I came in 2nd during that game... And I was like: "So?" Did I have FUN? Not really... I'd rather play "TradeWorlds" ... or maybe "Hero Realms" - TBD!

X3M
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lewpuls wrote:Three kinds of

lewpuls wrote:
Three kinds of games and game fans: math, people, story

Is the game math?
Design to let the opponent be the puzzle, not the game.
Thus mix math and people; to make a story.

***

Imho, the math can have 2 faces. Deterministic and Randomness.

The determinism is good for making puzzles.
The randomness is good for introducing risk.

You need to include people for both faces.

How fast, good, can a person solve the puzzle?
How willingly is a person to take the risk?

I guess your comment can go either way with this. When I go math, I choose the latter, not the puzzle.

***

1 question though.
Where does skill fit in? People?

john smith
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"Math is hard"A large

"Math is hard"

A large portion off all the bogged down complexity chatter comes to this. That's why we have calculators, PC's and consoles.

"Stuck between worlds"
I see allot of board game design talk where this appears to be the case. They want al the automation of a Computer on a baordgames. The close you will come to that is a heavily scripted Tablet top simulator mod or an App like Monopoly on your phone.
You can't fit a square peg in a round hole. If you want automation, then Video games are your destination.

"Fun is Relative"
A fact many in Board game Design conversations do not accept. If I had a dime for everybody that told me what is fun,Id be rich.

http://www.mathgoodies.com/articles/math_anxiety.html
You can get paralyzed by trying to jump through hoops for people that advise on what direction the market goes or. Or get obsessed with "complexity". But there's likely a market for anything you make to some degree. If you let Math Anxiety rule, You will have allot harder time of it.

john smith
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"Where does skill fit

"Where does skill fit in?"
Another theory my room mate has. That people so do not like to be beaten. "Simulations" are thought to require skill. If the game is all about random chance then your opponent doesn't beat you, some random bad luck does. Much easier on the eqo. I believe that's why wargames have pushed this chaos theory on their games so hard.

X3M
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john smith wrote:"Where does

john smith wrote:
"Where does skill fit in?"
Another theory my room mate has. That people so do not like to be beaten. "Simulations" are thought to require skill. If the game is all about random chance then your opponent doesn't beat you, some random bad luck does. Much easier on the eqo. I believe that's why wargames have pushed this chaos theory on their games so hard.

This is true.
But what skills does a player need in a wargame?
Finding the best chances and work on that, is my pick.
Further, good troop placement if the game allows for it.
And math helped me balancing that.

Math also can help finding the chaos factor more easily, if you will. I always felt it was a bit too low for my game. I had an sd of 1.11 yet needed more. It got lifted to 1.49 after changing the dice roll.
It might sound too technical. But basicly it means. I found a way to shift the determinism in the battle more to a "second round needs new consideration", type of game.
The whole game changed regarding strategies. And it was thanks to math to pick the right die rolls. Having a number of playtests regarding this matter would have taken years?

larienna
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Quote:But what skills does a

Quote:
But what skills does a player need in a wargame?

Games with very high randomness can be as skillfull as game with no randomness. Experience will tell the players which move, formation, position, strategy is more effective than others. Because both type of games is less or not influenced by luck.

---------------------------------------
From another thread:

I think I found a good analogy

Video game mechanics is like a text file. You can Add, remove and change text without worrying much.

Board game design is like a compressed text file. It's much more complicated to modify the content since a word could be spread around the file, or linked to other letters. While new additions must be compressed first before beign inserted.

john smith
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Well wargaming used to

Well wargaming used to require knowledge of the armies and the History . I don't think that's as much a part of it now.

They are more like Fictional War movies as opposed to hypothetical battle analysis as they used to be.

They went backwards to the exploit the rules tactics. So I supposed detailed rules knowledge is the best skill.

questccg
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Where is the FUN???

john smith wrote:
Well wargaming used to require knowledge of the armies and the History...

But I fail to see what "re-creation" of existing battles do anything towards a game experience. If party "X" won battle "Y", why would you want to re-create that battle -- with the same outcome?

I guess I fail to see the FUN in it... You already know who will win and it's just a matter of semantics to go through the motions ... so to speak.

X3M
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That is why I like fictional

That is why I like fictional battles.
New units.
New story.
New mechanics.
New rules.
New RPS if given.
And the outcome of a battle is unknown.

FrankM
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Hey, if *I* was in charge they would have won!

questccg wrote:
john smith wrote:
Well wargaming used to require knowledge of the armies and the History...

But I fail to see what "re-creation" of existing battles do anything towards a game experience. If party "X" won battle "Y", why would you want to re-create that battle -- with the same outcome?

I guess I fail to see the FUN in it... You already know who will win and it's just a matter of semantics to go through the motions ... so to speak.


Never really got into this genre, but one of the things that makes a battle exciting is that it could have turned out differently. If the historical conflict was really lopsided, all you really need are asymmetric goals.

OGRE was historical, right? :-)

larienna
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I think historical settings

I think historical settings should only be a setting, which means a set of constraint due to historical and technological reasons.

But there are many war games, like war riven war games, that tend to put players on train rails forcing the game to happen almost how it did historically. That, from my point of view, if very boring because it makes all games alike.

FrankM
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Is it considered a "game"?

larienna wrote:
I think historical settings should only be a setting, which means a set of constraint due to historical and technological reasons.

But there are many war games, like war riven war games, that tend to put players on train rails forcing the game to happen almost how it did historically. That, from my point of view, if very boring because it makes all games alike.


This sounds more like a reenactment, which some people find fun for reasons that elude me.

The "modeling" in that case would be making the outcome come out "correct" despite any randomness in the mechanics.

X3M
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FrankM wrote:larienna wrote:I

FrankM wrote:
larienna wrote:
I think historical settings should only be a setting, which means a set of constraint due to historical and technological reasons.

But there are many war games, like war riven war games, that tend to put players on train rails forcing the game to happen almost how it did historically. That, from my point of view, if very boring because it makes all games alike.


This sounds more like a reenactment, which some people find fun for reasons that elude me.

The "modeling" in that case would be making the outcome come out "correct" despite any randomness in the mechanics.


There are some games that do this, but they are being adjusted for a fair chance on both sides. This means that both players have the same chance in winning. While the scenario is an replay of historical battle's.

Only those, I find interesting. Since you can "try" to re-enact the event. Or try something different to see what might happen. The best games imho are those where the player can find a better strategy than what happened.

lewpuls
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"Fun" is a mostly-useless word

The point is, what's "fun" (a word I do not use) differs immensely for fans of different kinds of games. Fun is purely personal, subjective. You can pay attention only to your personal idea of fun, but that makes it hard to discuss game design, does it not?

questccg wrote:
lewpuls wrote:
Three kinds of games and game fans: math, people, story...

I really don't care about Math, People or Story. In my mind it kind of goes like this: is the game FUN to play??? Usually this means cool and innovative mechanics... Some "power struggles" between players... And a tiny bit of randomness "to make it just a little less predictable".

I don't like purely deterministic combat. Maybe Magic has the monopoly on this... And the mechanics work real well with the cards... But in most cases I don't want the cards to 100% dictate the outcome of a battle. I want some dice action to take into account "other" variables that come into play.

But Magic may be deterministic, there are a BUNCH of "take-that" cards you can play to offset the determinism and make it much LESS predictable. All those +1/+1 and Flying, etc. cards make it more variable who will win the battle. And so I really LIKE "take-that" cards (Magic Instant cards). And if you have those cards to affect the outcome of a battle.

So I don't care about "Math", "people" (somewhat: is everyone enjoying the game? I would hope to play with people who are open-minded... Not like "I don't like games with dice...") and "story" (is something that I would like to get more of - be more involved in creating my OWN stories for people to enjoy!)

I played "Metro" at a game night. And we were all laying tracks to earn points. I came in 2nd during that game... And I was like: "So?" Did I have FUN? Not really... I'd rather play "TradeWorlds" ... or maybe "Hero Realms" - TBD!

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