ssho88
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March 1st, 2021 at 10:06:10 PM permalink
Hi,


How to find covariance of a game ?

There is an example how to find covariance of Blackjack at this link https://wizardofodds.com/games/blackjack/variance/

Firstly, find total variance of N hands per round, and use the formula,

Total variance of N hands per round= N* Variance of 1 hand per round+ N * (N-1) * covariance.

Fro example, variance of 1 hand per round = 1.31 and Total variance of 2 hands per round(through simulations) = 3.565, N = 2,

So, 3.565 = 2 * 1.31 + 2(1)* covariance

covariance = (3.565 - 2.62) /2 = 0.4725.

My question : Is simulations the only way to find total variance of N hands per round ?




Regards

ssho88
ThatDonGuy
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March 2nd, 2021 at 7:11:27 AM permalink
My gut answer is, if you can calculate the variance of 1 hand per round, you can calculate the variance of multiple hands per round, but the number of possible deals for even 2 hands per round may be overwhelming.
Wizard
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March 2nd, 2021 at 1:01:07 PM permalink
I did my covariance articles, including that on video poker by simulation.

Perhaps in multi-card keno one could do an exact analysis. Perhaps I'll work on that.
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CrystalMath
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March 2nd, 2021 at 2:30:22 PM permalink
Quote: Wizard

I did my covariance articles, including that on video poker by simulation.

Perhaps in multi-card keno one could do an exact analysis. Perhaps I'll work on that.



But in keno, it really depends on the overlap of the marked spots. I know people who mark 5 or 6 spots the same on 20 cards and then use 1 or 2 other spots to spread around the screen. In this case, the covariance is really high. Conversely, you could do a quick pick which would give much lower covariance.
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ssho88
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March 2nd, 2021 at 4:41:31 PM permalink
Thanks for your replies.
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March 2nd, 2021 at 6:13:58 PM permalink
Quote: CrystalMath

Quote: Wizard

I did my covariance articles, including that on video poker by simulation.

Perhaps in multi-card keno one could do an exact analysis. Perhaps I'll work on that.



But in keno, it really depends on the overlap of the marked spots. I know people who mark 5 or 6 spots the same on 20 cards and then use 1 or 2 other spots to spread around the screen. In this case, the covariance is really high. Conversely, you could do a quick pick which would give much lower covariance.



I was once out with my boss and we were gambling pretty heavy as partners on video poker. We were down a couple thousand dollars and he asked me for $100 to play video keno. As a numbers guy I was offended that he would play a 9% hold game. He put in his $100 and played four card keno with the same seven numbers on all four cards. He hit for $28,000 and just laughed at me for not giving him $100. He then proceeded to ask me to sign the W2-G and I got the last laugh and it only cost me $14,000.
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USpapergames
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March 2nd, 2021 at 11:59:42 PM permalink
Quote: ssho88

Hi,


How to find covariance of a game ?

There is an example how to find covariance of Blackjack at this link https://wizardofodds.com/games/blackjack/variance/

Firstly, find total variance of N hands per round, and use the formula,

Total variance of N hands per round= N* Variance of 1 hand per round+ N * (N-1) * covariance.

Fro example, variance of 1 hand per round = 1.31 and Total variance of 2 hands per round(through simulations) = 3.565, N = 2,

So, 3.565 = 2 * 1.31 + 2(1)* covariance

covariance = (3.565 - 2.62) /2 = 0.4725.

My question : Is simulations the only way to find total variance of N hands per round ?




Regards

ssho88



What a great question. The answer is no. I don't use computers to solve any math for me except computation & I still always get the numbers right. What your asking for is extremely difficult to teach and requires years of game analyze to identify the mathematical relationships between game mechanics and rules. I can analyze ANY game as long as I am provided the data for it (video games can hide the math behind them very well). Just some examples of the types of data I can get on anything related to: Variance, Probability, Volatility, Standard Deviation, Game Theory, Expected Revenue, & believe it or not even a Skill, Strategy, Luck ratio analysis which I invited since I made some major discoveries in the science of game design. No joke, I can tell you exactly how much luck there is in any game the only thing is I do all my math work by hand that its not worth the effort for me to over analyze. If the game is simple then yes getting the exact value of luck in the game could be important. But any good game is going to be difficult for any player to play perfectly so the amount of perceived luck in a game is often much higher than the actual amount of luck. Which is why I've learned that the ability to calculate estimates that are within your acceptable tolerance is the better approach in the long run. 1st, focus on the game analysis skills then I'll teach you some tricks for getting accurate estimates that way you will intuitively be able to get a sense of any games variance just from experience because your not spending a lot of time of getting the exact numbers. Game design is like engineering, it only needs to be 97% right to get it into production.
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March 3rd, 2021 at 7:02:54 AM permalink
I am interested in this covariance question as well.

In BJ, the dealer's hand is nominally 1/2 of what determines the outcome so the covariance when playing multiple hands against a dealer must be high. I'm not sure what the metric is for covariance, though.

I also struggle with questions such as: if you randomly partition n objects into 4 piles, what is the variance in the size of the piles, or variance in the average deviation of the piles. Clearly the size of the 4th pile is determined by the size of the three previous piles, so there is covariance in this problem.
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marcel55
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March 3rd, 2021 at 9:16:33 AM permalink
Quote: Wizard

I did my covariance articles, including that on video poker by simulation.

Perhaps in multi-card keno one could do an exact analysis. Perhaps I'll work on that.


Well I check that out but there is something missing I think so.
USpapergames
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March 3rd, 2021 at 10:50:18 AM permalink
Quote: gordonm888

I am interested in this covariance question as well.

In BJ, the dealer's hand is nominally 1/2 of what determines the outcome so the covariance when playing multiple hands against a dealer must be high. I'm not sure what the metric is for covariance, though.

I also struggle with questions such as: if you randomly partition n objects into 4 piles, what is the variance in the size of the piles, or variance in the average deviation of the piles. Clearly the size of the 4th pile is determined by the size of the three previous piles, so there is covariance in this problem.



It's funny you mention that because combinatorics plays a HUGE role in determining variance and covariance.
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March 3rd, 2021 at 11:30:29 AM permalink
In doing work on covariance in multi-card keno, I discovered an error in my page on the same in video poker. I just completely redid that page. I welcome all to have a new look.
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USpapergames
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March 3rd, 2021 at 1:02:26 PM permalink
If I ever make enough money to get some free time I'll review all your game analysis & PM you if I discover any errors just because we are friends. On a side note, I shouldn't even be mentioning this but I feel that finding the "true variance" or a game is more important than the actually variance of a game. This term "true variance" is a phase that I coined but the definition of "true variance" has been discussed by severally game designers but no established terms has ever been used. But for game analysis the "true variance" is actually much more important than the actual variance. An example of true variance would be how the game Tic Tac Toe has an actual variance of 255,168 but a "true variance" is only 6!
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March 3rd, 2021 at 1:27:19 PM permalink
Quote: USpapergames

... I shouldn't even be mentioning this but I feel that finding the "true variance" or a game is more important than the actually variance of a game. This term "true variance" is a phase that I coined but the definition of "true variance" has been discussed by severally game designers but no established terms has ever been used. But for game analysis the "true variance" is actually much more important than the actual variance. An example of true variance would be how the game Tic Tac Toe has an actual variance of 255,168 but a "true variance" is only 6!



I'll bite. I guess I'm not in the loop about "true variance" as no game inventor has mentioned it to me yet. Why don't we start with an easy example. What is the true variance of the field bet in craps? Assume a total of 12 pays 3 to 1.
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USpapergames
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March 3rd, 2021 at 1:42:35 PM permalink
Quote: Wizard

I'll bite. I guess I'm not in the loop about "true variance" as no game inventor has mentioned it to me yet. Why don't we start with an easy example. What is the true variance of the field bet in craps? Assume a total of 12 pays 3 to 1.



Great question, true variance only exist when there are dominate strategies in games. So for example, the Shannon number which is an estimate of the actual variance of chess is 10¹²⁰ but the "true variance" of chess is far lower since there are so many moves that competent players will never make (such as fools mate with white moving their pawns to F3 & G4) which makes the true variance of chess always a fraction for the total possible games. It's difficult to classify true variance with a house game since all bets are supposed to result in bad outcomes for the player. So I guess the true variance could be considered 0 since no player should make any bets but if you wanted to justify true variance as a form of maximizing your betting opportunity at a negative sum game then you would say the true variance would be to bet the minimum pass or come with all hard way bets. I hope I got that betting system right, its been a long time since I've played Craps. So you would say the actually variance of craps would be all the dice outcomes with all the different betting options and true variance as all the dice outcomes with just the dominate betting strategies, which would be the total amount of ways its possible to bet the hard ways.

P.S. It's VERY important as a game designers to understand that there really are only 6 different games that can actually be played of Tic Tack Toe, if there is any other game outside of the 6 then a player missed up and wasn't competent when playing.

P.P.S. I almost forgot to include 1 last very important criteria for true variance, you only count the games that are unique, all equivalent games are excluded! So for example if the player makes their 1st move in the center for tic tack toe, all corner moves are identical in board states & are just counted as 1 unique game state.

So I guess the true variance of craps would be just the different ways to bet the hard ways since I don't think there is a better bet than that. So you can bet on 4, 5, 6, 8, 9, 10 and as stated earlier you don't count equivalents. so then its the amount of combinations of ways to bet the hardways which are 6, so craps has a true variance of 216, which are the 6 (hardway selections) x 36 (Dice Outcomes). Since players can only bet 1 hardway, 2 hardways, to 6.

If you really think about it, knowing the actual variance of most games is not really beneficial information in the vast majority of games.

P.P.P.S. I forgot to include the 36 dice outcomes, sometimes I make myself look so stupid :/
Last edited by: USpapergames on Mar 3, 2021
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March 3rd, 2021 at 2:41:18 PM permalink
Quote: USpapergames

So I guess the true variance could be considered 0 since no player should make any bets




Direct: https://www.youtube.com/watch?v=uOoXwxqeVzg
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USpapergames
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March 3rd, 2021 at 2:48:23 PM permalink
Quote: Wizard



OMG SEE I love the movie War Games!!!!!!!! THE BEST MOVE IN NUCULAR WAR IS TO MAKE NO MOVES!!!!!! ASBOSULTLY CORRECT!!!!!!!
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unJon
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March 3rd, 2021 at 3:10:29 PM permalink
Quote: USpapergames

If I ever make enough money to get some free time I'll review all your game analysis & PM you if I discover any errors just because we are friends. On a side note, I shouldn't even be mentioning this but I feel that finding the "true variance" or a game is more important than the actually variance of a game. This term "true variance" is a phase that I coined but the definition of "true variance" has been discussed by severally game designers but no established terms has ever been used. But for game analysis the "true variance" is actually much more important than the actual variance. An example of true variance would be how the game Tic Tac Toe has an actual variance of 255,168 but a "true variance" is only 6!



Iíll bite farther. Why isnít the true variance of tic tac toe 0? Since all 6 games that should be played end in a tie. So the true outcome is 100% tie there is no true variance.
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March 3rd, 2021 at 3:14:07 PM permalink
Quote: unJon

Iíll bite farther. Why isnít the true variance of tic tac toe 0? Since all 6 games that should be played end in a tie. So the true outcome is 100% tie there is no true variance.



I think USPG will say there are various ways for two perfect players to tie, as evidenced in War Games.
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USpapergames
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March 3rd, 2021 at 3:21:15 PM permalink
Quote: Wizard

I think USPG will say there are various ways for two perfect players to tie, as evidenced in War Games.



Someone is a quick learner ;) Couldn't have said that better myself!

The true variance of craps is 0 but if the player is forced to play then it's 216.
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March 3rd, 2021 at 3:29:14 PM permalink
Quote: Wizard

I think USPG will say there are various ways for two perfect players to tie, as evidenced in War Games.



That is a definition of variance (forget true variance, stick to actual variance) that doesnít align with common usage I think. Go back to craps and instead of a field bet use a place the 6 bet. There are infinite ways that the place bet can win (next roll 6; next series of rolls 2, 6; next series of rolls 2, 3, 6; etc.) and infinite ways to lose. Is the actual variance of the place 6 bet then infinite?
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USpapergames
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March 3rd, 2021 at 3:44:11 PM permalink
Quote: unJon

That is a definition of variance (forget true variance, stick to actual variance) that doesnít align with common usage I think. Go back to craps and instead of a field bet use a place the 6 bet. There are infinite ways that the place bet can win (next roll 6; next series of rolls 2, 6; next series of rolls 2, 3, 6; etc.) and infinite ways to lose. Is the actual variance of the place 6 bet then infinite?



Ah, I think we might finally be getting to the core of the thread discussion for Variance = Luck. I'm afraid it might take an essay of writing to definitively teach these concepts, but trust me they are worth defining and will change how you perceive game engines. the vast majority of the game terminology that I use has roots to mathematics. But variance and luck are seldom ever defended in game design books which is just mind boggling to me.
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teliot
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March 3rd, 2021 at 4:13:51 PM permalink
Quote: USpapergames

Ah, I think we might finally be getting to the core of the thread discussion for Variance = Luck. I'm afraid it might take an essay of writing to definitively teach these concepts, but trust me they are worth defining and will change how you perceive game engines. the vast majority of the game terminology that I use has roots to mathematics. But variance and luck are seldom ever defended in game design books which is just mind boggling to me.

By game design books I assume you mean either Dan's book or my book since those are the only two books on the topic. I cover variance in great detail in my book.
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March 3rd, 2021 at 4:17:01 PM permalink
I'm going to help define variance but I know its going to lead to a length discussion but if I don't nobody will understand what I'm talking about :/ Does anyone know what the root to the word variance is? It's from the word variation. In mathematics we use the word variables which is also derived from the word variation. Basically the definition of variance is the sum of all possible games that can be played under the rules and mechanics of a game. That's how math defines variables and that's how variance has been defined ever since it was 1st applied to games. Problem is not many game designers are mathematicians & the term was applied from game designers that were mathematicians. So the bias of these game designers that didn't understand the math have been passed down in definitions for generations :/

Many people fail to apply player actions to games but player actions result in different game states so they defiantly are a part of variance but most people just figure out the total sum of outcomes of the game engine(s) and this is a mistake.
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USpapergames
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March 3rd, 2021 at 4:21:15 PM permalink
Quote: teliot

By game design books I assume you mean either Dan's book or my book since those are the only two books on the topic. I cover variance in great detail in my book.



I'd love to read it. Were can I find it? And your absolutely correct but I was actually talking about game design books in general. There are plenty of game design books that are as good as Dan or better but Dan's book is the best for casino game design. But if you want to succeed at game design you need to understand the core science to games as a whole. Having a solid foundation of the science for game design has done wonders for helping me analyze games.
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teliot
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March 3rd, 2021 at 4:24:30 PM permalink
Quote: USpapergames

I'd love to read it. Were can I find it? And your absolutely correct btw :)

you can find a used copy on Amazon.
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unJon
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March 3rd, 2021 at 4:25:56 PM permalink
I see. You are using a word that has two meanings depending on the field of study. In a game, itís the experience that matters so you use ďvarianceĒ to mean all the different ways the game could play out.

For gamblers wondering what the ideal bet size should be as a fraction of their bankroll, they use variance of outcomes in the statistical sense. It doesnít really matter all the ways that the game plays out, all that matters is the distribution of outcomes as the bet resolves.

I wouldnít necessarily say one use of the word is correct and the other incorrect. Itís just two different contexts.
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USpapergames
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March 3rd, 2021 at 4:28:39 PM permalink
Quote: teliot

you can find a used copy on Amazon.



I will defiantly have to get a copy. I just checked out your website. Nice job documenting that chess tournament.
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March 3rd, 2021 at 4:31:09 PM permalink
Quote: unJon

I see. You are using a word that has two meanings depending on the field of study. In a game, itís the experience that matters so you use ďvarianceĒ to mean all the different ways the game could play out.

For gamblers wondering what the ideal bet size should be as a fraction of their bankroll, they use variance of outcomes in the statistical sense. It doesnít really matter all the ways that the game plays out, all that matters is the distribution of outcomes as the bet resolves.

I wouldnít necessarily say one use of the word is correct and the other incorrect. Itís just two different contexts.



Wow, I swear I have never heard the term variance ever applied in that manor. Sir, you are talking about game volatility which is a completely different subject :(
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unJon
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March 3rd, 2021 at 4:35:32 PM permalink
Quote: USpapergames

Wow, I swear I have never heard the term variance ever applied in that manor. Sir, you are talking about game volatility which is a completely different subject :(

Agree itís a different subject. But not a different word apparently. Variance to most people on this board is most commonly defined to mean the square of the standard deviation of a probability distribution.
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March 3rd, 2021 at 4:37:26 PM permalink
Quote: unJon

Agree itís a different subject. But not a different word apparently. Variance to most people on this board is most commonly defined to mean the square of the standard deviation of a probability distribution.



Yes, that's variance in statistics. Ok, now I see what the problem is! You are comparing the variances of a players expected return as if that's the variance of the game when the game is obviously more than just financial return lol. Let's make this clear, I'm not talking about variance of your bank roll, I'm talking about variance of the game.
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March 3rd, 2021 at 4:43:09 PM permalink
Quote: USpapergames

Yes, that's variance in statistics. Ok, now I see what the problem is! You are comparing the variances of a players expected return as if that's the variance of the game when the game is obviously more than just financial return lol. Let's make this clear, I'm not talking about variance of your bank roll, I'm talking about variance of the game.



I know. Thatís what I said above. Itís now very clear to me why you and lots of other people were having the variance vs luck debate. You were mostly talking past each other by meaning different things with the word ďvariance.Ē
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March 3rd, 2021 at 4:43:17 PM permalink
Quote: USpapergames

I will defiantly have to get a copy. I just checked out your website. Nice job documenting that chess tournament.

the way you wrote about it, it sounded like you had read all the books already. You made it sound like you know precise details about what all the books covered on the topic of variance. Now it turns out there's two books and you haven't read one of them. Maybe you haven't read either.

When it comes to you telling me what I have in my book I think it's time to hold you accountable just a little bit.
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March 3rd, 2021 at 4:48:43 PM permalink
Quote: teliot

the way you wrote about it, it sounded like you had read all the books already. You made it sound like you know precise details about what all the books covered on the topic of variance. Now it turns out there's two books and you haven't read one of them. Maybe you haven't read either.

When it comes to you telling me what I have in my book I think it's time to hold you accountable just a little bit.



I'm not going to make any judgement about the knowledge that is within your books until I read them, promise ;) It was Noah Dean that introduced me to Dan's book and back then I thought there was virtually no info on the subject lol. Who knows maybe your books have all the answers and you beat me to market before I could publish my own.
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March 3rd, 2021 at 4:51:32 PM permalink
Quote: unJon

I know. Thatís what I said above. Itís now very clear to me why you and lots of other people were having the variance vs luck debate. You were mostly talking past each other by meaning different things with the word ďvariance.Ē



Yep, your right. Makes complete sense now. I wish I could have known Mission's definition 1st before making that thread :/
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March 3rd, 2021 at 5:00:36 PM permalink
Quote: unJon

I know. Thatís what I said above. Itís now very clear to me why you and lots of other people were having the variance vs luck debate. You were mostly talking past each other by meaning different things with the word ďvariance.Ē



Wait, wait, wait, Stop! How were you not confused about the tic tack toe game? How could you possibly apply the meaning of a players bankroll fluctuations to a game that has absolutely no expected return since its not a gambling game and if it was its not in the rules and certainly isn't going to have whole number values like 6 lol.
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March 3rd, 2021 at 5:15:34 PM permalink
Quote: USpapergames

Wait, wait, wait, Stop! How were you not confused about the tic tack toe game? How could you possibly apply the meaning of a players bankroll fluctuations to a game that has absolutely no expected return since its not a gambling game and if it was its not in the rules and certainly isn't going to have whole number values like 6 lol.

Reread my first post in the thread. I was confused and I asserted I thought the true variance should be zero.
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March 3rd, 2021 at 5:37:53 PM permalink
Quote: unJon

Reread my first post in the thread. I was confused and I asserted I thought the true variance should be zero.



Your right, I completely forgot that you said the variance was zero lol.
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March 3rd, 2021 at 6:11:40 PM permalink
So I guess I will give you guys some examples of variance and luck in games. Its a rather basic example but just from this basic example you can see that luck and variance are usually correlated but there is not cause link between the 2 and you can have games were you change the luck and don't effect the variance and vice versa.

So lets play a game called DIE. The rules are that the player can bet on any single dice outcome & if the dice outcome matches they make 6x their return and if not then they lose their bet. The mechanics of the game are that the player gets to roll a single dice (using rules that eliminate dice control like a craps table or dice cup). So the variance of the game is 6 x 6 & the luck in the game is 1/6. Now lets use that same example but we are going to make 1 modification. We are going to change the probability outcomes of the dice (we could save we shaved a corned of the dice or what ever excuse to make the distribution probabilities unequal) so that 1 is 1.5x more likely to occur than 2-5 & 6 is .5 as likely to occur. Now the luck in the game is considerably different at 1/4 since every player should be betting on outcome 1 since its the most likely to occur BUT variance has not changed at all since there are still only 6 player actions and 6 game engine outcomes.
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USpapergames
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March 3rd, 2021 at 6:16:23 PM permalink
Anyone interested in helping me publish my game design book. I want to call it the "Game Designer's Bible".
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SOOPOO
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March 3rd, 2021 at 6:29:40 PM permalink
Quote: USpapergames

Anyone interested in helping me publish my game design book. I want to call it the "Game Designer's Bible".

. I am sure there are dozens of publishers you could pay to have your book published. Iím not sure how much it will cost you but shouldnít be hard to find a publisher.
USpapergames
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March 3rd, 2021 at 6:41:59 PM permalink
Quote: SOOPOO

. I am sure there are dozens of publishers you could pay to have your book published. Iím not sure how much it will cost you but shouldnít be hard to find a publisher.



Sorry, my lpd kicked in again. I need an editor or maybe even a ghost writer. I think I'm using the right words this time :/

I might just need a Co-Writer? I feel so ignorant on this subject like I have no clue what I need, too embarrassing.
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USpapergames
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March 3rd, 2021 at 6:54:54 PM permalink
BTW I was waiting for someone to say that it was impossible for me to calculate any game variance since the variance of games like GO and Chess have not been figured out yet. My reply was going to be that I "could" figure out the variance to those games if I was given enough time ;)
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USpapergames
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March 3rd, 2021 at 7:05:55 PM permalink
https://drive.google.com/file/d/1utfiho2tuzLeC1AhzUfqt3g-HT0mzoiG/view?usp=sharing

Check out my 1,774 unread emails and growing everyday. Can't keep up, sometimes I see the emails and read them and sometimes I don't use my email for days and don't know what to do about it :/
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Wizard
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USpapergames
March 3rd, 2021 at 7:48:44 PM permalink
Quote: USpapergames

It's funny you mention that because combinatorics plays a HUGE role in determining variance and covariance.



I was just curious -- that is 29 posts for you so far today and still counting.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
USpapergames
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March 3rd, 2021 at 8:03:22 PM permalink
Quote: Wizard

I was just curious -- that is 29 posts for you so far today and still counting.



It's all about quality over quantity ;)
Math is the only true form of knowledge
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