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

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

Quote:WizardI 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.

Quote:CrystalMathQuote:WizardI 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.

Quote:ssho88Hi,

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.

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.

Quote:WizardI 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.

Quote:gordonm888I 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.