Interesting problem statement and good analysis. If I am understanding correctly then this is continuous space discrete time. How might the problem change if time was also continuous, meaning card-draw would be integration of some random continuous function, rather than discrete samples?
I can't seem to access the server with the article right now, but generally the easiest way to get there is as the limit of the discrete case as the timestep δ -> 0 and the payoff scaled down so they still add up to the same macro conditions.
I suspect it would be the same but less interesting, though. The smaller you make the increments, the closer the optimal stopping point is to 21. At the limit, the stopping point would be 21, and -- I think -- all N players plus dealer would have an 1/(N+1) chance of reaching it.
Basically, without jumps, it just becomes a situation with all players waiting for a continuous counter to reach 21, and then stopping precisely as it does. The one who was lucky to have the fastest counter (on average) wins. (With jumps as in discrete Blackjack, the fastest counter not necessarily the best!)
What might make it more interesting is if you did introduce discontinuities in the otherwise continuous process. So basically each player is fed with what looks like the absolute values of stock price movements, and has to stand somewhere. That should be easy to simulate and maybe even fun to play!
If there wouldn’t be a delay between observing the current value and stopping (assuming a reasonably well-behaved distribution P), that problem is trivial: stop when you reach the threshold.
So, you would have to add some time delay between observation of a value and actual stopping.
Intuitively, I can think of two stopping strategies that might be optimal: call ‘stop’ when the expected value at stop is that ≈ 0.5706 or call stop when the probability of overshooting it is 50%.
I’m fairly sure that intuition is wrong in general, though.
What is going on? I remember seeing this several days ago, which the search feature confirms[1], but now, it appears again with the same story URL but is now only two hours old? With all comments forward-dated too?
so even if everything in that paper is VALID mathematically,
in reality it will not work because blackjack in casinos is DIFFERENT game then one that is played in this paper.
you're right! The article is not about the usual blackjack but a continuous version of the game, however I find interesting to understand this kind of problems. And maybe some day the casinos will start playing this version of the game ;)
There are still many Vegas casinos where you get 3/2 odds on blackjack and play with a six deck shoe. Don’t give business to the strip CSM and 6/5 games if you don’t like it.
There's still single and double deck blackjack too. Treasure Island on the strip has double deck and a lot of casinos on Fremont, with El Cortez being my favorite, has single deck. Both are 3/2.
In Vegas in 2021, I was surprised to see double or even (rarely) single 0 roulette tables mostly empty, right next to full triple-0 tables with the same min bet. Players seemed to understand the game too, so I guess the better table had just opened up and they didn't feel like moving.
Please read the actual article, which about a mathematical game using a continuous random distribution. Other than the name, it is very dissimilar to the Casino game.
19 comments
[ 3.9 ms ] story [ 58.5 ms ] threadI suspect it would be the same but less interesting, though. The smaller you make the increments, the closer the optimal stopping point is to 21. At the limit, the stopping point would be 21, and -- I think -- all N players plus dealer would have an 1/(N+1) chance of reaching it.
Basically, without jumps, it just becomes a situation with all players waiting for a continuous counter to reach 21, and then stopping precisely as it does. The one who was lucky to have the fastest counter (on average) wins. (With jumps as in discrete Blackjack, the fastest counter not necessarily the best!)
What might make it more interesting is if you did introduce discontinuities in the otherwise continuous process. So basically each player is fed with what looks like the absolute values of stock price movements, and has to stand somewhere. That should be easy to simulate and maybe even fun to play!
So, you would have to add some time delay between observation of a value and actual stopping.
Intuitively, I can think of two stopping strategies that might be optimal: call ‘stop’ when the expected value at stop is that ≈ 0.5706 or call stop when the probability of overshooting it is 50%.
I’m fairly sure that intuition is wrong in general, though.
[1] https://hn.algolia.com/?q=continuous+blackjack
99.99% of youtube channels dealing with counting cards are SCAM.
Just kiding, nice article.