Blackjack Paradox

[andyfox]

If you're correct on question one, that your edge on your hand one is the same as his was on his hand one, wouldn't the answer to question two be that you have the same edge throughout? Even though he has an edge on you on his hands 2, 3, and 4 (your hands 1, 2 and 3), you both have the same edge on the first, second, third, and fourth hands you both play, even though those are different hands for each of you.

Five hands are dealt. On hand one, you don't play. On hand two, you have the edge he had on hand one. On hand three, you have the edge he had on hand two. On hand four, you have the edge he had on hand three. On hand five, which he doesn't play, you have the edge on hand four. So while you might both play the same dealt hand differently, the aggregate edge is the same?

[PairTheBoard]

mmorpg has the right idea and I believe I've seen a calculation of the "Effect" somewhere in the literature. A deck which has had a hand played is different than a fresh deck. How the hand is played can have an Effect on likely count.

For example, suppose the First Hand played was not a regular BlackJack Hand but a SklanskyBJ hand where the rules are that the player keeps hitting until the count goes negative - maybe he wins if he takes fewer than 3 cards or something. Clearly you would not want to step into that deck after 1 hand to play a regular Blackjack hand.

Of course the First Hand was not a SklanskyBJ hand but a regular Blackjack hand. Still, there's no reason to think that play of a regular Blackjack hand might not have a similiar Effect, although it's not clear whether the Effect would bias the remaining deck to a positive or negative count. While mmorpg has the right idea I'm thinking he may have the bias backwards. If good cards come out the Hand tends to end right away, as in TT vs TT, AT vs xx, thus leaving a negative count. If small cards come out early in the Hand, making a temporary positive count, the Rules force both Dealer and Player to take more cards thus diluting the positive count and bringing the count down on average. It looks to me that the Effect is to bias the Count to the negative side.

The Ace cuts both ways though and would tend to quash the Effect. Although it's a Good Card and can end the hand quickly, it can also act as a small card. When acting as a small card it forces hits on an Unfavorable Deck - one that's missing an Ace - thus diluting the Deck's Unfavorability.

A computer Simulation should be able to answer the question definitively.

PairTheBoard

[Mickey Brausch]

[ QUOTE ]
Taking this a step further, say the player already there is identically skilled and playing with the same rules. So he has to leave one hand early.

[/ QUOTE ] I think you mean that the other guy (call him Joe) has to leave the table after his four rounds. (Or he can stay at the table but he won't give out any information or advice.) I'm pretty sure this is what you mean.

[ QUOTE ]
Question Two: Are both of you playing with the exact same edge? Even though you will often be playing hands two and three assuming different counts. In fact you might sometimes have the exact same hands and play them differently? If so how could this be?

[/ QUOTE ]Yes, the exact same edge, over the long run.

The cut card effect is nullified, so an excess number of small cards does not mean less rounds dealt, nor vice versa. Each of us will be seeing exactly four rounds. It's as if we sit down to play at two different tables where the same cards are dealt. But for me Round 1 is already in the deck and I play rounds 2,3,4,5 (four rounds in all). While Joe plays Rounds 1,2,3,4 (a total of four rounds, too) and then they shuffle on him.

We will have the same edge but, of course, that doesn't mean that we will be playing each and everyone of our hands the same way! (I will have missed, for example the two Fives that were dealt in Round 1. But the two Fives I missed are gonna be reconciled against the two Aces of my round which he will miss. See, for me, two Fives will be still in the deck, while in reality they're not gonna be dealt. For him, two Aces will be still in the deck, while in reality, they're not gonna be dealt --they are too deep.)

Notice that the chronological order of "my" round and "Joe's" round are completely unimportant. We are dealing strictly with information here, or the lack of it, at the time of the player's playing decision.

Is this what you're asking, David ?

Mickey Brausch

[T50_Omaha8]

I'm no blackjack expert, but here's my stab:

Isn't a player's edge during the rest of a deck roughly a function of the number of exposed cards? If more cards are exposed, the more opporunities a player has to deviate from basic startegy profitably (even if it means getting up and leaving the table). Assume this function is f(x) and is nonstrictly increasing.

Assuming an average of n cards is revealed in the average blackjack hand, the first player has the following expected advantage function:
First hand: f(0)
Second hand: f(n)
Third hand: f(3n) (since the other player has now sat down and exposed n more cards)
Fourth hand: f(5n)

This assumes his first hand was played ALONE.

Now consider hero, player 2, who plays his first three hands with player 1:
First hand: f(0)
Second hand: f(2n)
Third hand: f(4n)
Fourth hand: f(6n)

Thus player 2's edge is greater than or equal to that of player 1 for each hand, and a latecomer advantage exists.

And the fact that any number of cards is missing from the deck does NOT affect hero's expectation positively or negatively given that the cards removed are random and variable.

[PairTheBoard]

btw, I was once banned from the Tropicana for Flat Betting their single deck game when they had the Double after Splits Strip Rules with a slight edge off the top for a Basic Strategy player. They pushed me back after I made a Double on Soft 18 play. However the real reason may have been because I was playing so slow, trying to perfect a perfect recall of all the cards with advanced use of Effects of Removal for strategy decisions. Also I failed to give up my seat to a returning little old rich lady sucker. Also I had a nasty boil on the back of my neck at the time. Also I'd been extremely lucky on a daily basis for quite a while.

PairTheBoard

[NewTeaBag]

<u>Assumptions</u>
1) Advantage is gained solely through viewed cards.
2) Advantage is only used on the following hand
3) # Viewed cards is a sum of player and dealer cards
(Each segment of a hand (either dealer or player) is represented by an X)
4) All viewed cards are of equal value in adding to the advantage.

Hand 1
Player 1 Adv = 0X

Hand 2
Player 1 Adv = 2X
Player 2 Adv = 0X

Hand 3
Player 1 Adv = 5X
Player 2 Adv = 3X

Hand 4
Player 1 Adv = 8x
Player 2 Adv = 6x

Hand 5
Player 2 Adv = 9x

Sum of Player 1 Adv = 0X + 2X + 5X + 8X = 15X
Sum of Player 2 Adv = 0X + 3X + 6X + 9X = 18X

This seems a clear advantage to be player 2, vice player 1 from a card exposure (useful and actable information) standpoint. Both, your net and avg actable information is higher.

[Mickey Brausch]

[ QUOTE ]
If you're correct on question one, that your edge on your hand one is the same as his was on his hand one, wouldn't the answer to question two be that you have the same edge throughout? Even though he has an edge on you on his hands 2, 3, and 4 (your hands 1, 2 and 3), you both have the same edge on the first, second, third, and fourth hands you both play, even though those are different hands for each of you.

[/ QUOTE ] Me and the other player (call him Joe) are playing with the same edge in the long run but we do not necessarily have the same edge on each and every round! Joe might have for example a greater edge in Round 2 of game k, e.g. Joe is doubling down on A8v4 on account of what he saw in Round 1, while, me, I'm merely standing, this being my first round. However, the cards Joe sees in his last round, which is Round 4, are used as information by me in my last round, Round 5.

Here is how to picture this: Me and Joe are playing at different tables, in different casinos even. The order of the cards in both our deck is exactly the same every time. At Joe's table, the game is played regularly, with four rounds dealt. At my table, the dealer takes a number of cards equal to the number of cards dealt at Joe's table in Joe's first round, places them at the bottom of the deck and starts dealing to me. I too get four rounds.

Do I have a different edge than Joe ? Of course not.

Mickey Brausch

[PairTheBoard]

[ QUOTE ]
T50_Omaha8 -
And the fact that any number of cards is missing from the deck does NOT affect hero's expectation positively or negatively given that the cards removed are random and variable

[/ QUOTE ]

That's exactly the problem. The cards removed from the first hand are NOT random. They are affected by how the hand is played. If the hand is played according to SklanskyBJ rules then clearly what ends up getting played is not random. It always produces a remaining deck with a negative count. A player stepping into a deck with a Negative Count - one where High Cards have been removed - is going to have a disadvantage not only on the First Hand he plays but overall on all 4 hands that he plays. He's playing Blackjack with a stacked deck.

While it's not as easy to see that Play of a Regular Blackjack Hand has an Effect biasing the Deck Count as it is seeing that play of a SklanskyBJ hand does, I'd bet a dollar that a computer simulation will show that such an Effect does indeed exist for play of a Regular Blackjack Hand.

The original Player playing off the top of the Deck will also have an overall average disadvantage on the last 3 hands he plays. But he makes up for it by having an edge on those 1st hands that produce a negative count for the remaining deck. **** At this point I have to say that my thinking appears to be producing a paradox. If the deck on average becomes unfavorable after the first hand then it seems to imply that the first hand must be enjoying an average edge. But that doesn't make sense. Might have to think some more about this. ****

PairTheBoard

[T50_Omaha8]

My quote:
[ QUOTE ]
And the fact that any number of cards is missing from the deck does NOT affect hero's expectation positively or negatively given that the cards removed are random and variable

[/ QUOTE ]
Sometimes a bunch of kings will be taken off the deck, sometimes a bunch of threes will. It does not affect our expectation, however, since all the cases are equally likely. Your argument is akin to saying that the probability of flopping a flush is 1 if the first three cards are spades and 0 if the first three cards aren't.

[ QUOTE ]
Assumptions
1) Advantage is gained solely through viewed cards.
2) Advantage is only used on the following hand
3) # Viewed cards is a sum of player and dealer cards
(Each segment of a hand (either dealer or player) is represented by an X)
4) All viewed cards are of equal value in adding to the advantage.


[/ QUOTE ]
Good job...I forgot to consider dealer's hands in my analysis. I'm not sure, however, that you can necessarily sum the advantage over the course of the four hands--that assumes that the expectation advantage gained by seeing cards not only varies based on the number of cards exposed, but also that it necessarily varies linearly based on the number of cards exposed, resulting in the ability to add positive expectation 'units' over the course of the hand. Your assumptions do provide that the function is increasing, however, so you can show that player 2's expected advantage for HIS nth hand (ie player 1's (n+1)th hand) is at least as large as for player 1's nth hand.

And everyone should note that none of this is bound to happen--these are all EXPECTED EXPECTATION advantages. For example, the first hand dealt to player 1 could plausibly reveal every 2 and 3 in the deck, giving him a huge edge over the house, and player 2's not knowing this would mean his advantage would be significantly lowered throughout the course of HIS four hands. That's one possible scenario.

But over the long run, since player 2 tends to get to see more total cards before making his decisions, he should have an edge over player 1.

[Keepitsimple]

Lets say so many hands are played that there are only 10 cards left in the deck.

You assume that the cards left are random. Then any card you see has a greater impact on the remaining cards than if there were more cards left.

Extrapolated, I therefor think its better to play after one hand is played.