Blackjack Paradox

[David Sklansky]

You are a pro who specializes beating single decks with good rules by flat betting and deviating from basic strategy according to the count. A certain casino lets you play four hands and then shuffles.

One day you walk up to a table where someone has already played one hand that you haven't seen. The pit boss tells you that you can play four hands from this point. (Not because he saw high cards come out. He wasn't even watching.

Question One: If you can't ask anyone what cards came out on the first hand ,is this just as good for you as playing starting from hand one, and getting four hands? In spite of the fact that your playing strategy will assume a count that will often be wrong?

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.

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?

Note: Please disregard the effects of going second. Assume in fact that strategy changes will be based only on cards seen before the hand started. I'm dealing with a general principle here.

[HP]

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Question One: If you can't ask anyone what cards came out on the first hand ,is this just as good for you as playing starting from hand one, and getting four hands? In spite of the fact that your playing strategy will assume a count that will often be wrong?

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yeah

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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 ]

each players edge for the whole 4-hand round will be the same. Though the first player will have a greater edge than the second player on hands #2, #3, and #4. The first player makes less money on hand #1 than the second player makes on hand #5

[felson]

i agree with HP.

i believe that we need an additional assumption -- that the expected number of cards played during the first, unseen hand, is equal to the expected number of cards played during hands 2,3,4. otherwise player 2 will have more information on the average than player 1. so we need someone else to step in and play a hand during hand 1, then leave the table.

[mmorpg]

Interesting question.

Here is my theory which is most likely wrong, but since I am dying to know the right answer someone might feel the need to correct me.

Q1
Deck is more likely to have lost 'rubbish' cards than good cards so on avarage your edge will be slightly better than what the count implies.

Assuming they are both sober.
Player 2 will have the better edge since there are less cards in the deck, but he wont know the true count and might play a hand wrong.


:0)

[BeerMoney]

This seems like asking if the chances of getting pocket aces differs at a HE table by how many people are playing.

[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

[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]

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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

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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:
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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

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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.

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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.


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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.