ONE PLAYER : A better proof is the following. Assume that two players both play flatbetting Basic Strategy for n rounds, starting with a full shoe, wheren is sufficiently large to guarantee that they will play past areshuffle. Let the expected per-round profit of player 2 under theseconditions be g. Therefore, the expected profit of player 2 afterplaying n games is: n * gNow assume that they start their n rounds after 1 round has been dealtfrom the deck and that, knowing the set of cards already dealt, theexpected profit of player 2 for the remaining r rounds until reshufflingis g', where g' is not equal to g. The expected profit of player 2 isnowthe sum of the profit from the r rounds before the next reshuffle, andthe (n - r) rounds after: Profit1 = r * g' + (n - r) * gNow, look what happens if player 1 decides to change hisstrategy to always stand, until the next reshuffle, where he revertsto Basic Strategy. The only thing that changes is the number of roundsbefore the next reshuffle, which is now r', and we can assume that thiscan be greater than r. The expected profit per round of player 2 isstill g' before the next reshuffle, and g after, thanks to the TrueCountTheorem. So the expected profit of player 2 over the n games is now: Profit2 = r' * g' + (n - r') * gwhich is different from when player 1 plays pure Basic Strategy (Profit1 ONE PLAYER : above). The actual difference between the two is (r' - r) * (g' - g),which is nonzero if g is not equal to g' and r is not equal to r'.I'm just curious, but how does player 2's choice affect player 1?If the count is low and player 1 takes a card, that could make the countzero or even lower. This could likely make player 2 draw more cards, if hisstrategy is based on the count. If the count is high, and player 1 stands sothe count remains high, player 2 should double more often.Can you give us a chart to play against? I would really like to see one.(I'm not slamming you if you can't. I can't make one right now either.)"Hit your 12 against the dealer's 4 if there are four or more players at thetable and you are seated third or later."Seems to me you'd have to know the other players' strategies, and some playscannot be predicted.Right now I would bet this won't help at the table, but I could be wrong. ONE PLAYER : The numbers 1 and 2 are just to distinguish between the player whochanges strategy and the one who doesn't. It could have beenreversed, with player 2 changing strategy and affecting player 1.now you can go back to your spamming for your site, which incidentallydoesn't appear to work at all in the world's most popular browser (IE5),althought the dhtml tricks are cute.the *expected values* of Profit1 and Profit2 are equal, provided thatPlayer1's decision of what strategy to play is independent of the count. ithink you realize this, but its not clear from your post.if, however, he always hits in low counts and always stands (or better yet,sits out) in high counts, he will help Player2's EV. this concept has beenknown for some time. i think its been called "card eating". the inverseproposition can happen when 2 counters are playing independently at thesame time. if Player1 wongs out on low counts, Player2 is stuck playing outthe negative end of the shoe by himself. i've seen people in this newsgroupget really pissed off when this happens to them ONE PLAYER : your last point is interesting... given a marginal play in a high count,there is a small bias towards standing (or equivalently towards hitting ina low count) in the EV which isn't usually accounted for. i'd be verysurprised if you could find a practical scenario where this bias would havean effect on basic strategy, especially since high-count plays are natuallymore hit-oriented than low-count plays anyway. also, this effect is onlyrelevant to games with a cut-card.Thank you (and Chris in a previous post) for recognizing what I wassaying. I didn't think it was that difficult a concept, so I guess mypowers of explanation in previous posts were lacking. Anyway, youare right, I realize that only play that is correlated with carddistribution (e.g., the count) will affect another player.Yes, I don't know if this small bias would affect normal countingstrategy. However, I can imagine a scenario where it might, byusing a particular team strategy that increases profit without drawingunwanted attention (no leaving/entering the table, and no changingbet size). Imagine 2 players at the table, where the first is flatbetting table minimum, say $1, and the other, flat betting tablemaximum, say $50. Sometimes, the first player might choose asub-optimal strategy that, for instance, costs him 1% (-1 cent)relative to the correct strategy, but because it eats up a card (oravoids eating a card), it helps out player 2 by, say 0.1 %(+5 cents). Thus player 2 is using player 1 as a cheap sourceof card eating when the count is negative, and avoiding theattention that comes with leaving the table or changing bets. ONE PLAYER : If a player always leaves the table when the countat the beginning of a round drops below a certainlevel, then yes most definitely the 'strategy' ofthis player will affect that of all others.This player will cause the other players to haveto play through more unfavorable rounds.OK, 'always stand' strategy?When is anyone in their right mind going to choosean 'always stand' strategy?Why not just have the second player leave the tablewhen the count goes positive?The first player would then enjoy more favorable hands.Doug wrote:SECTION SIXTEEN: IF THE COUNTERMEASURES ARE USED TOTALLY AT THE DISCRETIONOF THE CASINOS, THEN SUCH COUNTERMEASURES WILL (1) ALTER THE GAME (2)DECEIVE THE PATRONS (3) FAIL TO NOTIFY THE PATRONS OF THE APPLICABLE RULESOF THE GAME. "As noted in the proposal, these rules do not affect the ability orprobability of any player to win at the game of blackjack; they simplyaffect the profitability of the game to a card counter who knows that thecards remaining in the blackjack shoe are favorable to the player. Ingeneral, the rules will not affect the average blackjack player or even theskilled basic strategy player because these players do not vary their wagersbased upon values of the cards remaining in the shoe (because they areunknown) and because such players rarely spread their wagers within a widerange."
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