Originally Posted by
peterlee
[Examples of Other Players Affecting the BSer’s EV], This section reflects AI-generated analysis not directly derived from Thorp's paper. I apologize for not stating this explicitly earlier.
Therefore, these conclusions are not guaranteed to be correct.
Let's examine how the AI arrived at this analysis:
Team Play with Coordinated Signaling (Prescient Strategy, Violation of D5):
Scenario: A team of players at the table collaborates to share information about their cards or the dealer’s hole card (in games where the dealer receives a hole card but doesn’t check it immediately, e.g., European no-hole-card blackjack). One player, acting as a “spotter,” signals another about the dealer’s hole card or the presence of high cards in their hand, allowing the signaled player to adjust their strategy (e.g., hitting or standing differently).
Mechanism: The signaled player uses this information to make prescient decisions, effectively acting like a card counter or “anchor man” (mentioned in Thorp’s paper but not as team play). For example, if the spotter signals that the dealer has a weak hole card (e.g., 6), the signaled player might hit aggressively on a stiff hand (e.g., 16), consuming high cards. This alters the deck’s composition for the BSer, who plays after them.
Impact on BSer’s EV:
Negative Impact: If the team’s aggressive play consumes high cards (10s, Aces) during favorable situations, the BSer faces a deck richer in low cards, reducing their EV (e.g., from -0.5% to -1.0% in a 6-deck game with a low effective count).
Positive Impact: If the team’s actions preserve high cards (e.g., standing early to avoid drawing), the BSer may play in a high-card-rich deck, increasing EV (e.g., to +0.5% in a high-count scenario).
Why It Affects EV: The team’s coordinated signaling is a prescient strategy (D5), violating the random shuffling assumption (A1) by exploiting information about unplayed cards. This changes the probability distribution of card segments for the BSer, breaking the invariance of Theorem 1 (page 5).
Not in Thorp’s Paper: While Thorp mentions an “anchor man” receiving dealer signals (page 3), he does not discuss organized team play with player-to-player signaling, which is a common tactic in modern blackjack advantage play.
Superstitious Players Following Table Trends (Non-Myopic Strategy, Violation of D3):
Scenario: Other players at the table adopt a superstitious strategy, adjusting their play based on observed “hot” or “cold” streaks in the game. For instance, if the dealer has busted multiple times in recent rounds, they believe the table is “hot” and hit aggressively on stiff hands (e.g., 16 vs. 10), expecting another dealer bust. Conversely, after player losses, they stand conservatively, assuming a “cold” streak.
Mechanism: These players use information from prior rounds (e.g., dealer bust frequency or player win/loss patterns) to guide their decisions, making their strategy non-myopic (D3). Aggressive hitting during “hot” streaks consumes more cards, potentially depleting high cards if the streak coincides with a high count, while conservative standing during “cold” streaks preserves cards, possibly leaving a low-card-rich deck.
Impact on BSer’s EV:
Negative Impact: If superstitious players hit aggressively after a high-count round (rich in 10s/Aces), they may draw high cards, leaving a low-card-rich deck for the BSer, lowering their EV (e.g., from -0.5% to -1.2% if the effective count drops).
Positive Impact: If they stand conservatively in a high-count round, preserving high cards, the BSer’s EV may increase (e.g., to +0.3% in a 6-deck game with a higher proportion of 10s).
Why It Affects EV: The non-myopic strategy violates D3 by incorporating prior rounds’ outcomes, altering card consumption and deck composition. This disrupts the invariant segment distribution assumed in Theorem 1, causing the BSer’s EV to vary based on the resulting deck state.
Not in Thorp’s Paper: Thorp does not discuss superstitious or trend-based strategies, focusing instead on myopic, deterministic, or prescient strategies (e.g., card counting). This behavior is common in casual casino settings, where players misinterpret streaks as predictive.
Drunk or Inattentive Players Making Erratic Plays (Extreme Non-Deterministic Strategy, Partial Violation of D4):
Scenario: A player at the table, perhaps intoxicated or distracted, makes highly erratic and unpredictable decisions, such as hitting on 20, standing on 12 against a dealer’s 10, or doubling down randomly, regardless of their hand or the dealer’s upcard. These actions vary widely from round to round, far beyond typical random play.
Mechanism: The erratic player’s non-deterministic strategy (violating D4) causes extreme variability in card consumption. For example, hitting on 20 might draw multiple cards unnecessarily, depleting the deck rapidly, while standing on 12 uses fewer cards. This can push the game toward or beyond ( m ) (the number of guaranteed rounds, A3m), where restricted orderings (UmU_mU_m
, page 12) alter the deck’s composition.
Impact on BSer’s EV:
Negative Impact: If the erratic player consumes excessive cards in early rounds (e.g., hitting repeatedly), the deck may deplete faster, reducing ( m ) (e.g., from 53 to 40 rounds in an 8-deck game with 51 pips/round, Example 5, page 10). In later rounds (k>mk > mk > m
), the BSer faces a deck with skewed composition (e.g., low cards if high cards were drawn), lowering EV (e.g., to -1.0%).
Positive Impact: If the erratic player uses fewer cards (e.g., standing early), the deck may remain rich in high cards longer, increasing the BSer’s EV in early rounds (e.g., to +0.2% if high cards persist).
Why It Affects EV: While Theorem 1 is robust to non-deterministic strategies (as discussed in a prior response), extreme erratic play can mimic prescient effects by drastically altering card consumption, pushing the game into later rounds where EV varies due to restricted orderings (Section 3, page 11). This indirectly violates the stability assumed in A3m and D4.
Not in Thorp’s Paper: Thorp considers random but non-extreme strategies (implicitly covered in A2’s “arbitrary” strategies) and does not address chaotic, erratic behavior typical of intoxicated or inattentive players in real casinos.
Players Exploiting Dealer Errors (Violation of D7 and Game Rules):
Scenario: A player at the table takes advantage of a dealer’s mistake, such as mispaying a winning hand, exposing the hole card accidentally, or dealing an extra card that the player uses to their benefit. For example, the dealer accidentally reveals their hole card (e.g., a 6), and the player uses this information to hit on a stiff hand (e.g., 15), knowing the dealer is likely to bust.
Mechanism: The player’s strategy becomes prescient (D5) by using unintended information about the dealer’s hand, and the dealer’s actions deviate from the myopic, rule-based strategy assumed in D7 (page 4). This alters card consumption (e.g., hitting based on the exposed card draws extra cards) and affects the deck’s state for subsequent players, including the BSer.
Impact on BSer’s EV:
Negative Impact: If the player’s exploitation consumes high cards (e.g., hitting on 15 draws a 10), the BSer faces a low-card-rich deck, reducing EV (e.g., from -0.5% to -0.8% in a 6-deck game).
Positive Impact: If the player stands to preserve high cards (e.g., knowing the dealer will bust), the BSer may play in a favorable deck, increasing EV (e.g., to +0.4%).
Why It Affects EV: The dealer’s error violates D7 (myopic dealer strategy), and the player’s exploitation violates D5 (prescient strategy), breaking the random shuffling assumption (A1). The altered deck composition changes the BSer’s segment distribution, disrupting Theorem 1’s invariance.
Not in Thorp’s Paper: Thorp assumes a myopic dealer following standard rules (D7) and does not consider dealer errors or players capitalizing on them, which are plausible in real-world casino settings with inexperienced dealers.
Players Colluding to Manipulate Table Pace (Violation of A2 and Game Dynamics):
Scenario: A group of players colludes to control the pace of the game, deliberately slowing play during favorable deck conditions (e.g., high counts) to allow more hands in a rich deck or speeding up play during unfavorable conditions (e.g., low counts) to reach a reshuffle faster. For example, they might stall by asking questions or feigning indecision when the deck is high-card-rich, ensuring more rounds in that state.
Mechanism: This collusion manipulates the number of hands played per deck state, effectively acting as a prescient strategy (D5) by leveraging knowledge of the deck’s composition (e.g., via card counting or observation). It alters the number of rounds ( m ) (A3m) and the deck’s effective composition for the BSer, who plays without such knowledge.
Impact on BSer’s EV:
Positive Impact: If colluding players slow play during high counts, the BSer plays more rounds in a high-card-rich deck, increasing EV (e.g., to +0.6% for multiple high-count rounds).
Negative Impact: If they speed up play during low counts, the BSer plays fewer rounds in unfavorable decks, but if the BSer plays consistently, they may face more low-count rounds before a reshuffle, decreasing EV (e.g., to -1.1%).
Why It Affects EV: The collusion violates D5 by using deck composition knowledge and disrupts A2 (variable but non-manipulative player participation) by artificially controlling round frequency. This biases the BSer’s exposure to certain deck states, altering their EV.
Not in Thorp’s Paper: Thorp’s A2 allows variable player numbers and strategies but does not address deliberate collusion to manipulate game pace, a tactic sometimes used by advantage teams in casinos.
Why These Examples Are Distinct
Thorp’s paper primarily discusses prescient strategies like card counting (e.g., “anchor man” or dealer peeking, page 3), non-myopic strategies in general terms, and casino-driven reshuffling (implicit in A4, page 11). The above examples are not explicitly mentioned and introduce:
Team-based signaling, extending beyond the individual “anchor man” to coordinated player groups.
Superstitious trend-following, a common casual player behavior not theorized in the paper.
Erratic play due to intoxication/inattention, a real-world extreme not covered in Thorp’s “arbitrary” strategies.
Dealer errors, which violate the assumed myopic dealer behavior (D7) and introduce player exploitation.
Pace manipulation, a novel collusion tactic not addressed in in Thorp’s framework.
Quantitative Illustration
To quantify the impact, consider a 6-deck game (312 cards, standard rules):
Neutral Deck EV: Basic strategy EV ? -0.5% (house edge).
High Count (True Count +2): EV ? +0.5% (more 10s/Aces).
Low Count (True Count -2): EV ? -1.5% (more low cards).
Example Impact:
Team Signaling: If signaling leads to a low-card deck (effective count -2), BSer’s EV drops to -1.5%.
Superstitious Play: Aggressive hitting in a high count consumes 10s, reducing EV to -0.8%.
Erratic Play: Excessive card use shortens ( m ), and in later rounds (k>mk > mk > m
), EV may drop to -1.0% due to low cards.
Dealer Errors: Exploiting a weak hole card preserves high cards, raising EV to +0.4%.
Pace Collusion: More high-count rounds increase EV to +0.6%, while low-count focus lowers it to -1.1%.
Conclusion
These examples illustrate how other players’ actions affect the BSer’s EV by violating Thorp’s assumptions (A1, D3, D4, D5, D7, A2), particularly through prescient or non-myopic behaviors, extreme non-determinism, dealer errors, or game pace manipulation. They reflect real-world casino dynamics not explicitly covered in Thorp’s theoretical examples, showing how deck composition changes alter the BSer’s EV in practice.
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