Physisorption: Difference between revisions
en>Wotnow m Format citation to facilitate access to online preview |
|||
| Line 1: | Line 1: | ||
{{More footnotes|article|date=October 2007}} | |||
In the card game of [[poker]], a '''bluff''' is a bet or raise made with a hand which is not thought to be the best hand. ''To bluff'' is to make such a bet. The objective of a bluff is to induce a fold by at least one opponent who holds a better hand. The size and frequency of a bluff determines its profitability to the ''bluffer''. By extension, the term is often used outside the context of poker to describe the act of making threats one cannot execute. | |||
== Pure bluff == | |||
A '''pure bluff''', or '''stone-cold bluff''', is a bet or raise with an inferior hand that has little or no chance of improving. A player making a pure bluff believes he can win the pot only if all opponents fold. The [[pot odds]] for a bluff are the ratio of the size of the bluff to the pot. A pure bluff has a positive [[expected value|expectation]] (will be profitable in the long run) when the probability of being called by an opponent is lower than the pot odds for the bluff. | |||
For example, suppose that after all the cards are out, a player holding a [[Glossary of poker terms#busted|busted]] [[Draw (poker)|drawing]] hand decides that the only way to win the pot is to make a pure bluff. If the player bets the size of the pot on a pure bluff, the bluff will have a positive expectation if the probability of being called is less than 50%. Note, however, that the opponent may also consider the pot odds when deciding whether to call. In this example, the opponent will be facing 2-to-1 pot odds for the call. The opponent will have a positive expectation for calling the bluff if the opponent believes the probability the player is bluffing is at least 33%. | |||
==Semi-bluff==<!-- This section is linked from [[List of poker terms]] --> | |||
In games with multiple betting rounds, to bluff on one round with an inferior or drawing hand that might improve in a later round is called a '''semi-bluff'''. A player making a semi-bluff can win the pot two different ways: by all opponents folding immediately or by catching a card to improve the player's hand. In some cases a player may be on a draw but with odds strong enough that he is favored to win the hand. In this case his bet is not classified as a semi-bluff even though his bet may force opponents to fold hands with better current strength. | |||
For example, a player in a [[stud poker]] game with four spade-suited cards showing (but none among their downcards) on the penultimate round might raise, hoping that his opponents believe he already has a flush. If his bluff fails and he is called, he still might be dealt a spade on the final card and win the [[Showdown (poker)|showdown]] (or he might be dealt another non-spade and try his bluff again, in which case it is a ''pure bluff'' on the final round rather than a semi-bluff). | |||
== Bluffing circumstances == | |||
Bluffing may be more effective in some circumstances than others. Bluffs have a higher expectation when the probability of being called decreases. Several game circumstances may decrease the probability of being called (and increase the profitability of the bluff): | |||
* Fewer opponents who must fold to the bluff. | |||
* The bluff provides less favorable pot odds to opponents for a call. | |||
* A [[Poker jargon#scare card|scare card]] comes that increases the number of superior hands that the player may be perceived to have. | |||
* The player's betting pattern in the hand has been consistent with the superior hand they are representing with the bluff. | |||
* The opponent's betting pattern suggests the opponent may have a marginal hand that is vulnerable to a greater number of potential superior hands. | |||
* The opponent's betting pattern suggests the opponent may have a [[Draw (poker)|drawing]] hand and the bluff provides unfavorable pot odds to the opponent for [[Poker jargon#chase|chasing]] the draw. | |||
* Opponents are not irrationally committed to the pot (see [[sunk cost fallacy]]). | |||
* Opponents are sufficiently skilled and paying sufficient attention. | |||
The opponent's current state of mind should be taken into consideration when bluffing. Under certain circumstances external pressures or events can significantly impact an opponent's decision making skills. | |||
== Optimal bluffing frequency == | |||
If a player bluffs too infrequently, observant opponents will recognize that the player is betting for [[Value (poker)|value]] and will call with very strong hands or with [[Draw (poker)|drawing]] hands only when they are receiving favorable [[pot odds]]. If a player bluffs too frequently, observant opponents '''''snap off''''' his bluffs by calling or re-raising. Occasional bluffing disguises not just the hands a player is bluffing with, but also his legitimate hands that opponents may think he may be bluffing with. [[David Sklansky]], in his book ''The Theory of Poker'', states "Mathematically, the optimal bluffing strategy is to bluff in such a way that the chances against your bluffing are identical to the pot odds your opponent is getting." | |||
Optimal bluffing also requires that the bluffs must be performed in such a manner that opponents cannot tell when a player is bluffing or not. To prevent bluffs from occurring in a predictable pattern, [[game theory]] suggests the use of a randomizing agent to determine whether to bluff. For example, a player might use the colors of his hidden cards, the second hand on his watch, or some other unpredictable mechanism to determine whether to bluff. | |||
'''Example ([[Texas Hold'em]])''' | |||
Again, let us return to the examples in The Theory of Poker: | |||
<blockquote> | |||
when I bet my $100, creating a $300 pot, my opponent was getting 3-to-1 odds | |||
from the pot. Therefore my optimum strategy was . . . [to make] the odds against | |||
my bluffing 3-to-1. | |||
</blockquote> | |||
Since the dealer will always bet with (nut hands) in this situation, he should bluff with (his) "Weakest hands/bluffing range" 1/3 | |||
of the time in order to make the odds 3-to-1 against a bluff.<ref>http://docs.google.com/viewer?a=v&q=cache:x_ORRoe0q_gJ:www.swansonsite.com/W/instructional/game_theory.pdf+jason+swanson+poker&hl=en&gl=us&sig=AHIEtbQiHN7aarn_e0BmkBDxV_AceJ4cbA</ref> | |||
'''Ex:''' | |||
On the last betting round (river), Worm has been betting a "semi-bluff" drawing hand with: A♠ K♠ on the board: | |||
10♠ 9♣ 2♠ 4♣ | |||
against Mike's A♣ <span style="color:red;">10♦</span> hand. | |||
The river comes out: | |||
2♣ | |||
The pot is currently 30 dollars, and Worm is contemplating a 30 dollar bluff on the river. If Worm does bluff in this situation, he is giving Mike 2-to-1 [[pot odds]] to call with his two pair (10's and 2's). | |||
'''Important:''' In these [[hypothetical]] circumstances, Worm will have the nuts 50% of the time, and be on a busted draw 50% of the time. Worm will bet the nuts (100%) of the time, and bet with a bluffing hand (using [[Strategy (game theory)|mixed optimal strategies]]): | |||
<math>x = s/(1+s)</math> <ref>The Mathematics of Poker, Bill Chen and Jerrod Ankenman</ref> | |||
Where s is equal to the percentage of the pot that Worm is bluff betting with and x is equal to the percentage of busted draws Worm should be bluffing with to bluff optimally. | |||
Pot = 30 dollars. | |||
Bluff bet = 30 dollars. | |||
s = 30(pot) / 30(bluff bet) = 1. | |||
Worm should be bluffing with his busted draws: | |||
x = <math>1/(1+s) = 50%</math> Where s = 1 | |||
''Assuming 4 trials'', Worm has the nuts 2 times, and has a busted draw 2 times. (EV = [[Expected Value]]) | |||
{| class="wikitable" | |||
|- | |||
! Worm bets with the nuts (100% of the time) | |||
! Worm bets with the nuts (100% of the time) | |||
! Worm bets with a busted draw (50% of the time) | |||
! Worm checks with a busted draw (50% of the time) | |||
|- | |||
| Worm's EV = 60 dollars | |||
| Worm's EV = 60 dollars | |||
| Worm's EV = 30 dollars (if Mike folds) and -30 dollars (if Mike calls) | |||
| Worm's EV = 0 dollars (since he will neither win the pot, nor lose 30 dollars on a bluff) | |||
|- | |||
| Mike's EV = -30 dollars (because he would not have won the original pot, but lost to Worm's value bet on the end) | |||
| Mike's EV = -30 dollars (because he would not have won the original pot, but lost to Worm's value bet on the end) | |||
| Mike's EV = 60 dollars (if he calls, he'll win the whole pot, which includes Worm's 30 dollar bluff) and 0 dollars (if Mike folds, he can't win the money in the pot) | |||
| Mike's EV = 30 dollars (assuming Mike checks behind with the winning hand, he will win the 30 dollar pot) | |||
|} | |||
Under the circumstances of this example: Worm will bet his nut hand 2 times, for every one time he bluffs against Mike's hand (assuming Mike's hand would lose to the nuts and beat a bluff). This means that (if he called all three bets) Mike would win 1 time, and lose two times, and would break even against 2-to-1 pot odds. This also means that Worm's odds against bluffing is also 2-to-1 (since he will value bet twice, and bluff once). | |||
Say in this example, Worm decides to use the second hand of his watch to determine when to bluff (50% of the time). If the second hand of the watch is between 1 and 30 seconds, Worm will check his hand down (not bluff). If the second hand of the watch is between 31 and 60 seconds, Worm will bluff his hand. Worm looks down at his watch, and the second hand is at 45 seconds, so Worm decides to bluff. Mike folds his two pair saying, "the way you've been betting your hand, I don't think my two pair on the board will hold up against your hand." Worm takes the pot by using optimal bluffing frequencies. | |||
'''Please note:''' This example is meant to illustrate how optimal bluffing frequencies work. Because it was an example, we assumed that Worm had the nuts 50% of the time, and a busted draw 50% of the time. In real game situations, this is not usually the case. | |||
The '''purpose''' of optimal bluffing frequencies is to make the opponent (mathematically) [[Principle of indifference|indifferent]] between calling and folding. Optimal bluffing frequencies are based upon game theory and the [[Nash Equilibrium]], and ''assist'' the player using these strategies to become [[unexploitable]]. By bluffing in optimal frequencies, you will typically end up breaking even on your bluffs (in other words, optimal bluffing frequencies ''are not'' meant to generate positive expected value from the bluffs alone). Rather, optimal bluffing frequencies allow you to gain ''more'' value from your "value bets," because your opponent is indifferent between calling or folding when you bet (regardless to whether it's a value bet or a bluff bet).<ref>The Mathematics of Poker, Bill Chen and Jerrod Ankenman</ref> | |||
== Bluffing in other games == | |||
Although bluffing is most often considered a poker term, similar tactics are useful in other games as well. In these situations, a player makes a play that shouldn't be profitable unless an opponent misjudges it as being made from a position capable of justifying it. Since a successful bluff requires deceiving one's opponent, it occurs only in games where the players conceal information from each other. In games like chess and backgammon where both players can see the same board, they should simply make the best legal move available. Examples include: | |||
*[[Contract Bridge]]: [[Psychic bid]]s and [[Glossary of contract bridge terms#F|falsecards]] are attempts to mislead the opponents about the distribution of the cards. A risk (common to all bluffing in partnership games) is that a bluff may also confuse the bluffer's partner. Psychic bids serve to make it harder for the opponents to find a good contract or to accurately place the key missing cards with a defender. Falsecarding (a tactic available in most trick taking card games) is playing a card that would naturally be played from a different hand distribution in hopes that an opponent will wrongly assume that the falsecarder made a natural play from a different hand and misplay a later trick on that assumption. | |||
*[[Stratego]]: Much of the strategy in Stratego revolves around identifying the ranks of the opposing pieces. Therefore depriving your opponent of this information is valuable. In particular, the "[[Shoreline Bluff]]" involves placing the flag in an unnecessarily vulnerable location in hopes that the opponent won't look for it there. | |||
*[[Spades]]: In late game situations, it is useful to bid a nil even if it cannot succeed.<ref>http://masterspades.home.att.net/stout3.htm</ref>{{dead link|date=December 2012}} If the third seat bidder sees that making a natural bid would allow the fourth seat bidder to make an uncontestable bid for game, he may bid nil even when it has no chance of success. The last bidder then must choose whether to make his natural bid (and lose the game if the nil succeeds) or to respect the nil by making a riskier bid that allows his side to win even if the doomed nil is successful. If he chooses wrong and both teams miss their bids, the game continues. | |||
*[[Scrabble]]: Scrabble players will sometimes deliberately play a phony hoping the opponent doesn't challenge it. Bluffing in Scrabble is a bit different from the other examples. Although Scrabble players do conceal their tiles, they have little opportunity to make significant deductions about their opponent's tiles, and even less opportunity to spread disinformation about them. Bluffing by playing a phony is instead based on assuming players have imperfect knowledge of the acceptable word list. | |||
==In Popular Culture== | |||
*In [[Stardust Crusaders]], the third volume of the [[Jojo's Bizarre Adventure]] series, [[Jotaro Kujo]] defeated gambler (and cheater) D'Arby in a game of poker without looking at his cards and raising higher bets on the souls of his closer associates. It was revealed that Jotaro's hand was weaker than everyone's expectations that caused D'Arby to fold and go into a nervous breakdown. | |||
== See also == | |||
* [[Poker jargon]] | |||
* [[Slow play (poker)|Slow play]] | |||
{{colend}} | |||
== Notes == | |||
<references/> | |||
==References== | |||
* {{cite book | title = The Theory of Poker | author = [[David Sklansky]] | publisher = [[Two Plus Two Publishing|Two Plus Two Publications]] | year = 1987 | isbn = 1-880685-00-0}} | |||
* {{cite book | title = Tournament Poker for Advanced Players | author = [[David Sklansky]] | publisher = Two Plus Two Publications | year = 2001 | isbn = 1-880685-28-0}} | |||
* {{cite book | title = Hold 'em Poker for Advanced Players | author = [[David Sklansky]] and [[Mason Malmuth]] | publisher = Two Plus Two Publications | year = 1988 | isbn = 1-880685-22-1}} | |||
* {{cite book | title = Harrington on Hold'em: Expert Strategy For No-Limit Tournaments; Volume I: Strategic Play | author = [[Dan Harrington]] and [[Bill Robertie]] | year = 2004 | publisher = Two Plus Two Publications | isbn = 1-880685-33-7}} | |||
* {{cite book | title = Harrington on Hold'em: Expert Strategy For No-Limit Tournaments; Volume II: The Endgame | author = [[Dan Harrington]] and [[Bill Robertie]] | year = 2005 | publisher = Two Plus Two Publications | isbn = 1-880685-35-3}} | |||
* {{cite book | title = The Mathematics of Poker | author = [[Bill Chen]], Jerrod Ankenman}} | |||
== External links == | |||
* [http://www.cardschat.com/semi-bluff.php Semi Bluff] | |||
* [http://www.learnhowtoplaypoker.org/Bluffing101.aspx Bluffing 101] | |||
{{DEFAULTSORT:Bluff (Poker)}} | |||
[[Category:Deception]] | |||
[[Category:Poker gameplay and terminology]] | |||
[[Category:Poker strategy]] | |||
Revision as of 09:54, 23 January 2014
Template:More footnotes In the card game of poker, a bluff is a bet or raise made with a hand which is not thought to be the best hand. To bluff is to make such a bet. The objective of a bluff is to induce a fold by at least one opponent who holds a better hand. The size and frequency of a bluff determines its profitability to the bluffer. By extension, the term is often used outside the context of poker to describe the act of making threats one cannot execute.
Pure bluff
A pure bluff, or stone-cold bluff, is a bet or raise with an inferior hand that has little or no chance of improving. A player making a pure bluff believes he can win the pot only if all opponents fold. The pot odds for a bluff are the ratio of the size of the bluff to the pot. A pure bluff has a positive expectation (will be profitable in the long run) when the probability of being called by an opponent is lower than the pot odds for the bluff.
For example, suppose that after all the cards are out, a player holding a busted drawing hand decides that the only way to win the pot is to make a pure bluff. If the player bets the size of the pot on a pure bluff, the bluff will have a positive expectation if the probability of being called is less than 50%. Note, however, that the opponent may also consider the pot odds when deciding whether to call. In this example, the opponent will be facing 2-to-1 pot odds for the call. The opponent will have a positive expectation for calling the bluff if the opponent believes the probability the player is bluffing is at least 33%.
Semi-bluff
In games with multiple betting rounds, to bluff on one round with an inferior or drawing hand that might improve in a later round is called a semi-bluff. A player making a semi-bluff can win the pot two different ways: by all opponents folding immediately or by catching a card to improve the player's hand. In some cases a player may be on a draw but with odds strong enough that he is favored to win the hand. In this case his bet is not classified as a semi-bluff even though his bet may force opponents to fold hands with better current strength.
For example, a player in a stud poker game with four spade-suited cards showing (but none among their downcards) on the penultimate round might raise, hoping that his opponents believe he already has a flush. If his bluff fails and he is called, he still might be dealt a spade on the final card and win the showdown (or he might be dealt another non-spade and try his bluff again, in which case it is a pure bluff on the final round rather than a semi-bluff).
Bluffing circumstances
Bluffing may be more effective in some circumstances than others. Bluffs have a higher expectation when the probability of being called decreases. Several game circumstances may decrease the probability of being called (and increase the profitability of the bluff):
- Fewer opponents who must fold to the bluff.
- The bluff provides less favorable pot odds to opponents for a call.
- A scare card comes that increases the number of superior hands that the player may be perceived to have.
- The player's betting pattern in the hand has been consistent with the superior hand they are representing with the bluff.
- The opponent's betting pattern suggests the opponent may have a marginal hand that is vulnerable to a greater number of potential superior hands.
- The opponent's betting pattern suggests the opponent may have a drawing hand and the bluff provides unfavorable pot odds to the opponent for chasing the draw.
- Opponents are not irrationally committed to the pot (see sunk cost fallacy).
- Opponents are sufficiently skilled and paying sufficient attention.
The opponent's current state of mind should be taken into consideration when bluffing. Under certain circumstances external pressures or events can significantly impact an opponent's decision making skills.
Optimal bluffing frequency
If a player bluffs too infrequently, observant opponents will recognize that the player is betting for value and will call with very strong hands or with drawing hands only when they are receiving favorable pot odds. If a player bluffs too frequently, observant opponents snap off his bluffs by calling or re-raising. Occasional bluffing disguises not just the hands a player is bluffing with, but also his legitimate hands that opponents may think he may be bluffing with. David Sklansky, in his book The Theory of Poker, states "Mathematically, the optimal bluffing strategy is to bluff in such a way that the chances against your bluffing are identical to the pot odds your opponent is getting."
Optimal bluffing also requires that the bluffs must be performed in such a manner that opponents cannot tell when a player is bluffing or not. To prevent bluffs from occurring in a predictable pattern, game theory suggests the use of a randomizing agent to determine whether to bluff. For example, a player might use the colors of his hidden cards, the second hand on his watch, or some other unpredictable mechanism to determine whether to bluff.
Example (Texas Hold'em)
Again, let us return to the examples in The Theory of Poker:
when I bet my $100, creating a $300 pot, my opponent was getting 3-to-1 odds from the pot. Therefore my optimum strategy was . . . [to make] the odds against my bluffing 3-to-1.
Since the dealer will always bet with (nut hands) in this situation, he should bluff with (his) "Weakest hands/bluffing range" 1/3 of the time in order to make the odds 3-to-1 against a bluff.[1]
Ex: On the last betting round (river), Worm has been betting a "semi-bluff" drawing hand with: A♠ K♠ on the board:
10♠ 9♣ 2♠ 4♣ against Mike's A♣ 10♦ hand.
The river comes out:
2♣
The pot is currently 30 dollars, and Worm is contemplating a 30 dollar bluff on the river. If Worm does bluff in this situation, he is giving Mike 2-to-1 pot odds to call with his two pair (10's and 2's).
Important: In these hypothetical circumstances, Worm will have the nuts 50% of the time, and be on a busted draw 50% of the time. Worm will bet the nuts (100%) of the time, and bet with a bluffing hand (using mixed optimal strategies):
Where s is equal to the percentage of the pot that Worm is bluff betting with and x is equal to the percentage of busted draws Worm should be bluffing with to bluff optimally.
Pot = 30 dollars. Bluff bet = 30 dollars.
s = 30(pot) / 30(bluff bet) = 1.
Worm should be bluffing with his busted draws:
Assuming 4 trials, Worm has the nuts 2 times, and has a busted draw 2 times. (EV = Expected Value)
| Worm bets with the nuts (100% of the time) | Worm bets with the nuts (100% of the time) | Worm bets with a busted draw (50% of the time) | Worm checks with a busted draw (50% of the time) |
|---|---|---|---|
| Worm's EV = 60 dollars | Worm's EV = 60 dollars | Worm's EV = 30 dollars (if Mike folds) and -30 dollars (if Mike calls) | Worm's EV = 0 dollars (since he will neither win the pot, nor lose 30 dollars on a bluff) |
| Mike's EV = -30 dollars (because he would not have won the original pot, but lost to Worm's value bet on the end) | Mike's EV = -30 dollars (because he would not have won the original pot, but lost to Worm's value bet on the end) | Mike's EV = 60 dollars (if he calls, he'll win the whole pot, which includes Worm's 30 dollar bluff) and 0 dollars (if Mike folds, he can't win the money in the pot) | Mike's EV = 30 dollars (assuming Mike checks behind with the winning hand, he will win the 30 dollar pot) |
Under the circumstances of this example: Worm will bet his nut hand 2 times, for every one time he bluffs against Mike's hand (assuming Mike's hand would lose to the nuts and beat a bluff). This means that (if he called all three bets) Mike would win 1 time, and lose two times, and would break even against 2-to-1 pot odds. This also means that Worm's odds against bluffing is also 2-to-1 (since he will value bet twice, and bluff once).
Say in this example, Worm decides to use the second hand of his watch to determine when to bluff (50% of the time). If the second hand of the watch is between 1 and 30 seconds, Worm will check his hand down (not bluff). If the second hand of the watch is between 31 and 60 seconds, Worm will bluff his hand. Worm looks down at his watch, and the second hand is at 45 seconds, so Worm decides to bluff. Mike folds his two pair saying, "the way you've been betting your hand, I don't think my two pair on the board will hold up against your hand." Worm takes the pot by using optimal bluffing frequencies.
Please note: This example is meant to illustrate how optimal bluffing frequencies work. Because it was an example, we assumed that Worm had the nuts 50% of the time, and a busted draw 50% of the time. In real game situations, this is not usually the case.
The purpose of optimal bluffing frequencies is to make the opponent (mathematically) indifferent between calling and folding. Optimal bluffing frequencies are based upon game theory and the Nash Equilibrium, and assist the player using these strategies to become unexploitable. By bluffing in optimal frequencies, you will typically end up breaking even on your bluffs (in other words, optimal bluffing frequencies are not meant to generate positive expected value from the bluffs alone). Rather, optimal bluffing frequencies allow you to gain more value from your "value bets," because your opponent is indifferent between calling or folding when you bet (regardless to whether it's a value bet or a bluff bet).[3]
Bluffing in other games
Although bluffing is most often considered a poker term, similar tactics are useful in other games as well. In these situations, a player makes a play that shouldn't be profitable unless an opponent misjudges it as being made from a position capable of justifying it. Since a successful bluff requires deceiving one's opponent, it occurs only in games where the players conceal information from each other. In games like chess and backgammon where both players can see the same board, they should simply make the best legal move available. Examples include:
- Contract Bridge: Psychic bids and falsecards are attempts to mislead the opponents about the distribution of the cards. A risk (common to all bluffing in partnership games) is that a bluff may also confuse the bluffer's partner. Psychic bids serve to make it harder for the opponents to find a good contract or to accurately place the key missing cards with a defender. Falsecarding (a tactic available in most trick taking card games) is playing a card that would naturally be played from a different hand distribution in hopes that an opponent will wrongly assume that the falsecarder made a natural play from a different hand and misplay a later trick on that assumption.
- Stratego: Much of the strategy in Stratego revolves around identifying the ranks of the opposing pieces. Therefore depriving your opponent of this information is valuable. In particular, the "Shoreline Bluff" involves placing the flag in an unnecessarily vulnerable location in hopes that the opponent won't look for it there.
- Spades: In late game situations, it is useful to bid a nil even if it cannot succeed.[4]Template:Dead link If the third seat bidder sees that making a natural bid would allow the fourth seat bidder to make an uncontestable bid for game, he may bid nil even when it has no chance of success. The last bidder then must choose whether to make his natural bid (and lose the game if the nil succeeds) or to respect the nil by making a riskier bid that allows his side to win even if the doomed nil is successful. If he chooses wrong and both teams miss their bids, the game continues.
- Scrabble: Scrabble players will sometimes deliberately play a phony hoping the opponent doesn't challenge it. Bluffing in Scrabble is a bit different from the other examples. Although Scrabble players do conceal their tiles, they have little opportunity to make significant deductions about their opponent's tiles, and even less opportunity to spread disinformation about them. Bluffing by playing a phony is instead based on assuming players have imperfect knowledge of the acceptable word list.
In Popular Culture
- In Stardust Crusaders, the third volume of the Jojo's Bizarre Adventure series, Jotaro Kujo defeated gambler (and cheater) D'Arby in a game of poker without looking at his cards and raising higher bets on the souls of his closer associates. It was revealed that Jotaro's hand was weaker than everyone's expectations that caused D'Arby to fold and go into a nervous breakdown.
See also
Notes
- ↑ http://docs.google.com/viewer?a=v&q=cache:x_ORRoe0q_gJ:www.swansonsite.com/W/instructional/game_theory.pdf+jason+swanson+poker&hl=en&gl=us&sig=AHIEtbQiHN7aarn_e0BmkBDxV_AceJ4cbA
- ↑ The Mathematics of Poker, Bill Chen and Jerrod Ankenman
- ↑ The Mathematics of Poker, Bill Chen and Jerrod Ankenman
- ↑ http://masterspades.home.att.net/stout3.htm
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534