One possible algorithm for shuffling cards without the use of a trusted third party is to use a commutative encryption scheme. A commutative scheme means that if some data is encrypted more than once, the order in which you decrypt this data will not matter.
Example: Alice has a plaintext message. She encrypts this, producing a garbled ciphertext which she gives this to Bob. Bob encrypts the ciphertext again, using the same scheme as Alice but with another key. When decrypting this double encrypted message, if the encryption scheme is commutative, it will not matter who decrypts first.
The algorithm
An algorithm for shuffling cards using commutative encryption would be as follows:
Alice and Bob agree on a certain “deck” of cards. In practice, this means they agree on a set of numbers or other data that each represents a card.
Alice picks an encryption key A and uses this to encrypt each card of the deck.
Alice shuffles the cards.
Alice passes the encrypted and shuffled deck to Bob. With the encryption in place, Bob cannot know which card is which.
Bob picks an encryption key B and uses this to encrypt each card of the encrypted and shuffled deck.
Bob shuffles the deck.
Bob passes the double encrypted and shuffled deck back to Alice.
Alice decrypts each card using her key A. This still leaves Bob’s encryption in place though so she cannot know which card is which.
Alice picks one encryption key for each card (A1, A2, etc) and encrypts them individually.
Alice passes the deck to Bob.
Bob decrypts each card using his key B. This still leaves Alice’s individual encryption in place though so he cannot know which card is which.
Bob picks one encryption key for each card (B1, B2, etc) and encrypts them individually.
Bob passes the deck back to Alice.
Alice publishes the deck for everyone playing (in this case only Alice and Bob, see below on expansion though).
The deck is now shuffled.
This algorithm may be expanded for an arbitrary number of players. Players Carol, Dave and so forth need only repeat steps 2-4 and 8-10.
During the game, Alice and Bob will pick cards from the deck, identified in which order they are placed in the shuffled deck. When either player wants to see their cards, they will request the corresponding keys from the other player. That player, upon checking that the requesting player is indeed entitled to look at the cards, passes the individual keys for those cards to the other player. The check is to ensure that the player does not try to request keys for cards that does not belong to that player.
Example: Alice has picked cards 1 to 5 in the shuffled deck. Bob has picked cards 6 to 10. Bob requests to look at his alotted cards. Alice agrees that Bob is entitled to look at cards 6 to 10 and gives him her individual card keys A6 to A10. Bob decrypts his cards by using both Alice’s keys and his own for these cards, B6 to B10. Bob can now see the cards. Alice cannot know which cards Bob has because she does not have access to Bob keys B6 to B10 which are required to decrypt the cards.
Weakness
Depending on the deck agreed upon, this algorithm may be weak. When encrypting data, certain properties of this data may be preserved from the plaintext to the ciphertext. This may be used to “tag” certain cards. Therefore, the parties must agree on a deck where no cards have properties that are preserved during encryption.
A scratchcard is a small piece of card where an area has been covered by a substance that cannot be seen through, but can be scratched off. Under this area are concealed the items/pictures that must be ‘found’ in order to win.
The generic scratchcard requires the player to match three of the same prize amounts. If this is accomplished, they win that amount. Other scratchcards involve matching symbols, pictures or words.
Scratchcards are a very popular form of gambling due to their low cost. However, the low cost to buy a scratchcard is offset by the smaller prizes, compared to casino jackpots or lottery wins.
Unbeatable; often said of a hand that a player knows cannot be beaten under the circumstances of play. See also “lock”, “nuts”.
implied odds
Similar to pot odds, but including future bets one can reasonably expect to win. I only had 3-to-1 pot odds for a 5-to-1 draw, but I knew if I made it I’d get two extra bets from Jim, so I called for the implied odds.
improve
To achieve a better hand than one currently holds by adding or exchanging cards as provided in the rules of the game being played. I didn’t think Paula was bluffing, so I decided not to call unless I improved on the draw.
inside, inside straight
A draw to a straight with a single missing rank in the interior, for example, 8-9-J-Q, seeking a 10. Sometimes used to describe a one-end straight, which is mathematically equivalent. Also “belly buster”, “gutshot”. Compare “bobtail”, “open-ender”.
insurance
A “business” deal in which players agree to split or reduce a pot (roughly in proportion to the chances of each of them winning) with more cards to come rather than playing out the hand, or else a deal where one player makes a side bet against himself with a third party to hedge against a large loss. This is usually done when one player is all in.
in the bushes, in the weeds
A player sandbagging is said to be “in the bushes” during the time he is quietly checking and calling while others bet aggressively. He will eventually “decloak”.
in the middle
In a game with multiple blinds, an incoming player may sometimes be allowed to post the blinds “in the middle” (that is, out of their normal order) rather than having to wait for them to pass.
A player being whipsawed is said to be “in the middle”.
in the money
To place high enough in a tournament to get prize money.
in turn
A player, or an action, is said to be in turn if that player is expected to act next under the rules. Jerry said “check” while he was in turn, so he’s not allowed to raise.
irregular declaration
An action taken by a player in turn that is not a straightforward declaration of intent, but that is reasonably interpreted as an action by other players, such as rapping the table to signify “check”, or pointing a thumb up to signify “raise”. House rules or dealer discretion may determine when such actions are meaningful and/or binding.
irregularity
Any of a number of abnormal conditions in play, such as unexpectedly exposed cards, that may call for corrective action.
isolation
To play aggressively in order to drive out all but one specific opponent who you believe to be weak. I reraised to isolate Diane because she’s been playing junk all day.
From 1971 on, all WSOP events have been tournaments with cash prizes. In 1973 a new event, five-card stud, was added to the main event of No Limit Texas Hold ‘em. Since then new events have been added and removed. In 2006 there will be 42 events at the WSOP, including poker games like Omaha and Razz as well as events only for seniors and women. Event winners get, in addition to their prize money, the coveted golden bracelet.
Doyle Brunson (nicknamed “Texas Dolly”) and Johnny Chan have each won ten bracelets, while Phil Hellmuth has nine. Doyle’s son, Todd Brunson, won a bracelet in a pot-limit Omaha event in 2005, making them the first father/son combo to win at least one event at the WSOP. Also, actress Jennifer Tilly became the second non-poker celebrity to win a WSOP event when she won the Women’s No-Limit Texas Hold-’Em event in 2005. French singer/actor Patrick Bruel won a Limit Hold’em championship in 1998.
The number of participants in the WSOP has grown every year, and in recent years the growth has exploded. In 2000 there were 4,780 entrants in the various events, but in 2005, the number rose to over 23,000 players. In the main event alone, participants grew from 839 in 2003, to 2,576 in 2004, to 5,619 in 2005. For the 2006 main event, a cap of 8,000 players has been established. Much of this growth can be attributed to the WSOP airing on ESPN and the World Poker Tour being shown on the Travel Channel, along with other USA television over-the-air and cable networks such as Fox Sports Net and their “Poker Superstars” series, Bravo with the “Celebrity Poker Showdown” series and GSN with their “Poker Royale” series, as well as the boom in online poker cardrooms on the World Wide Web.
Like most tournaments, the sponsoring casino takes a “rake” (a percentage of between six and ten percent, depending on the buy-in amount) from the buy-ins and distributes the rest, hence the prize money increases with more players. In the 2005 main event $52,818,610 (US) in prize money was distributed, including a $7.5 million first prize. Subtracting the $10,000 buy-ins, over $47 million was won by 560 players in the event. Carl Ygborn finished “on the bubble” (in 561st place), and Harrah’s gave him a free entry into the 2006 Main Event.
One event, that was scheduled for Biloxi, Mississippi, was canceled after the Grand Casino Biloxi, which was scheduled to host the event, suffered major damage from Hurricane Katrina. The Rio also hosted the 2006 World Series of Poker, which began on June 25 with satellite events and formally began the day after with the annual Casino Employee event, won in 2006 by Chris Gros. 2006 featured the “Tournament of Champions” on June 25 and 26, won by Mike Sexton. Various events led up to the main event, which was held from July 28 until August 10. The first prize of $12 million was awarded to Jamie Gold.
Caribbean Stud Poker differs slightly in the United Kingdom, and most parts of Europe, from the US. The game is officially known as “Casino Five Card Stud Poker”, and not all casinos have the jackpot prize. Those which do have the prize, usually the large chain groups, officially call the game “Casino Jackpot Five Card Stud Poker”. In both instances, the game is commonly referred to as “Casino Stud Poker”.
The basic rules are the same in the UK as the US, although the payouts differ – the maximum bet is generally £100 on the ante and £200 on the raise, and all payouts are paid on the raise, meaning the maximum payout can potentially be £10,000 (a Royal Flush pays at the same odds, 50:1, as a Straight Flush).
Casinos offering the jackpot generally have the card shuffled by a card shuffling machine – the cards are then removed and dealt out by the dealer, or croupier. Independent and small casinos generally have the croupier shuffle the cards by hand.
British casinos do not use the chip dropper system; instead, a £1 chip is placed on a small plastic circle on the table, which lights up. The croupier then presses a button on a panel infront of them, which keeps the lights lit up once the chips are removed. The dealer removes the chips, and can then tell which players are playing the jackpot game and which are not.
If the dealer does not show an Ace/King, hands playing the jackpot must be turned over, face up, and shown to the dealer and table. If the player is not playing the jackpot prize, the cards are not shown.
The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713. In this letter, Waldegrave provides a minimax mixed strategy solution to a two-person version of the card game le Her. It was not until the publication of Antoine Augustin Cournot’s Researches into the Mathematical Principles of the Theory of Wealth in 1838 that a general game theoretic analysis was pursued. In this work Cournot considers a duopoly and presents a solution that is a restricted version of the Nash equilibrium.
Although Cournot’s analysis is more general than Waldegrave’s, game theory did not really exist as a unique field until John von Neumann published a series of papers in 1928. These results were later expanded in the 1944 book The Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern. This profound work contains the method for finding optimal solutions for two-person zero-sum games. During this time period, work on game theory was primarily focused on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.
In 1950, the first discussion of the Prisoner’s dilemma appeared, and an experiment was undertaken on this game at the RAND corporation. Around this same time, John Nash developed a definition of an “optimum” strategy for multiplayer games where no such optimum was previously defined, known as Nash equilibrium. This equilibrium is sufficiently general, allowing for the analysis of non-cooperative games in addition to cooperative ones.
Game theory experienced a flurry of activity in the 1950s, during which time the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed. In addition, the first applications of Game theory to philosophy and political science occurred during this time.
In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium (later he would introduce trembling hand perfection as well). In 1967, John Harsanyi developed the concepts of complete information and Bayesian games. He, along with John Nash and Reinhard Selten, won The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel (also known as The Nobel Prize in Economics) in 1994.
In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionary stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge[5] were introduced and analyzed.
In 2005, game theorists Thomas Schelling and Robert Aumann won the Nobel Prize in Economics. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, developing an equilibrium coarsening correlated equilibrium and developing extensive analysis of the assumption of common knowledge.
Guts is quite different from most other poker games (in fact classifying it as a poker game at all is somewhat questionable). Rather than the customary rounds of betting followed by a single showdown, guts features multiple rounds, each of which consist of the decision to be “in” or “out”, and each of which contains a showdown. Only the players who stay “in” participate in the showdown. In the most common version, the player who stays in with the best hand receives the current pot, while all other players who stayed in must match the pot. (For example, if the pot is $5 and three people stay in, then one player will receive the $5 pot and two players will be forced to add $5 each to the pot, thus doubling it.) Then the hand is re-dealt, and all players (even those who were “out” in the last round) can participate again. The game ends when only a single player has the guts to stay “in”, and thus the pot is taken without replenishment.
Each player’s hand usually consists of a reduced poker hand of either 2 or 3 cards. The cards are ranked as in regular 5-card poker, but in some variations straights and flushes count and in some they do not.
Another variation is for three-card guts. The hands are ranked as follows: Three of a kind, straight flush, straight, flush, pair. Each player receives two cards face down. In turn, each player declares whether they’re in or out. If they’re in, they receive their third card face up. The dealer declares last; if no other player has stayed in, then the dealer must have a pair or better to win the pot. Another variation is for the other players to have another chance to declare and challenge the dealer. With this variation, there is no requirement for the dealer’s hand; if no one challenges him, the dealer wins.
Declaring “in” or “out” is similar to declaring high or low in high-low games. Each player takes a chip, places their hands under the table, and either places the chip in one fist or not. Each player then holds their closed fist above the table, and the players simultaneously open their hands to reveal their decision (a chip represents “in”, an empty hand represents “out”).
Because the pot can double (or more) each round, the stakes can grow exponentially, and pots of 50 or 100 times the original ante are not unheard of.
There are many variations. Sometimes only the single player with the worst hand (who stayed in) must add to the pot, but they must double the pot rather than match it. In an especially vicious variation, nobody wins the pot unless nobody else stays in. This can degenerate quickly, when one player must add a large amount to the pot, and decides to stay in until he wins it back. Thus the game continues indefinitely, with one player continually adding larger and larger amounts to the pot. The pot may grow so big that no player has enough cash to match it, leading to arguments about how to end the game. (This variation is not recommended when playing among friends. Often this variation is abandoned after the first really big pot leads to conflict.)
One solution to the exponentially growing pots is to cap them at 50x or 100x the ante. That is, if there are 5 players with an ante of $1, the pot started at $5. If there were 3 doublings, the pot is now at $40. Suppose the “cap the pot at $50″ rule were in force. Then, if another doubling occurred, each loser would pay $40, but the pot would now be at $50 and the extra $30 would be set aside as the ante once there’s a hand with a winner and no loser.
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One of the most popular games in Australian casinos is a Texas hold ‘em variant called “Manila” (also called “Seven-up” in some places). It is played with a Stripped deck in which all cards below the rank of 7 are removed (leaving 32 cards). Each player is dealt two private cards, and a single community card is dealt face up, followed by the first betting round. Then a second community card is followed by a second round, a third community card and a third round, and fourth community card and a fourth round, and finally a fifth community card, fifth betting round, and showdown. On showdown, unlike Texas hold ‘em (and more like Omaha), each player makes the best hand he can from both of his hole cards with exactly three of the five community cards.
Because of the stripped deck, a flush beats a full house. Also, an ace may not be played low for a straight (that is, the hand A-7-8-9-10 is not a straight in Manila). Manila and its variants are rarely played high-low split (in fact, very few stripped deck games are ever played low).
Common variations involve dealing three cards to each player, one of which can either be discarded at some point (like Pineapple, above), or else held to the end, but maintaining the requirement that each player play exactly two of his own cards with exactly three of the board. The three-card variant is sometimes played with 6s being restored to the deck, making it 36 cards.
Six-card stud is usually played as identical to seven-card stud, except that the last face-up round is removed (Thus it is two down, three up, one down). It can also be played as 1-4-1, where the first betting round occurs after only two cards are dealt (one down and one up). This latter form more closely resembles five-card stud with an extra downcard.
A variation called Alligator stud starts with one hole card and one upcard, followed by a first betting round; then two upcards are dealt to each player followed by a second betting round; then a fourth upcard and betting round, and finally a fifth upcard and betting round. This game plays well at no limit and pot limit. The same game, but with each player initially dealt two downcards and one upcard, and restricted to using only one of his downcards in his final hand, is called Zanetti stud.
