Game theory

There are two types of game theory: 1) working out how to win, lose or draw a game played for entertainment; or 2) applying the theory of a game to real life.

The latter meaning is covered by game theory as a branch of mathematics, operations research and economics, analyzing interactions with formalized incentive structures ("games") - whether purposeful games, or battles, or accidental games. The predicted and actual behavior of individuals in these games are studied, as well as optimal strategies . Seemingly different types of interactions can be characterized as having similar incentive structures, thus all being examples of one particular game.

Some theories seeks to find rational strategies in situations where the outcome depends not only on one's own strategy and "market conditions", but upon the strategies chosen by other players with possibly different or overlapping goals. It also finds wider application in fields such as political science and military strategy.

An example of the application of game theory to real life is the prisoner's dilemma as popularized by mathematician Albert W. Tucker; it has many implications for the nature of human cooperation. Biologists have used game theory to understand and predict certain outcomes of evolution, such as the concept of evolutionarily stable strategy introduced by John Maynard Smith in his essay Game Theory and the Evolution of Fighting. See also Maynard Smith's book Evolution and the Theory of Games.

Note it is difficult to apply game theory to life because everybody wants different things out of life-someone may go all out for a little piece of money, others may want huge amounts with little effect, and yet others wish for things as well as money.

Other branches of mathematics, in particular probability, statistics and linear programming, are commonly used in conjunction with game theory to analyse games.

The mechanisms of some games such as snakes and ladders, or ludo, depend very heavily on random inputs, to the extent that game theory cannot usefully analyse them. It can only theorise on strategies as strategic choices.

The difference between a rule (or law) and a theory

Technically speaking, there is no difference, but a rule tends to be more fundamental to playing the game. For instance in chess saying that you need to take as many pieces as possible is a rule, that you should start with say the Bishop's Gambit is a theory. Note too that rules tend to be more useful in playing the game. Theories (and this includes scientific theories like e=mc²) may be debunked later on. However in life rules too may sometimes be debunked.

Types of games and examples

Game theory classifies games into many categories that determine which particular methods can be applied to solving them (and indeed how one defines "solved" for a particular category). Some common categories are:

Zero-sum games are those in which the total benefit to all players in the game adds to zero (or more informally put, that each player benefits only at the expense of others). Chess and Poker are zero-sum games, because one wins exactly the amount one's opponents lose. Business, politics and the prisoner's dilemma, for example, may be condisered are non-zero-sum games because some outcomes are good for all players or bad for all players. It is easier, however, to analyze a zero-sum game, and it turns out to be possible to transform any game into a zero-sum game by adding an additional dummy player often called "the board," whose losses compensate the players' net winnings.

A convenient way to represent a game is given by its payoff matrix. Consider for example the two-player zero-sum game with the following matrix:

 

                                 Player 2 

                     Action A    Action B    Action C

           Action 1    30         -10          20 
 Player 1
           Action 2    10          20         -20

 

This game is played as follows: the first player chooses one of the two actions 1 or 2, and the second player, unaware of the first player's choice, chooses one of the three actions A, B or C. Once these choices have been made, the payoff is allocated according to the table; for instance, if the first player chose action 2 and the second player chose action B, then the first player gains 20 points and the second player loses 20 points. Both players know the payoff matrix and attempt to maximize the number of their points. What should they do?

Player 1 could reason as follows: "with action 2, I could lose up to 20 points and can win only 20, while with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, player 2 would choose action C (negative numbers in the table are good for him). If both players take these actions, the first player will win 20 points. But how about if player 2 anticipates the first player's reasoning and choice of action 1, and deviously goes for action B, so as to win 10 points? Or if the first player in turn anticipates this devious trick and goes for action 2, so as to win 20 points after all?

The fundamental and surprising insight by John von Neumann was that probability provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. The probabilities are computed so as to maximize the expected point gain independent of the opponent's strategy; this leads to a linear programming problem with a unique solution for each player. This method can compute provably optimal strategies for all two-player zero-sum games.

For the example given above, it turns out that the first player should chose action 1 with probability 57% and action 2 with 43%, while the second player should assign the probabilities 0%, 57% and 43% to the three actions A, B and C. Player one will then win 2.85 points on average per game.

Non Zero-Sum game The most famous example of a non-zero-sum game is the Prisoner's dilemma, as mentioned above. Any gain by one player does not necessarily correspond with a loss by another player. The 'kill or be killed' business ideal are non zero-sum games. For example, a business contract ideally is a positive-sum game, where each side is better off than if they didn't have the contract. Most games that people play for recreation are zero-sum.

Cooperative games are those in which the players may freely communicate among themselves before making game decisions and may make bargains to influence those decisions. Monopoly can be a cooperative game, while the Prisoner's dilemma is not. However, Monopoly is a zero-sum game as there can be only one winner, whereas the Prisoner's dilemma is a non-zero-sum game. Most of life can be described as a cooperative game, because we normally cooperate against our opponents.

Complete information games are those in which each player has the same game-relevant information as every other player. Chess and the Prisoner's dilemma are complete-information games, while Poker is not. Not much of life can be described as complete information game.

Risk aversion

For the above example to work, the participants in the game have to be assumed to be risk neutral. This means that, for example, they would value a bet with a 50% chance of receiving 20 'points' and a 50% chance of paying nothing as being worth 10 points. However, in reality people are often risk averse and prefer a more certain outcome - they will only take a risk if they expect to make money on average. Subjective expected utility theory explains how a measure of utility can be derived which will always satisfy the criterion of risk neutrality, and hence is suitable as a measure for the payoff in game theory.

One example of risk aversion can be seen on Game Shows. For example, if a person has a 1 in 3 chance of winning $50,000, or can take a sure $10,000, many people will take the sure $10,000.

Games and numbers

John Conway developed a notation for certain games and defined several operations on those games, originally in order to study Go endgames. In a surprising connection, he found that a certain subclass of these games can be used as numbers, leading to the very general class of surreal numbers.

History

Though touched on by earlier mathematical results, modern game theory became a prominent branch of mathematics in the 1940s, especially after the 1944 publication of The Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. This profound work contained the method for finding optimal solutions for two-person zero-sum games alluded to above.

Around 1950, John Nash developed a definition of an "optimum" strategy for multi player games where no such optimum was previously defined, known as Nash equilibrium. This concept was further refined by Reinhard Selten. These men were awarded The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel in 1994 for their work on game theory, along with John Harsanyi who developed the analysis of games of incomplete information.

Conway's number-game connection was found in the early 1970s.

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

Trick-taking games

Trick-taking games are card games in which play is divided into multiple rounds called tricks, during which each player plays one card from his hand, and the rules of the game determine which player wins that trick based on the cards played. Play ends when all players have played their cards. The object of such games varies; it is often to win the most tricks, but it may instead be to avoid winning tricks, to win exactly a certain number of tricks, or to acquire (or not acquire) certain cards. One might also include in this category other games such as the Chinese Tien Gow, played with dominoes.

A common feature of trick-taking games is the concept of following suit, in which each player is constrained in which card he may play by the obligation to match the suit of the first card played in that trick, called the lead, if he can. Another feature common to many games is the concept of trump (from the French triomphe, although the idea probably originated in Italy), in which special cards (sometimes all the cards of a certain suit) are designated to outrank all other cards played. In general, the player who wins the trick is the player who played the highest trump, or, if no trump is played, the player who played the highest card in the suit that was led. In some games players are obligated to play a higher card (and/or trump the suit to win, if they do not have the suit led) if possible. In most games the player who won the previous trick has to lead on the next one.

Popular trick-taking games include Ambition, Whist, Bridge, Euchre, Pinochle, Skat, Tarocchi, Hearts, Spades, Pitch (card game), Napoleon, Sheepshead, 500, Ninety-nine, Tarocchini, and Forty-five.

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

Contract bridge play

Two partnerships of two players each are needed to play bridge. The four players sit around a table with partners opposite one another. The compass directions are often used to refer to the four players, aligned with their seating pattern. Thus, South and North form one partnership and East and West form the other.

A session of bridge consists of many deals (also called hands or boards); the game play of each deal consists of four phases: the deal, the bidding (or auction), the play of the cards, and scoring.

The goal is to achieve as high a numerical score as possible with the given cards. The score is affected by two principal factors: the number of tricks bid in the auction, and the number of tricks taken during play, where the latter must be higher than or equal to the former. Broadly said, the highest score is achieved when the number of tricks won is equal to (or close to) the number that was bid, so there is incentive to the players to accurately bid the number of tricks that their hands are capable of delivering. Thus, in the bidding stage, the pairs compete to see who proposes the highest number of tricks (and associated trump suit), and the side who wins the bidding must then fulfill that bargain by taking at least the specified number of tricks in play. The number of tricks bid and the trump suit (or lack thereof) are referred to as a contract. If the side who wins the auction then takes the contracted number of tricks (or more), it is said to have fulfilled the contract and is awarded points; otherwise, the contract is said to be defeated and points are awarded to their opponents.

Dealing

The game is played with one complete deck of 52 cards. One of the players is the dealer. In rubber bridge (or other "friendly" games), the cards are shuffled and the dealer distributes all the cards clockwise one at a time, starting with his left-hand opponent and ending with himself, so each player receives a hand of thirteen cards. At the same time, for convenience, the dealer's partner usually shuffles a second deck, to be ready for use on the following deal. The deal rotates clockwise, so the dealer's left-hand opponent will deal next.

In duplicate bridge, the hands are shuffled only once, at the beginning of the tournament, and dealt clockwise one at a time (there are also special machines for pre-dealing on large tournaments), and placed into bridge boards. At each subsequent table, each player pulls his cards from the board and counts them to ensure that the deal has not been corrupted. Unlike in other trick-taking games, the players do not throw their cards to the middle of the table in each trick; instead, each player keeps his played cards before him, to allow the completed deal to be returned to the board unaltered.

The auction

To prepare for the play of the cards, the auction phase determines several things: the contract, which consists of the trump suit and the intended number of tricks; which partnership will play for the contract; and which of the players in that partnership will play the hand. In addition, doubling and redoubling may occur, which represents a "raising of the stakes" when the played hand is scored.

During the auction, each player makes a call at his turn, which consists of any one of the following:

• Pass
• Make a new bid
• Double or Redouble

The auction consists of each player making a call, starting with the dealer and continuing clockwise until three players in a row have passed after any bid. (The word "bid" is also often used informally in place of "call".)

A player may always pass when it is his or her turn.

A bid specifies how many tricks the bidder believes that he can take using his hand and his partner's hand, and with which suit as trump. Any bid starts with the assumption that the bidder can make at least six tricks, called book, plus the stated number of additional tricks. So the bid includes a level (from one to seven, representing how many tricks beyond six the bidder proposes to make) and a denomination (also called strain), which is either a suit or "no trump." For instance, "3 hearts" suggests that his partnership can take nine tricks (book plus three) with hearts as the trump suit.

A player may bid at his turn as long as the bid is higher than the most recent bid. A bid is considered higher if it specifies either a higher level or the same level but with a higher-ranking suit. The denominations are ordered, from lowest to highest, as clubs (♣), diamonds (♦), hearts (♥), spades (♠), and no trump (NT). Thus, after a bid of 3♥, bids of 2♠ or 3♣ are illegal, but 3♠ or 4♦ are legal.

If the most recent bid was made by the opponents, a player may "double" that bid if his partner has not already done so. This essentially states that the player is so confident that the opponents cannot make their bid during play that the player is willing to double their score if they do (and the penalty if they do not). If the most recent bid was made by the player or the player's partner, and it has been doubled by an opponent but not yet redoubled by the player's partner, the player may "redouble," further increasing the potential score or penalty.

The auction ends either if all four players pass initially (in which case the hand is not played or scored) or when three players pass in a row after any bid(s) have been made. The last bid becomes the contract, and its denomination determines whether there will be a trump suit, and if so, what it is. The pair that did not win the contract is called the defense. The pair that made the last bid is divided further: the player who first made a bid in the strain of the final contract becomes the declarer and his or her partner becomes the dummy. For example, suppose West is the dealer and the bidding goes:

South	West	North	East
	pass	1♥	pass
1♠	pass	2♦	double
3♠	pass	4♠	pass
pass	pass

Then East and West would be the defenders, South would be the declarer (since South was the first to bid spades), North would be the dummy, and spades would be the trump suit.

The play of the hand

The play of the hand is similar to other trick-taking games. To summarize, the play consists of thirteen tricks, each trick consisting of one card played from each of the four hands. The first card played in a trick is called the lead, and each player plays a card sequentially around the table clockwise. Any card may be selected as the lead, but the remaining hands must follow suit (meaning, they must play a card in the same suit as the lead), unless they have no more cards of that suit, in which case any card may be played. The hand that plays the highest card in the suit of the lead wins the trick, unless any of the cards are in the trump suit, in which case the hand that plays the highest trump card wins the trick. (Aces are high in bridge, followed by Kings, then Queens, and so on, with 2s the lowest card in each suit.) The hand that wins each trick plays the lead card of the next trick, until all the cards are played.

The first lead, called the opening lead, is made by the defender to the left of the declarer. After the opening lead is played, the dummy lays his entire hand face up on the table. The declarer is thereafter responsible for selecting cards to play from the dummy's hand at the dummy's turn, and from his own hand at his turn. The defenders each choose the cards to play from their own hands. The player who is dummy has practically no rights and must not interfere with the play; (s)he may only play cards from the dummy hand at declarer's order (so that the declarer does not have to lean over the table).

In the end, the goal for each pair is to take as many tricks as possible together (it doesn't matter which player takes them). However, the level of the contract makes a more relevant specific target: the number (level) of the contract is the number of odd tricks the declarer must take, that is, the number of tricks beyond 6. Thus, the declarer is always attempting to take at least a majority of the tricks. In the example above, the declarer must manage to take 10 tricks—6 (assumed) + 4 (bid)—with spades as trump, to make the contract. Success in this goal is rewarded by points in the scoring phase for the declarer's side. If the declarer fails to make the contract, the defenders are said to have set or defeated the contract, and are rewarded points for doing so.

Scoring

When the declarer makes the contract, the declarer's side receives points for:

• The contract bid and made
• Overtricks (tricks taken over the contract level)
• Other bonuses

When the declarer fails to make the contract, the defending pair receives points for undertricks – the number of tricks by which declarer fell short of the goal.

Most bidding revolves around efforts to bid and make a game. Because of the structure of bonuses, certain bid levels are given special significance. The most important level is game, which is any contract whose bid trick value is 100 or more points. Game level varies by the suit, since different suits are worth different amounts in scoring. The game level for no trump is 3 (9 tricks), the game level for hearts or spades (major suits) is 4 (10 tricks), and the game level for clubs or diamonds (minor suits) is 5 (11 tricks). Slam is any contract on level 6 or 7, and it is given very large bonuses.

There are two important variations in bridge scoring: rubber scoring and duplicate scoring. They share most features, but differ how the total score is accumulated. In rubber bridge, points for each pair are tallied either "above the line" or "below the line". In duplicate bridge, all the points are accumulated and present a single score, expressed as a positive number (sum of trick points and bonus points) to the winning pair, and by implication, as a negative number to the opponents. "Chicago" bridge is a form of friendly game which uses duplicate scoring, that is, a set consists of four deals with different vulnerabilities (whether a team has already made game), and every deal is scored as a single number.

In duplicate bridge, the same hand is played unchanged across two or more tables and the results are compared using various methods. The differences are expressed in matchpoints or IMPs. They are summed for every pair for every board they play, and the pair with highest total score becomes the winner of the tournament. Thus, even with bad cards, a pair can win the tournament if it has bid better and played better than the other players who played the same set of cards.

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

History of the contract bridge

A number of card games similar to whist can be traced all the way back to the early 16th century. They were all trick-taking games with a variety of variations. Whist became the dominant form, and enjoyed a loyal following for centuries.

According to the Oxford English Dictionary, the word bridge is the English pronunciation of biritch, an older name of the game of unknown Middle Eastern origin; the oldest known rule book, from 1886, calls it "Biritch, or Russian Whist". The OED reports speculation that the word may come from a Turkish term bir-üç, or "one-three", supposedly referring to the one exposed and three concealed hands.) This game, known today by the retronyms bridge-whist and straight bridge, became popular in the United States and the UK in the 1890s.

Biritch featured several significant developments from Whist: the trump suit was either chosen by the dealer, or he could pass the choice to his partner; there was a call of no trumps; and the dealer's partner laid his cards on the table as dummy to be played by the dealer. It also featured other characteristics found in modern bridge: points scored above and below the line; game was 3NT, 4H and 5D (although 8 club tricks and 15 spade tricks were needed!); the score could be doubled and redoubled; there were slam bonuses.

In 1904 auction bridge arose where the players bid in a competitive auction to decide the contract and declarer. The object became to make at least as many tricks as were contracted for and penalties were introduced for failing to do so.

The modern game of contract bridge was the result of innovations to the scoring of auction bridge made by Harold Stirling Vanderbilt and others. The most significant change was that only tricks contracted for were counted below the line towards game and for slam bonuses, which resulted in bidding becoming much more challenging and interesting. Also new was the concept of vulnerability to make it more expensive to sacrifice to protect the lead in a rubber, and the various scores were adjusted to produce a more balanced game. Vanderbilt wrote down his rules in 1925, and within a few years contract bridge had so supplanted other forms of the game that "bridge" became synonymous with "contract bridge."

These days most bridge played is tournament bridge.

Tournaments

Tournaments were possible because of duplicate bridge, a variation of the game where many sets of players play with the same hands. Duplicate had occasionally been used for whist matches, as early as 1857. For some reason, duplicate was not thought to be suitable for bridge, and so it wasn't until the 1920s that (auction) bridge tournaments became popular.

In 1925 when contract bridge first evolved, bridge tournaments were becoming popular, but the rules were somewhat in flux, and several different organizing bodies were involved in tournament sponsorship: the American Bridge League (formerly the American Auction Bridge League, which changed its name in 1929), the American Whist League, and the United States Bridge Federation. In 1935, the first officially recognized world championship was held. By 1937, however, the American Contract Bridge League had come to power (a union of the ABL and the USBF), and it remains the principal organizing body for bridge tournaments in North America. In 1958, the World Bridge Federation was founded, as bridge had become an international activity.

Today, the ACBL has over 160,000 members and runs 1100 tournaments per year with 3200 officially-associated bridge clubs.

Bidding boxes and bidding screens

Bidding box

In tournaments, "bidding boxes" are frequently used. A bidding box is a box of cards, each bearing the name of one of the legal calls in bridge. A player wishing to make a call displays the appropriate card from the box, rather than making a verbal declaration. This prevents unauthorized information from being conveyed via voice inflection. In top national and international events, "bidding screens" are used. These are diagonal screens which are placed across the table, preventing a player from seeing his partner during the game.

Important Bridge Players

Terence Reese
Charles Goren
Samuel Stayman
Ely Culbertson
Oswald Jacoby
Helen Sobel Smith
Easley Blackwood Sr.
Giorgio Belladonna
Benito Garozzo
Bob Hamman
Omar Sharif
Jeff Meckstroth
Eric Rodwell

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

Bridge Game Strategy - Bidding systems and conventions

Much complexity in bridge arises from the difficulty of successfully arriving at a good final contract in the auction. This is a fundamentally difficult problem: the two players in a partnership must try to communicate enough information about their hands to ultimately arrive at a makeable contract, but the information they can exchange is restricted in two ways:

• Information may only be passed by the calls made and later by the cards played, and not by any other means.
• The agreed-upon meaning of all information passed must be available to the opponents.

A bidding system is the typical solution to this problem: each player evaluates his or her own hand and makes bids to give or request information from their partner, with the goal of eventually arriving at an ideal contract. Bids, doubles, redoubles, and even passes can be either natural or conventional. A natural bid is a proposal to reach a contract in the named suit. A conventional bid is an attempt to communicate, offering and/or asking for information about the partnerships' hands, that is not intended to be a proposal for the final contract. A wide variety of bidding systems have been developed over the course of the 20th century. However, most modern systems have well-established common ground.

First of all, a fairly universal system of high card points is used to give a basic evaluation of the strength of a hand. Aces are counted as 4 points, kings as 3, queens as 2, and jacks as 1 point; therefore, the deck contains 40 points. 26 points shared between partners is considered sufficient for a partnership to bid, and make, game in a major or in no trump. In addition, the distribution of the cards in a hand into suits may also contribute to the strength of a hand and be counted as distribution points. Because 26 points is usually considered sufficient to make game, 13 points in one hand is considered sufficient to open the bidding (that is, make the first bid in the auction), by bidding 1 of a suit.

A one no trump opening bid reflects a hand that has relatively balanced suits and high cards, and usually refers to a hand with 15-17 high card points. In some systems the number of points expected from a 1NT opening bid changes, but it always refers to a relatively narrow range of points.

Opening bids of 2 or higher are reserved for two types of bids: unusually strong bids and preemptive bids. Unusually strong bids communicate an especially high number of points; the availability of unusually strong bids allows a player with a weak hand to safely pass when their partner opens the bidding at one of a suit. Preemptive bids are often made with weak hands that especially favor a particular suit. For instance, with a hand of ♠ AK98742 ♥ 73 ♦ 42 ♣ 76, an opening bid of 3♠ is a very reasonable sacrificial bid, designed to make it difficult for the opposing team to determine a contract for themselves (which is good here, since they are likely to have the bulk of the points).

Most systems include the weak two bid convention, in which opening bids of 2♥, 2♦, or 2♠ are reserved for preemptive bids, while 2♣ is used for very strong hands. This is a first example of a conventional bid: an opening bid of 2♣ in no way suggests 2♣ as a final contract: indeed, in these systems 2♣ may be bid without any clubs.

Another common convention is the 5-card major convention, in which an opening bid of 1♥ or 1♠ promises at least 5 cards in that suit. This leads to some awkward bids, for instance, when a player has four cards in each major, and is forced to open the bidding with 1 of a 3-card minor suit.

Doubles are sometimes used in bidding conventions. A natural, or penalty double, is one used to try to gain extra points when the defenders are confident of setting (defeating) the contract. The most common example of a conventional double is the takeout double of a low-level bid, implying support for the unbid suits and asking partner to choose one of them.

There are many other conventions. Some of the most famous are Stayman, Jacoby transfers and Blackwood.

Bidding systems depart from these basic ideas in varying degrees. Standard American, for instance, is a collection of conventions designed to bolster the accuracy and power of these basic ideas, while Precision Club is a highly conventional system that uses the 1♣ opening bid for strong hands (but sets the threshold rather lower than most other systems) and requires many other changes in order to handle other situations. Many experts today use a system called 2/1 game forcing. In the UK, Acol is the standard system. There are even a variety of techniques used for hand evaluation. The most basic is the Milton Work point count, but this is sometimes augmented by other guidelines such as losing trick count, law of total tricks or Zar Points.

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

Play techniques in bridge game

Terence Reese, a prolific author of bridge books, points out that there are only four ways of taking a trick by force, and two of these are very easy:

• playing a high card that no one else can beat
• trumping an opponent's high card
• establishing long cards (the last cards in a suit will take tricks if the opponents don't have the suit and are unable to trump)
• playing for the opponents' high cards to be in a particular position (if their ace is in front of your king, your king may take a trick)

Nearly all trick-taking techniques in bridge can be reduced to one of these four methods.

The optimum play of the cards can require much thought and experience, and is too complicated to describe in a short article. However, some basic ideas of probability may be considered:

Some of the most important probabilities have to do with the position of high cards.

• The probability that a given opponent holds one particular card, e.g. the king: 50%
• The probability that a given opponent holds two particular cards, e.g. the king and the queen: approximately 25%
• The probability that a given opponent holds at least one of two particular cards, e.g. the king or the queen: approximately 75%

When developing long cards, it is important to know the likelihood that the opponents' cards in the suit are evenly divided between them. Generally speaking, if they hold an even number of cards, they are unlikely to be exactly divided; if the opponents have an odd number in the suit, the cards will probably be divided as evenly as possible. For example, if declarer and dummy have eight trumps between them, the opponents' trumps are probably (68% chance) divided 3-2 (one opponent with three trumps, the other with two) and trumps can be drawn in three rounds. If declarer is trying to play with a seven card trump suit, it is more likely that the outstanding trumps are divided 4-2 (48%) than that the cards are evenly divided 3-3 between the opponents (36%).

Basic techniques by declarer

When new to the game, a player should be familiar with these strategies for playing the hand:

• trumping
• crossruff
• establishing long suits
• finesse
• holdup (mostly at NT contracts)
• managing entries
• drawing trumps

Advanced techniques by declarer

Someone who plays regularly in tournaments should be familiar with these concepts:

• counting the hand (tracking the distribution of suits and high cards in the opponents' hands using inferences from the bidding and play)
• coup
• duck
• dummy reversal
• endplay
• principle of restricted choice
• safety play
• squeeze

Basic techniques by defenders

• opening lead
• when to lead trump

Advanced techniques by defenders

• avoiding an endplay or squeeze
• counting the hand (tracking the distribution of suits and high cards in the unseen hands using inferences from the bidding and play)
• false carding
• opening lead—using information from auction
• signaling
• uppercut

Example

		♠	A6
		♥	KQ1053
		♦	83
		♣	AJ85
♠	10954	N W         E S		♠	KQ872
♥	96			♥	A2
♦	KQ9			♦	J42
♣	K964			♣	1072
		♠	J3
		♥	J874
		♦	A10765
		♣	Q3

The cards are dealt as in the diagram, and South is the dealer. As neither South nor West have sufficient high card strength to open the bidding, North opens with the bid of 1♥, which denotes a long suit and at least 12 high card points. East overcalls with 1♠, South supports partner's suit with 2♥, and West also supports spades with 2♠. North inserts a game try of 3♣, inviting the partner to bid the game of 4♥ with good club support and overall values, and South complies, having extra values in form of ♦A, fourth trump, and doubleton Queen of clubs. The bidding was:

West	North	East	South
			Pass
Pass	1♥	1♠	2♥
2♠	3♣	Pass	4♥
Pass	Pass	Pass

In bidding, North-South were trying to investigate if their cards are worthy for making a game, which yields bonus points if bid and made. East-West were competing with spades, hoping to play a contract in spades at a low level. 4♥ is the final contract, 10 tricks being required for N-S to make with hearts as trumps.

West (left of North, who is the declarer, having been first to bid hearts) has to make the opening lead and chooses the King of spades, playing it face down. After that, South lies his cards on the table and becomes dummy; West turns his leading card face up, and the declarer makes a plan of playing: the bottom line is, since he has to concede trump ace, a spade, and a diamond, he must not lose a trick in clubs.

After a while, the declarer dictates South to play a small spade. West plays low (small card) and North takes the ♠A, gaining the lead. He proceeds by drawing trumps, leading the ♥K. West takes his Ace and cashes the ♠Q. Since he may not continue spades for fear of a ruff and discard, he plays a diamond. Declarer ducks from the table, and East scores the ♦Q. Not having anything better to do, he returns the remaining trump, taken in North's hand. North enters the dummy using ♦A, and leads ♣Q in an attempt to finesse West's King. West covers with the King, North takes the Ace, and proceeds by caching now high ♣J, then ruffs a small club with a dummy's trump. He ruffs a diamond in hand for an entry back, and ruffs the last club in dummy. Finally, he claims the remaining tricks by showing his hand, as it now contains only high trumps and there's no need to continue the play.

(The trick-by-trick notation can be also expressed using a table, but textual explanation is usually preferred, for reader's convenience. Plays of small cards or discards are not explicated, unless they were important for the outcome).

North-South have scored the required 10 tricks, and their opponents took the remaining 3. The contract is fulfilled, and North enters +620 for his side (North-South are in charge for bookkeeping in duplicate tournaments) in the traveling sheet. Every player returns his own cards into the board, and the next deal is played.

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

Auction bridge

The card game auction bridge was developed from straight bridge and was a predecessor to contract bridge. Around the same time five hundred was created by the United States Playing Card Company in 1904.

The main difference between auction bridge and contract bridge is that in auction bridge a game is scored whenever the required number of tricks (9 in No Trump, 10 in Hearts or Spades, 11 in Clubs or Diamonds) is scored. In contract bridge the number of points from tricks taken past the bid do not count towards making a game. Because of this, accurate bidding becomes much more important in contract bridge: partners have to use the bidding to tell each other what their suits and strengths are, so a judgement can be made as to what the chances are of making a game.

Play

The bidding, play and laws were the same as contract bridge.

Scoring

A scoring table for Auction Bridge, from the Official Rules of Card Games, 1973 is as follows:

Odd-tricks: no trumps are worth 10; spades 9; hearts 8; diamonds 7; clubs 6.

Game was 30 points, and only odd-tricks counted towards game. The first side to win two games won the rubber and scored a 250 point bonus.

Each under-trick was worth 50 points to the opponents.

Small slam was worth 50 points; grand slam was worth 100 points.

Honours were scored as follows: 4 trump honours in one hand 80; 5 trump honours or 4 aces in no trumps in one hand 100. For an addition honour in partner's hand, or for 3 or more honours divided between both hands 10 each.

Contracts could be doubled and redoubled, which doubled or quadrupled the odd-trick and under-trick amounts. In addition there was a bonus of 50 points for making a doubled contract and for each over-trick, this was doubled if the contract was redoubled.

Links

• Rules of Card Games: Bridge

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

Beer card

The beer card or the 7 of diamonds is a card in the card game of bridge which is given a special importance in popular bridge sub-culture. The "beer card rule" is not an official part of the rules of bridge but it is played commonly in universities in the United Kingdom and elsewhere.

The basic rule is that, if a player wins the last trick of the hand with the 7 of diamonds, his partner must buy them a pint of beer. The additional requirements vary depending whether the beer card trick winner is the declarer or one of the defenders. For the declarer, the requirements are that:

• Must make contract,
• Must win last trick with the ♦7,
• Diamonds must not be trumps (though some people play that only diamond part scores are excluded),
• Must take a justifiable line on the contract to win as many tricks as possible (i.e. not lose tricks to setup the beer or in order to keep the 7 until the last trick),

For a defender, the requirements are that:

• Contract must be defeated
• Must win last trick with the beer card
• Diamonds must not be trumps
• Must try to win as many tricks as possible (i.e. not lose tricks to setup the beer or in order to keep the 7 until the last trick)

If the contract is doubled then two beers are earned. If the contract is redoubled then four beers are earned.

Example

♠	7
♥	Q832
♦	AKQT9
♣	Q76
N S
♠	Q832
♥	AK
♦	J732
♣	AK5

South plays in the inferior contract of three notrump, against which the opponents cash the first four spade tricks. To maximize the chance of getting a beer, declarer must discard two top diamond honors and a small club from dummy. If the diamonds do not break 4-0, it's straightforward to cash nine winners, ending with the beer card. If the diamonds don't break, there's a chance that a defender will be pseudosqueezed and choose to discard a diamond. For declarer to discard three diamond honors risks losing the contract unncessarily, and so forfeits the beer, even if diamonds turn out to break normally.

 

Links

• NSW Bridge Association
• Edinburgh University Bridge Club

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.