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.

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Italians playing cards, Sancai-type bowl, Northern Italy, mid-15th century.

In the late 1300s, the use of playing cards spread rapidly across Europe. The first widely accepted references to cards are in 1371 in Spain, in 1377 in Switzerland, and, in 1380, they are referenced in many locations including Florence, Paris, and Barcelona. A Paris ordinance dated 1369 does not mention cards; its 1377 update includes cards. In the account-books of Johanna, duchess of Brabant, and her husband, Wenceslaus of Luxemburg, there is an entry dated May 14, 1379 as follows: “Given to Monsieur and Madame four peters, two forms, value eight and a half moutons, wherewith to buy a pack of cards”. An early mention of a distinct series of playing cards is the entry of Charles or Charbot Poupart, treasurer of the household of Charles VI of France, in his book of accounts for 1392 or 1393, which records payment for the painting of three sets or packs of cards, which were evidently already well known.

It is clear that the earliest cards were executed by hand, like those designed for Charles VI. However, this was quite expensive, so other means were needed to mass-produce them. It is possible that the art of wood engraving, which led to the art of printing, developed because of the demand for implements of play. If the assumption is true that the cards of that period were printed from wood blocks, the early card makers or cardpainters of Ulm, Nuremberg, and Augsburg, from about 1418 to 1450, were most likely also wood engravers.

Many early woodcuts were colored using a stencil, so it would seem that the art of depicting and coloring figures by means of stencil plates was well known when wood engraving was first introduced. No playing cards engraved on wood exist whose creation can be confirmed as early 1423 (the earliest-dated wood engraving generally accepted). However, in this period professional card makers were established in Germany, so it is probable that wood engraving was employed to produce cuts for sacred subjects before it was applied to cards, and that there were hand-painted and stencilled cards before there were wood engravings of saints. The German Brief maler or card-painter probably progressed into the wood engraver; but there is no proof that the earliest wood engravers were the card-makers.

The Europeans experimented with the structure of playing cards, particularly in the 1400s. Europeans changed the court cards to represent European royalty and attendants, originally “king”, “chevalier”, and “knave” (or “servant”). Queens were introduced in a number of different ways. In an early surviving German pack (dated in the 1440s), Queens replace Kings in two of the suits as the highest card. Throughout the 1400s, 56-card decks containing a King, Queen, Knight, and Valet were common. Suits also varied; many makers saw no need to have a standard set of names for the suits, so early decks often had different suit names (though typically 4 suits). The cards manufactured by German printers used the suits of hearts, bells, leaves, and acorns still present in Eastern and Southeastern German decks today used for Skat and other games. Later Italian and Spanish cards of the 15th century used swords, batons, cups, and coins. It is likely that the Tarot deck was invented in Italy at that time, though it is often mistakenly believed to have been imported into Europe by Gypsies. While originally (and still in some places, notably Europe) used for the game of Tarocchi, the Tarot deck today is more often used for cartomancy and other occult practices. This probably came about in the 1780s, when occult philosophers mistakenly associated the symbols on Tarot cards with Egyptian hieroglyphs.

The four suits (hearts, diamonds, spades, clubs) now used in most of the world originated in France, approximately in 1480. These suits have generally prevailed because decks using them could be made more cheaply; the former suits were all drawings which had to be reproduced by woodcuts, but the French suits could be made by stencil. The trèfle, so named for its resemblance to the trefoil leaf, was probably copied from the acorn; the pique similarly from the leaf of the German suits, while its name derived from the sword of the Italian suits. It is not derived from its resemblance to a pike head, as commonly supposed. In England the French suits were used, and are named hearts, clubs (corresponding to trèfle, the French symbol being joined to the Italian name, bastoni), spades (corresponding to the French pique, but having the Italian name, spade=sword) and diamonds. This confusion of names and symbols is accounted for by Chatto thus:

“If cards were actually known in Italy and Spain in the latter part of the 14th century, it is not unlikely that the game was introduced into this country by some of the English soldiers who had served under Hawkwood and other free captains in the wars of Italy and Spain. However this may be, it seems certain that the earliest cards commonly used in this country were of the same kind, with respect to the marks of the suits, as those used in Italy and Spain.”

Court cards have likewise undergone some changes in design and name. Early court cards were elaborate full-length figures; the French in particular often gave them the names of particular heroes and heroines from history and fable. A prolific manufacturing center in the 1500s was Rouen, which originated many of the basic design elements of court cards still present in modern decks. It is likely that the Rouennais cards were popular imports in England, establishing their design as standard there, though other designs became more popular in Europe (particularly in France, where the Parisian design became standard). Rouen courts are traditionally named as follows: the kings of spades, hearts, diamonds, and clubs are David, Alexander, Caesar, and Charles (Charlemagne), respectively. The knaves (or “jacks”; French “valet”) are Hector (prince of Troy), La Hire (comrade-in-arms to Joan of Arc), Ogier (a knight of Charlemagne), and Judas Maccabeus (who led the Jewish rebellion against the Syrians). The queens are Pallas (warrior goddess; equivalent to the Greek Athena or Roman Minerva), Rachel (biblical mother of Joseph), Argine (the origin of which is obscure; it is an anagram of regina, which is Latin for queen), and Judith (from Book of Judith). Parisian tradition uses the same names, but assigns them to different suits: the kings of spades, hearts, diamonds, and clubs are David, Charles, Caesar, and Alexander; the queens are Pallas, Judith, Rachel, and Argine; the knaves are Ogier, La Hire, Hector, and Judas Maccabee. Oddly, the Parisian names have become more common in modern use, even with cards of Rouennais design.

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The origin of playing cards is obscure, but it is almost certain that they began in China after the invention of paper. Ancient Chinese “money cards” have four “suits”: coins (or cash), strings of coins (which may have been misinterpreted as sticks from crude drawings), myriads of strings, and tens of myriads. These were represented by ideograms, with numerals of 2-9 in the first three suits and numerals 1-9 in the “tens of myriads”. Wilkinson suggests in The Chinese origin of playing cards that the first cards may have been actual paper currency which were both the tools of gaming and the stakes being played for. The designs on modern Mahjong tiles and dominoes likely evolved from those earliest playing cards. The Chinese word p’ai is used to describe both paper cards and gaming tiles.

The time and manner of the introduction of cards into Europe are matters of dispute. The 38th canon of the council of Worcester (1240) is often quoted as evidence of cards having been known in England in the middle of the 13th century; but the games de rege et regina there mentioned are now thought to more likely have been chess. If cards were generally known in Europe as early as 1278, it is very remarkable that Petrarch, in his dialogue that treats gaming, never once mentions them. Boccaccio, Chaucer and other writers of that time specifically refer to various games, but there is not a single passage in their works that can be fairly construed to refer to cards. Passages have been quoted from various works, of or relative to this period, but modern research leads to the supposition that the word rendered cards has often been mistranslated or interpolated.

It is likely that the ancestors of modern cards arrived in Europe from the Mamelukes of Egypt in the late 1300s, by which time they had already assumed a form very close to those in use today. In particular, the Mameluke deck contained 52 cards comprising four “suits”: polo sticks, coins, swords, and cups. Each suit contained ten “spot” cards (cards identified by the number of suit symbols or “pips” they show) and three “court” cards named malik (King), nā’ib malik (Viceroy or Deputy King), and thānī nā’ib (Second or Under-Deputy). The Mameluke court cards showed abstract designs not depicting persons (at least not in any surviving specimens) though they did bear the names of military officers. A complete pack of Mameluke playing cards was discovered by L.A. Mayer in the Topkapi Sarayi Museum, Istanbul, in 1939; this particular complete pack was not made before 1400, but the complete deck allowed matching to a private fragment dated to the twelfth or thirteenth century. There is some evidence to suggest that this deck may have evolved from an earlier 48-card deck that had only two court cards per suit, and some further evidence to suggest that earlier Chinese cards brought to Europe may have travelled to Persia, which then influenced the Mameluke and other Egyptian cards of the time before their reappearance in Europe.

It is not known whether these cards influenced the design of the Indian cards used for the game of Ganjifa, or whether the Indian cards may have influenced these. Regardless, the Indian cards have many distinctive features: they are round, generally hand painted with intricate designs, and comprise more than four suits (often as many as twelve).

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