Goofspiel

Last updated
Goofspiel
Royal flush.jpg
Each player is dealt a full suit of cards and chooses which card to play each round
Other namesThe Game of Pure Strategy, GOPS, Psychological Jujitsu
Players2+
Setup timeShort
Playing timeShort
ChanceNone
Age range8+
SkillsTactics, psychology

Goofspiel (also known as The Game of Pure Strategy, GOPS or Psychological Jujitsu [1] ) is a card game for two or more players. It was invented by Merrill Flood while at Princeton University in the 1930s, [2] and Alex Randolph describes a similar game as having been popular with the 5th Indian Army during the Second World War. [3]

Contents

The game is simple to learn and play, but has some degree of strategic depth. It is commonly used as an example of multi-stage simultaneous move game in game theory and artificial intelligence.

Game play

Goofspiel is played using cards from a standard deck of cards, and is typically a two-player game, although more players are possible. [4] Each suit is ranked A (low), 2, ..., 10, J, Q, K (high).

One suit is singled out as the "prizes"; each of the remaining suits becomes a hand for one player, with one suit discarded if there are only two players, or taken from additional decks if there are four or more. The prizes are shuffled and placed between the players with one card turned up.

Play proceeds in a series of rounds. The players make sealed bids for the top (face up) prize by selecting a card from their hand (keeping their choice secret from their opponent). Once these cards are selected, they are simultaneously revealed, and the player making the highest bid takes the competition card. Rules for ties in the bidding vary, possibilities including the competition card being discarded, or its value split between the tied players (possibly resulting in fractional scores). [1] Some play that the current prize "rolls over" to the next round, so that two or more cards are competed for at once with a single bid card.

The cards used for bidding are discarded, and play continues with a new upturned prize card.

After 13 rounds, there are no remaining cards and the game ends. Typically, players earn points equal to sum of the ranks of cards won (i.e. ace is worth one point, 2 is two points, etc., jack 11, queen 12, and king 13 points). Players may agree upon other scoring schemes.

Mathematical analysis

Goofspiel (or variants of it) has been the subject of mathematical study. For example, Sheldon Ross considered the case when one player plays their cards randomly, to determine the best strategy that the other player should use. [5] Using a proof by induction on the number of cards, Ross showed that the optimal strategy for the non-randomizing player is to match the upturned card, i.e. if the upturned card is the Jack, they should play their Jack, etc. In this case, the expected final score is 59½ - 31½, for a 28-point win. [5]

In 2012 Glenn Rhoads and Laurent Bartholdi found a Nash equilibrium in mixed strategies for the game as defined by Ross, where the payoff players maximize is the point difference in scores rather than the probability of winning, using linear and dynamic programming. [6] A Nash equilibrium strategy is not necessarily the best strategy, only one that does the best if the other player uses the strategy that the Nash equilibrium assigns to them.

Strategy

Any pure strategy in this game has a simple counter-strategy where the opponent bids one rank higher, or as low as possible against the King bid. As an example, consider the strategy of matching the upturned card value mentioned in the previous section. The final score will be 78 - 13 with the King being the only lost prize. [6]

In general, making a very low bid can be advantageous if the player has correctly guessed that the opponent is making a high bid; despite losing a (presumably high-scoring) prize, the player gains an advantage in bidding power that can last for multiple turns. In the variant in which tie bids cause prizes to accumulate, the player with a bidding advantage might make bids that are more likely to tie, knowing that they can then use their uncontested high-bid card to win the accumulated group.

See also

Related Research Articles

<span class="mw-page-title-main">Pinochle</span> Card game

Pinochle, also called pinocle or penuchle, is a trick-taking, Ace-Ten card game typically for two to four players and played with a 48-card deck. It is derived from the card game bezique; players score points by trick-taking and also by forming combinations of characters into melds. It is thus considered part of a "trick-and-meld" category which also includes the game belote. Each hand is played in three phases: bidding, melds, and tricks. The standard game today is called "partnership auction pinochle".

<span class="mw-page-title-main">Trick-taking game</span> Type of card game

A trick-taking game is a card or tile-based game in which play of a hand centers on a series of finite rounds or units of play, called tricks, which are each evaluated to determine a winner or taker of that trick. The object of such games then may be closely tied to the number of tricks taken, as in plain-trick games such as contract bridge, whist, and spades, or to the value of the cards contained in taken tricks, as in point-trick games such as pinochle, the tarot family, briscola, and most evasion games like hearts. Trick-and-draw games are trick-taking games in which the players can fill up their hands after each trick. In most variants, players are free to play any card into a trick in the first phase of the game, but must follow suit as soon as the stock is depleted. Trick-avoidance games like reversis or polignac are those in which the aim is to avoid taking some or all tricks.

<span class="mw-page-title-main">500 (card game)</span> Rules of 500 card game

500 or Five Hundred is a trick-taking game developed in the United States from Euchre. Euchre was extended to a 10 card game with bidding and a Misere contract similar to Russian Preference, producing a good cut-throat three player game like Preference and a four player game played in partnerships like Whist which is the most popular modern form, although with special packs it can be played by up to six players. It arose in America before 1900 and was promoted by the US Playing Card Company, who copyrighted and marketed a deck with a set of rules in 1904. The US Playing Card Company released the improved Avondale scoring table to remove bidding irregularities in 1906. 500 is a social card game and was highly popular in the United States until around 1920 when first auction bridge and then contract bridge drove it from favour. It continues to be popular in Ohio and Pennsylvania, where it has been taught through six generations community-wide, and in other countries: Australia, New Zealand, Canada and Shetland. Despite its American origin, 500 is the national card game of Australia.

<span class="mw-page-title-main">Euchre</span> Card game for two teams of two players

Euchre or eucre is a trick-taking card game commonly played in Australia, Canada, Great Britain, New Zealand, and the Midwestern United States. It is played with a deck of 24, 28, or 32 standard playing cards. There are normally four players, two on each team, although there are variations for two to nine players.

<span class="mw-page-title-main">Spades (card game)</span> Card game

Spades is a trick-taking card game devised in the United States in the 1930s. It can be played as either a partnership or solo/"cutthroat" game. The object is to take the number of tricks that were bid before play of the hand began. Spades is a descendant of the Whist family of card games, which also includes Bridge, Hearts, and Oh Hell. Its major difference as compared to other Whist variants is that, instead of trump being decided by the highest bidder or at random, the Spade suit always trumps, hence the name.

Ninety-nine is a card game for 2, 3, or 4 players. It is a trick-taking game that can use ordinary French-suited cards. Ninety-nine was created in 1967 by David Parlett; his goal was to have a good 3-player trick-taking game with simple rules yet great room for strategy.

<span class="mw-page-title-main">Forty-fives</span> Trick-taking card game

Forty-fives is a trick-taking card game that originated in Ireland. The game is popular in many communities throughout Atlantic Canada as well as the Gaspé Coast in Québec. Forty-fives is also played in parts of Massachusetts and southern New Hampshire in New England, United States, as well as in the South Island of New Zealand.

<span class="mw-page-title-main">Ombre</span> Trick-taking card game

Ombre or l'Hombre is a fast-moving seventeenth-century trick-taking card game for three players and "the most successful card game ever invented."

Rubber bridge is a form of contract bridge played by two competing pairs using a particular method of scoring. A rubber is completed when one pair becomes first to win two games, each game presenting a score of 100 or more contract points; a new game ensues until one pair has won two games to conclude the rubber. Owing to the availability of various additional bonus and penalty points in the scoring, it is possible, though less common, to win the rubber by amassing more total points despite losing two games out of three. Rubber bridge involves a high degree of skill but there is also a fair amount of luck involved in who gets the best cards. A popular variation of rubber bridge is known as Chicago.

Kuhn poker is a simplified form of poker developed by Harold W. Kuhn as a simple model zero-sum two-player imperfect-information game, amenable to a complete game-theoretic analysis. In Kuhn poker, the deck includes only three playing cards, for example a King, Queen, and Jack. One card is dealt to each player, which may place bets similarly to a standard poker. If both players bet or both players pass, the player with the higher card wins, otherwise, the betting player wins.

Pedro is an American trick-taking card game of the All Fours family based on Auction Pitch. Its most popular variant is known as Cinch, Double Pedro or High Five which was developed by Pedro Carlos of San Juan Bautista, California in 1880. Carlos also enlarged the game Pitch Seven Up by adding the five spot of trumps. and soon regarded as the most important American member of the All Fours family. Although it went out of fashion with the rise of Auction Bridge, it is still widely played on the western coast of the United States and in its southern states, being the dominant game in some locations in Louisiana. Forms of the game have been reported from Nicaragua, the Azores, Niobe NY, Italy and Finland. The game is primarily played by four players in fixed partnerships, but can also be played by 2–6 individual players.

<span class="mw-page-title-main">Briscola</span> Card game

Briscola is one of Italy's most popular games, together with Scopa and Tressette (Tresette). A little-changed descendant of Brusquembille, the ancestor of Briscan and Bezique, Briscola is a Mediterranean trick-taking, Ace-Ten card game for two to six players played with a standard Italian 40-card deck. The game can also be played with a modern Anglo-French deck, without the eight, nine and ten cards. With three or six players, twos are removed from the deck to ensure the number of cards in the deck is a multiple of the number of players; a single two for three players and all four twos for six players. The four and six-player versions of the game are played as a partnership game of two teams, with players seated such that every player is adjacent to two opponents.

<span class="mw-page-title-main">Bid whist</span> Card game

Bid whist is a partnership trick-taking variant of the classic card game whist. As indicated by the name, bid whist adds a bidding element to the game that is not present in classic whist. Bid whist, along with spades, remains popular particularly in U.S. military culture and a tradition in African-American culture.

Pitch is the American version of the English trick-taking game of Blind All Fours which, in turn, is derived from classic All Fours. Historically, Pitch started as "Blind All Fours", a very simple All Fours variant that is still played in England as a pub game. The modern game involving a bidding phase and setting back a party's score if the bid is not reached came up in the middle of the 19th century and is more precisely known as Auction Pitch or Setback.

Sueca is a 4 player-partnership point trick-taking card game of the Ace-Ten family, and a popular variant of the Bisca card game. The game is played in Portugal, Brazil, Angola and other Portuguese communities. Its closest relative is the very similar German game Einwerfen.

<span class="mw-page-title-main">Xactika</span> Card game

Xactika is a card game for two to ten players created by Set Enterprises in 2002. The trick-taking game is played with a dedicated deck of 81 cards, in which the object is to obtain the greatest number of points after eight rounds of play. Points are awarded by taking the exact number of tricks that one bids before each round.

Smear is a North-American trick-taking card game of the All Fours group, and a variant of Pitch (Setback). Several slightly different versions are played in Michigan, Minnesota, Northern and Central Iowa, Wisconsin and also in Ontario, Canada.

<span class="mw-page-title-main">Euchre variants</span>

The card game of Euchre has many variants, including those for two, three, five or more players. The following is a selection of notable Euchre variants.

Russian Schnapsen, Thousand Schnapsen, 1000 or Tysiacha is a trick-taking game of the Ace-Ten family for three players, the aim of which is to score over 1000 points to win the game. It is a variant of the popular Austrian game of Schnapsen. Like its parent, Russian Schnapsen features "marriages" which are worth extra points.

<span class="mw-page-title-main">1000 (card game)</span>

1000 is an easy-to-learn card game for two or three players. Its simple rules make it suitable for players to quickly become familiar with the basic concepts of trick-taking and trump-based card games. The name is taken from the score at the end of the game.

References

  1. 1 2 Fixx, James F. (1972). Games for the Superintelligent. Fawcett Popular Library. p. 39.
  2. Tucker, Albert W. (1985). "The Princeton Mathematics Community in the 1930s: Merrill Flood". The Trustees of Princeton University. Archived from the original on 10 March 2015. Retrieved 26 February 2015.
  3. Parlett, David (2000). The Penguin Encyclopedia of Card Games. Penguin. p. 393. ISBN   9780140280326.
  4. "GOPS" in The Very Best Two Player Card Games, PlayingCardDecks, 5 October 2021. Retrieved 13 October 2021.
  5. 1 2 Ross, Sheldon M. (September 1971). "Goofspiel -- The Game of Pure Strategy". Journal of Applied Probability. Journal of Applied Probability, Vol. 8, No. 3. 8 (3): 621–625. doi:10.2307/3212187. JSTOR   3212187. Archived from the original on 24 September 2017.
  6. 1 2 Rhoads, G. C.; Bartholdi, L. (2012). "Computer Solution to the Game of Pure Strategy". Games. 3 (4): 150–156. arXiv: 1202.0695 . doi: 10.3390/g3040150 .