Goofspiel

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Goofspiel
Players 2+
Age range 8+
Setup time Short
Playing time Short
Random chance Low
Skill(s) required Tactics, psychology

Goofspiel, also known as The Game of Pure Strategy or GOPS, is a card game for two or more players. It was invented by Merrill Flood while at Princeton University.[1] It is simple to learn and play, but has some degree of strategic depth. It is commonly used as an example multi-stage simultaneous move game in game theory and artificial intelligence.

Game play[edit]

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

  1. One suit is singled out as "competition suit" (in this explanation, we use the spades suit); 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 spades are shuffled and placed between the players with one card turned up.
  2. Play proceeds in a series of rounds. The players make "closed bids" for the top (face up) spade 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). Some play that the current spade may "roll over" to the next round, so that two or more cards are competed for at once with a single bid card.
  3. The cards used for bidding are discarded, and play continues with a new upturned spade card.
  4. 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 being worth 11, Queen 12, and King worth 13 points). Players may agree upon other scoring schemes.

Mathematical analysis[edit]

Goofspiel (or variants of it) has been the subject of mathematical study. For example, Sheldon Ross considered the case when one player plays his cards randomly, to determine the best strategy that the other player should use.[2] Using a proof by induction on the number of cards, Ross showed that the optimal strategy for the second (non-randomizing) player is to match the upturned card, i.e. if the upturned card is the Jack, he should play his Jack, etc. In this case, the expected winnings of player two is 28 points.[2]

The game as defined by Ross, where payoffs are point differences, was solved using linear programming and dynamic programming in 2012.[3]

Strategy of GOPS[edit]

Any pure strategy in this game has a simple counter-strategy where the opponent bids one rank higher and as low as possible against the King. 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 trick.

If a high-scoring card is turned early in the game, it may profit to play a low card. If the opponent wins the trick with a high play, then this advantage can be valuable later on, especially if a tie is played for more than one card. In this case it is sometimes possible to play for a tie, try to continue the tie, and win the game on a single trick.

Poker analogy / betting variant[edit]

If a Goofspiel player can convince the opponent that he (or she) has chosen a high card to win the trick, then the opponent is likely to discard a low card on a losing proposition. The trick can then be won with a lower play, maintaining the strength of the hand. Bluffing thus becomes an important aspect of the game. An option that increases the importance of bluffing is to give each player the chance to modify their play before the reveal. Another option to emphasize the similarity to poker is to add a round of betting after the card choice but before the reveal. The betting provides additional clues as to whether each play is likely to be strong or weak, and after the betting ends, each player is given another chance to change his (or her) play. The final choice of play will be influenced by the size of the "pot" for each card.

Stock market analogy[edit]

GOPS is an exact model of activity on the stock market from day to day. For the market case, the objective is to buy lower than some other buyer, and sell higher than some other seller. The "other" here represents some sort of aggregate of all other participants. The type of calculation of what the market will do next is exactly the same as that of anticipating what the other player will do, based on the history of the play.

References[edit]

  1. ^ Tucker, Albert W.. "Merrill Flood (Interview): The Princeton Mathematics Community in the 1930s". "Transcript #30". 
  2. ^ a b Ross, Sheldon M. (Sep 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. 
  3. ^ Rhoads, G. C.; Bartholdi, L. (2012). "Computer Solution to the Game of Pure Strategy". Games 3 (4): 150–156. doi:10.3390/g3040150. 

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