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# Goofspiel

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. It is simple to learn and play, but has a large degree of tactical depth. It is popular with game theorists because, as a game of pure strategy, it is susceptible to analysis.

## 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. 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.

## Variant with perfect information

If all spades are arranged face up (in order) from the start of the game, then Goofspiel becomes a game of perfect information. Even if this is not the case, a player knows all the cards held by her opponent(s) — this makes it atypical for a card game.

## 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 his cards randomly, to determine the best strategy that the other player should use. 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.