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. 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. 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. 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. References External links Complete goofspiel analysis GOPS page at Pagat.com Implementating GOPS in various style of programming Card games introduced in the 1930s Comparing card games Year of introduction missing Games of mental skill