real-world players play according to this distribution when the number of players in the auction is low. Closely related is the Lowest Unique Positive Integer (LUPI) game, studied by Östling et al (2011). This simplification removes the requirement that the winner pays the successful bid value, so the game is simply to select the lowest unique positive integer, with the value of that integer having no further consequence. In practice, because the value of the bid in the auction game is usually negligible compared to the prize, in those circumstances the strategy for a full auction is essentially identical to that of the simpler LUPI game. A daily LUPI game called "Limbo" was offered by the Swedish state gaming company Svenska Spel between January and March 2007, attracting on average about 50,000 players. Östling et al give a method for calculating the Nash equilibrium distribution for the game, shown on the right for the case of N=100 independent entries. This is also the probability of that integer winning the game, if all the players are following the distribution to pick their integer. A striking feature is the existence of a maximum integer, above which the strategy recommends a weighting of zero. In the case here for . Above this number, the chance that the game has not been won by a lower integer falls below , so that even if one knew with certainty that one would be the only player with such a number, the chance of winning would still be less than one could achieve with a lower number. Below this threshold the Nash distribution is constructed to give each player the same chance of winning, regardless of which number they choose. This chance is a combination of two factors: first, there must be no other player who has chosen the same number; and second, there must be no lower number that has already won the game. Together these lead to the distinctive shape of the curve. At lower numbers, the chance of an even lower number having already won the game is smaller; but this is compensated by players being more likely to choose the number, and so it is less likely to be unique. These factors exactly cancel each other out, giving each player the same chance of winning whichever number they choose; but more such players are likely to choose the lower number, so these