numbers are more likely to win, following the shape of the curve shown. The size of the two factors can be estimated as follows, for reasonably large values of : If all players play according to the Nash distribution, the number of times a particular integer is picked should follow a Poisson distribution, with a probability that no other player will have picked the number. The chance of a lower number having already won the game is , Combining these leads to the equation: where is the chance each player has of winning. Using the estimate then gives the formula The cut-off value will thus be a little above . Deviations from the Nash distribution Considering data from the Swedish "Limbo" game, Östling et al found that players had rapidly adapted to avoid high numbers, above the Nash cut-off, when these did not win. Similarly a marked initial excess of very low numbers also disappeared, as players became more aware of the pattern of typical winning numbers. However, the range of numbers backed by most players did not extend quite as high as the Nash equilibrium would predict. This appears to be a self-perpetuating feature, since if players do not back the numbers at the higher end of the Nash range, such numbers are less likely to win, and therefore players will continue to be disinclined to back them. Östling et al found that a model based on a mixture of iterative solutions of increasing depth was able to reproduce the observed distribution quite well. Looking at auction data, Pigolotti et al found behaviour in auctions with smaller numbers of players () to be in "striking" agreement with the predicted Nash distribution, in particular accurately matching the position of the sharp cut-off. Östling et al found similarly in a LUPI game recreated for the study with an average of 27 players. However, agreement deteriorated as the size of the auction increased, until in auctions with the observed pattern of bids was better matched by an exponential distribution. Pigolotti et al suggest that the larger auctions had attracted more players with less knowledge of the pattern of typical winning bids; and that some players may have had a reluctance against making higher bids. As a result, other players bidding in the middle of the Nash range sometimes had a probability of success over 10 times the level they would have had,