If a rational player reasonably believes that other players will not follow the chain of elimination described above, it would be rational for him/her to guess a number above 0. Even perfectly rational players playing in such a game should not guess 0 unless they know that the other players are rational as well and that all players' rationality is common knowledge. This game illustrates the difference between perfect rationality of an actor and the common knowledge of rationality of all players. Rationality versus common knowledge of rationality 19,196 people participated and the prize was 5000 Danish kroner. When performed among ordinary people it is usually found that the winner's guess is much higher than 0: the winning value was found to be 21.6 in a large online competition organized by the Danish newspaper Politiken. This game is a common demonstration in game theory classes, where even economics graduate students fail to guess 0. (In this way, it is a lopsided version of the so-called 'consensus game', where one wins by being in the majority.) In this case, all integers except 0 and 1 vanish it becomes advantageous to select 0 if one expects that at least 1 / 4 of all players will do so, and 1 otherwise.

$pick a number between 1 and 3 joke$

This degeneration does not occur in quite the same way if choices are restricted to, for example, the integers between 0 and 100. All players selecting 0 also happens to be the Pareto optimal solution. This process will continue until all numbers above 0 have been eliminated. Once these strategies are eliminated for every player, any guess above 44 + 4 / 9 is weakly dominated for every player since no player will guess above 66 + 2 / 3, and 2 / 3 of 66 + 2 / 3 is 44 + 4 / 9. Guessing any number that lies above 66 + 2 / 3 is weakly dominated for every player since it cannot possibly be 2 / 3 of the average of any guess. This equilibrium can be found by iterated elimination of weakly dominated strategies. However, there is a unique pure strategy Nash equilibrium. In this game there is no strictly dominant strategy. 3 Rationality versus common knowledge of rationality.

Pick a number between 1 and 3 joke