Equilibrium Analysis
In this game there is no strictly dominant strategy. However, there is a unique pure strategy Nash equilibrium. This equilibrium can be found by iterated elimination of weakly dominated strategies. Guessing any number that lies above 66.67 is weakly dominated for every player since it cannot possibly be 2/3 of the average of any guess. These can be eliminated. Once these strategies are eliminated for every player, any guess above 44.44 is weakly dominated for every player since no player will guess above 66.67 and 2/3 of 66.67 is approximately 44.44. This process will continue until all numbers above 0 have been eliminated.
This degeneration does not occur in quite the same way if choices are restricted to, for example, the integers between 0 and 100. 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 if otherwise. (In this way, it is a lopsided version of the so-called "consensus game", where one wins by being in the majority.)
Read more about this topic: Guess 2/3 Of The Average
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