In this paper we provide a full characterization of the pure-strategy Nash Equilibria for the p-Beauty Contest Game when we restrict player’s choices to integer numbers. Opposed to the case of real number choices, equilibrium uniqueness may be lost depending on the value of p and the number of players: in particular, as p approaches 1 any symmetric profile constitutes a Nash Equilibrium. We also show that any experimental p-Beauty Contest Game can be associated to a game with the integer restriction and thus multiplicity of equilibria becomes an issue. Finally, we show that in these games the iterated deletion of weakly dominated strategies may not lead to a single outcome while the iterated best-reply process always does (though the outcome obtained depends on the initial conditions).
