First-price auction with discrete values
The suggested pure strategies do not create an equilibrium, because if $X_2=X_1=2$, there is a tie. Whichever tie-breaking rule is employed, one of the players would want to deviate to $\beta_i=1+\epsilon$ to avoid the tie and win with certainty. If tie breaking is always in favor of the second player, player 1 deviates to $0+\epsilon$.

Discrete values in this setting do suggest a mixed strategy equilibrium (non-degenerate), because the probability of a tie in valuations is non-zero. This is why the pure strategy equilibrium from the continuous case is missing here.

The more general case of this problem is in Doni &amp; Menicucci (2013).