Incentive compatibility: Weakly dominant strategy versus Nash equilibrium?
A Nash equilibrium that consists of weakly dominant strategies is a stronger solution concept than a NE itself. 

Consider the following simple matrix game where best replies have been marked with *

\begin{array}{c|cc}
P1/P2&amp;\text{left}&amp;\text{right}\\
\hline
\text{Up}&amp;1^*,1^*&amp;0^*,0\\
\text{Down}&amp;0,0^*&amp;0^*,0^*
\end{array}

Both Up and left are weakly dominant strategies. They are always at least as good as Down and right respectively. That does not rule out, however that there are no other Nash equilibria as the game shows. The combination (Down, right) is also a Nash equilibrium because there is no gain from unilateral deviation. By deleting weakly dominated strategies we lose the latter Nash equilibrium so a Nash equilibrium that consists of weakly dominant strategies is a stronger solution concept.