Incentive compatibility: Weakly dominant strategy versus Nash equilibrium? When it comes to proving that a mechanism e.g. auction is incentive compatible this is the approach I'm using:

I break down all the cases that might happen if the agent reports an untruthful value to achieve higher utility. Then I show that by lying about the valuation an agent will achieve a lower or at best the same utility as of truthful valuation reporting. I call this weakly dominant strategy truthfulness.

Now my question is that what is the difference between weakly dominant strategy and Nash equilibrium? I know that a strong dominant strategy equilibrium is stronger than the nash equilibrium. But what about the weakly dominant strategy 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.