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?