In these classes, we studied dominance. Recall that a dominated strategy is one where there is some other strategy that does at least as well (and sometimes strictly better) regardless of the strategy chosen by the rival.
One can use this concept repeatedly to "dominance solve" some games, but each round of dominance requires ever greater levels of common knowledge of rationality. For instance, the equilibrium for the beauty contest can be solved using infinite rounds of iteration of dominated strategies. But, as you saw earlier, that prediction fares poorly. A good rule of thumb is the deletions up to about two rounds are same, but not thereafter.
We then applied this concept to Vickrey auctions. These are auctions satisfying Vickrey's law: You pay the amount of the externality you inflict on others. This is a version of the It's a Wonderful Life Principle. First, compute the values to all other players if you were not present. Then compute their values when you are. If you pay this amount in winning an item or items, then it is a dominant strategy to bid truthfully. This is a powerful insight for designing proper incentives.
In practice, there are two limitations: 1. It doesn't work well when there are a large number of options with synergies between them. 2. It doesn't work well when fairness is an important consideration. For instance, we saw how Vickrey auctions can lead to situations where the high bidder pays less (or even nothing) for an item while a lower bidder ends up paying more for the same item.