I have some confusions about the concept of "efficiency" in auction theory. One interpretation is that an auction is efficient if it maximizes the social-welfare. But social-welfare is not well defined (or if you can point me to a formal definition of social welfare) and its interpretation seems to vary in different scenarios.

Another interpretation of "efficiency" is that an auction is efficient if each item goes to the highest bidder. This one is easy to understand than the first interpretation. But when it comes to multi-unit auctions in which items can also be sold in bundles, this interpretation also puzzles me. For example, consider the following auction. The table lists the valuation of three bidders for two items and their bundle.

```
A B {A, B}
bidder 1 3 0 0
bidder 2 0 3 0
bidder 3 0 0 5
```

So in this case, bidder 1 the highest bidder for $A$, bidder 2 is the highest bidder for $B$, and bidder 3 is the highest bidder for package $\{A, B\}$. If we use the first interpretation, then the efficient outcome is to assign $A$ to 1 and assign $B$ to 2; but if we use the second interpretation, assigning $\{A, B\}$ to bidder 3 seems also be efficient, isn't it?