# Concept of efficiency in auctions

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?

• I think you're sneaking in an allocation constraint in there. If the bundle is not divisible, then giving it to 3 is the best under both criteria since 5>3+0+0. If you can split {A,B}, then dividing it between 1 & 2 makes social welfare (3+3)=6>5. Apr 23, 2013 at 18:52
• Usually, efficiency defined as the case when the person who values the good most is the one who gets it. In this case, it seems to be more complex than that. The allocation constraint mentioned by D. Masterov seems to create a case where there might be more than one efficient allocation. This is not unusual. An allocation is Pareto efficient if you cannot make any one person better off without hurting one of the others. Remember also that you can apply several different types of social welfare functions: utilitarian, Rawlsian, some concave alternative, etc.. Apr 24, 2013 at 8:20

In auction theory, we usually assume that all bidders have quasilinear preferences. If $A$ is the set of all possible allocations, the utility of bidder $i$ can be written as $u_i(A)+x$ where $x$ can be interpreted as "money". So one can directly compare utilities using the good $x$ as a yardstick. The allocation $a^*$ is then efficient if it maximizes $\sum_i u_i(a)$.