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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?

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closed as off-topic by Ricardo Andrade, Stefan Kohl, Lucia, Yemon Choi, Chris Godsil Dec 31 '14 at 13:00

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Ricardo Andrade, Stefan Kohl, Lucia, Yemon Choi, Chris Godsil
If this question can be reworded to fit the rules in the help center, please edit the question.

Efficiency means that there is no unexploited opportunity to make everyone better off via some combination of reallocation and side payments. Giving both goods to bidder 3 is inefficient because it misses the opportunity to transfer good A to bidder 1, transfer good B to bidder 2, and have each of bidders 1 and 2 make a $2.75 side payment to bidder 3. That said, this question is clearly off topic here. – Steven Landsburg Apr 23 '13 at 16:09
PS --- and of course your example goes away if you assume free disposal (so that the value of A is always greater than or equal to the value of $\{A,B\}$). – Steven Landsburg Apr 23 '13 at 18:37

Pareto-efficiency means that there's no way to reallocate goods and make everyone better off. This is a weak notion of efficiency in general. For example, suppose we cannot require payments, and just have to allocate a single good to someone. Then, if you give the good to anyone, that would be Pareto-efficient.

However, in auction theory it is usually assumed that the utility function takes the form $u(x)-t$, where $x$ is the allocation and $t$ is the amount paid. Under this assumption it is possible to make "interpersonal comparisons of utility", so that Pareto-efficiency becomes equivalent to maximizing the sum of $u_i(x)$ over all agents $i$.

Here's how this works: suppose we have a single good and we are giving the good to someone who is not the person with the highest utility for it, so we're not maximizing the sum. Then we can give the good to someone who values it more, but make that new person getting the good pay the old person an amount that makes him just as good. This transfer makes everyone better off.

To sum up: In auctions, Pareto-efficiency means the choice of $x$ maximizes $\sum_i u_i(x)$.

So the proper notion of efficiency here is Pareto-efficiency. Under this, your conclusion that giving $\{A,B\}$ to bidder 3 is efficient is not right. You could reallocate the respective goods to players 1 and 2, make them pay 2.7, and take the good out of the hands of bidder 3 but make him receive 5.4, and this would make everyone better off.

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