Can someone explain Cremer-Mclean's astonishing result in auction theory?

In mechanism design/auction theory, there is a famous result by Cremer and Mclean that if agents'/bidders' valuations are even slightly correlated, then all the surplus can be extracted by the principal/auctioneer. This is what makes auction theory without i.i.d. valuations an unpopular research topic even though it's interesting from a practical point of view.

My question is: Can someone explain to me why the Cremer-Mclean result is true?

The original paper is Econometrica 1998, "Full extraction of the surplus in Bayesian and dominant strategy auctions" by Cremer and Mclean.

There is an explanation beginning on page 151 in Vijay Krishna's Auction Theory textbook.

Solution 1:

Suppose you and I are consultants for competing oil companies. We independently estimate the value of some new drilling site, and recommend our employers make bids based upon our estimates. Since some amount of error is involved, but our estimates are correlated (we are experts, after all!), if I can observe the bid your company makes before my company makes its bid, I can get a sense of whether or not I'm over- or under-estimating the value of the new site.

The problem is that we each only have one estimate. If the seller adopts an auction style that permits me to see your bid (i.e., not sealed-bid), I get additionally information about the distribution of probable values for the new drill site.

We're interested in discovering something about the random variable $t$, the distribution of estimates of the value of the new site. Suppose I have the correct estimate (the true value of the new site, $T$) by accident, and call my estimate $t_i = T$. Suppose further your estimate $t_j$ is higher than $T$. If I observe your bid, I might conclude that the true value actually lies somewhere between $t_i$ and $t_j$, so I tell my boss to bid $(t_i+t_j)/2$. This sort of "watering down" is desirable in order to avoid the "winner's curse". This is where "Bayesian" comes into the jargon of the paper. I obtain new information, your estimate of the value, and use it to update by estimation of the value. After the auction, my company starts drilling, only to discover the true value of production is $T<(t_i+t_j)/2$, so we've overbid.

So, thinking about surplus extraction, the seller of the new field wants to encourage high bidding, or to discourage shading. Discouraging shading is another way of saying "encouraging revelation of your true valuation $t$." This is what Krishna calls "truth-telling". Suppose you value the asset at $t_j$, and you bid $b_j$. Your goal is to maximize $t_j-b_j$, or to bid as low as possible. Ideally, you'd bid $\epsilon$ above the second highest valuation, $t_{j-1}$. This isn't truth-telling.

If you don't know the true value, as in this example, the seller can use other bidders' estimates to reduce your shading. You wouldn't ever bid more than your estimate of the value for the asset, so $b_j \leq t_j$. The seller wants to force you to reveal your estimate, in other words to make you bid your value, or make $b_j=t_j$.

I haven't worked through the full Bayesian-Nash equilibrium, so I can't explain why the seller is successfully able to push $b_j$ toward $t_j$, but I hope this helps!

Solution 2:

Let me try to explain in rough terms why the Cremer-Mclean's result is true. Lets assume that thare are $n$ bidders and $B$ possible values for the item for each bidder. For each bidder $i$ we compute what is his expected gain $g(i,v(i))$ in a second price auction conditioned on his value $v(i)$. We can also compute the expected value $V(j;i,v(i))$ of the item for bidder $j$ conditioned on the value for $i$ being $v(i)$.

We want to express $g(i)$ as a linear combination

(*) $a(1)V(1,i,v(i))+a(2)V(2,i,v(i))+...+a(B)V(n,i,v(i))$, (The sum is over all integers from 1 to $n$ except $i$)

So that the coefficients do not depend on the value $v(i)$!

This is possible if the dependencies among the agents bids are of sufficiently general type. (Namely, certain matrix described this system of equations has a full rank.)

Now, you can charge bidder $i$ entry price which is $a(1)v(1)+a(2)v(2)+...+a(n)v(n)$, (The sum is again over all integers from $1$ to $n$ except $i$)

This way the expected entry price is precisely the expected gain given the bidder's bid but it is computed as a linear combinations of the other bids.

Solution 3:

The main idea behind the Cremer-McLean result is the following:

1. When agents values are correlated, after an agent learns her own value is $v$, her posterior on the other agent's values is distinct from the case where it is some over value $v'$.

2. Given that the two values $v$ and $v'$ induce different posteriors on the other agents' values, we can potentially concoct a bets on the realization of the other agents' values for each of $v$ and $v'$ such that each prefers their own bet. (And these bets are revenue-neutral for the auction.)

3. Scale the magnitudes of these bets so that the incentive to obtain the right bet exceeds the incentive to get a good deal in the auction. Now in the auction the winner can be charged her value.

The conditions under which Cremer-McLean holds specify the conditions under which these bets exist.