My collaboration with Paul fortuitously combined game-theoretic and economic perspectives. It began one afternoon in the fall of 1979 (shortly after Paul joined MEDS from Stanford and I moved to MEDS from Cowles) when I stopped by Paul�s office to mention that a colleague at Yale and I had worked out the (unique) equilibrium for a two-risk-neutral-bidders, first-price, common-value auction in which one bidder was strictly better informed than the other. What I thought was of interest was that the two bidders� strategies at equilibrium generated the same bid distribution.

Paul asked about the bidders' expected revenues at equilibrium. I replied that the bidder with no private information had an expected payoff of 0. Paul smiled, nodded, and pointed out that this 0-payoff result was actually a special case of a theorem in his dissertation.

If V is the value of the item to either bidder, and X is the informed bidder�s private information, the expected payoff for the informed bidder is ∫ F(h)⋅(1-F(h)) dh , where F(⋅) is the distribution of H = E[V|X].

Paul looked at this for only a few moments before observing that the concavity of the integrand in F meant, via Jensen�s Inequality, that making X less or more informative concerning V should lower or raise, respectively, the informed bidder�s expected payoff. And there, in a matter of a few minutes, was the heart of our first two joint papers.

Paul joined in as coauthor of �Competitive Bidding and Proprietary Information� [1], where his 0-payoff theorem served to simplify the derivation of the equilibrium (and where, picking up on the next topic below, we first discussed �point-opening� as a technique for unifying auctions with continuously-distributed signals and auctions with atoms in the signal distributions). Separately, we wrote �The Value of Information in a Sealed-Bid Auction� [2], where we ultimately (using the notion of �affiliation�) were able to prove the first of our �Honesty is (under appropriate conditions) the best (information-release by the seller) policy� theorems.

A few days later, I dropped by Paul�s office again to offer a new perspective on strategies for games with incomplete information. At a Cowles seminar the previous spring, John Reilly had discussed the mixed-strategy equilibrium to the common-value version of the War of Attrition, and the symmetric pure-strategy equilibrium to the private-value version. It seemed to me that, as the information structure of a game collapsed (from continuous distributions of types on shrinking supports, to a single type), a sequence of equilibria in pure strategies should in some way converge to a mixed strategy equilibrium in the limit. This suggested that a strategy be defined �distributionally,� as a joint distribution on the product of a player�s type and action spaces, with a marginal distribution equal to the original type distribution.

Again, Paul responded immediately: �Hmmm. The original type distribution is tight, so (if the action space is compact) the set of distributional strategies is tight, and by Prohorov�s Theorem is a compact metric space in the weak topology. Then Glicksberg�s existence theorem should give us a general existence proof!� Actually, it took Paul more than a few minutes to say that, since I had to stop him several times to explain what the heck he was talking about! I had to confess to a personal failing: I hadn�t memorized all of Billingsley�s *Convergence of Probability Measures*, as Paul obviously had.
This led to �Distributional Strategies for Games with Incomplete Information� [3], featuring a proof of the first relatively general existence theorem for equilibria of normal-form games with continuous type distributions.

As well, in his dissertation Paul had explored ways to topologize the information structure of games. That work, combined with use of the weak topology on distributional strategies, enabled us (also in [3]) to prove a convergence (of equilibria) theorem that covered the War of Attrition as a special case, and a theorem showing that the set of pure strategies is dense in the space of all distributional strategies (generalizing the point-opening idea in the previous auction paper [1]).

While work on these topics continued to percolate, we decided to see if we could make progress on the fundamental open challenge in auction theory: To extend the known results for auctions in independent private-value settings to settings more closely related to financial markets, where there is an aspect of quality uncertainty to which all bidders are exposed.

We began working with the model Paul had used in his dissertation: A state variable determined the common value (to all bidders) of the item being sold, and the bidders had private signals independently conditioned on that state variable. We struggled with this model into the year-end holiday season with little progress. Fortunately, this time the game-theoretic approach proved to be the key to success: One afternoon (while we were working together in Paul�s dining room), I commented that, in the normal form, the state variable would integrate out of the problem, and suggested that we remove it from the initial formulation. We started off instead with simply a joint distribution of signals, and a payoff function for each bidder that depended only on his signal, and symmetrically on the signals of the other bidders.

This reformulation exposed the affiliation property as the critical monotonicity property required for our analyses, and made the comparison of seller�s revenues across alternative auction formats somewhat simpler. Today, of course, we have intuitive arguments which make the Linkage Principle easy to understand. It�s striking to recall that when the original revenue ordering result (between first-price and second-price auctions) first arose in a blackboard analysis, it took us a moment of thought to realize in which direction we�d shown that the ordering held!

A notable aspect of the revenue-ordering result was that it was *not* linked to there being a common-value (e.g., state-based) element in the economic environment. Indeed, the same ordering holds in affiliated private-value models.

The resulting paper, �A Theory of Auctions and Competitive Bidding� [4], was a joy to work on together. We frequently debated terminology and phrasing, beginning with the very first word of the title: A humble �A Theory ...�, or an aggressively grandiose �The Theory ...�? Given all of the intriguing work that has followed, we fortunately chose �A�.

Across our joint work, two particular results that came mostly from Paul continue to delight me.

The first (from [4]) is the typical revenue superiority of entry fees over reserve prices, which is a delicate result since the mere existence of equilibria in auctions with entry fees is a difficult subject.

The other is an unpublished working paper illustrating via de Finetti�s Theorem that an infinite sequence of exchangeable affiliated random variables must consist of conditionally (on a state variable) independent signals where the conditional distributions have the monotone likelihood ratio property. (This neatly ties together the general symmetric auction model, and the state-based model.)

[1] Engelbrecht-Wiggans, Richard & Milgrom, Paul R. & Weber, Robert J., 1983.
"**Competitive bidding and proprietary information**,"
Journal of Mathematical Economics,
Elsevier, vol. 11(2), pages 161-169, April.

[2] Milgrom, Paul & Weber, Robert J., 1982.
"**The value of information in a sealed-bid auction**,"
Journal of Mathematical Economics,
Elsevier, vol. 10(1), pages 105-114, June. Reprinted in *The Economic Theory of Auctions I* (ed. P. Klemperer), Edward Elgar Publishing Ltd., 2000.

[3] Milgrom, Paul R. & Weber, Robert J., 1985.
"**Distributional strategies for games with incomplete information**, Mathematics of Operations Research , INFORMS, vol. 10(4), pages 619-632, November.

[4] Milgrom, Paul R. & Weber, Robert J., 1982.
"**A theory of auctions and competitive bidding**,"
Econometrica,
Econometric Society, vol. 50(5), pages 1089-1122, September. Reprinted in *The Economic Theory of Auctions I* (ed. P. Klemperer), Edward Elgar Publishing Ltd., 2000.