This is the oldest paper I plan on posting, and it’s a far more unconventional selection than the other logical candidates for a first paper — von Neumann and Morgenstern’s Games and Economic Behavior, which is virtually unreadable, and Nash’s 1950 paper, which would be useful only really as a non-constructive foil.

Instead, I think that the distinction should go to Eisenberg and Gale’s four-page gem from 1959, which considers how a collection of individual probabilities could be pooled into a single probability estimate. The paper is written in a rigorous but conversational manner that has aged well, making it an enjoyable short read more than fifty years later. But what really sets it apart is this:

[The existence of a solution] can be proved by means of fixed-point theorems. We prefer, however, to prove existence in an elementary manner using a variational method which seems to be of interest in itself.

This “variational method” defined the solution to the consensus problem as the solution to a convex optimization, which is so much more heartening to a computer scientist than seeing the solution as the product of a Brouwer fixed point.

Old paper, new results

Eisenberg and Gale’s paper was brought to the forefront by recent work by Kamal Jain and Vijay Vazirani. Essentially, because the solution of the E-G process can be expressed as a convex optimization, and convex optimization is computationally easy, things that “behave like” E-G can be solved well in practice. In the face of a plethora of recent results showing that Economic thinking about equilibria has been built on the unstable ground of computational intractability, it’s nice to have some classic equilibrium-finding algorithms that actually work in the real world.

Finally, Manski’s result on equilibrium in prediction markets — which implies the long-shot bias that shows up in actual market data — can be derived in a straightforward manner from E-G, reading the paper from the perspective of beliefs being continuously distributed.

Eisenberg and Gale — Consensus of Subjective Probabilities: The Pari-mutuel Method (1959)