From time to time we work on projects that would benefit from a free lightweight pure Java linear programming library. That is a library unencumbered by a bad license, available cheaply, without an infinite amount of file format and interop cruft and available in Java (without binary blobs and JNI linkages). There are a few such libraries, but none have repeatably, efficiently and reliably met our needs. So we have re-packaged an older one of our own for release under the Apache 2.0 license. This code will have its own rough edges (not having been used widely in production), but I still feel fills an important gap. This article is brief introduction to our WVLPSolver Java library. Continue reading
von Neumann and Morgenstern’s “Theory of Games and Economic Behavior” is the famous basis for game theory. One of the central accomplishments is the rigorous proof that comparative “preference methods” over fairly complicated “event spaces” are no more expressive than numeric (real number valued) utilities. That is: for a very wide class of event spaces and comparison functions “>” there is a utility function u() such that:
a > b (“>” representing the arbitrary comparison or preference for the event space) if and only if u(a) > u(b) (this time “>” representing the standard order on the reals).
However, an active reading of sections 1 through 3 and even the 2nd edition’s axiomatic appendix shows that the concept of “events” (what preferences and utilities are defined over) is deliberately left undefined. There is math and objects and spaces, but not all of them are explicitly defined in term of known structures (are they points in R^n, sets, multi-sets, sums over sets or what?). The word “event” is used early in the book and not in the index. Axiomatic treatments often rely on intentionally leaving ground-concepts undefined, but we are going to work a concrete example through von Neumann and Morgenstern to try and illustrate a bit more of the required intuition and deep nature of their formal notions of events and utility. I also will illustrate how, at least in discussion, von Neuman and Morgenstern may have held on to a naive “single outcome” intuition of events and a naive “direct dollars” intuition of utility despite erecting a theory carefully designed to support much more structure. This is possible because they never have to calculate in the general event space: they prove access to the preference allows them to construct the utility funciton u() and then work over the real numbers. Sections 1 through 3 are designed to eliminate the need for a theory of preference or utility and allow von Neuman and Morgenstern to work with real numbers (while achieving full generality). They never need to make the translations explicit, because soon after showing the translations are possible they assume they have already been applied. Continue reading