Ordinal notions of submodularity

Ordinal notions of submodularity

We consider several ordinal formulations of submodularity, defined for arbitrary binary relations on lattices. Two of these formulations are essentially due to David Kreps (A Representation Theorem for “Preference for Flexibility”, Econometrica, 1979) and one is a weakening of a notion due to Paul Milgrom and Chris Shannon (Monotone Comparative Statics, Econometrica, 1994). We show that any ref...

On Ordinal VC-Dimension and Some Notions of Complexity

We generalize the classical notion of VC-dimension to ordinal VC-dimension, in the context of logical learning paradigms. Logical learning paradigms encompass the numerical learning paradigms commonly studied in Inductive inference. A logical learning paradigm is defined as a set W of structures over some vocabulary, and a set D of first-order formulas that represent data. The sets of models of...

Kant's Four Notions of Freedom

Submodularity and the evolution of Walrasian behavior

Vega-Redondo (1997) showed that imitation leads to the Walrasian outcome in Cournot Oligopoly. We generalize his result to aggregative quasi-submodular games. Examples are the Cournot Oligopoly, Bertrand games with differentiated complementary products, CommonPool Resource games, Rent-Seeking games and generalized Nash-Demand games. JEL-Classifications: C72, D21, D43, L13.

Polyhedral aspects of Submodularity, Convexity and Concavity

The seminal work by Edmonds [9] and Lovász [39] shows the strong connection between submodular functions and convex functions. Submodular functions have tight modular lower bounds, and a subdifferential structure [16] in a manner akin to convex functions. They also admit polynomial time algorithms for minimization and satisfy the Fenchel duality theorem [18] and the Discrete Seperation Theorem ...