Tuesday, January 21, 2014

John Nash and riemannian embedding (memorandum)

NOTICE
  • DO NOT TRUST THE FOLLOING
HISTORICAL FACTS
  • John Nash earned his degree with game theory paper
  • it largely solved Francis Edgeworth's conjecture on economic competence and its equilibrium (it had been also vaguely stated by Cournot)
  • von Neumann had low opinion on his result: "That's trivial, you know. That's just a fixed point theorem."
  • Nash turned to differential geometry and studied riemannian embedding
  • Then he went mad and could not continue his prominent career
QUESTION
  • Do Nash's two principal research areas are not internally related?
SPECULATION
  • Economic system could be a (near-) manifold in economic "spacetime"
  • Approach to equilibrium (relaxation, or tatonnement) is performed in time, but not in same time
  • Economic system as reference frame have time other than wallclock (physical time)
  • above all is global view
  • local view should be told with incommutable operators
  • two players have strategies, some could be commutable (their payoff is independent each other),  but almost all should be incommutable
  • farther they apart, less they interact
  • amount of their interaction is curvature of the two-person universe
  • and it approaches to zero as they get "wealthier"

Saturday, January 18, 2014

Stochastic Process is Topology

NOTICE
  • DO NOT TRUST THE FOLLOWING
OBSERVATIONS
  • branch pattern of infinite length is stochastic process
  • so stochastic process is topology
  • topology have its shape, order type
  • order types are ordinals
  • ordinals made metamathematical (with sufficiently strong reflection principle) are forcing notions
  • Lebesgue used order types to define integration
  • set theorists related enumeration of integration pattern and determinacy
WHAT WE NEED TO BELIEVE IN COMPLETENESS OF PROBABILITY?
  • just an assumption saying unmeasurable set would not appear
  • regularity assumptions on time (transversality, no ponzi, ..)
  • require smoothness of time and economic spacetime hypermanifold
  • set-theoretic axioms (stronger reflection principle)
  • meta set-theoretic axiom sufficient to proof Con(ZF+AD)

Friday, January 17, 2014

Continuity of Equilibrium Hypermanifold in Economic Spacetime (memorandum)

NOTICE
  • DO NOT TRUST THE FOLLOWING
MOTIVATION
  • Existence of Walrasian equilibrium manifold depends on continuous contraction mapping in n-space of goods with non-negativity of price and quantity
  • Think in n+1-"spacetime" and see evolution along time or convergence to above manifold at a point of Ramsey's "bliss"
CONTINUITY OF TIME
  • I mean dt/dx in n+1-spacetime, or in adequately blown-ups of such spacetime
  • In other words, regularity of stochastic process
  • In other words, completeness of expectation
  • (added) dynamic constraints like transversality and no Ponzi game must be related to continuity of time
PROBLEM OF JUMP VARIABLES AND DIVERGENT PATH
  • "Walrasian equilibrium" in spacetime needs continuity of time
  • If no continuity of time is assured, generally divergent paths exist
  • ex-ante vs ex-post (realized state of economy is always continuous evolution of the past in ex-post)
  • how we treat ex-ante irregular changes as extra dimension added to continuous economic spacetime?
STRATEGIC FOUNDATION OF ECONOMIC DYNAMICS
  • existence of pay-off in infinitely repeated games follows from axiom of determinacy (Mycielski-Steinhaus(1962))
  • such axiomatics support is really needed? one serious problem is AD contradicts AC so it hurts Walrasian equilibrium thru Uzawa equivalence