Phase Transition in a Bounded Rational Speculative and Hedging Model

A ‘bounded rational’ overlay is constructed for a model of an interaction between two players who speculate on oil and the U.S. dollar, subject to financial transaction taxes. This model also has two types of operators: a real economic subject (Air) and an investment bank (Bank). Many investment operators (banks) are also considered. Their behavior equilibrates much more quickly, as they react to the move of Air. In this sense, Air is an acting external agent (such as with an external magetic field in a magnetic system), whereas the random component of the bounded rational behavior of banks is ‘annealed’ (i.e., averaged out before Air makes its next transaction). Under certain conditions for the model, the equilibrium measure for the bank agents after Air has played its strategy, is a Gibbs measure from statistical mechanics. The Gibbs measure arises since the interactions between operators are that for a bounded-rational Potential game. When Air makes no purchases of oil futures as a hedge, two Nash equilibriums exist for the bank agents due to a market exchange symmetry in the potential. However, a unique Nash equilibrium exists for the bank agents when Air makes a purchase. This is a result of Air’s purchase breaking a symmetry of the potential. The existence of multiple equilibriums in this two-market model is in the spirit of the Sonnenschein-Mantel-Debreu theorem, and is associated with phase transitions in statistical mechanics. Finally, we show that there is a phase transition. When the longer-term investing Air remains in the oil futures market (nonzero field), the speculating Bank agents prefer the oil spot market and their behavior is more stable, as is reflected by finite spatial volatility. Bank agents also prefer the oil spot market immediately after a relatively slow divestment of Air from the oil futures market. When Air makes no purchases of oil futures as a hedge (zero field), then as Bank agents become more rational (thermal cooling), they will “crowd” or “herd” their preferences into one market or the other at a critical temperature, which is spontaneous symmetry breaking of the market exchange symmetry. Thus we see that spontaneous symmetry breaking refines the potential and signifies the emergence of a market preference among speculators. The spatial volatility diverges at this critical temperature, indicating a less stable situation when longerterm investing is absent. We express our graditude to Alessia Donato for typesetting and editing.