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## "Phase Transitions in Economic Models: Hot and Cold Market Equilibria in Bounded-Rational Potential Games," November 7, 12:00pm, MH 606

Mike Campbell, CSUF

In economic “games” for which there exists a potential (Lloyd Shapley & Dev Monderer), a dynamical model for which each agent’s strategy adjustment follows the gradient of the potential (perfectly rational part that wants to maximize payoff) along with a normally distributed random perturbation (part that considers errors in judgment, miscalculations, emotional bias, etc), is shown to equilibrate to a Gibbs measure for a finite number of agents. There is a variable in front of the random perturbation that allows us to give more or less influence to the “irrational” part of the decision.

For a finite number of agents, there is always a unique Gibbs equilibrium. When an infinite number of agents is considered, more than one equilibrium measure may occur, which is the analogy of a phase transition in statistical mechanics (similar to water changing to ice or steam). Here, the variable that allows us to adjust the influence of the “non-rational” element of decisions is related to “temperature” in statistical mechanics.

The standard Cournot model of an oligopoly is shown not to have a phase transition, as it is equivalent to a continuum version of the Curie-Weiss model. However, when there is increased local competition among agents, a phase transition will likely occur. In the case of a simple discrete model (where an agent either buys or sells a good), if the oligopolistic competition has power-law falloff and there is increased local competition among agents, then the model has a rich phase diagram with an "antiferromagnetic" checkerboard state, striped states and maze-like states with varying widths, and finally a "paramagnetic" unordered state.

Such phases have economic implications as to how agents compete given various restrictions on how goods are distributed. The standard Cournot model corresponds to a uniform distribution of goods, whereas the power-law variations correspond to goods for which the distribution is more localized.