1. A method for deriving a combinatorial characterization of an equilibrium in an exchange arena in order to establish a price at which a good can be sold, wherein first and second participants in the exchange arena each have an initial amount of money and goods to exchange, each participant has a utility function representing consumption of goods, the money and goods are exchanged, and an assignment of the price to the good at the equilibrium maximizes a mathematical product of the utility functions, the method comprising: selecting a first test point in a convex set, wherein each point in the convex set describes a price assignment and the first test point has an associated Eisenberg-Gale objective function; selecting a second test point in the convex set, wherein the second test point represents a price assignment associated with a transfer of all of the first participant's goods to the second participant; on a line between the first test point and the second test point, calculating a derivative of the Eisenberg-Gale objective function from the first test point in the direction of the second test point, wherein the derivative is positive at the first test point; if the derivative remains positive along the line from the first test point to the second test point, then selecting the assignment of price represented by the second test point; and if the derivative equals zero somewhere on the line thereby representing an equilibrium, then selecting the assignment of price represented by the point on the line where the derivative equals zero.