1. A tangible computer-readable storage medium for deriving a combinatorial characterization of an equilibrium in an exchange arena, wherein first and second participants in the exchange, each participant has a utility function representing consumption of goods, the money and goods are exchange, and an allocation of a gamut of resources at the equilibrium maximizes a mathematical product of the utility functions, the tangible computer-readable storage medium storing computer-executable instructions for causing a computer to perform the steps comprising: selecting a first test point in a convex set, wherein each point in the convex set describes a respective allocation of resources and the first test point has an associated Eisenberg-Gale objective function, and wherein the derivative of the Eisenberg-Gale objective function at the first test point is positive; selecting a second test point in the convex set, wherein the second test point represents an allocation of resources associated with a transfer of all of the first participant's goods to the second participant; on a line from the first test point to the second test point, calculating the 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 indicating the allocation of resources represented by the second test point; and if the derivative equals zero somewhere on the line from the first test point to the second test point, thereby representing the equilibrium, then indicating the allocation of resources represented by the point on the line from the first test point to the second test point where the derivative equals zero; wherein the Eisenberg-Gale objective function is represented as: ∑ i ⁢ m i ⁢ log ( ∑ j ⁢ u ij ⁡ ( x ij + ɛ ⁢ ⁢ z ij ) ) wherein i represents the i th participant; j represents the j th good; m j represents the initial amounts of money of the i participants; u ij represents the utility functions of the i participants with respect to the j goods: and z ij represents (x ij ′−x ij ) where x ij ′represents an assignment in equilibrium of good j with respect to participant i, and x ij represents an assignment not in equilibrium of good j with respect to participant i; such that goods are transferred from the first participant to the second participant, at least some goods are transferred from the second participant to the first participant in exchange, and ∈ represents a positive constant.