Time polynomial arrow-debreu market equilibrium
First Claim
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;
wherein i represents the ith participant;
j represents the jth good;
mj represents the initial amounts of money of the i participants;
uij represents the utility functions of the i participants with respect to the j goods; and
zij represents (xij′
−
xij) where xij′
represents an assignment in equilibrium of good j with respect to participant i, and xij 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.
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Abstract
A concept for providing a process and apparatus for allocating a gamut of assets/resources across a spectrum of consumers is described. The concept includes an apparatus for allocating resources across a spectrum of users. The apparatus includes one or more processors and a memory coupled to the one or more processors. The memory is configured to store data representative of characteristics and capabilities of the resources and describing needs of the spectrum. The memory further includes computer-readable code configured to cause the one or more processors to perform acts of: estimating current requests from the spectrum for the resources; comparing the current requests to the capabilities and characteristics; and allocating the resources with respect to the requests in conformance with a convex program implementation of Arrow-Debreu theory.
117 Citations
8 Claims
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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:
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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;
wherein i represents the ith participant;
j represents the jth good;
mj represents the initial amounts of money of the i participants;
uij represents the utility functions of the i participants with respect to the j goods; and
zij represents (xij′
−
xij) where xij′
represents an assignment in equilibrium of good j with respect to participant i, and xij 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.- View Dependent Claims (2, 3)
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4. An apparatus for allocating a gamut of resources across a spectrum of users in an exchange arena, wherein the users each have an initial amount of money and goods to exchange, each user has a utility function representing consumption of goods, and the money and goods are exchanged, comprising:
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at least one processor; and a memory coupled to the processor, wherein the memory stores data representing the resources and multiple needs of the spectrum of the users, and the memory further includes computer-readable instructions capable of causing the processor to perform the steps comprising; estimating current requests from the spectrum of the users for the resources; comparing the current requests to the data; and allocating the resources with respect to the current requests according to a combinatorial characterization of an equilibrium in the exchange arena, wherein the combinatorial characterization determines when an assignment of the resources between the users provides, in response to allocating the resources, an increase in a product of utility functions associated respectively with the users, wherein the allocating includes; selecting a first test point in a convex set describing a plurality of assignments, wherein 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; locating a second point in the convex set, wherein the second point represents an exchange of all the money and goods between the users; testing the derivative of the Eisenberg-Gale objective function at the first test point to determine when the derivative is positive along a line from the first test point to the second point; if the derivative remains positive along the line from the first test point to the second point, then indicating the assignment represented by the second point; and if the derivative for at least one point along a length of the line from the first test point to the second point is not positive, then finding a third point along the line from the first test point to the second point at which the derivative has a value of zero, representing the equilibrium, and indicating the assignment represented by the third point; wherein the Eisenberg-Gale objective function is represented as;
wherein i represents the ith user;
j represents the jth good;
mj represents the initial amounts of money of the i users;
uij represents the utility functions of the i users with respect to the j goods; and
zij represents (xij′
−
xij) where xij′
represents an assignment in equilibrium of good j with respect to user i, and xij represents an assignment not in equilibrium of good j with respect to user i;
such that ∈
represents a positive constant.- View Dependent Claims (5, 6, 7, 8)
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Specification