Decentralized Resource Allocation Mechanisms in Networks Tudor Stoenescu Information Science and Technology Caltech
Organization of the Talk � Major issues of resource allocation in networks � Overview of fundamental issues in decentralized resource allocation � Development of two network pricing mechanisms � Implementation in networks � Conclusions
Motivation � Integrated services networks support the delivery of a variety of services to their users � Diversity of information imposes different requirements on the delivery methods – (audio, video, file transfer)
Challenge � Design of resource allocation strategies which guarantee the delivery of different services, each with its own Quality of Service (QoS) requirement, maximize some performance criterion (e.g. network's utility to its users) and satisfy the network’s informational constraints – Issue: Compatibility with individual objectives
Key Network Features Informationally decentralized system formed by two types of agents: � Users � Network
Users’ Informational Constraints � Preferences over the set of services offered by the network are private information. – Preferences are expressed by a utility function � Users are unaware as well as uninterested in the delivery method used for the requested services � Users are unaware of the other users requesting services from the network
Network Informational Constraints � Network manager knows the network topology and the network's resources – link capacities, buffer size � Network manager is unaware of the number of users that may request services, as well as the users' utilities
Decentralization of information
Major issue � If information were centralized one could use Math Programming methods
Major issue � If information were centralized one could use Math Programming methods � But it is not…
Major issue � If information were centralized one could use Math Programming methods � But it is not… � Can we find ways of implementing the centralized design and still satisfy the informational constraints?
Major issue � If information were centralized one could use Math Programming methods � But it is not… � Can we find ways of implementing the centralized design and still satisfy the informational constraints? � If we find a method of implementing the centralized design, can we guarantee that the agents will follow this method?
Organization of the Talk � Major issues of resource allocation in networks � Overview of fundamental issues in decentralized resource allocation � Development of two network pricing mechanisms � Implementation in networks � Conclusions
Decentralized Resource Allocation Background � Early 1800’s (beginning of the socialist debate) � Late 1800’s – Walrasian school (Pareto, Barone,…) � World War I – German economy � von Mises – economic calculation (1920’s) � Socialist economists of the 1930’s (Taylor, Dickinson, Lange, Lerner,…) � von Hayek – rebuttal to the socialist arguments
von Hayek’s arguments regarding the weakness of socialist economies � Amount of information exchange and calculation needed by a central-control system to determine an optimal resource allocation may be too great. � Incentives provided by the market economy could not be reproduced by any socialist system.
Mechanism Design � Realization Theory – Informational efficiency – Complexity of information processing � Implementation Theory
Mechanism Design (Realization Theory) E
Mechanism Design (Realization Theory) E A
Mechanism Design (Realization Theory) π E A
Mechanism Design (Realization Theory) π E A µ h M
Mechanism components � E – Environment � A – Action Space � M – Message Space � π – Goal correspondence � µ – Equilibrium message correspondence � h – Outcome function
Requirements For each element of the environment there exist a 1. non-empty set of feasible actions.
Requirements For each element of the environment there exist a 1. non-empty set of feasible actions. For each element of the environment the set of 2. feasible actions satisfying the goal correspondence π is non-empty.
Requirements For each element of the environment there exist a 1. non-empty set of feasible actions. For each element of the environment the set of 2. feasible actions satisfying the goal correspondence π is non-empty. The actions generated by π also satisfy some sort 3. of optimality criteria.
Requirements For each element of the environment there exist a 1. non-empty set of feasible actions. For each element of the environment the set of 2. feasible actions satisfying the goal correspondence π is non-empty. The actions generated by π also satisfy some sort 3. of optimality criteria. Requirements 1-3 are constraints on the problem type considered.
Requirements ∈ µ ≠ ∅ , ( ) For all e E e 4.
Requirements ∈ µ ≠ ∅ , ( ) For all e E e 4. µ ⊆ π ∀ ∈ ( ( )) ( ), (non-wastefulness) h e e e E 5.
Requirements ∈ µ ≠ ∅ , ( ) For all e E e 4. µ ⊆ π ∀ ∈ ( ( )) ( ), (non-wastefulness) h e e e E 5. A mechanism satisfying requirements 1 - 5 is called goal realizing .
Requirements Unbiasedness - mechanism should not favor one 6. group of agents over another.
Requirements Unbiasedness - mechanism should not favor one 6. group of agents over another. 7. Essential single-valued - for any environment, the rules of the process leads the system to a uniquely determined allocation
Requirements Unbiasedness - mechanism should not favor one 6. group of agents over another. 7. Essential single-valued - for any environment, the rules of the process leads the system to a uniquely determined allocation A mechanism satisfying requirements 4 through 7 is called satisfactory .
Requirements 8. Privacy preserving - all the agents generate their equilibrium messages based only on their own information about the environment.
Requirements 8. Privacy preserving - all the agents generate their equilibrium messages based only on their own information about the environment. 9. Spot threadedness of µ - the correspondence has a continuous selection around every point in the domain
Requirements 8. Privacy preserving - all the agents generate their equilibrium messages based only on their own information about the environment. 9. Spot threadedness of µ - the correspondence has a continuous selection around every point in the domain 0.a 11 a 12 a 13 … 0.a 21 a 22 a 23 … 0.a 11 a 21 a 31 a 12 a 22 a 32 … 0.a 31 a 32 a 33 …
Requirements 8. Privacy preserving - all the agents generate their equilibrium messages based only on their own information about the environment. 9. Spot threadedness of µ - the correspondence has a continuous selection around every point in the domain A mechanism satisfying requirements 8 and 9 is called regular .
Desired property � A mechanism is said to be informationally efficient if it is goal realizing and regular and it has a message space of a dimensionality which is minimal among all the other goal realizing and regular mechanisms
Implementation Theory � Studies the constrains on the design of mechanisms imposed by the divergence of individual incentives from the performance objective � Question: – Can we design a noncooperative game that implements the social choice rule in some sort of equilibrium messages (Nash, Bayesian, subgame perfect, undominated strategies, etc.) ?
Implementation Issues π A E = × × × ... E E E E 1 2 N
Implementation Issues π A E h M = × × × ... E E E E 1 2 N = × × × ... M M M M 1 2 N
Implementation Issues π A E R h M ( ) = × × × ... E E E E = * * * * ( ) : ( ), ( ),..., ( ) m e m e m e m e 1 2 N 1 2 N = × × × ( ) ... M M M M = 1 2 * * * * * ( ) : ( ),..., ( ), ( ), ( ),..., ( ) N m e m e m e m e m e m e − − i + 1 1 1 i i i N m ∈ M i i ( ) ( ) ∀ ∈ ∀ ∈ * * ( ) ( ) ( ), [ 1 , 2 ,..., ] h m e R e h m e m i N e E − i i i
Implementation Issues π A E R h M ( ) = × × × ... E E E E = * * * * ( ) : ( ), ( ),..., ( ) m e m e m e m e 1 2 N 1 2 N = × × × ( ) ... M M M M = 1 2 * * * * * ( ) : ( ),..., ( ), ( ), ( ),..., ( ) N m e m e m e m e m e m e − − i + 1 1 1 i i i N m ∈ M i i ( ) ( ) ∀ ∈ ∀ ∈ * * ( ) ( ) ( ), [ 1 , 2 ,..., ] h m e R e h m e m i N e E − i i i ⊆ π ∀ ∈ ( ( )) ( ), h R e e e E
Organization of the Talk � Major issues of resource allocation in networks � Overview of fundamental issues in decentralized resource allocation � Development of two network pricing mechanisms � Implementation in networks � Conclusions
Mechanism design in the context of networks � Studied two problems – Unicast with routing and QoS requirements – Multi-rate multicast
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