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Mechanisms for a Spatially Distributed Market. Moshe Babaioff, Noam Nisan and Elan Pavlov School of Computer Science and Engineering Hebrew University of Jerusalem, Israel ACM Conference on Electronic Commerce (EC'04) May 17-20, 2004, New York City. Spatially Distributed Market (SDM).
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Mechanisms for a Spatially Distributed Market Moshe Babaioff, Noam Nisan and Elan Pavlov School of Computer Science and Engineering Hebrew University of Jerusalem, Israel ACM Conference on Electronic Commerce (EC'04) May 17-20, 2004, New York City
Spatially Distributed Market (SDM) • A single good is manufactured and consumed in many different locations. • Transportation of the good from one location to the other incurs a cost. • Integration of electronic markets for a single good in different locations to one global market. • Determine the production, consumption and exchange relationships between the markets. • Determine the market prices.
Talk Structure • Spatially Distributed Market • SDM with non-strategic agents • Efficiency by Minimum Cost Circulation • Two Welfare Theorems. • SDM with strategic agents • VCG mechanism characterization. • Computationally efficient, budget balanced mechanism with high efficiency.
S B S B S B S B M1 M2 5 20 12 8 3 2 1 4 M3 6 1 M4 7 4 3 SDM Model Buyer’s value 2 9 Unit shipment cost Seller’s cost Utility maximizing agents with quasi-linear utility function (value-payment)
S B S B S B S B M1 M2 M3 M4 Allocations 5 20 2 9 12 8 3 2 1 4 6 1 7 4 3 Allocation – a set T of trading agents and a shipment vector x, which is materially balanced. Allocation value – total value to the agents minus the shipment cost. V(A) = (20+12) – (6+1) – (4+2) Goal: find an efficient allocation
,flow S B S B S B M1 M2 ,1 M3 M4 (-9,1 ) ,1 ,1 (1,1 ) Efficiency by Min-Cost-Circulation (cost,capacity ) S B 5 20 2 9 12 8 4 3 1 2 (1,∞ ) 6 1 7 4 3 The minimal cost circulation is the efficient allocation Sink
Spatial Price Equilibrium (SPE) Prices are per market not per agent. Definition: An allocation A=(T,x) and vector of market prices p are in a Spatial Price Equilibrium (SPE) iff the allocation and price vector satisfy the following equilibrium conditions: • Each agent buy/sell a unit iff the price in its market is attractive to him. • For any edge between markets (Mi,Mj)E: • pi+ ci,jpj. • If xi,j > 0 then pi+ ci,j =pj
The Two Welfare Theorems Theorem: • (First Welfare Theorem for SDM) If an allocation and a price vector are in a SPE, then the allocation is efficient. • (Second Welfare Theorem for SDM) If an allocation is efficient, then there exists a price vector such that the allocation and the price vector are in a SPE.
VCG SDM mechanisms • Each agent’s cost/value for the good is a private information. Mechanism: Given reported values, define an allocation ruleand a payment rule. • Desired Economic properties: • Incentive Compatibility (IC) in dominant strategies • Individual Rationality (IR) • Efficiency • Ex post (weakly) Budget Balanced (BB) – the sum of payments including the transportation costs is non-negative. • Also Desired: Computational efficiency.
S B S B S B M1 M2 (1,∞,1) (1,∞) (-1,1) M3 M4 (-9,1,1) (9,1) (1,1,1) (-1,1) The residual graph (cost,capacity,flow) S B 20 2 9 12 8 6 1 7 3 Sink
VCG payments characterization Theorem: The VCG payments for trading agents are calculated using the residual graph of the optimal flow : • Any trading seller in market Mi receives the distance from the sink to Mi. • Any trading buyer in market Mj pays the distance from Mi to the sink, negated. computationally efficient algorithm for VCG. Theorem: • The VCG payment from a trading buyer is the minimal equilibrium price in her market. • The VCG payment to a trading seller is the maximal equilibrium price in her market
VCG mechanism – a drawback • Fact: The VCG mechanism is IR, IC and efficient but not Budget Balanced. • It is impossible to have all 4 properties [Myerson & Satterthwaite, 1983] • We would like to build a mechanism that is IR and IC. We trade some (little) efficiency to get budget balance. • Efficiency reduced by Trade Reduction.
The Trade Reduction Mechanism • We define the Reduced Residual Graph (a sub graph of the residual graph). • Allocation: Remove the minimal positive cycle from each Commercial Relationship Component (CRC) (A set of markets with direct or indirect trade). • Payments: By distances in the reduced residual graph.
TRM Properties Theorem: The TRM mechanism has a polynomial running time, and is IR, IC and BB. The efficiency loss of the mechanism in each CRC is at most one over the trade size in the CRC. Formally, if the efficient allocation is non empty Where || is the size of trade in CRC , and is the set of all CRCs.
Extensions • We look at two extensions of the model: • Agents with multi unit demand/supply with convex valuation. • Carriers bidding for shipment privilege. • We characterize the VCG mechanism for these models.
Future Work • Extend the TRM mechanism to a model of convex multi unit demand/supply. • Design IR, IC, BB and highly efficient mechanisms for a global Logistics Market, where the transportation between the markets is controlled by strategic carriers.
Summary • SDM with non strategic agents: • Reduction to (convex) minimum cost flow. • Two Welfare Theorems. • SDM with strategic agents: • Double characterization of the VCG payments: • As distances in the residual graph of a minimum cost flow computational efficiency. • As extreme equilibrium prices. • TRM mechanism that is a computationally efficient, IR, IC, Budget Balanced and highly efficient auction for SDM.
