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.

SDM Model 2 9 S B Unit shipment cost Seller’s cost Buyer’s value M1M1 M2M2 M3M3 M4M4 Utility maximizing agents with quasi-linear utility function (value-payment)

Allocations 2 9 S B M1M1 M2M2 M3M3 M4M4 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

Efficiency by Min-Cost-Circulation (cost, capacity ) 2 9 S B (1, ∞ ) Sink (-9, 1 ) (1, 1 ) M1M1 M2M2 M3M3 M4M4 1,flow,1 The minimal cost circulation is the efficient allocation

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 (M i,M j )  E : p i + c i,j  p j. If x i,j > 0 then p i + c i,j =p j

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.

SDM with strategic agents

SDM mechanisms Each agent’s cost/value for the good is a private information. Mechanism: Given reported values, define an allocation rule and 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. VCG

SDM VCG payments characterizations

The residual graph (cost, capacity, flow) 2 9 S B (1, ∞, 1)(1, ∞, 1) Sink (9, 1)(9, 1) M1M1 M2M2 M3M3 M4M4 (1, 1, 1)(1, 1, 1)(-1, 1) (-9, 1, 1) (1, ∞)(1, ∞) (-1, 1)

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 M i receives the distance from the sink to M i. –Any trading buyer in market M j pays the distance from M i 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 (TRM) and its properties.

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.

Thank you!