1. Optimal Clearing of Supply/Demand Curves Ankur Jain, Irfan Sheriff, Shashidhar Mysore {ankurj, isheriff, shashimc}@cs.ucsb.edu Computer Science Department UC Santa Barbara T. Sandholm and S. Suri. Optimal clearing of supply/demand curves. In AAAI-02 workshop on Agent-Based Technologies for B2B Electronic Commerce, Edmonton, Canada, 2002.

2. Market Clearing Preliminaries Supply Curve  Suppose seller has Q identical units to sell.  Supply curve denotes the supply at price p as s(p).  Upward sloping curve. Price Quantity p s(p).

3. Market Clearing Preliminaries… Demand Curves  Suppose there are N buyers – each with a demand curve.  Demand at price p is d(p).  Downward sloping curves. Price Quantity p d(p)

4. Auctions, reverse auctions and exchanges  Forward Auctions – One Seller, Multiple buyers.  Reverse Auction – One buyer, Multiple sellers.  Exchanges – Multiple buyers, Multiple sellers. Variations  Piecewise linear vs. Linear curves.  Non-Discriminatory (uniform) vs. Discriminatory pricing.

5. Main Results Market typeCurve type Computational complexity Non discriminatory markets Piecewise LinearO( nk log(nk) ) Discriminatory markets LinearO( n logn ) Discriminatory markets Piecewise LinearNP-Complete

6. Objective  To study optimal clearing of supply/demand curves with multiple indistinguishable units such that the auctioneer’s profit is maximized.

7. Market clearing algorithms This project involves the implementation of the following algorithms :  Non discriminatory (ND) auctions.  ND reverse auctions.  ND exchanges.  Discriminatory reverse auctions.  Discriminatory auctions.

8. ND Auctions  Uniform clearing price, One seller - multiple buyers. n curves, with maximum k linear pieces each.  Feasibility Σ s i (p* ask ) = Σ d j (p* bid ).  Goal is to maximize p* bid ( Σ d j (p* bid )) – p* ask ( Σ s i (p* ask )) Steps - Compute Aggregate curve. For each linear piece solve for maximum revenue auction. Suppose p* bid is the unit price with maximum revenue. Clear each buyer at p* bid i.e., d(p* bid ) units. Price Quantity s1 d1 d2 P* bid

9. ND Auctions Price Quantity Strategy – Sort nk breakpoints O(nklog(nk)) Aggregate – Linesweep O(nk) Envelope – Linesweep O(nk) Decompose into K trapezoids, find maximum revenue O(K) Clear each buyer at p* bid Profit

10. ND Reverse auction  Uniform clearing price, One buyer multiple sellers - n curves, with maximum k pieces each  Maximize third party (who runs the market) profit Strategy – Sort nk breakpoints O(nk log(nk)) Aggregate – Linesweep O(nk) Envelope – Linesweep O(nk) Decompose into K trapezoids, find maximum revenue O(K) Clear each seller at p* ask Profit Price Quantity

11. ND Exchanges  Maximize third party (who runs the market) profit  Feasibility – Σ s i (p* ask ) = Σ d j (p* bid )  Goal is to maximize p* bid ( Σ d j (p* bid )) – p* ask ( Σ s i (p* ask )) Strategy – Sort nk breakpoints O(nk log(nk)) Aggregate – Linesweep O(nk) Envelope – Linesweep O(nk) Decompose into K trapezoids, find maximum revenue O(K) Clear each seller at p* ask, buyer at p* bid Price Quantity Profit

12. Discriminatory Reverse Auction  Non uniform clearing price, multiple sellers - One buyer - wants to buy Q units  Minimize total cost for the buyer Clearing Problem Sellers have upward sloping supply curve, Minimize s. t. Using Lagrangian Multipliers

13. Discriminatory Reverse Auction … Strategy – Arrange sellers by their minimum feasible price (b i /a i ) – O(nlogn) Incrementally add sellers and check for feasibility and minimum cost constraint O(1) Suppose minimum total cost occurs with S i sellers, solve for clearing price and quantity for each seller. Price QuantityQ

14. Discriminatory Auction  Non uniform clearing price, One seller – Multiple buyers (has Q units)  Maximize total revenue for the seller Each buyer is represented by a downward sloping demand curve, Maximize s. t. Unconstrained solution – Sell exactly ½ b i units to buyer i. If Q < ½ Σb i then Using Lagrangian Multipliers

15. Discriminatory Auction … Strategy – Initialize (p i,q i ) = b i /2a i, b i /2) If Σ(q i ) <=Q, done Choose l with min p i (say p l ), increase each bid’s unit price by p l, if feasible, compute lagrangian and output each buyer’s quantity Otherwise, remove buyer l from the market and repeat above steps Price QuantityQ

16. Re-cap of Main Results Market typeCurve type Computational complexity Non discriminatory markets Piecewise LinearO( nk log(nk) ) Discriminatory markets LinearO( n logn ) Discriminatory markets Piecewise LinearNP-Complete

17. Demo …