Competitive Bidding in a Certain Class of Auctions

1. Competitive Bidding in a Certain Class of Auctions Mathias JohanssonUppsala University, Dirac ResearchSweden Mathias Johansson

2. Problem background • Fixed pricing is used today in mobile data networks (such as GSM, 3G) • Could automatic mechanisms for dynamic pricing be used to obtain a desired service level? • Why shouldn’t prices reflect supply/demand? • Radio channels fluctuate strongly and unpredictably Mathias Johansson

3. Problem background • Quality of service (QoS) requirements differ among users • Typically defined in terms of throughput and delay requirements • Streaming media requires tight service levels • An automatic auctioning procedure could be used in order to obtain desired QoS Mathias Johansson

4. The rules of the auction • The resource can be used by only one user at a time • For each user the resource carries a certain utility, the user-specific capacity of the resource • Each user submits one sealed bid to the auctioneer, stating • the price the user is willing to pay per unit resource, and • the user’s capacity Mathias Johansson

5. Auction set-up • The winning total bid (bid per unit resource times the capacity) obtains the resource for a specific period of time • This process is repeated many times • Different bidders may have different capacities • Bids and capacities of other users are hidden • Future capacities are typically uncertain at the time of the bid Mathias Johansson

6. What’s the best bid? • Each user determines the probability for winning and a loss function reflecting the value of the resource. • Clearly, some information of past auctions must be given to the users: • The average winning price-capacity product will be announced along with its sample variance over a given time • This is announced after every lth auction Mathias Johansson

7. User u’s probability for winning • Let qu be the bid per unit resource of user u, cu the corresponding capacity • If v is the user with the largest bid-capacity product among all users except u, the probability for winning is P(qvcv < qucu | cu,qu,I) • If cu is uncertain, we must also marginalize over cu Mathias Johansson

8. User u’s probability for winning • The probability for user u to win is thuswhere y = cvqv. • But how do we assign P(y|cu,qu, I)? Mathias Johansson

9. Probability assignments • P(cu | I) • We assume that cu can only take one of K possible values. • Assuming that our only further information is a past history of how many times the K different levels have ocurred, Laplace’s rule of succession applies: Mathias Johansson

10. Probability assignments • P(y |cu qu I) • Only the mean winning bid-capacity product and its sample variance is known • According to the maximum entropy principle, we assign a Gaussian distribution. • Note! The announced information regards all users, but y should not include user u • A correction is made by subtracting the contributions from u’s wins. Mathias Johansson

11. Typical loss functions • Constant throughput: • A user wants u resource units per time unit:where xu (qu) is the obtained throughput • Price-performance ratio: • A user may want to raise her bid if that results in a substantially better throughput (i.e. stockpiling when capacity is cheap) Mathias Johansson

12. Price-performance criterion • A possible formalization isi.e. a price increase of 1 unit is ok if the throughput increases by a factor of a. If the throughput is less than b, the resource is of no value. Mathias Johansson

13. Expectations and computations • The expected ”constant throughput” loss is • The expected ”price-performance” loss is approximated by Mathias Johansson

14. Examples • 4 users • Every 20th time unit, mean-variance information is broadcast and bids are updated • 4 different possible capacities, c={0, 74, 92, 106} • Rate probabilities are updated by Laplace’s rule (rates are generated as Gaussian numbers with avg 80, std dev 20) • All users have a maximum bid per unit = 5. Mathias Johansson

15. Example 1 • Constant rate loss for all users: • 1 = 15, 2 = 20, 3 = 20, 4 = 30, • Resulting average throughput over 600 time units (30 price updates): • x1 = 14, x2 = 21, x3 = 21, x4 = 33, • More competitive setting: • 1 = 15, 2 = 20, 3 = 25, 4 = 30, • x1 = 13, x2 = 19, x3 = 26, x4 = 31, and the average paid price per bit nearly doubles Mathias Johansson

16. Example 2 • Price-performance loss for user 1 and constant rate loss for users 2-4: • 2 = 10, 3 = 20, 4 = 20, • Resulting average throughput: • x1 = 34, x2 = 11, x3 = 21, x4 = 21. Mathias Johansson

17. Evolution of bids Throughput per time unit Mathias Johansson

18. Price-to-throughput ratio Mathias Johansson

19. Comments • Bidding can be used to satisfy QoS demands • But what are the long-term customer reactions? • Other types of bidding situations call for Bayesian treatment! • Challenge (MaxEnt 2007?): What is the expected winning bid in the sale of an apartment given knowledge of existing bid history? • Previous bids: 500k, 550k, and 565k • How would you, as a devoted Bayesian, bid? Mathias Johansson