MaxEnt Mathias Johansson1 Competitive Bidding in a Certain Class of Auctions Mathias Johansson Uppsala University, Dirac Research Sweden

MaxEnt Mathias Johansson2 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

MaxEnt Mathias Johansson3 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

MaxEnt Mathias Johansson4 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

MaxEnt Mathias Johansson5 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

MaxEnt Mathias Johansson6 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

MaxEnt Mathias Johansson7 User u’s probability for winning Let q u be the bid per unit resource of user u, c u 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(q v c v < q u c u | c u,q u,I) If c u is uncertain, we must also marginalize over c u

MaxEnt Mathias Johansson8 User u’s probability for winning The probability for user u to win is thus where y = c v q v. But how do we assign P(y|c u,q u, I)?

MaxEnt Mathias Johansson9 Probability assignments P(c u | I) –We assume that c u 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:

MaxEnt Mathias Johansson10 Probability assignments P(y |c u q u 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.

MaxEnt Mathias Johansson11 Typical loss functions Constant throughput: –A user wants  u resource units per time unit: where x u (q u ) 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)

MaxEnt Mathias Johansson12 Price-performance criterion A possible formalization is i.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.

MaxEnt Mathias Johansson13 Expectations and computations The expected ”constant throughput” loss is The expected ”price-performance” loss is approximated by

MaxEnt Mathias Johansson14 Examples 4 users Every 20 th 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.

MaxEnt Mathias Johansson15 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): –x 1 = 14, x 2 = 21, x 3 = 21, x 4 = 33, More competitive setting: –  1 = 15,  2 = 20,  3 = 25,  4 = 30, –x 1 = 13, x 2 = 19, x 3 = 26, x 4 = 31, and the average paid price per bit nearly doubles

MaxEnt Mathias Johansson16 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: –x 1 = 34, x 2 = 11, x 3 = 21, x 4 = 21.

MaxEnt Mathias Johansson17 Evolution of bids Throughput per time unit

MaxEnt Mathias Johansson18 Price-to-throughput ratio

MaxEnt Mathias Johansson19 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?