Competitive Bidding in a Certain Class of Auctions

2006 Mathias Johansson 1

MaxEnt 2006

Competitive Bidding in a Certain Class of Auctions

Mathias JohanssonUppsala University, Dirac Research

Sweden


MaxEnt 2006

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 2006

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 2006

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 2006

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 2006

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 2006

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


MaxEnt 2006

User u’s probability for winning

• The probability for user u to win is thus

where y = cvqv.

• But how do we assign P(y|cu,qu, I)?


MaxEnt 2006

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:


MaxEnt 2006

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.


MaxEnt 2006

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)


MaxEnt 2006

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 2006

Expectations and computations

• The expected ”constant throughput” loss is

• The expected ”price-performance” loss is approximated by


MaxEnt 2006

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.


MaxEnt 2006

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


MaxEnt 2006

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.


MaxEnt 2006

Evolution of bids

Throughput per time unit


MaxEnt 2006

Price-to-throughput ratio


MaxEnt 2006

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?

Competitive Bidding in a Certain Class of Auctions

Documents

Transcript of Competitive Bidding in a Certain Class of Auctions