MultiUnit Auctions with Budget Limits

1
Multi-Unit Auctions with Budget Limits

• Shahar Dobzinski, Ron Lavi, and Noam Nisan

2
Valuations

• How can we model
• An advertising agency is given a budget of
  1,000,000
• A daily budget for online advertising
• I am paying up to 200 for a TV.
• The starting point of all auction theory is the
  valuation of the single bidder.
• The quasi-linear model (my utility) (my
  value) (my price)

3
Budgets

• Approximation in the quasi-linear setting
• Define v(S) min( v(S), budget )
  Mehta-Saberi-Vazirani-Vazirani,
  Lehamann-Lehmann-Nisan
• Doesnt really capture the issue.
• E.g., marginal utilities.

4
Our Model

• Utility of winning a set of items S and paying p
• If pb v(S) p
• If pgtb -8 (infeasible)
• Inherently different from the quasi linear
  setting.
• Maximizing social welfare does not make sense.
• What to do with bidder with large value and small
  budget?
• VCG doesnt work.
• The usual characterizations of truthful
  mechanisms do not hold anymore.
• E.g., cycle montonicity, weak monotonicity, ...
• ...

5
Previous Work

• Budgets are central element in general
  equilibrium / market models
• Budgets in auctions -- economists
• Benot-Krishna 2001, Chae-Gale 1996, 2000, Maskin
  2000, Laffont-Robert 1996, few more
• Analysis/comparison of natural auctions
• Budgets in auctions CS
• Borgs et al 2005 Design auctions with good
  revenue
• Feldman et al. 2008, Sponsored search auctions
• This work design efficient auctions
• Again, what is efficiency if bidders have budget
  limits?
• But we also discuss revenue considerations.

6
Multi-Unit Auctions with Budgets

• m identical indivisible units for sale.
• Each bidder i has a value vi for each unit and
  budget limit bi.
• Utility of winning x items and paying p
• If pbi xvi-p
• If pgtbi -8 (infeasible)
• In the divisible setting we have only one unit.
• The value of i for receiving a fraction of x is
  xvi.
• We want truthful mechanisms.
• The vis and the bis are private information.

7
What is Efficiency?

• Minimal requirement Pareto
• Usually means that there is no other allocation
  such that all bidders prefer.
• Instead of the standard definition, we use an
  equivalent definition (in our setting) no trade.
• Dfn an allocation and a vector of prices
  satisfy the no-trade property if all items are
  allocated and there is no pair of bidders (i,j)
  such that
• Bidder j is allocated at least one item
• vigtvj,
• Bidder i has a remaining budget of at least vj

8
Main Theorem

• Theorem There is no truthful Pareto-optimal
  auction.
• the bis and the vis are private.
• Positive News
• Nice weird auction when bi's are public
  knowledge.
• Uniqueness implies main theorem.
• Obtains (almost) the optimal revenue.

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Ausubel's Clinching Auction

• Ascending auction implementation of VCG prices
• Increase p as long as demand gt supply.
• Bidder i clinches a unit at price p if (total
  demand of others at p) lt supply, and pay for the
  clinched unit a price of p.
• Reduce the supply.
• Ausubel This gives exactly VCG prices, ends in
  the optimal allocation, hence truthful.

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The Adaptive Clinching Auction (approx.)

• The demand of i at price p depends on the
  remaining budget
• If pvi min(remaining items,floor(remaining
  budget /p)), else 0.
• The auction
• Increase p as long as demand gt supply.
• Bidder i clinches a unit at price p if (total
  demand of others at p) lt supply, and pay for the
  clinched unit a price of p.
• Reduce the supply.
• Not truthful in general anymore!
• Theorem The mechanism is truthful if budgets are
  public, the resulting allocation is
  Pareto-efficient, and the revenue is close to the
  optimal one.
• Theorem The only truthful and pareto optimal
  mechanism.

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Example

• 2 bidders, 3 items.
• v1 5, b1 1 v2 3, b27/6

of 1
of 2
of 2
0
0
3
3
7
/
6
3
1
0

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Truthfulness

• Basic observation the only decision of the
  bidder is when to declare I quit.
• Because the demand (almost) doesnt depend on the
  value
• If pvi min( of remaining items,
  floor(remaining budget/p) )
• Else 0
• No point in quitting after the time
• Until pvi the auction is the same.
• The player can only lose from winning items when
  pgtvi.
• No point in quitting ahead of time.
• The auction is the same until the bidder quits.
• The bidder might win more items by staying.

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Pareto-Efficiency

• We need to show that the no-trade condition
  holds.
• Lemma (no proof) The adaptive clinching auction
  always allocates all items.
• Consider bidder j who clinched at least one item.
  Let the highest price an item was clinched by
  bidder j be p (so vjp).
• Let the total number of items demanded by the
  others at price p be qp.
• There are exactly qp items left after j clinches
  his item.
• There are at least qp items left after j clinches
  his item (by the definition of the auction).
• There cannot be more items left since all items
  are allocated at the end of the auction, but j is
  not allocated any more items, and the demand of
  the others cannot increase.
• Hence each bidder is allocated the items he
  demands at price p.
• At the end of the auction a player that have a
  valuegtp, have a remaining budgetltpvj.

14
Revenue

• Dfn The optimal revenue (in the divisible case)
  is the revenue obtained from the monopolist
  price. Borgs et al
• The monopolist price the price p the maximizes
  p(fraction of the good sold).
• Dfn Bidder dominance amaxi((fraction sold to i
  at the monopolist price)/(total fraction sold at
  the monopolist price)
• Borgs et al there is a randomized mechanism
  such that If a approaches 0 then the revenue
  approaches the optimum.
• Some improved bounds by Abrams.
• Thm The revenue obtained by the adaptive
  clinching auction is (1-a) of the optimum.
• Efficiency and revenue, simultaneously!

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Revenue (cont.)

• Let the optimal monopolist price be p.
• Well prove that the adaptive clinching auction
  sells all the good at price at least (1-a)p
• Well show that at price (1-a)p, for each bidder
  i, the total demand of the others is more than 1.
• So for each fraction x we get at least x(1-a)p.
• Lemma WLOG, at price p all the good is
  allocated.
• If bigtvi, then done. Else, the demand of each
  bidder is bi/p, hence the price can be reduced
  until all the good is allocated while still
  exhausting all budgets of demanding bidders.
• Fix bidder i, at price p the demand of the others
  is at least (1-a). The demand of each bidder is
  bi/p, so in price (1-a)p the total demand of the
  other is 1.

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Summary

• Auction theory needs to be extended to handle
  budgets.
• We considered a simple multi-unit auction
  setting.
• Bad news no truthful and pareto-efficient
  auction.
• Good news with public budgets, there is a unique
  truthful and pareto-efficient auction
• (almost) optimal revenue.
• Whats next?
• Relax the pareto efficiency requirement
• Approximate pareto efficiency? Randomization?
• Other settings
• Combinatorial auctions? Sponsored Search?

17
Two bidders, b1b21

• One divisible good
• The following auction is IC Pareto
• If min(v1,v2)1 use 2nd price auction
• Else, assuming 1ltv1ltv2
• x1 ½ 1/(2v1v1) , p11-1/v1
• x2 ½ 1/(2v1v1) , p21

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Two bidders, b11, b28

• One divisible good.
• The following auction is IC Pareto
• If min(v1,v2)1 use 2nd price auction
• Else, if 1ltv1ltv2
• x1 0
• x2 1 , p21ln(v1)
• Else, if 1ltv2ltv1
• x11/v2, p11
• x2 1-1/v2, p2ln(v2)

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Warm Up Market Equilibrium

• One divisible good.
• A competitive equilibrium is reached at price p
• If the total demand at price p is 1.
• Each bidder gets his demand at price p.
• Demand of i at price p is
• If pvi min(1,bi/p)
• Else 0

20
Warm Up Market Equilibrium

• At equilibrium, p(?bi), xibi/(?bi)
• Sum over i's with vip
• Pareto
• We need to verify that the no-trade condition
  holds.
• Ascending auction implementation
• Increase p as long as supplyltdemand
• Allocate demands at price p
• Observation truthful if viltltbi or vigtgtbi
• If budgets dont matter or values dont matter