Multi-Unit Auctions with Budget Limits Shahar Dobzinski, Ron Lavi, and Noam Nisan

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)

Budgets  “Approximation” in the quasi-linear setting  Define: v’(S) = min( v(S), budget ) Mehta-Saberi-Vazirani-Vazirani, Lehamann-Lehmann-Nisan  Doesn’t really capture the issue.  E.g., marginal utilities.

Our Model  Utility of winning a set of items S and paying p:  If p≤b : v(S) – p  If p>b : -∞ (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 doesn’t work.  The usual characterizations of truthful mechanisms do not hold anymore.  E.g., cycle montonicity, weak monotonicity,... ...

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.

Multi-Unit Auctions with Budgets  m identical indivisible units for sale.  Each bidder i has a value v i for each unit and budget limit b i.  Utility of winning x items and paying p:  If p≤b i : xv i -p  If p>b i : -∞ (infeasible)  In the divisible setting we have only one unit.  The value of i for receiving a fraction of x is xv i.  We want truthful mechanisms.  The v i ’s and the b i ’s are private information.

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  v i >v j,  Bidder i has a remaining budget of at least v j

Main Theorem Theorem: There is no truthful Pareto-optimal auction.  the b i’ s and the v i ’s are private. Positive News:  Nice weird auction when b i 's are public knowledge.  Uniqueness implies main theorem.  Obtains (almost) the optimal revenue.

Ausubel's Clinching Auction  Ascending auction implementation of VCG prices:  Increase p as long as demand > supply.  Bidder i clinches a unit at price p if (total demand of others at p) < 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.

The Adaptive Clinching Auction (approx.)  The “demand of i at price p” depends on the remaining budget:  If p≤v i : min(remaining items,floor(remaining budget /p)), else: 0.  The auction:  Increase p as long as demand > supply.  Bidder i clinches a unit at price p if (total demand of others at p) < 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.

Example  2 bidders, 3 items.  v 1 = 5, b 1 = 1; v 2 = 3, b 2 =7/6 Items of2 Items of1 Items avail DemandBudgetDemandBudget of 1 p 00337/ /6211/ /6215/ /607/ 7/ /47/ of 1 of 2

Truthfulness  Basic observation: the only decision of the bidder is when to declare “I quit”.  Because the demand (almost) doesn’t depend on the value  If p≤v i : min(# of remaining items, floor(remaining budget/p) )  Else: 0  No point in quitting after the time  Until p=v i the auction is the same.  The player can only lose from winning items when p>v i.  No point in quitting ahead of time.  The auction is the same until the bidder quits.  The bidder might win more items by staying.

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 v j ≥p).  Let the total number of items demanded by the others at price p be q p.  There are exactly q p items left after j clinches his item.  There are at least q p 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 value>p, have a remaining budget<p≤v j.

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  =max i ((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  approaches 0 then the revenue approaches the optimum.  Some improved bounds by Abrams.  Thm: The revenue obtained by the adaptive clinching auction is (1-  ) of the optimum.  Efficiency and revenue, simultaneously!

Revenue (cont.)  Let the optimal monopolist price be p.  We’ll prove that the adaptive clinching auction sells all the good at price at least (1-  )p  We’ll show that at price (1-  )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-  )p.  Lemma: WLOG, at price p all the good is allocated.  If b i >v i, 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-  ). The demand of each bidder is b i /p, so in price (1-  )p the total demand of the other is 1.

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.  What’s next?  Relax the pareto efficiency requirement  Approximate pareto efficiency? Randomization?  Other settings  Combinatorial auctions? Sponsored Search?

Two bidders, b 1 =b 2 =1  One divisible good  The following auction is IC + Pareto:  If min(v 1,v 2 )≤1 use 2 nd price auction  Else, assuming 1<v 1 <v 2:  x 1 = ½ – 1/(2v 1 v 1 ), p 1 =1-1/v 1  x 2 = ½ + 1/(2v 1 v 1 ), p 2 =1

Two bidders, b 1 =1, b 2 =∞  One divisible good.  The following auction is IC + Pareto:  If min(v 1,v 2 )≤1 use 2 nd price auction  Else, if 1<v 1 <v 2:  x 1 = 0  x 2 = 1, p 2 =1+ln(v 1 )  Else, if 1<v 2 <v 1 :  x 1 =1/v 2, p 1 =1  x 2 = 1-1/v 2, p 2 =ln(v 2 )

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 p≤v i : min(1,b i /p)  Else: 0

Warm Up: Market Equilibrium  At equilibrium, p=(∑b i ), x i =b i /(∑b i )  Sum over i's with v i ≥p  Pareto  We need to verify that the “no-trade” condition holds.  Ascending auction implementation:  Increase p as long as supply<demand  Allocate demands at price p  Observation: truthful if v i >b i  If “budgets don’t matter” or “values don’t matter”