Multi-Unit Auctions with Budget Limits

Multi-Unit Auctions with Budget Limits ShaharDobzinski, 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 vi for each unit and budget limit bi. • Utility of winning x items and paying p: • If p≤bi : xvi-p • If p>bi : -∞ (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 vi’s and the bi’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 • vi>vj, • Bidder i has a remaining budget of at least vj

Main Theorem Theorem:There is no truthful Pareto-optimal auction. • the bi’s and the vi’s are private. Positive News: • Nice weird auction when bi'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≤vi : 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. • v1 = 5, b1 = 1; v2 = 3, b2=7/6 p Budget Demand Budget Demand Items Items Items of 1 avail of 1 of 2 of 1 of 2 of 2 0 + 1 3 7 / 6 3 3 0 0 1 / 3 + 1 2 7 / 6 3 3 0 0 5 / 12 + 1 2 5 / 6 1 2 0 1 7 / 12 + 7 / 12 0 5 / 6 1 1 1 1 7 / 12 1 / 4 0 1 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≤vi : min(# of remaining items, floor(remaining budget/p) ) • Else: 0 • No point in quitting after the time • Until p=vi the auction is the same. • The player can only lose from winning items when p>vi. • 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 vj≥p). • 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 value>p, have a remaining budget<p≤vj.

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 a=maxi((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!

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-a)p • We’ll 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 bi>vi, 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.

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, b1=b2=1 • One divisible good • The following auction is IC + Pareto: • If min(v1,v2)≤1 use 2nd price auction • Else, assuming 1<v1<v2: • x1= ½ – 1/(2•v1•v1) , p1=1-1/v1 • x2= ½ + 1/(2•v1•v1) , p2=1

Two bidders, b1=1, b2=∞ • One divisible good. • The following auction is IC + Pareto: • If min(v1,v2)≤1 use 2nd price auction • Else, if 1<v1<v2: • x1= 0 • x2= 1 , p2=1+ln(v1) • Else, if 1<v2<v1: • x1=1/v2, p1=1 • x2= 1-1/v2, p2=ln(v2)

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≤vi : min(1,bi/p) • Else: 0

Warm Up: Market Equilibrium • At equilibrium, p=(∑bi), xi=bi/(∑bi) • Sum over i's with vi≥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 vi<<bi or vi>>bi • If “budgets don’t matter” or “values don’t matter”

Multi-Unit Auctions with Budget Limits