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 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

Itemsof2

Itemsof1

Itemsavail

DemandBudget DemandBudgetof 1

p

00337/6310+

00337/6211/3+

10215/6215/12+

11115/607/127/12+

2101/47/12

of 1 of 2 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≤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>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 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 =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 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 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-). The demand of each bidder is bi/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, 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”