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.



The usual characterizations of truthful mechanisms do not
hold anymore.


What to do with bidder with large value and small budget?
VCG doesn’t work.
...
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:



Design auctions with “good” revenue
Feldman et al. 2008, Sponsored search auctions
This work: design efficient auctions
Borgs et al 2005


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:



In the divisible setting we have only one unit.


If p≤bi : xvi-p
If p>bi : -∞ (infeasible)
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 notrade 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
of 1
0+
1
3
7/6
1/3+
1
2
5/12+
1
7/12+
7/12
7/12
Demand
of 1
Budget Demand
of 2
of 2
Items
avail
Items
of 1
Items
of 2
3
3
0
0
7/6
3
3
0
0
2
5/6
1
2
0
1
0
5/6
1
1
1
1
0
1
2
1/4
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



No point in quitting after the time



If p≤vi : min(# of remaining items, floor(remaining budget/p) )
Else: 0
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



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.


The monopolist price: the price p the maximizes
p*(fraction of the good sold).
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



Lemma: WLOG, at price p all the good is allocated.


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.
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”