Economics and Computer Science
Auctions
CS595, SB 213
Xiang-Yang Li
Department of Computer Science
Illinois Institute of Technology
Auction One Item
Auctions
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Methods for allocating goods, tasks, resources...
Participants: auctioneer, bidders
Enforced agreement between auctioneer & winning bidder(s)
Easily implementable e.g. over the Internet
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Many existing Internet auction sites
Auction (selling item(s)): One seller, multiple buyers
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Reverse auction (buying item(s)): One buyer, multiple sellers
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E.g. selling a bull on eBay
E.g. procurement
We will discuss the theory in the context of auctions, but same theory applies
to reverse auctions
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at least in 1-item settings
Auction settings
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Private value : value of the good depends only on the
agent’s own preferences
 E.g. cake which is not resold or showed off
Common value : agent’s value of an item determined
entirely by others’ values
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E.g. treasury bills
Correlated value : agent’s value of an item depends
partly on its own preferences & partly on others’
values for it
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E.g. auctioning a transportation task when bidders
can handle it or reauction it to others
Auction protocols: All-pay
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Protocol: Each bidder is free to raise his bid. When no bidder is willing
to raise, the auction ends, and the highest bidder wins the item. All
bidders have to pay their last bid
Strategy: Series of bids as a function of agent’s private value, his prior
estimates of others’ valuations, and past bids
Best strategy: ?
In private value settings it can be computed (low bids)
Potentially long bidding process
Variations
 Each agent pays only part of his highest bid
 Each agent’s payment is a function of the highest bid of all agents
E.g. CS application: tool reallocation [Lenting&Braspenning ECAI-94]
Auction protocols: English
(first-price open-cry = ascending)
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Protocol: Each bidder is free to raise his bid. When no bidder is
willing to raise, the auction ends, and the highest bidder wins the
item at the price of his bid
Strategy: Series of bids as a function of agent’s private value, his
prior estimates of others’ valuations, and past bids
Best strategy: In private value auctions, bidder’s dominant strategy
is to always bid a small amount more than current highest bid, and
stop when his private value price is reached
 No counterspeculation, but long bidding process
Variations
 In correlated value auctions, auctioneer often increases price at
a constant rate or as he thinks is appropriate
 Open-exit: Bidder has to openly declare exit without re-entering
possibility => More info to other bidders about the agent’s
valuation
Auction protocols:
First-price sealed-bid
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Protocol: Each bidder submits one bid without knowing
others’ bids. The highest bidder wins the item at the price
of his bid
 Single round of bidding
Strategy: Bid as a function of agent’s private value and his
prior estimates of others’ valuations
Best strategy: No dominant strategy in general
 Strategic underbidding & counterspeculation
 Can determine Nash equilibrium strategies via
common knowledge assumptions about the probability
distributions from which valuations are drawn
Strategic underbidding in first-price sealed-bid
auction
N risk-neutral bidders
Common knowledge that their values are drawn independently, uniformly in [0,
vmax]
Claim: In symmetric Nash equilibrium, each bidder i bids bi = b(vi) = vi (N-1) / N
Proof. First divide all bids by vmax so bids were in effect drawn from [0,1]. We
show that an arbitrary agent, agent 1, is motivated to bid
b1 = b(v1) = v1 (N-1) / N given that others bid b(vi) = vi (N-1) / N
Prob{b1 is highest bid} =  Pr{b1 > bi}
= Pr{b1 > v2 (N-1)/N}  …  Pr{b1 > vN (N-1)/N}
= Pr{b1 > v2 (N-1)/N)}N-1 = Pr{b1 N / (N-1) > v2}N-1 = (b1 N / (N-1))N-1
E[u1|b1] = (v1-b1) Prob{b1 is highest bid} = (v1-b1) (b1 N / (N-1))N-1
dE[u1|b1] / db1 = (N/(N-1))N-1 (-N b1N-1 + v1 (N-1) b1N-2) = 0
<=> N b1N-1 = v1 (N-1) b1N-2 | divide both sides by b1N-2  0
N b1 = v1 (N-1)
<=> b1 = v1 (N-1) / N
QED
Strategic underbidding in first-price
sealed-bid auction
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Example
 2 risk-neutral bidders: A and B
 A knows that B’s value is 0 or 100 with
equal probability
 A’s value of 400 is common knowledge
 In Nash equilibrium, B bids either 0 or
100, and A bids 100 +  (winning more
important than low price)
Auction protocols:
Dutch (descending)
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Protocol: Auctioneer continuously lowers the price until a
bidder takes the item at the current price
Strategically equivalent to first-price sealed-bid protocol in
all auction settings
Strategy: Bid as a function of agent’s private value and his
prior estimates of others’ valuations
Best strategy: No dominant strategy in general
 Lying (down-biasing bids) & counterspeculation
 Possible to determine Nash equilibrium strategies via
common knowledge assumptions regarding the
probability distributions of others’ values
 Requires multiple rounds of posting current price
Dutch flower market, Ontario tobacco auction, Filene’s
basement, Waldenbooks
Dutch (Aalsmeer) flower auction
Auction protocols: Vickrey
(= second-price sealed bid)
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Protocol: Each bidder submits one bid without knowing (!) others’
bids. Highest bidder wins item at 2nd highest price
Strategy: Bid as a function of agent’s private value & his prior
estimates of others’ valuations
Best strategy: In a private value auction with risk neutral bidders,
Vickrey is strategically equivalent to English. In such settings,
dominant strategy is to bid one’s true valuation
 No counterspeculation
 Independent of others’ bidding plans, operating environments,
capabilities...
 Single round of bidding
Widely advocated for computational multiagent systems
Old [Vickrey 1961], but not widely used among humans
Revelation principle --- proxy bidder agents on www.ebay.com,
www.webauction.com, www.onsale.com
Vickrey auction is a special case of Clarke
tax mechanism
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Who pays?
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The bidder who takes the item away from
the others (makes the others worse off)
Others pay nothing
How much does the winner pay?
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The declared value that the good would
have had for the others had the winner
stayed home = second highest bid
Results for private value auctions
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Dutch strategically equivalent to first-price sealed-bid
Risk neutral agents => Vickrey strategically equivalent to
English
All four protocols allocate item efficiently
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English & Vickrey have dominant strategies => no effort
wasted in counterspeculation
Which of the four auction mechanisms gives highest
expected revenue to the seller?
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(assuming no reservation price for the auctioneer)
Assuming valuations are drawn independently & agents are
risk-neutral
The four mechanisms have equal expected revenue!
Revenue equivalence theorem (II)
Proof sketch. We show that expected payment by an arbitrary bidder i is the same in
both equilibria. By revelation principle, can restrict to Bayes-Nash incentive-compatible
direct revelation mechanisms.
So, others’ bids are identical to others’ valuations.
ti = expected payment by bidder (expectation taken over others’ valuations)
By choosing his bid bi, bidder chooses a point on this curve
(we do not assume it is the same for different mechanisms)
ui = vi pi - ti <=> ti = vi pi - ui
ti(pi*(vi))
vi
pi*(vi)
utility increases
pi = probability of winning (expectation taken over others’ valuations)
dti(pi*(vi)) / dpi*(vi) = vi Integrate both sides from pi*(vi) to pi*(vi): ti(pi*(vi)) - ti(pi*(vi)) =
=
vivi s dp *(s)
i
pi*(vi)pi*(vi) v (q) dq
i
Since the two equilibria have the same allocation probabilities yi(v1, … v|A|) and every bidder reveals
his type truthfully, for any realization vi, pi*(vi) has to be the same in the equilibria. Thus the RHS
is the same. Now, since ti(pi*(vi)) is same by assumption, ti(pi*(vi)) is the same. QED
Revenue equivalence ceases to hold if
agents are not risk-neutral
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Risk averse bidders:
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Dutch, first-price sealed-bid ≥ Vickrey, English
Risk averse auctioneer:
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Dutch, first-price sealed-bid ≤ Vickrey, English
Results for non-private value auctions
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Dutch strategically equivalent to first-price sealed-bid
Vickrey not strategically equivalent to English
All four protocols allocate item efficiently
Winner’s curse
 Common value auctions:
v˜1  E[v | vˆ1,b(vˆ2 )  b( vˆ1 ),...,b( vˆN )  b(vˆ1 )]
Agent should lie (bid low) even in Vickrey & English
Revelation to proxy bidders?
Thrm (revenue non-equivalence ). With more than 2
bidders, the expected revenues are not the same:
English ≥ Vickrey ≥ Dutch = first-price sealed bid
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Results for non-private value auctions
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Common knowledge that auctioneer has private info
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Q: What info should the auctioneer release ?
A: auctioneer is best off releasing all of it
 “No news is worst news”
 Mitigates the winner’s curse
Results for non-private value auctions.
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Asymmetric info among bidders
 E.g. 1: auctioning pennies in class
 E.g. 2: first-price sealed-bid common value
auction with bidders A, B, C, D
 A & B have same good info. C has this & extra
signal. D has poor but independent info
 A & B should not bid; D should sometimes
=> “Bid less if more bidders or your info is worse”
 Most important in sealed-bid auctions & Dutch
Vulnerability to bidder collusion
[even in private-value auctions]
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v1 = 20, vi = 18 for others
Collusive agreement for English: e.g. 1 bids 6, others bid
5. Self-enforcing
Collusive agreement for Vickrey: e.g. 1 bids 20, others bid
5. Self-enforcing
In first-price sealed-bid or Dutch, if 1 bids below 18, others
are motivated to break the collusion agreement
Need to identify coalition parties
Vulnerability to shills
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Only a problem in non-private-value settings
English & all-pay auction protocols are vulnerable
 Classic analyses ignore the possibility of shills
Vickrey, first-price sealed-bid, and Dutch are not
vulnerable
Vulnerability to a lying auctioneer
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Truthful auctioneer classically assumed
In Vickrey auction, auctioneer can overstate 2nd highest
bid to the winning bidder in order to increase revenue
Bid verification mechanisms, e.g. cryptographic
signatures
 Trusted 3rd party auction servers (reveal highest bid to
seller after closing)
In English, first-price sealed-bid, Dutch, and all-pay,
auctioneer cannot lie because bids are public
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Auctioneer’s other possibilities
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Bidding
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Seller may bid more than his reservation price because
truth-telling is not dominant for the seller even in the
English or Vickrey protocol (because his bid may be
2nd highest & determine the price) => seller may
inefficiently get the item
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In an expected revenue maximizing auction, seller sets a
reservation price strategically like this [Myerson 81]
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Auctions are not Pareto efficient (not surprising in light of
Myerson-Satterthwaite theorem)
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Setting a minimum price
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Refusing to sell after the auction has ended
Undesirable private information revelation
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Agents strategic marginal cost information revealed
because truthful bidding is a dominant strategy in
Vickrey (and English)
 Observed problems with subcontractors
First-price sealed-bid & Dutch may not reveal this info as
accurately
 Lying
 No dominant strategy
 Bidding decisions depend on beliefs about others
Untruthful bidding with local uncertainty even in
Vickrey
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Uncertainty (inherent or from computation limitations)
Many real-world parties are risk averse
Computational agents take on owners preferences
Thrm [Sandholm ICMAS-96]. It is not the case that in a
private value Vickrey auction with uncertainty about an
agent’s own valuation, it is a risk averse agent’s best
(dominant or equilibrium) strategy to bid its expected value
Higher expected utility e.g. by bidding low
Wasteful counterspeculation
Thrm [Sandholm ICMAS-96]. In a private value Vickrey auction
with uncertainty about an agent’s own valuation, a risk neutral
agent’s best (deliberation or information gathering) action can
depend on others.
E.g. two bidders (1 and 2) bid for a good.
v1 uniform between 0 and 1; v2 deterministic, 0 ≤ v2 ≤ 0.5
Agent 1 bids 0.5 and gets item at price v2:
1
pdf
gain
loss
v2
]   v1  v2 dv1 
nopay
1
v1
E[1
0
1
 v2
2
Say agent 1 has the choice of paying c to find out v1. Then
agent 1 will bid v1 and get the item iff v1 ≥ v2 (no loss
possibility, but c1 invested)
]  c   v1  v 2 dv
pay
1
E[
v2
E[1pay]  E[ nopay
]  v 2  2c
1
Sniping
= bidding very late in the auction
in the hopes that other bidders do
not have time to respond
Especially an issue in electronic auctions with
network lag and lossy communication links
[from Roth & Ockenfels]
Sniping
Amazon auctions give automatic extensions, eBay does not
Antiques auctions have experts
[from Roth & Ockenfels]
Sniping
[from Roth & Ockenfels]
Mobile bidder agents in eMediator
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Allow user to participate while disconnected
Avoid network lag
Put expert bidders and novices on an equal footing
Full flexibility of Java (Concordia)
Template agents through an HTML page for nonprogrammers
 Information agent
 Incrementor agent
 N-agent
 Control agent
 Discover agent
Mobile bidder agents in eMediator
Mobile bidder agents in eMediator...
Conclusions on 1-item auctions
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Nontrivial, but often analyzable with reasonable
effort
 Important to understand merits & limitations
 Unintuitive protocols may have better properties
 Vickrey auction induces truth-telling & avoids
counterspeculation (in limited settings)
Choice of a good auction protocol depends on the
setting in which the protocol is used
Multi-item auctions & exchanges
(multiple distinguishable items for sale)
According to Tuomas Sandholm
Computer Science Department
Carnegie Mellon University
Multi-item auctions
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Auctioning multiple distinguishable items when bidders have
preferences over combinations of items
Example applications
 Allocation of transportation tasks
 Allocation of bandwidth
 Dynamically in computer networks
 Statically e.g. by FCC
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Manufacturing procurement
Electricity markets
Securities markets
Liquidation
Reinsurance markets
Retail ecommerce: collectibles, flights-hotels-event tickets
Resource & task allocation in operating systems & mobile
agent platforms
Inefficient allocation in interrelated auctions
Prop. [Sandholm ICMAS-96]. If agents with deterministic valuations
treat Vickrey auctions of interdependent goods without lookahead
regarding later auctions, and bid truthfully, the resulting allocation
may be suboptimal
Optimal allocation:
Agent 1 handles
both tasks
t1
1.0
Agent 2
Agent 1
0.5
t2
0.5
t1 auctioned first
Agent 1 bids c1({t1}) = 2
Agent 2 bids c2({t1}) = 1.5
t1 allocated to Agent 2
t2 auctioned next
Agent 1 bids c1({t2}) = 1
Agent 2 bids c2({t2}) = 1.5
t2 allocated to Agent 1
(or Agent 2 bids
c2({t1,t2}) - c2({t1}) = 1
=> either agent may get t2)
Lying in interrelated auctions
Prop. [Sandholm ICMAS-96]. If agents with deterministic valuations
treat Vickrey auctions of interdependent goods with full lookahead
regarding later auctions, their dominant strategy bids can differ from
the truthful ones of the corresponding isolated auctions
In the second auction (of t2)
Agent 1 bids c1({t1, t2}) - c1({t1}) = 0 if it
t1
has t1, and c1({t2}) = 1 if not.
Agent 2 bids c2({t1, t2}) - c2({t1}) = 1 if it
1.0
has t1, and c2({t2}) = 1.5 if not.
So, t1 is worth 1.5 to Agent 1 in the
Agent 2
Agent 1
t2
second auction (worth 0 to Agent 2)
So, in the first auction (of t1)
0.5
0.5
Agent 1 bids c1({t1}) - 1.5 and wins
Lookahead requires counterspeculation
Powerful contracts, decommitting, recontracting
Protocol design for multi-item auctions
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Sequential auctions
 How should rational agents bid (in equilibrium)?
 Full vs. partial vs. no lookahead
 Need normative deliberation control methods
 Inefficiencies can result from future uncertainties
Parallel auctions
 Inefficiencies can still result from future uncertainties
 Postponing & minimum participation requirements
 Unclear what equilibrium strategies would be
Methods to tackle the inefficiencies
 Backtracking via reauctioning (e.g. FCC [McAfee&McMillan96])
 Backtracking via leveled commitment contracts
[Sandholm&Lesser95,96][Sandholm96][Andersson&Sandholm98a,b]
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Breach before allocation
Breach after allocation
Protocol design for multi-item auctions
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Combinatorial auctions [Rassenti,Smith&Bulfin82]...
 Bidder’s perspective
 Reduces the need for lookahead
#items valuation calculations
 Potentially 2
 Automated optimal bundling of items
 Auctioneer’s perspective:
 Label bids as winning or losing so as to maximize
sum of bid prices (= revenue  social welfare)
 Each item can be allocated to at most one bid
#bids
 Exhaustive enumeration is 2
Space of allocations
Level
(4)
{1}{2}{3}{4}
{1},{2},{3,4}
{1},{2,3,4}
{3},{4},{1,2}
{1,2},{3,4}
{1},{3},{2,4}
{2},{1,3,4}
{2},{4},{1,3}
{1,3},{2,4}
{1},{4},{2,3}
{3},{1,2,4}
{2},{3},{1,4}
{1,4},{2,3}
{1,2,3,4}
#partitions is (#items#items/2), O(#items#items)
[Sandholm et al. 98]
Another issue: auctioneer could keep items
{4},{1,2,3}
(3)
(2)
(1)
Dynamic programming for winner determination

Uses (2#items), O(3#items) operations independent of #bids
 (Can trivially exclude items that are not in any bid)
 Does not scale beyond 20-30 items
1
1,2
2
1,3
3
2,3
[Rothkopf et al.95]
1,2,3
NP-completeness
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NP-complete [Karp 72]
 Weighted set packing
Polynomial time approximation algorithms
with worst case guarantees
value of optimal allocation
k=
value of best allocation found
General case
 Cannot be approximated to k = #bids1-  (unless
probabilistic polytime = NP)
 Proven in [Sandholm 99] using [Håstad96]
 Best known approximation gives
k  O(#bids / (log #bids)2 ) [Haldorsson98]
Polynomial time approximation algorithms
with worst case guarantees
Special cases
 Let  be the max #items in a bid: k= 2 / 3 [Haldorsson SODA-98]
 Bid can overlap with at most  other bids:
k= min( (+1) / 3 , (+2) / 3,  / 2 )
[Haldorsson&Lau97;Hochbaum83]
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k= sqrt(#items) [Haldorsson99]
k= chromatic number / 2 [Hochbaum83]
 k=[1 + maxHG minvH degree(v) ] / 2 [Hochbaum83]
 Planar: k=2 [Hochbaum83]
So far from optimum that irrelevant for auctions
Probabilistic algorithms?
New special cases, e.g. based on prices [Lehmann et al. 01]
Restricting the allowable combinations
Restricting the allowable combinations that can be bid
on to get polytime winner determination [Rothkopf et al.95]
1
2
6
3
5
4
|set|  2
1
O(#items2)
or
O(#items3)
2 3 4
5 6 7
or |set| > #items / c
O(nlargec-1 #items3)
NP-complete already
if 3 items per bid are
allowed
Gives rise to the same economic inefficiencies
that prevail in noncombinatorial auctions
Search algorithm for optimal / anytime
winner determination

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Capitalizes on sparsely populated space of bids
Generates only populated parts of space of allocations
Highly optimized
First generation algorithm scaled to hundreds of items &
thousands of bids [Sandholm IJCAI-99]
Second generation algorithm [Sandholm&Suri AAAI-00, Sandholm et
al. IJCAI-01]
First generation search algorithm: branch-on-items
Bids:
1
2
1,2
3
4
5
3,5 3
1,2
1,3,5
1,4
4
4
2,5
3,5
5
1,3,5
2
4
1
1,4
2,5
3
2
2,5
3,5
3
5
3
4
2
3,5
3
4
4
5
First generation search algorithm: branch-on-items


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Depth first search
Thrm. Need only consider children that include the item with the
smallest index among the items that are not on the path [Sandholm
IJCAI-99]
Insert dummy bid for price 0 for each single item that has no bids
=> allows bid combinations that would not cover the item
Generates each allocation of positive value once, others not generated
Complexity

Let b = #bids that contain a particular item

O(b#items) leaves

Actually O((#bids / #items)#items) leaves

Can be used as an anytime algorithm or sped up 2 orders of
magnitude by using IDA*
2nd generation search algorithm: Branching on bids
Search tree
E.g. bids
A={1,2}
B={2,3}
C={3}
D={1,3}
A
{(A,B),(A,D)}
{(B,C),(B,D),(C,D)}
C
IN
B
A
Bid graph G
D
C
{(A,B),(A,D)}
IN
OUT
C
OUT
B
{(B,C),(B,D)}
{(C,D)}
B
D
C
{(B,C),(B,D)}
IN
OUT
C
D
C
{(C,D)}
{(C,D)}
IN
OUT
D
IN
D
OUT
2nd generation search algorithm: Branching on bids

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O((#bids / #items +1)#items)
Follows principle of least commitment
O(#remaining neighbors) addition & deletion of bids in G
f* = value of best solution found so far
g = sum of prices of bids that are IN on path
h = value of linear programming relaxation of remaining
problem
Upper bounding: Prune the path when g+h ≤ f*


Storing seed solutions in LP
Usually need not solve LP to completion
Structural improvements
Optimum reached faster & better anytime performance

Always branch on a bid j that maximizes e.g. pj / |Sj| (presort)

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Lower bounding: If g+L>f*, then f*g+L
Identify decomposition of bid graph in O(|E|+|V|) time & exploit


More sophisticated bid-ordering heuristics in Sandholm et al IJCAI-01 paper
Pruning across subproblems (upper & lower bounding) by using f* values of
solved subproblems and h values of yet unsolved ones
Forcing decomposition by branching on an articulation bid


All articulation bids can be identified in O(|E|+|V|) time
Could try to identify combinations of bids that articulate (cut sets)
Price-Based Vs. Articulation Based Bid
Ordering

For any scheme that picks a set
maximizes  (pS ) for any given positive
function  ( S ) and any scheme that picks
an articulation bid if one exists, there are
instances where the former leads to
fewer search nodes and instances where
the later leads to fewer.
i
i
i
Exploiting tractable cases at search
nodes

Never branch on short bids with 1 or 2 items
 Short bids cause most complexity in search [Sandholm IJCAI-99]
 At each search node, we solve short bids from bid graph
separately
3
 O(#short bids ) time using maximal weighted matching
[Edmonds 65; Rothkopf et al 98]
NP-complete even if only 3 items per bid allowed
Dynamically delete items included in only one bid


Exploiting tractable cases at search nodes


At each search node, use a polynomial algorithm if remaining bid graph only
contains interval bids

Ordered list of items: 1..#items

Each bid is for some interval [q, r] of these items

Rothkopf et al. 98 presented O(#items2) DP algorithm

Our DP algorithm is O(#items + #bids)
 Bucket sort bids in ascending order of r
 opt(i) is the optimal solution using items 1..i
 opt(i) = max b in bids whose last item is i {pb + opt(qb-1), opt(i-1)}
Identifying linear ordering
A
B
2, 4, 6
1, 2, 4, 5, 7
1, 3, 5, 7
6
C
B
A
1, 3, 7, 8
D
D
4
2
5
1
7
3
8
Can be identified in O(|E|+|V|) time [Korte & Mohring SIAM-89]
Interval bids with wraparound can be identified in O(#bids2) time [Spinrad
SODA-93] and solved in O(#items (#items + #bids)) time using our DP while DP
of Rothkopf et al. is O(#items3)


C
Preprocessors
[Sandholm IJCAI-99]

Only keep highest bid for each combination that has received bids

Superset pruning





E.g. {1,2,3,4}, $10 is pruned by {1,3}, $7 and {2,4}, $6
For each bid (prunee), use same search algorithm as main search,
except restrict to bids that are subsets of prunee
Terminate the search and prune the prunee if f* ≥ prunee’s price
Only consider bids with ≤ 30 items as potential prunees
Tuple pruning






E.g. {1,2}, $8 and {3,4}, $3 are not competitive together given {1,3},
$7 and {2,4}, $6
Construct virtual prunee from pair of bids with disjoint item sets
Use same pruning algorithm as superset pruning
If pruned, insert an edge into bid graph between the bids
O(#bids2 cap #items)
O(#bids3 cap #items) for pruning triples, etc.
 More complex checking required in main search
Generalizations of combinatorial auctions







Free disposal
Substitutability
Multiple units of each item
Combinatorial exchanges (= many-to-many auctions)
Reservation prices
 On items
 On combinations
 With substitutability
Combinatorial reverse auctions
Combinations of these generalizations
Generalization: substitutability
[Sandholm IJCAI-99]



What if agent 1 bids
 $7 for {1,2}
 $4 for {1}
 $5 for {2} ?
Bids joined with XOR
 Allows bidders to express general preferences
 Groves-Clarke pricing mechanism can be applied to make truthful
bidding a dominant strategy
#items-1 combinations
 Worst case: Need to bid on all 2
OR-of-XORs bids maintain full expressiveness & are more concise
 E.g. (B2 XOR B3) OR (B1 XOR B3 XOR B4) OR ...
 Our algorithm applies (simply more edges in bid graph => smaller
search space, but usually slower)
 Preprocessors do not apply
 Short bid technique & interval bid technique do not apply
eMediator electronic commerce server








eAuctionHouse
 Customizable auctions, millions of types
 Traditional auction types
 All the generalized combinatorial auctions & exchanges
 Price-quantity graph bidding
 (Bidding with general utility functions)
 Expert system to help auctioneer select an auction type
 Mobile agents
Leveled commitment contract optimizer
eExchangeHouse
(Coalition formation support)
(Voting server)
(Meta-auction)
(Reputation databases & algorithms)
(Collaborative filtering & recommender systems)
Multi-unit auctions & exchanges
(multiple indistinguishable units of one item
for sale)
Auctions with multiple indistinguishable
units for sale

Examples





IBM stocks
Barrels of oil
Pork bellies
Trans-Atlantic backbone bandwidth from NYC to
Paris
…
Multi-unit auctions: pricing rules



Auctioning multiple indistinguishable units of an item
Naive generalization of the Vickrey auction: uniform price auction

If there are k units for sale, the highest k bids win, and each bid pays the
(k+1)st highest price

Demand reduction lie [Crampton&Ausubel 96]:
 k=5
 Agent 1 values getting her first unit at $9, and getting a second unit is
worth $7 to her
 Others have placed bids $2, $6, $8, $10, and $14
 If agent 1 submits one bid at $9 and one at $7, she gets both items, and
pays 2 x $6 = $12. Her utility is $9 + $7 - $12 = $4
 If agent 1 only submits one bid for $9, she will get one item, and pay $2.
Her utility is $9-$2=$7
IC mechanism that is Pareto efficient and ex post individually rational

Clarke tax. Agent i pays a-b



b is the others’ sum of winning bids
a is the others’ sum of winning bids had i not participated
What about revenue (if market is competitive)?
In all of the curves together
Multi-unit reverse auctions with supply
curves

Same complexity results apply as in
auctions



O(#pieces log #pieces) in nondiscriminatory
case with piecewise linear supply curves
NP-complete in discriminatory case with
piecewise linear supply curves
O(#agents log #agents) in discriminatory
case with linear supply curves
Multi-unit exchanges
• Multiple buyers, multiple sellers, multiple
units for sale
• By Myerson-Satterthwaite thrm, even in 1unit case cannot obtain all of
• Pareto efficiency
• Budget balance
• Individual rationality (participation)
Screenshot from
eMediator
[Sandholm AGENTS-00]
Supply/demand curve bids
Quantity
Aggregate supply
Aggregate demand
profit
psell
pbuy
Unit price
profit = amounts paid by bidders – amounts paid to
sellers
Can be divided between buyers, sellers & market maker
One price for everyone (“classic partial equilibrium”):
profit = 0
One price for sellers, one for buyers ( nondiscriminatory pricing ):
profit > 0
Nondiscriminatory vs. discriminatory pricing
Aggregate demand
Quantity
Supply of agent 1
Supply of agent 2
psell pbuy
Unit price
One price for sellers, one for buyers
( nondiscriminatory pricing ):
profit > 0
p2sell
pbuy
p1sell
One price for each agent
( discriminatory pricing ):
greater profit
Shape of supply/demand curves


Piecewise linear curve can approximate any
curve
Assume





Each buyer’s demand curve is downward sloping
Each seller’s supply curve is upward sloping
Otherwise absurd result can occur
Aggregate curves might not be monotonic
Even individuals’ curves might not be
continuous
Pricing scheme has implications on
time complexity of clearing



Piecewise linear curves (not necessarily continuous) can
approximate any curve
Clearing objective: maximize profit
Thrm. Nondiscriminatory clearing with piecewise linear
supply/demand: O(p log p)

p = total number of pieces in the curves
Pricing scheme has implications on
time complexity of clearing


Thrm. Discriminatory clearing with piecewise linear supply/demand:
NP-complete
Thrm. Discriminatory clearing with linear supply/demand: O(a log
a)



a = number of agents
These results apply to auctions, reverse auctions, and exchanges
So, there is an inherent tradeoff between profit and computational
complexity
Approximating Optimal Auctions
What and Why
•We will discuss the issue of revenue maximization,
also known as optimal auction design.
•It is a subject of long and intensive research in
microeconomics.
•We will look for an approximation.
Notations
• [ n ] = { 0 , 1 , 2 , .. , n}
• Wi = {1, 1 + ε , 1 + 2 ε , … , 2 , 2 + ε , … , h } : The possible
types (valuations ) of each agent.
•Φ = A distribution over the type space.
• Rm = The revenue of the auction m = The expected payment
Definitions
An Auction: A pair of function (k,p) such that:
• K : W [n] is an allocation algorithm determining who wins
the object (a zero – no winner).
• P : W R is a payment function determining how much the
winner must pay.
Definitions more
A Valid Auction: An auction the satisfies both:
• Individual Rationality (IR): The profit of a truth telling agent
is always non – negative: p(w) ≤ wk(w).
• Incentive Compatibility (IC): Truth-telling is a dominant
strategy for each agent.
C – Approximation: An Auction m is a C-approximation over Φ
1
R

R.
if for every valid auction v’,
m
c
If c=1, the auction is optimal.
The problem to solve
An Algorithm with the following characteristics:
Input:
• One item to sell.
• A probability distribution over the type space.
• Constant C.
Output:
• An auction.
Restrictions:
• Auction is a C-approximation optimal auction.
• Both Algorithm and auction are polytime.
Some Simple Examples
Suppose Alice wishes to sell a house to either Bob1 or Bob2, for
prices in the range [0,100].
Let’s look at a few simple connections:
• Independent Valuations: Both v1 and v2 are uniform in [0,100].
Good: Second price auction.
Better: Second price auction with reserve price 50.
More Examples
• Correlation: v1 is uniform in [0,100]. v2 = 2v1.
Bob1 is always rejected.
Optimal: P = twice the lower bid.
• Anti - Correlation: v1 is uniform in [0,100]. v2 = 100 - v1.
Optimal: P = The maximum of (w,100-w) where w is the
lower bid.
1-Lookead Auction
The 1 – lookahead auction computes, based on
declarations from the non-highest bidders, a price p1:
p1  p1 (w2 ,..., wn )
That maximizes it’s revenue from agent1 (according
to 1 ).
1
1
w

p
If
than agent1 wins, and pays p1.
Otherwise, nobody wins.
One Short Theorem
Theorem: the 1-lookahead auction is a 2-approximation.
Sketch Of Proof:
• It satisfies IR and IC, therefore a valid auction.
• It’s a 2-approximation auction:
splitting R ' to two cases:
R '1
and R'2 , and showing that :
R  R '1 and R  R '2
• The approximation ratio of 2 is tight.
Example why it is tight
Agent2’s type is fixed to 1.
v1 is determined acording to:


Pr v1  k 

1
h
1
k h
1
h
k 1  
The optimal revenue is about 2.
Our auction generates a revenue of about 1.
Computing the Auction
When we have a polytime algorithm that can compute, given a
price k and valuations (v2,…,vn), the probability:

Pr v1  k (v 2 ,..., v n )

We can simply try for all possible k’s and choose the one that
maximizes:

k  Pr v1  k (v 2 ,..., v n )

If h is large, we can, for some α, try only the cases:
(v2, α·v2, α2·v2,…,h), and we will get a α-approximation of the
optimal price.
Another Definition
Vickrey Auction With Reserved Price:
Let r  0 . It is the following the auction:
If v1 < r, all agents are rejected.
Otherwise, agent1 wins and pays max(v2,r).
Proposition
Their exists a price r, such that the Vickrey auction with
reserved price r is a 2log(h) approximation.
Proof:
Given a distribution d, v d  is the expectation of v1.
1
Look at intervals [2i,2i+1). (log(h) such intervals).
Ii is the interval that contributes most to v d  .
1
Take r = 2i.
The revenue:
1
1
1
Rd  
 v d  
 ROPT d 
2 log h 
2 log h 
K-lookahead auction
Let  be the conditional distribution


 v1 ,..., v k v k 1 ,..., v n

The K-lookahead auction is the optimal auction on agents (1,…,k)
according to  .
Obviously, at least a 2 – approximation.
The approximation ratio is tight!
Example why it is tight
Three agents, k = 2.
Agent3’s type is always 1.
Agent2’s type is uniformly drawn from 1  j    where
j  1,2,..., log h
The probability of the type of agent1 is determined by
agent2’s type. If v 2  1  j   ,then v1  2 j 1 with
probability 2
1
j 1
, and v  1   j  1  
1
Our auction’s revenue is around
1
with probability 1 
1
log h 
.
j
2
A better auction: Asks agent1 for
. If v1  2 j , sells to
agent3 for the price 1. Revenue – around 2.
1
2 j 1
.
Another Theorem
Theorem: If (v1,…,vn) are independent, the k-lookahead auction is
a
k 1
k
-approximation.
Sketch Of Proof:
Fix the (n-k) lowest valuations (agents k+1,…,n).
Aopt is the optimal auction, R is our revenue, Ropt the optimal revenue.
k 1
mk 1 the optimal revenue from agents (k+1,…,n). mk 1  v
For j  k , mj is the contribution of agent j to Ropt.
Ropt   m j
j
Case I: for all j  k , mk 1  m j .
Proof of Theorem
Case II: Not all j  k , mk 1  m j .
Let
ĵ denote the agent with minimal mj: mk 1  m ˆj
Pretend he declared vk+1, and run Aopt on it.
If any of the (n-k) won, sell to agent ĵ for v k+1.
Now, mk 1  m ˆj .
Because the distributions are independent, the distributions of the
other agents don’t change.
k
R  R opt  mk 1  R opt  m ˆj   m j 
R opt
k 1
j  ˆj
Conclusions
• We showed a simple 2-approximation. (1 – lookahead auction).
•It can be computed in polytime if there are polytime algorithms
computing the distribution Φ.
• We showed an improvement of that auction – to improve the
k 1
k
approximation ratio to
, but only under the
assumption that the valuations are independent.