Economics and Computer ScienceAuctions

CS595, SB 213

Xiang-Yang Li Department of Computer Science

Illinois Institute of Technology

Auction One Item

Auctions

Methods for allocating goods, tasks, resources... Participants: auctioneer, bidders Enforced agreement between auctioneer & winning bidder(s) Easily implementable e.g. over the Internet

Many existing Internet auction sites Auction (selling item(s)): One seller, multiple buyers

E.g. selling a bull on eBay

Reverse auction (buying item(s)): One buyer, multiple sellers E.g. procurement

We will discuss the theory in the context of auctions, but same theory applies to reverse auctions

at least in 1-item settings

Auction settings

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 E.g. treasury bills

Correlated value : agent’s value of an item depends

partly on its own preferences & partly on others’

values for it E.g. auctioning a transportation task when bidders

can handle it or reauction it to others

Auction protocols: All-pay

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)

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

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 biddersCommon 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) / NProof. 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) b1

N-2) = 0

<=> N b1N-1 = v1 (N-1) b1

N-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

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)

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)

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

Who pays? The bidder who takes the item away

from the others (makes the others worse off)

Others pay nothing How much does the winner pay?

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

Dutch strategically equivalent to first-price sealed-bid Risk neutral agents => Vickrey strategically equivalent to

English All four protocols allocate item efficiently

(assuming no reservation price for the auctioneer) English & Vickrey have dominant strategies => no effort

wasted in counterspeculation Which of the four auction mechanisms gives highest

expected revenue to the seller? Assuming valuations are drawn independently & agents are

risk-neutral The four mechanisms have equal expected revenue!

Revenue equivalence theorem (II)

dti(pi*(vi)) / dpi*(vi) = vi Integrate both sides from pi*(vi) to pi*(vi): ti(pi*(vi)) - ti(pi*(vi)) = pi*(vi)pi*(vi) vi(q) dq =

vivi

s dpi*(s)

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.

pi = probability of winning (expectation taken over 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)

pi*(vi)

ti(pi*(vi))

ui = vi pi - ti <=> ti = vi pi - ui

utility increasesvi

Since the two equilibria have the same allocation probabilities y i(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

Risk averse bidders: Dutch, first-price sealed-bid ≥ Vickrey, English

Risk averse auctioneer: Dutch, first-price sealed-bid ≤ Vickrey, English

Results for non-private value auctions

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:

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

˜ v 1 E[v | ˆ v 1,b(ˆ v 2 ) b( ˆ v 1 ),...,b( ˆ v N ) b(ˆ v 1)]

Results for non-private value auctions

Common knowledge that auctioneer has private info 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.

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]

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

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

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

Auctioneer’s other possibilities

Bidding

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 In an expected revenue maximizing auction, seller sets a

reservation price strategically like this [Myerson 81] Auctions are not Pareto efficient (not surprising in light of

Myerson-Satterthwaite theorem)

Setting a minimum price

Refusing to sell after the auction has ended

Undesirable private information revelation

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

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 ≤ 0.5Agent 1 bids 0.5 and gets item at price v2:

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 c invested)

E[1nopay] v1 v2d

0

1

v1 1

2 v2

v1

pdf

v2

loss gain

1

E[1pay] c v1 v2d

v 2

1

v E[1pay] E[1

nopay] v2 2c

Sniping

= bidding very late in the auction in the hopes that other bidders do not have

time to respondEspecially an issue in electronic

auctions with network lag and lossy communication links

[from Roth & Ockenfels]

Sniping

[from Roth & Ockenfels]

Amazon auctions give automatic extensions, eBay does notAntiques auctions have experts

Sniping

[from Roth & Ockenfels]

Mobile bidder agents in eMediator

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 non-

programmers 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

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 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

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

t1 auctioned firstAgent 1 bids c1({t1}) = 2Agent 2 bids c2({t1}) = 1.5t1 allocated to Agent 2

t2 auctioned nextAgent 1 bids c1({t2}) = 1Agent 2 bids c2({t2}) = 1.5t2 allocated to Agent 1(or Agent 2 bids c2({t1,t2}) - c2({t1}) = 1=> either agent may get t2)

t2

t1

Agent 1 Agent 2

0.5 0.5

1.0

Optimal allocation: Agent 1 handlesboth tasks

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 has t1, and c1({t2}) = 1 if not.

Agent 2 bids c2({t1, t2}) - c2({t1}) = 1 if it

has t1, and c2({t2}) = 1.5 if not.

So, t1 is worth 1.5 to Agent 1 in the

second auction (worth 0 to Agent 2)

So, in the first auction (of t1)Agent 1 bids c1({t1}) - 1.5 and wins

t2

t1

Agent 1 Agent 2

0.5 0.5

1.0

Lookahead requires counterspeculationPowerful contracts, decommitting, recontracting

Protocol design for multi-item auctions

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]

Breach before allocation Breach after allocation

Protocol design for multi-item auctions

Combinatorial auctions [Rassenti,Smith&Bulfin82]... Bidder’s perspective

Reduces the need for lookahead Potentially 2#items valuation calculations

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 Exhaustive enumeration is 2#bids

Space of allocations

#partitions is (#items#items/2), O(#items#items) [Sandholm et al. 98]Another issue: auctioneer could keep items

{1}{2}{3}{4}

{1},{2},{3,4} {3},{4},{1,2} {1},{3},{2,4} {2},{4},{1,3} {1},{4},{2,3} {2},{3},{1,4}

{1},{2,3,4} {1,2},{3,4} {2},{1,3,4} {1,3},{2,4} {3},{1,2,4} {1,4},{2,3} {4},{1,2,3}

{1,2,3,4}

Level

(4)

(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

2

3

1,2

1,3

2,3

1,2,3

[Rothkopf et al.95]

NP-completeness

NP-complete [Karp 72] Weighted set packing

Polynomial time approximation algorithms with worst case guarantees

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]

value of optimal allocationk = value of best allocation found

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]

k= sqrt(#items) [Haldorsson99]

k= chromatic number / 2 [Hochbaum83]

k=[1 + maxHG minvH 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

1

2

3

4

5

6

1 2 3 4 5 6 7

|set| 2

or |set| > #items / c

O(#items2) or O(#items3)

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

Restricting the allowable combinations that can be bid on to get polytime winner determination [Rothkopf et al.95]

Search algorithm for optimal / anytime winner determination

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 3 4 5 1,2 1,3,5 1,4 2,5 3,5

5

1,2 1,3,5 1,4 1

3,5 3 2 2,5 2 22,5

4 4 4 3 3,5 3 3 3,5 3

5 5 4 4 4

First generation search algorithm: branch-on-items

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

A

C B

C

D

A DB

C

IN OUT

IN

IN

OUT

OUT

IN OUT

C

IN OUT

B

C

D

Bid graph GSearch tree

C

D

D

{(A,B),(A,D)} {(B,C),(B,D),(C,D)}

{(A,B),(A,D)}

{(B,C),(B,D)} {(C,D)}

{(B,C),(B,D)}

{(C,D)} {(C,D)}

E.g. bidsA={1,2}B={2,3}C={3}D={1,3}

2nd generation search algorithm: Branching on bids

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 improvementsOptimum reached faster & better anytime performance

Always branch on a bid j that maximizes e.g. pj / |Sj| (presort) More sophisticated bid-ordering heuristics in Sandholm et al IJCAI-01 paper

Lower bounding: If g+L>f*, then f*g+L Identify decomposition of bid graph in O(|E|+|V|) time & exploit

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 for any given positive function 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

S

p

)( iS

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

O(#short bids 3) 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

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)

2, 4, 6

1, 2, 4, 5, 7 1, 3, 7, 8

1, 3, 5, 7

AB C

D 6 4 2 5 1 7 3 8

AB

C

D

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 Worst case: Need to bid on all 2#items-1 combinations

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 1-unit case cannot obtain all of

• Pareto efficiency• Budget balance• Individual rationality (participation)

Screenshot from eMediator[Sandholm AGENTS-00]

Supply/demand curve bids

profit = amounts paid by bidders – amounts paid to sellersCan be divided between buyers, sellers & market maker

Unit price

Quantity Aggregate supply Aggregate demand

One price for everyone (“classic partial equilibrium”):profit = 0

One price for sellers, one for buyers ( nondiscriminatory pricing ): profit > 0

profit

psell pbuy

Nondiscriminatory vs. discriminatory pricing

Unit price

Quantity

Supply of agent 1

Aggregate demand

Supply of agent 2

One price for sellers, one for buyers( nondiscriminatory pricing ): profit > 0

psell pbuy

One price for each agent ( discriminatory pricing ): greater profit

p1sell

pbuyp2sell

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

•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.

What and Why

• [ 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

Notations

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

C – Approximation: An Auction m is a C-approximation over Φ

if for every valid auction v’, .

If c=1, the auction is optimal.

Rc

Rm

1

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.

Definitions more

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.

The problem to solve

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.

Some Simple Examples

• 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.

• Correlation: v1 is uniform in [0,100]. v2 = 2v1.

Bob1 is always rejected.

Optimal: P = twice the lower bid.

More Examples

),...,( 211 nwwpp

111 pw

The 1 – lookahead auction computes, based on

declarations from the non-highest bidders, a price p1:

That maximizes it’s revenue from agent1 (according to ).

If than agent1 wins, and pays p1.

Otherwise, nobody wins.

1-Lookead Auction

Theorem: the 1-lookahead auction is a 2-approximation.

• It satisfies IR and IC, therefore a valid auction.

• It’s a 2-approximation auction:

splitting to two cases:

and , and showing that :

and

'R

1'R 2'R

1'RR 2'RR

• The approximation ratio of 2 is tight.

Sketch Of Proof:

One Short Theorem

Agent2’s type is fixed to 1.

v1 is determined acording to:

The optimal revenue is about 2.

Our auction generates a revenue of about 1.

kv1Pr h

1hk

1kh

11

Example why it is tight

When we have a polytime algorithm that can compute, given a price k and valuations (v2,…,vn), the probability:

We can simply try for all possible k’s and choose the one that maximizes:

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.

),...,(Pr 21 nvvkv

),...,(Pr 21 nvvkvk

Computing the Auction

Vickrey Auction With Reserved Price:

Let . It is the following the auction:

If v1 < r, all agents are rejected.

Otherwise, agent1 wins and pays max(v2,r).

0r

Another Definition

Their exists a price r, such that the Vickrey auction with reserved price r is a 2log(h) approximation.

Proof:

Given a distribution d, is the expectation of v1.

Look at intervals [2i,2i+1). (log(h) such intervals).

Ii is the interval that contributes most to .

Take r = 2i.

The revenue:

dv1

dv1

dRh

dvh

dR OPTlog2

1

log2

1 1

Proposition

Let be the conditional distribution

The K-lookahead auction is the optimal auction on agents (1,…,k) according to .

nkk vvvv ,...,,..., 11

Obviously, at least a 2 – approximation.

The approximation ratio is tight!

K-lookahead auction

Three agents, k = 2.

Agent3’s type is always 1.

Agent2’s type is uniformly drawn from where

The probability of the type of agent1 is determined by

agent2’s type. If ,then with

probability , and with probability .

Our auction’s revenue is around .

A better auction: Asks agent1 for . If , sells to

agent3 for the price 1. Revenue – around 2.

12

1j

j1 hj log,...,2,1

jv 12 11 2 jv

111 jv 12

11

j

hlog

11

j2 jv 21

Example why it is tight

Theorem: If (v1,…,vn) are independent, the k-lookahead auction is

a -approximation.k

k 1

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.

the optimal revenue from agents (k+1,…,n).

For , mj is the contribution of agent j to Ropt.

Case I: for all , .

11

k

k vm1km

kj

j

jopt mR

kj jk mm 1

Another Theorem

Case II: Not all , .

Let denote the agent with minimal mj:

Pretend he declared vk+1, and run Aopt on it.

If any of the (n-k) won, sell to agent for v k+1.

Now, .

Because the distributions are independent, the distributions of the other agents don’t change.

kj jk mm 1

j jk mm ˆ1

j

jk mm ˆ1

opt

jjjjoptkopt R

k

kmmRmRR

1ˆˆ1

Proof of Theorem

• We showed a simple 2-approximation. (1 – lookahead auction).

• We showed an improvement of that auction – to improve the

approximation ratio to , but only under the assumption that the valuations are independent.

k

k 1

•It can be computed in polytime if there are polytime algorithms computing the distribution Φ.

Conclusions