Review of some of the course topics

1
Review of some of the course topics
2
Conclusions on game-theoretic analysis tools

• Different solution concepts
• For existence, use strongest equilibrium concept
• For uniqueness, use weakest equilibrium concept

3
Mechanism design
4
Goal of mechanism design

• Implementing a social choice function f(u1, ,
  uA) using a game
• Center auctioneer does not know the agents
  preferences
• Agents may lie
• Goal is to design the rules of the game (aka
  mechanism) so that in equilibrium (s1, , sA),
  the outcome of the game is f(u1, , uA)
• Mechanism designer specifies the strategy sets Si
  and how outcome is determined as a function of
  (s1, , sA) ? (S1, , SA)
• Variants
• Strongest There exists exactly one equilibrium.
  Its outcome is f(u1, , uA)
• Medium In every equilibrium the outcome is f(u1,
  , uA)
• Weakest In at least one equilibrium the outcome
  is f(u1, , uA)

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Revelation principle

• Any outcome that can be supported in Nash
  (dominant strategy) equilibrium via a complex
  indirect mechanism can be supported in Nash
  (dominant strategy) equilibrium via a direct
  mechanism where agents reveal their types
  truthfully in a single step

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Uses of the revelation principle

• Literal Only direct mechanisms needed
• Problems
• Strategy formulator might be complex
• Complex to determine and/or execute best-response
  strategy
• Computational burden is pushed on the center
  (assumed away)
• Thus the revelation principle might not hold in
  practice if these computational problems are hard
• This problem traditionally ignored in game theory
• Even if the indirect mechanism has a unique
  equilibrium, the direct mechanism can have
  additional bad equilibria
• As an analysis tool
• Best direct mechanism gives tight upper bound on
  how well any indirect mechanism can do
• Space of direct mechanisms is smaller than that
  of indirect ones
• One can analyze all direct mechanisms pick best
  one
• Thus one can know when one has designed an
  optimal indirect mechanism (when it is as good as
  the best direct one)

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Implementation in dominant strategies
Strongest form of mechanism design
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Implementation in dominant strategies

• Goal is to design the rules of the game (aka
  mechanism) so that in dominant strategy
  equilibrium (s1, , sA), the outcome of the
  game is f(u1, , uA)
• Nice in that agents cannot benefit from
  counterspeculating each other
• Others preferences
• Others rationality
• Others endowments
• Others capabilities

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Gibbard-Satterthwaite impossibility

• Thrm. If O 3 (and each outcome would be
  the social choice under f for some input profile
  (u1, , uA) ) and f is implementable in
  dominant strategies, then f is dictatorial

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(No Transcript)
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Special case where dominant strategy
implementation is possible Quasilinear
preferences -gt Clarke tax mechanism

• Outcome (x1, x2, ..., xk, m1, m2, ..., mA )
• Quasilinear preferences ui(x, m) mi vi(x1,
  x2, ..., xk)
• Utilitarian setting Social welfare maximizing
  choice
• Outcome s(v1, v2, ..., vA) maxx ?i vi(x1, x2,
  ..., xk)
• Agents payment mi ?j?i vj(s(v)) - ?j?i
  vj(s(v-i)) ? 0 is a tax
• Thrm Every agents dominant strategy is to
  reveal preferences truthfully
• Intuition Agent internalizes the negative
  externality he imposes on others by affecting the
  outcome
• Agent pays nothing if he does not change the
  outcome
• Example k1, x1joint pool built or not,
  mi
• E.g. equal sharing of construction cost -c / A

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Clarke tax mechanism

• Pros
• Social welfare maximizing outcome
• Truth-telling is a dominant strategy
• Feasible in that it does not need a benefactor
  (?i mi ? 0)
• Cons
• Budget balance not maintained (in pool example,
  generally ?i mi lt 0)
• Have to burn the excess money that is collected
• Thrm. Green Laffont 1979. Let the agents
  have arbitrary quasilinear preferences. No
  social choice function that is (ex post) welfare
  maximizing (taking into account money burning as
  a loss) is implementable in dominant strategies
• If there is some party that has no private
  information to reveal and no preferences over x,
  welfare maximization and budget balance can be
  obtained by having that partys payment be m0 -
  ?i1.. mi
• Auctioneer could be called agent 0
• Vulnerable to collusion
• Even by coalitions of just 2 agents

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Another approach for circumventing the
impossibility of dominant-strategy implementation

• Design the game so that (although manipulations
  exist), finding a beneficial manipulation is
  computationally so complex for an agent that the
  agent cannot do that
• E.g. Complexity of Manipulating Elections with
  Few Candidates Conitzer Sandholm AAAI-02
• E.g. Universal Voting Protocol Tweaks for Making
  Manipulation Hard Conitzer Sandholm IJCAI-03

14
Yet another approach for circumventing the
impossibility of dominant-strategy implementation

• Designing the mechanism automatically to the
  situation at hand
• Input is the probabilistic information that the
  center has about the agents
• Output is an optimal mechanism where the agents
  are motivated to reveal their preferences
  truthfully, and a social objective is satisfied
  to the optimal extent
• Advantages
• Can be used even without side payments
  quasilinear preferences
• Could achieve better outcomes than Clarke tax
  mechanism
• Circumvents impossibility in many cases
• Complexity of Mechanism Design Conitzer
  Sandholm UAI-02
• Designing a deterministic mechanism is
  NP-complete
• Designing a randomized mechanism is fast
• No loss in social objective, sometime a gain
• Both results also hold for Bayes-Nash
  implementation

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Auctioning one item

• Tuomas Sandholm
• Computer Science Department
• Carnegie Mellon University

16
Results for private value auctions

• Dutch strategically equivalent to first-price
  sealed-bid
• Risk neutral agents gt Vickrey strategically
  equivalent to English
• All four protocols allocate item efficiently
• (assuming no reservation price for the
  auctioneer)
• English Vickrey have dominant strategies gt 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!

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Revenue equivalence theorem

• Even more generally Thrm.
• Assume risk-neutral bidders, valuations drawn
  independently from potentially different
  distributions with no gaps
• Consider two Bayes-Nash equilibria of any two
  auction mechanisms
• Assume allocation probabilities yi(v1, vA)
  are same in both equilibria
• Here v1, vA are true types, not revelations
• E.g., if the equilibrium is efficient, then yi
  1 for bidder with highest vi
• Assume that if any agent i draws his lowest
  possible valuation vi, his expected payoff is
  same in both equilibria
• E.g., may want a bidder to lose pay nothing if
  bidders valuations are drawn from same
  distribution, and the bidder draws the lowest
  possible valuation
• Then, the two equilibria give the same expected
  payoffs to the bidders ( thus to the seller)

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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

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Optimal auctions

• Private-value auction with 2 risk-neutral bidders
• As valuation is uniformly distributed on 0,1
• Bs valuation is uniformly distributed on 1,4
• What revenue do the 4 basic auction types give?
• Can the seller get higher expected revenue?
• Is the allocation Pareto efficient?
• What is the worst-case revenue for the seller?
• For the revenue-maximizing auction, see
  Wolfstetters survey on class web page

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Vulnerability to bidder collusioneven 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

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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

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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

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Auctioneers 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) gt seller may inefficiently get the
  item
• In an expected revenue maximizing auction, seller
  sets a reservation price strategically like this
  Myerson 81
• So, auctions are not Pareto efficient
• Nor are any other mechanisms for this setting
  that are individually rational and budget
  balanced Myerson Satterthwaite 83
• Setting a minimum price (analogous)
• Refusing to sell after the auction has ended

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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

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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 agents own valuation, it is
  a risk averse agents best (dominant or
  equilibrium) strategy to bid its expected value
• Higher expected utility e.g. by bidding low

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Wasteful counterspeculation
Thrm Sandholm ICMAS-96. In a private value
Vickrey auction with uncertainty about an agents
own valuation, a risk neutral agents 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 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)
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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
• Winners 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

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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 winners curse

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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
• gt Bid less if more bidders or your info is
  worse
• Most important in sealed-bid auctions Dutch

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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

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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

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Exchanges

• markets with many buyers and many sellers
• Lets consider a 1-item 1-unit exchange first

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Exchange game in class

• 1 buyer, 1 seller, 1 good
• The agents valuations for the good are drawn
  uniformly from 0, 100. This is common
  knowledge
• The agents dont know each others valuations

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Does a good exchange mechanism exist ?

• E.g Keith is selling a car to Tuomas
• Both have quasilinear utility functions
• Each party knows his valuation, but not the
  others valuation
• Probability distributions of valuations are
  common knowledge
• Want a mechanism that is
• Budget balanced Keith gets what Tuomas pays
• Pareto efficient Car changes hands if and only
  if vbuyer gt vseller
• Individually rational Both Keith and Tuomas get
  higher expected utility by participating than not
• Thrm. Such a mechanism does not exist (even if
  randomized mechanisms are allowed)
  Myerson-Satterthwaite
• This impossibility is at the heart of more
  general exchange settings (NYSE, NASDAQ,
  combinatorial exchanges, ) !

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Multi-unit auctions exchanges (multiple
indistinguishable units of one item for sale)

• Tuomas Sandholm
• Carnegie Mellon University

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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 k1st highest price
• Demand reduction lie CramptonAusubel 96
• k5
• 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
  9-27
• Incentive compatible 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

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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)

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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
• 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

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Multi-item auctions exchanges (multiple
distinguishable items for sale)

• Tuomas Sandholm
• Carnegie Mellon University

40
Multi-item auctions

• Auctioning multiple distinguishable items when
  bidders have preferences over combinations of
  items complementarity substitutability
• 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

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Mechanism design for multi-item auctions

• Sequential auctions
• Impossbile to determine fbest strategy because
  game tree is huge
• 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
  McAfeeMcMillan96)
• Backtracking via leveled commitment contracts
  SandholmLesser95,96Sandholm96AnderssonSandh
  olm98a,b
• Breach before allocation
• Breach after allocation

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Mechanism design for multi-item auctions...

• Combinatorial auctions Rassenti,SmithBulfin82..
  .
• Bids can be submitted on combinations (bundles)
  of items
• Bidders perspective
• Avoids the need for lookahead
• (Potentially 2items valuation calculations)
• Auctioneers perspective
• Automated optimal bundling of items
• Winner determination problem
• 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 2bids

43
NP-completeness

• NP-complete Karp 72
• Weighted set packing

44
Polynomial-time approximation algorithm 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 IJCAI-99, AIJ-03 using
  Håstad96

45
Solving the winner determination problem when all
combinations can be bid onSearch algorithm for
optimal 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 SandholmSuri
  AAAI-00, Sandholm et al. IJCAI-01

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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

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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 2items-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 )

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Combinatorial reverse auction

• Example procurement in supply chains
• Auctioneer wants to buy a set of items (has to
  get all)
• Can take extras if there is free disposal
• Sellers place bids on how cheaply they are
  willing to sell bundles of items
• Thrm. Winner determination is NP-complete even in
  single-unit case with free disposal
• Thrm. Single unit case with free disposal is
  approximable
• k 1 log m (m largest number of items that
  any bid contains)
• Greedy algorithm Keep choosing bid with lowest
  price / items

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No free disposal

• Free disposal seller can keep items, buyers can
  take extras
• Free disposal has been assumed in the
  combinatorial auction literature so far
• In practice, freeness of disposal can vary across
  items bidders
• Without free disposal, the set of feasible
  solutions is same for combinatorial auctions
  reverse auctions
• Thrm. Even finding a feasible solution is
  NP-complete

50
Combinatorial exchange

• Example bid (buy 20 tons of water, sell 10 cubic
  meters of hydrogen, sell 5 cubic meters of
  oxygen, ask 500)
• Example application manufacturing where a
  participant bids for inputs outputs of a
  production plan simultaneously
• Label bids as winning or losing so as to maximize
  (revealed) surplus sum of amounts paid by
  bidders minus sum of amounts paid to bidders
• On each item, sell quantity ? buy quantity
• Equality if there is no free disposal
• Thrm. NP-complete even in the single-unit case
• Thrm. Inapproximable even in the single-unit case
• Could also maximize trading volume
• Thrm. Without free disposal, even finding a
  feasible solution is NP-complete (even in the
  single-unit case)

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Conclusions on generalization of combinatorial
auctions

• Generalizations of combinatorial auctions
• No free disposal
• Reverse auctions
• Exchanges
• Single- and multi-unit settings
• Theoretical results
• All these generalizations are NP-complete
• With free disposal
• Auction and exchanges are inapproximable
• Reverse auctions are approximable
• Even finding a feasible solution is NP-complete
  if XORs are allowed
• Without free disposal, even finding a feasible
  solution is NP-complete
• Experimental results
• Search does well on auctions at times even
  better on reverse auctions
• Search does well on single-unit exchanges,
  poorly on multi-unit exchanges
• Better algorithms needed
• Lack of free disposal makes the problem much
  harder

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Hot off the pressKothari, Suri Sandholm 2002

• Q How many bids have to be accepted fractionally
  (in worst case) so as to obtain maximum surplus
  in a multi-item multi-unit combinatorial exchange
  / combinatorial auction?
• Trivial answer bids
• A items (this is independent of units)
• Q How many bids have to be accepted fractionally
  (in worst case) so as to maximize liquidity in a
  multi-item multi-unit combinatorial exchange?
• Trivial answer bids
• A items 1 (this is independent of units)
• Q How complex is it to find such a solution?
• A Polynomial time fast

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Ascending multi-item auctions

• Increase prices until each item is demanded only
  once
• Item prices vs. bundle prices
• E.g. where there exist no appropriate item prices
• Discriminatory vs. nondiscriminatory prices

54
Automated bid elicitation

• in combinatorial auctions
• Conen Sandholm IJCAI-01 workshop
  ACM-Ecommerce-01 AAAI-02 Hudson Sandholm
  -02

55
Another complex problem in combinatorial
auctions The revelation problem

• Bidders may need to bid on all 2items
  combinations
• Need to compute the valuation for each
  combination
• Each valuation computation can be NP-complete
• For example if a carrier company bids on trucking
  tasks TRACONET Sandholm AAAI-93
• Need to communicate the bids
• Need to reveal the bids
• Loss of privacy strategic info

56
Approaches for tackling the revelation problem

• Classic single-shot full revelation mechanims
  (Vickrey-Clarke-Groves)
• Exponentially many valuations revealed
• Ascending mechanisms with price feedback
  (iBundle, Parkes et al 1999 , akBa Wurman et
  al. 2000 , etc.)
• Can save revelation
• Need exponential revelation in worst case Nisan
  2001
• Our new approach an elicitor agent
• Knows things that individual bidders dont
  (others bids so far)
• Asks non-redundant questions from bidders to
  focus their revelation
• Can save revelation
• Exponential revelation in worst case Nisan 2001
• Could be combined with price feedback mechanisms

57
Our Query Types for Elicitation

• Value information What is your valuation for
  bundle A? (Answer Exact or Bounds)
• Extensions
• More and more refined answers over times
• Bounds in the queries
• Order information Which bundle do you prefer, A
  or B?
• Rank information
• What is the rank of bundle b?
• What bundle is at rank x?
• Given bundle b, what is the next lower (higher)
  ranked bundle?
• We designed a host of elicitation algorithms that
  use these query types in different combinations
  and with different query policies

58
Example elicitation experiment with random
non-redundant value queries only
With free disposal
Number of bundle values asked / number of bundles
Random (nonredundant) elicitor
Best elicitor developed so far
Advantage of elicitation also holds as the number
of agents grows
59
Incentive compatibility

• Elicitors questions leak information about
  others preferences
• Can be made incentive compatible in weaker
  equilibrium notions
• Ask enough questions to determine Clarke tax
  prices (agents1 elicitors)
• Could interleave these extra questions with
  real questions
• To avoid lazyness Not necessary from an
  incentive perspective
• Agents dont have to answer the questions may
  answer questions that were not asked
• Unlike in ascending price feedback auction
  mechanisms

60
Side Constraints and Non-Price Attributes in
Markets
Tuomas Sandholm Carnegie Mellon
University Computer Science Department
Paper by Sandholm Suri 2001
61
Side constraints in markets

• Traditionally, markets (auctions, reverse
  auctions, exchanges) have been designed to
  optimize unconstrained economic value (Pareto
  efficiency/revenue)
• Side constraints are required in many practical
  markets (especially in B2B) to encode legal,
  contractual and business constraints
• Side constraints could be imposed by any party
• Sellers
• Buyers
• Auctioneer
• Market maker
• Side constraint have significant implications on
  the complexity of clearing the market

62
Outline

• Side constraints in non-combinatorial markets
• Side constraints in combinatorial markets
• Constraints under which the winner determination
  problem stays polynomial time solvable (if bids
  can be accepted partially)
• Constraints under which the winner determination
  problem is NP-complete even if bids can be
  accepted partially
• Constraints under which the winner determination
  problem is polynomial-time solvable even if bids
  have to be accepted entirely or not at all

63
Non-price attributes in markets

• Combinatorial markets exist (logistics.com,
  Bondconnect, FCC, CombineNet, ) and
  multi-attribute markets exist (Frictionless,
  Perfect, ), but have not been hybridized
• Here we propose a way to hybridize them
• Attribute types
• Attributes from outside sources, e.g., reputation
  databases
• Attributes that bidders fill into the partial
  item description
• Handling attributes in combinatorial auctions
  reverse auctions
• Attribute vector b
• Reweight bids, so p f(p, b)
• Side constraints could be specified on p or p
• Same complexity results on side constraints hold
• Attributes cannot be handled as a preprocessor in
  exchanges
• Buyers care which sellers goods come from vice
  versa
• Have to handle attributes as part of the main
  winner determination optimization problem

64
Conclusions

• Combinatorial markets are important now
  feasible
• Market types differ in clearing complexity
  approximability
• Expressive bidding language removes guesswork
  sets correct incentives
• Side constraints extend usability of dynamic
  pricing
• Allow the advantages of dynamic pricing while
  keeping the advantages of long-term contracts
• Different side constraints lead to different
  clearing complexity
• Can make problem harder or easier
• Even non-combinatorial markets become NP-complete
  to clear under natural side constraints