CS 15-892 Foundations of Electronic Marketplaces Summary

1
CS 15-892 Foundations of Electronic
MarketplacesSummary future research
directions

• Tuomas Sandholm
• Computer Science Department
• Carnegie Mellon University

2
Systems with self-interested agents
(computational or human)

• Mechanism (e.g., rules of an auction) specifies
  legal actions for each agent how the outcome is
  determined as a function of the agents
  strategies
• Strategy (e.g., bidding strategy) Agents
  mapping from known history to action
• Rational self-interested agent chooses its
  strategy to maximize its own expected utility
  given the mechanism
  gt strategic analysis required for
  robustness
  
  
  gt noncooperative game theory
• But computational complexity
• In executing the mechanism
• E.g. combinatorial auctions NP-complete
  inapproximable to clear
• In determining the optimal strategy
• E.g. NP-complete valuation calculations
• E.g. uncomputable best-response strategies in
  repeated games
• In executing the optimal strategy
• E.g. chess how much space needed to represent an
  optimal strategy?
• Has significant impact on prescriptions
• Has received little attention in game theory

3
Is there a desirable way to aggregate agents
truthful preferences?

• Set of outcomes O
• Each agent i has a most-to-least-preferred
  ordering Ri of O
• R (R1, R2, ... , RA )
• Social choice functional G (R, O ) R
• To avoid unilluminating technicalities in proof,
  assume Ri and R are strict total orders
• Some possible (weak) desiderata of G
• 1. Pareto principle If every agent prefers x to
  y , then x is preferred to y in R
• 2. Independence of irrelevant alternatives If x
  is preferred to y in G (R, O ), and if R is
  another preference profile s.t. each agents
  preference between x and y is the same as in R,
  then x is preferred to y in G (R, O )
• 3. Nondictatorship No agent is decisive for
  every pair of outcomes in O
• Arrows impossibility theorem If O 3, then
  no G satisfies desiderata 1-3
• The impossibility holds even if only the highest
  ranked outcome is sought
• Thrm. Let O 3. If a social choice function
  f R -gt outcomes is monotonic and Paretian, then
  f is dictatorial
• f is monotonic whenever x f(R) and x maintains
  its position in R, then f(R) x
• x maintains its position whenever x gti y gt x gt
  i y

4
Goal of mechanism design

• Implementing a social choice function f(R) using
  a game
• Actually, say we want to implement f(u1, , uA)
• Center auctioneer does not know the agents
  preferences
• Agents may lie
• unlike in the theory of social choice which we
  discussed in class before
• 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)

5
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

6
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)

7
Solution concepts from noncooperative game theory

• Tools for building robust, nonmanipulable systems
  with self-interested agents and different agent
  designers
• Different solution concepts
• For existence, use strongest equilibrium concept
• For uniqueness, use weakest equilibrium concept

8
Implementation in dominant strategies
Strongest form of mechanism design
9
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

10
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

11
Ways around the Gibbard-Satterthwaite
impossibility

• Use a weaker equilibrium notion
• In practice, agent might not know others
  revelations
• Design mechanisms where computing a beneficial
  manipulation is hard
• E.g. Manipulation by insincerely ranking the
  outcomes is NP-complete in second order
  Copeland voting mechanism Bartholdi, Tovey,
  Trick 1989
• Copeland score Number of competitors an outcome
  beats in pairwise competitions
• 2nd order Copeland Copeland, and break ties
  based on the sum of the Copeland scores of the
  competitors that the outcome beat
• Randomization
• Agents preferences have special structure

Need almost this much randomness
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Quasilinear preferences Groves 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)
• Thrm. Assume every agents utility function is
  quasilinear. A utilitarian social choice
  function f v -gt (s(v), m(v)) can be implemented
  in dominant strategies if mi(v) ?j?i vj(s(v))
  hi(v-i) for arbitrary function h

13
Uniqueness of Groves mechanism

• Thrm. Assume every agents utility function is
  quasilinear. A utilitarian social choice
  function f v -gt (s(v), m(v)) can be implemented
  in dominant strategies for all v A x O -gt R only
  if mi(v) ?j?i vj(s(v)) hi(v-i) for some
  function h

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Clarke tax pivotal mechanism

• Special case of Groves mechanism hi(v-i) -
  ?j?i vj(s(v-i))
• So, agents payment mi ?j?i vj(s(v)) - ?j?i
  vj(s(v-i)) ? 0 is a tax
• Intuition Agent internalizes the negative
  externality he imposes on others by affecting the
  outcome
• Agent pays nothing if he does not change
  (pivot) the outcome

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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 quasilinear preferences ui(x, m) mi
  vi(x) where vi(x) are arbitrary functions. 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
• E.g. auctioneer could be agent 0
• Might still not work if participation is
  voluntary
• Vulnerable to collusion
• Even by coalitions of just 2 agents

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Implementation in Bayes-Nash equilibrium
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Implementation in Bayes-Nash equilibrium

• Goal is to design the rules of the game (aka
  mechanism) so that in Bayes-Nash equilibrium (s1,
  , sA), the outcome of the game is f(u1, ,
  uA)
• Weaker requirement than dominant strategy
  implementation
• An agents best response strategy may depend on
  others strategies
• Agents may benefit from counterspeculating each
  others
• preferences
• rationality
• endowments
• capabilities
• Can accomplish more than under dominant strategy
  implementation
• E.g., budget balance Pareto efficiency (social
  welfare maximization) under quasilinear
  preferences

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Expected externality mechanism dAspremont
Gerard-Varet 79 Arrow 79

• Like Groves mechanism, but sidepayment is
  computed based on agents revelation vi ,
  averaging over possible true types of the others
  v-i
• 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)
• Others expected welfare when agent i announces
  vi is ?(vi) ?v-i p(v-i) ?j?i vj(s(vi , v-i))
• Measures change in expected externality as agent
  i changes her revelation
• Thrm. Assume quasilinear preferences and
  statistically independent valuation functions vi.
  A utilitarian social choice function f v -gt
  (s(v), m(v)) can be implemented in Bayes-Nash
  equilibrium if mi(vi) ?(vi) hi(v-i) for
  arbitrary function h
• Unlike in dominant strategy implementation,
  budget balance achievable
• Intuitively, have each agent contribute an equal
  share of others payments
• Formally, set hi(v-i) - 1 / (A-1) ?j?i
  ?(vj)
• Does not satisfy participation constraints (aka
  individual rationality constraints) in general
• Agent might get higher expected utility by not
  participating

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Myerson-Satterthwaite impossibility

• Avrim is selling a car to Tuomas, both are risk
  neutral, quasilinear
• Each party knows his own valuation, but not the
  others valuation
• The probability distributions are common
  knowledge
• Want a mechanism that is
• Ex post budget balanced
• Ex post Pareto efficient Car changes hands iff
  vbuyer gt vseller
• (Interim) individually rational Both Avrim and
  Tuomas get higher expected utility by
  participating than not
• Thrm. Such a mechanism does not exist (even if
  randomized mechanisms are allowed)
• This impossibility is at the heart of more
  general exchange settings (NYSE, NASDAQ,
  combinatorial exchanges, ) !

20
Auctioning one item
21
Auction settings

• Private value value of the good depends only on
  the agents own preferences
• E.g. cake which is not resold or showed off
• Common value agents value of an item
  determined entirely by others values
• E.g. treasury bills
• Correlated value agents 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

22
Common auction mechanisms

• First-price mechanisms First-price sealed-bid,
  Dutch
• Strategic underbidding in (Nash) equilibrium
• Second-price mechanisms English, Vickrey,
  Japanese ( open-exit)
• Truth-telling as a dominant strategy in
  private-value auctions
• If bidders know their own valuations

23
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!

24
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)

25
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

26
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

27
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
• Asymmetric info among bidders
• E.g. 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

28
Vulnerability to bidder collusion

• 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

29
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

30
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
• 3rd party auctionbots (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
• 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

32
Multi-unit auctions exchanges (multiple
indistinguishable units of one item for sale)
33
Auctions / reverse auctions / exchanges with
multiple indistinguishable units for sale

• Assume the supply/demand curves are reasonable
• Non-discriminatory pricing is O(N log N) to clear
  with piecewise linear supply/demand curves
• Discriminatory pricing is NP-complete to clear
  with piecewise linear supply/demand curves
• Discriminatory pricing is O(N log N) to clear
  with linear supply/demand curves

34
Multi-item auctions exchanges (multiple
distinguishable items for sale)
35
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
  McAfeeMcMillan96)
• Backtracking via leveled commitment contracts
  SandholmLesser95,96Sandholm96AnderssonSandh
  olm98a,b
• Breach before allocation
• Breach after allocation

36
Protocol design for multi-item auctions...

• Combinatorial auctions Rassenti,SmithBulfin82..
  .
• Bidders perspective
• Reduces the need for lookahead
• Potentially 2items valuation calculations
• Automated optimal bundling of items
• Auctioneers 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 2bids

37
Combinatorial markets can be complex to clear

• Optimal clearing NP-complete
• E.g. auctions reverse auctions
• Approximation is NP-complete
• E.g. auctions to bids1-? or items0.5-?
• E.g. reverse auctions to 1 ln(items that any
  one bid contains)
• E.g. multi-unit reverse auctions to 1 ln(units
  that any one bid contains)
• Finding a feasible solution is NP-complete
• E.g. reverse auctions with XOR-constraints
  (auctions with XORs are trivial)
• E.g. auctions, reverse auctions exchanges
  without free disposal
• However, can be solved fast in practice (at least
  for auctions reverse auctions) using modern
  search algorithms
• Cases with extreme special structure can be
  solved in provably polynomial time

38
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

39
Bidding languages for combinatorial markets

• Bundle bids
• ORs, XORs, OR-of-XORs Sandholm 99
• XOR-of-ORs, OR Nisan 00
• Logical connectives on subformulae with prices
  Boutilier Hoos 01
• Side constraints Sandholm et al 01
• Price quantity discounts / rebates Sandholm et
  al 01

40
Side constraints Sandholm Suri IJCAI-01
workshop on Distributed Constraint Reasoning

• Side constraints increase expressiveness make
  markets practical
• Noncombinatorial multi-item auctions are solvable
  in polynomial time
• Thrm. Budget constraints NP-complete
• Max number of items per bidder polynomial time
  Tennenholtz 00
• Thrm. Max winners NP-complete even if bids can
  be accepted partially
• Thrm. XORs NP-complete inapproximable even if
  bids can be accepted partially
• These results hold whether or not seller has to
  sell all items
• Combinatorial auctions are polynomial time if
  bids can be accepted partially
• Any side constraints from above make the problem
  NP-complete
• Also counting constraints
• Other constraints allow polynomial time clearing
• Cost constraints mutual business, trading
  volume, minorities,
• Unit constraints
• Some side constraints can make NP-hard
  combinatorial auction clearing easy !
• Results apply to exchanges reverse auctions also

41
Future research

• Expected revenue-maximizing multi-item auctions
• Dominant strategy equilibrium
• Bayes-Nash equilibrium
• May not be GVA, and may not be efficient
• Reserve price setting agent for GVA so as to
  maximize expected revenue (within GVA)
• Optimal auction design without prior knowledge of
  the valuation distribution
• Competitive analysis as in online algorithms
• Multi-unit case is partially solved by Hartline
  et al 01

42
Bidding Agents with Complex Valuation Problems in
Auctions

• Kate Larson and Tuomas Sandholm

43
Bidders may need to compute their valuations for
(bundles of) goods

• In many B2B applications, e.g.
• Vehicle routing problem in transportation
  exchanges
• Manufacturing scheduling problem in procurement
• Value of a bundle of items (tasks, resources,
  etc) value of solution
  with those items - value of solution without them

44
Performance profile tree
5
P(BA)
B
4
4
10
A
0
3
Solution quality
C
P(CA)
6
15
2
5
20

• Normative
• Allows conditioning on path of solution quality
• Allows conditioning on path of other solution
  features
• Allows conditioning on problem instance features
  (different trees to be used for different
  classes)
• Constructed from statistics on earlier runs

45
Theorems on strategic computing
Strategic computing ?
Counter-speculation by rational agents ?
Auction mechanism
Costly computing
Limited computing
yes
yes
yes
First price sealed-bid
Single item for sale
yes
yes
yes
Dutch (1st price descending)
no
Vickrey (2nd price sealed bid)
no
English (1st price ascending)
no
Generalized Vickrey On which ltbidder, bundlegt
pair to allocate next computation step ?
Multiple items for sale
If performance profiles are deterministic, only
weak strategic computing can occur ? New
normative deliberation control method uncovered a
new phenomenon
46
Future research

• In many B2B settings, automated bidders can
  compute valuations dynamically faster than humans
• Some future research directions
• Using our deliberation control method in systems
• Manufacturing planning, networks,
• New (market) mechanisms
• Game-theoretically engineered to work well under
  (different) models of bounded rationality
• Our results show that even the most common
  mechanism design principles (e.g., revelation
  principle) cease to hold
• Our normative deliberation control method basis
  for new design principles ?

47
Preference Elicitation in Combinatorial Markets

• Wolfram Conen Tuomas Sandholm

48
Another complex problem in (single-shot)
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

49
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
• Need exponential revelation in worst case Nisan
  2001
• Can be combined with price feedback mechanisms

50
Elicitor algorithms

• Query policy dependent elicitor algorithms
• Algorithm query policy are intertwined
• Based on search algorithms where each search step
  involves asking a bidder a question
• Policy independent elicitor algorithms
• General framework specific algorithms
• Can support any query policy
• Note Query policies are online control policies,
  i.e. contingency plans

51
Future research

• Good query policies
• Evaluating the elicitor
• Savings in revelation (how many queries needed ?)
• In general case / in cases with special
  preference structure
• Worst / average case
• Competitive analysis as in online algorithms
• Generalizing the elicitor
• To (combinatorial) exchanges
• To (combinatorial) markets with side constraints
• To (combinatorial) markets with multiattribute
  features

52
Thank you for your attention!

• Its been a fun course (at least for me -) )
• You have learned a LOT
• its time for final project presentations