Side Constraints and NonPrice Attributes in Markets

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Side Constraints and Non-Price Attributes in
Markets
Tuomas Sandholm Subhash Suri Carnegie
Mellon University University of California
Computer Science Department Santa
Barbara Dept of Computer Science
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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

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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

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Noncombinatorial auctions

• There are m items for sale
• Each bidder can submit any number of bids
• Each bid is for one item
• Without side constraints, winners can be
  determined in polynomial time by selecting the
  highest bid for each item separately

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Budget constraints in noncombinatorial auctions

• Thrm. If bidders can have budget constraints,
  revenue-maximizing winner determination is
  NP-complete
• Polynomial time (using LP) if bids can be
  accepted partially
• Proof is by reduction from PARTITION. PARTITION
  has m integers, and the question is whether they
  can be divided into two sets so that sum is same
  in each set. Create two bidders with same m bids
  (equal to the integers) and same budget
  constraint k. There is a solution of 2k iff
  there is a partition.
• Max number of items per bidder gt polynomial time
  ! Tennenholtz AAAI-00

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Max winners constraint in noncombinatorial
auctions

• Thrm. If there can be at most k winners,
  revenue-maximizing winner determination is
  NP-complete
• This holds even if bids can be accepted partially
  !
• Proof is by reduction from SET-COVER. SET-COVER
  has m ground items, and a list of sets of these
  items, and the question is whether the items can
  be covered with k sets. Corresponding to each
  set, generate a bidder who places a 1 bid for
  every item in the set. Now, there is a set cover
  of size k iff the auction has a solution with
  revenue m and max number of winners k.

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XOR constraints in noncombinatorial auctions

• In some auctions, bidders may want to submit XOR
  constraints between bids
• E.g. I want a Sony TV XOR an RCA TV
• Scenario bids (e.g., for restricted capacity
  settings)
• Under XOR-constraints , revenue-maximizing winner
  determination is NP-complete
• This holds even if bids can be accepted partially
  !
• Proof. Reduce INDEPENDENT-SET to this problem.
  For each vertex, generate an item and a 1 bid
  for it. Corresponding to each edge, insert an
  XOR-constraint between the bids. Now, the
  auction has a solution of revenue k iff there is
  an independent set of size k.

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Notes about generality

• The results from above hold whether or not the
  auctioneer has to sell all items
• They also hold if prices are restricted to be
  integers

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Combinatorial auction (CA)

• Can bid on combinations of items Rassenti,Smith
  Bulfin 82...
• Bidders perspective
• Allows bidder to express what she really wants
• No need for lookahead / counterspeculationing of
  items
• Auctioneers perspective
• Automated optimal bundling
• Binary winner determination problem
• Label bids as winning or losing so as to maximize
  sum of bid prices ( revenue)
• Each item can be allocated to at most one bid
• NP-complete Rothkopf et al 98, Karp 72
• Inapproximable Sandholm IJCAI-99 using Hastad
  99
• Fractional winner determination problem Bids
  can be accepted partially
• Polynomial time using LP

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Combinatorial reverse auction Sandholm, Suri,
Gilpin Levine AGENTS-01 workshop on Agents for
B2B

• Example procurement in supply chains
• Auctioneer wants to buy a set of items (has to
  get all) as cheaply as possible
• Sellers place bids on how cheaply they are
  willing to sell bundles of items
• Thrm. Binary clearing is NP-complete
• Thrm. Binary clearing is approximable
• k 1 log( largest items that any bid contains
  )
• Thrm. Even finding a feasible solution is
  NP-complete with XORs
• If seller(s) cannot keep items and buyer(s)
  cannot take extras, the set of feasible solutions
  becomes same for combinatorial auctions reverse
  auctions
• Thrm. Even finding a feasible solution is
  NP-complete
• Fractional clearing is polytime using LP

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Combinatorial exchange

• Each bid can buy some items, sell other items,
  and pay or request a payment
• Maximize surplus sum of accepted buy bid prices
  sum of accepted sell bid prices
• NP-complete and inapproximable in the binary case
• Polytime solvable via LP in the fractional case
• Our results hold for combinatorial markets
  (auctions, reverse auctions exchanges)
• We prove the negative results for auctions
  positive results for exchanges

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Side constraints in combinatorial markets

• Thrm. Practical side constraint classes under
  which the fractional case remains polytime
  solvable and the binary case remains NP-complete
• Cost constraints, e.g. mutual business, trading
  volume, minorities, long-term competitiveness via
  monopoly avoidance, risk hedging by requiring
  that at least k bidders get certain volume
• Unit constraints
• Absolute or compared to some group
• gt, lt, or
• Gross or net in exchanges

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Side constraints in combinatorial markets

• Thrm. Practical side constraint classes under
  which both the fractional and the binary case are
  NP-complete
• Counting constraints
• E.g. max winners
• gt there is no way to construct a counting gadget
  in LP
• XOR-constraints between bids
• Needed for full expressiveness gt inherent
  tradeoff between expressiveness and clearing
  complexity

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Need for XORs substitutability Sandholm
ICE-98, IJCAI-99, AIJ-01

• 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
• Risk free scenario bidding
• Clarke-Groves pricing mechanism can be applied to
  make truthful bidding a dominant strategy ?
  generalized Vickrey auction
• 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 ...
• Note coding XORs using dummy items Fujishima,
  Leyton-Brown, Shoham IJCAI-99 does not work in
  the fractional case

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Side constraints in combinatorial markets

• Thrm. Theoretical side constraint under which
  even the binary clearing problem becomes polytime
  solvable (the fractional case remains polytime
  solvable)
• Extreme equality each bid has to be accepted to
  the same extent

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Non-price attributes in markets

• Combinatorial markets exist (logistics.com,
  Bondconnect, FCC, ) 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 post-processor
  in exchanges
• Buyers care which sellers goods come from vice
  versa

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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
• Complexity is not an argument against (only)
  combinatorial markets