AdWords and Generalized On-line Matching

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AdWords and Generalized On-line Matching

• Aranyak Mehta et al.

Pieter Hooimeijer
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Problem Description

• Bidders have
• a daily budget
• a bid value associated with each keyword
• Search requests arrivein sequence (on-line)
• Goal maximize revenue

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Previous Work

• On-line bipartite matching
• budget 1 bids in 0, 1
• randomized algorithm KVV901 1 / e (lower
  bound)
• On-line b-matching
• budget b bids in 0, 1
• deterministic algorithm KP001 1 / e (lower
  bound)

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Naïve Algorithm

• Extend b-matching bids can be 0, 1, 2
• Algorithm
• assign query to the highest bidder
• break ties based on remaining budget
• This algorithm no longer has competitive ratio 1
  1/e

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Algorithm Example

• s p, q, p, p, r
• bA bB 3
• Optimal Solution R 6
• Online Solution R 5
• (we deplete As budget)

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Naïve Algorithm Performance
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Algorithm

• T(i) mi / bi (part of budget spent)
• ?(x) 1 - e-(1-x) (tradeoff function)
• Algorithm allocate next query q to bidder i so
  that
• ?(T(i)) bid
• is maximized.

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Tradeoff Function

• Use tradeoff revealing LPs to
• find ?(x) 1 - e-(1-x)
• Show that the algorithm has competitive ratio 1
  1 / e
• Similar to factor revealing LP(by some of the
  same authors)

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Analysis Idea

• Define an LP
• constraints are imposed by the algorithm
• objective function competitive ratio
• Problem exact constraints depend on ?
• Solution examine the dual LPs their constraints
  do not depend on ?
• There is a single vertex, y, at which the dual
  LPs reach their optimal value

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Adding Features

• The competitive ratio is not affected if
• Advertisers have different budgets
• Advertisers enter at different times
• More than one ad can be displayed per query

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Discussion

• Algorithm is more general than previous work
• Interesting use of LP duality
• Algorithm is somewhat resilient to gaming